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A behavioural modelling framework with a dynamic travel strategy path choice approach is presented for unreliable multiservice transit networks. The modelling framework is especially suitable for dynamic run-oriented simulation models that use subjective strategy-based path choice models. After an analysis of the travel strategy approach in unreliable transit networks with the related hyperpaths, the search for the optimal strategy as a Markov decision problem solution is considered. The new modelling framework is then presented and applied to a real network. The paper concludes with an overview of the benefits of the new behavioural framework and outlines scope for further research.

Transit network planning requires prediction of bus travel times, on-board loads, and other state variables representing system operations. One way to obtain such variables is to use simulation models [

In simulation models, a transit supply module is able to support detailed simulation of vehicles serving stops with a given schedule [

While the supply and demand-supply interaction components of transit simulation models are quite well defined in the literature [

According to the seminal paper by Spiess [

Two types of travel strategies can be considered from a modelling point of view. One is the

From this analysis of transit path choice modelling applied in simulation, the need arises to adopt in reproducing traveller behaviour not a hypothetical objective optimal strategy, but a subjective strategy-based approach, which is more realistic in relation to the cognitive and computational traveller’s capacities and obtained with a stochastic decision approach. This paper proposes such a type of subjective travel strategy approach, defining travellers’ utility as combinations of anticipated values through travellers’ parameters to estimate, moving from the first investigation performed by Nuzzolo and Comi [

The paper is structured as follows: Section

Let there be an origin-destination pair

Nguyen and Pallottino [

In general, two types of graph representation of a transit service network can be used:

Example of line and diachronic graphs.

Although this paper focuses on subjective optimal strategies, objective optimal strategy search methods are first analysed since such methods can suggest efficient search methods for the subjective case as well.

Path choice in an unreliable service network entails decision making without comprehensive knowledge of possible future evolution of all relevant factors. Hence the outcomes of any decision depend partly on randomness and partly on the agent’s decisions. Therefore, in this case a general

A Markov decision process (MDPs; [

where

An MDPs with a specified optimality criterion (hence forming a sextuple) is called a

Given a

the set

the state space set

an action

the change in the time of traveller location within the diversion node set consists in a Markov process;

the transition probabilities

the reward function

the optimal policy

To represent an MDPm, a

Example of run hyperpath with a diversion choice at origin.

As explored above, the search for an

In order to find the

In this paper, an approach is proposed which applies path choice behavioural modelling based on a dynamic subjective travel strategy and defined in the framework of a Markov decision problem. The proposed model, an advanced version of that presented in Comi and Nuzzolo [

Traveller behavioural assumptions are defined in the context of

an unreliable or stochastic and within-day dynamic transit service network with diversion nodes;

transit users who often travel on the origin-destination (O-D) pair (

subjective optimal strategy-based travel behaviour.

Given an O-D pair

Example of a master line hyperpath.

As a master line hyperpath

Given a

As a result of random service occurrences and traveller’s choices according to a diversion rule

Therefore, it can be assumed that travellers consider the average_{.}

We assume that a traveller

The proposed diversion rule

Given a diversion node

the anticipated utility

the

For example, the anticipated utility of link

Given subpath

Assuming that travellers use an exponential smoothing forecasting method [

The nodal anticipated utility

with

In the learning process, travellers search for the optimal weights

As an example of a diversion choice, consider the choice at origin

to identify, within the master line hyperpath, the set of diversion links with the root on

to associate an anticipated utility

for link

for link

to use the diversion link

Subsequently, at time

to consider the diversion link

to associate an

to compare the anticipated utilities of these diversion links;

to board run

if the traveller does not board run

The application of the presented model requires the knowledge of the following parameters:

Parameters

An application of the proposed path choice modelling, with a unique subjective optimal strategy and the same parameters

The service network (Figure

The application network [

As regards the master line hyperpath, according to the literature on choice set formation and as reviewed by Bovy [

The results entail the reproduction of an initial transient of about 60 days to set up the traveller's prior knowledge of path attributes and to reach an equilibrium state, followed by 30 replications of each simulation period, aiming to obtain statistically significant estimates of state variable expected values (i.e., confidence interval method with specified precision at 95%). Anticipated attributes are estimated assuming parameter

The assignment algorithm is coded in C++ and data are managed with a Postgres 9.1 DBMS. As the programming code is optimised to use the latest technologies in the field of multicore CPU processing, simulation times strictly depend on the CPU architecture (i.e., number of cores and processors) and on the operating system. Referring to the above-mentioned three-hour morning period of a workday (i.e., 7:00am - 10:00am), simulation takes 35 seconds on a computer with an Intel Core 2 Duo 3.33GHz, 8Gb RAM, running on Mac-OSX. This time is reduced to 12 seconds if we use a computer equipped with two Intel Core i7 293 GHz, 16Gb RAM, running on MS-Windows 7.

Four different coefficient variations of bus running times were used to consider different levels of service unreliability. The results (see Table

Application results.

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If the behavioural framework with the proposed diversion rule is applied, the subjective optimal strategy found at a diversion node can be considered as an approximate solution of a MDPm, where

the master hyperpath is found by considering quite simple logical and behavioural constraints (see Figure

the perceived shares of use of subpaths

the anticipated utilities are proxies for expected rewards (see (

the traveller, at each diversion node, considers as an action only that of choosing among all available diversion links. Referring to the example depicted in Figure

Example of reduced action tree of Figure

This paper sought to overcome some limits of transit path choice modelling, especially that concerning the use of an objective optimal travel strategy for multiservice stochastic networks, instead of subjective strategies. A path choice model was therefore developed by using a dynamic subjective travel strategy. Further, the model was defined in the framework of a Markov decision problem. The optimal subjective strategy can be considered as the solution of a simplified MDPm with approximate transition probabilities and approximate expected rewards. It takes into account service occurrences and the information provided to travellers and applies a diversion rule that considers some of the travellers’ cognitive limitations and simplifications.

Even if the proposed modelling framework requires several model parameters, the new opportunities resulting from the availability of a large quantity of data obtained from automated data collecting allow model parameter estimation and upgrading to be more easily achieved, for example, by using the reverse assignment method recalled in the paper. This same data availability helps to obtain new models of travel strategy generation for different categories of users, to be used as subjective travel strategies in assignment models. Therefore, the next steps in this research will be the setup and testing of an overall procedure, including inverse assignment parameter estimation, on the test network. In the near future, through a greater deployment of bidirectional communication between travellers and information centres, a suitable quantity of data will be available, making it possible, at least in theory, to calibrate not only individual model parameters, but also specific subjective strategy-based transit path choice models.

Further research should explore master line hyperpath modelling and the development of travel strategies within theories other than that of expected utility. In addition, the introduction of stochastic path choice models which take into account user perception errors and analyst modelling errors is another possible modelling improvement.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.