Journal of Advanced Transportation

Volume 2018, Article ID 8797328, 10 pages

https://doi.org/10.1155/2018/8797328

## A Subjective Optimal Strategy for Transit Simulation Models

Department of Enterprise Engineering, University of Rome Tor Vergata, Via del Politecnico 1, 00133 Rome, Italy

Correspondence should be addressed to Agostino Nuzzolo; ti.2amorinu.gni@olozzun

Received 16 February 2018; Revised 2 July 2018; Accepted 27 August 2018; Published 16 September 2018

Academic Editor: Monica Menendez

Copyright © 2018 Agostino Nuzzolo and Antonio Comi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1, 2] which reproduce interactions over time among travellers, transit vehicles, and sometimes also other vehicles sharing the right of way.

In simulation models, a transit supply module is able to support detailed simulation of vehicles serving stops with a given schedule [3], picking up, and dropping off passengers, while monitoring transit vehicles’ capacities and speeds. The simulation takes into account when passengers cannot board a vehicle because its capacity limit is already reached. Examples of supply model components are those of the simulators MATsim [3], BUSMEZZO [4], and DYBUS [2]. The simulators perform a within-day dynamic simulation. Each transit vehicle from the departure terminal to that of arrival is followed and, at each bus departure from a stop, the forecasted vehicle travel times, considering the irregularities of the transit services, are updated. Each traveller of a time-dependent origin-destination matrix is followed from origin to final destination, and dynamic routing is applied, taking into account real-time information on current and forecasted states of the transit network. Further, a day-to-day simulation with a traveller learning and forecasting process of service attributes allows a demand-supply equilibrium condition to be obtained.

While the supply and demand-supply interaction components of transit simulation models are quite well defined in the literature [4], traveller path choice modelling still presents its limits. A case that requires in-depth analysis is that of* multiservice stochastic (unreliable) service networks,* where at some bus stops more than one line is available to reach the destination and some path attributes (e.g., waiting time, on-board time, and on-board occupancy degree) are random variables.

According to the seminal paper by Spiess [5], in the case of multiservice stochastic networks, a stochastic decision approach should be considered, and* optimal travel strategy* modelling should be applied. In stochastic decision theory, an optimal strategy, detailed in Section 2 below, is the behaviour rule that travellers should follow to optimise the expected value of the experienced travel utility.

Two types of travel strategies can be considered from a modelling point of view. One is the* objective* (or* normative*) optimal strategy, which is the behaviour that travellers should follow to optimise the expected value of the experienced travel utility. A different question is the actual strategic behaviour,* subjective *(or* descriptive*) optimal strategy, which travellers adopt, with their cognitive constraints and own perceived path attributes. Drawing on data collected through new ticketing technologies, recent research confirms that, on unreliable transit networks with diversion nodes, subjective travel strategies are sometimes applied [6–11]. These subjective strategies can differ among travellers and very often differ from the objective optimal strategy. Therefore, in transit path choice modelling, a* subjective* optimal strategy should be used, in principle modelling each traveller or at least each traveller category. In practice, such an approach would be very complex, and therefore in the literature a unique optimal strategy is assumed valid for all travellers. Further, in order to determine the applied optimal strategy, until now two main approaches have been followed. In one approach, an* objective optimal strategy* is searched and adopted, such as the optimal strategy reported in Spiess and Florian [12], but in this way neglecting the travellers’ cognitive limitations and simplifications. The other, as in BUSMEZZO [4] and DYBUS [2], applies path choice random utility models, and the stochasticity of the services is hidden in the stochasticity of the path choice utilities.

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 [13]. The estimation process of such parameters is simplified by the new opportunities offered by big data collecting and processing, which allows effective reverse assignment procedures to be applied [14].

The paper is structured as follows: Section 2 analyses the travel strategy approach in unreliable transit networks and the related routes, while Section 3 considers the search for an optimal travel strategy as a solution to a Markov decision problem. Section 4 presents the proposed behavioural assumption framework and finally Section 5 reports some concluding remarks and future research perspectives.

#### 2. Transit Travel Strategy and Hyperpaths

Let there be an origin-destination pair* od* and an* unreliable* transit service network with* diversion nodes,* that is, nodes where choices are made among different subpaths. Because of transit service stochasticity, rather than relying on a* pretrip* selected single path from origin up to destination, users should adopt a travel strategy* ST* which is [5] a set of coherent behavioural decision rules* (diversion rules)* at diversion nodes, according to random service occurrences (e.g., random arrival times of buses at a stop, random transit vehicle crowding, failure to board, and so on), with the aim of minimising the expected travel cost or maximising the expected travel utility.

Nguyen and Pallottino [15] highlighted the underlying graph structure of Spiess’ basic strategy concept, introducing a graph-theoretic framework and the concept of* hyperpath, *which is an acyclic subnetwork, connecting the origin to the destination and including a subset of diversion nodes and a subset of diversion links. At each diversion node, the choice of diversion link depends on the occurrences of transit services and therefore there are certain probabilities for choosing a link among the alternative diversion links [16].

In general, two types of graph representation of a transit service network can be used:* line graph* and* run graph*. While nodes of a* line graph* (see Figure 1) have only spatial coordinates, in a* run graph* the nodes have space-time coordinates (*diachronic graph*). Hence, below we refer to two types of hyperpath representations:* line hyperpaths *and* run hyperpaths. *To each line hyperpath corresponds run hyperpaths with the same spatial nodes, but with different temporal coordinates for each spatial node.