# A Subjective Optimal Strategy for Transit Simulation Models

Authors: Agostino Nuzzolo; Antonio Comi
Journal: Journal of Advanced Transportation
 (2018)

---

## 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.


---

## Body

## 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 ofmultiservice 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, andoptimal 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 theobjective (ornormative) 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(ordescriptive) 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, asubjective 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, anobjective 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: Section2 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 pairod and anunreliable transit service network withdiversion nodes, that is, nodes where choices are made among different subpaths. Because of transit service stochasticity, rather than relying on apretrip selected single path from origin up to destination, users should adopt a travel strategyST 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 ofhyperpath,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 andrun graph. While nodes of aline graph (see Figure 1) have only spatial coordinates, in arun graph the nodes have space-time coordinates (diachronic graph). Hence, below we refer to two types of hyperpath representations:line hyperpathsandrun hyperpaths.To each line hyperpath corresponds run hyperpaths with the same spatial nodes, but with different temporal coordinates for each spatial node.Figure 1
Example of line and diachronic graphs.


## 3. Optimal Travel Strategy Search

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.


### 3.1. Objective Optimal Travel Strategies as Solutions to Markov Decision Problems

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 generaltheoretical framework forobjective optimal strategy search can be found in stochastic decision theory. If path choice is considered as decision making in a Markov decision process (MDPs), the Markov decision problem (MDPm) approach can be considered, as, for example, reported in Nuzzolo and Comi [17] and as summarised below for the reader's convenience.A Markov decision process (MDPs; [18]) can be defined by the quintuple (T;SSτ;Asτ;pτ[s′/s,a];rτ[s,a]), where(i)
T is aset of stagesτ at which the decision maker observes the state of the system and may make decisions,(ii)
SS is thestate space, where SSτ refers to the possible system states for a specific time τ,(iii)
Asτ is the set of possible actions that can be taken after observing states at time τ,(iv)
pτ[s′/s,a] are thetransition probabilities, determining how the system will move to the next state. In particular, pτ[s′/s,a] defines the transition to state s′ belonging to SSτ+1 at time τ+1 and, as a Markov process, only depends on states and chosen actiona at time τ.P is the set of transition probabilities,(v)
rτ[s,a] is thereward function, which determines the consequence for the decision maker’s choice of actiona while in states, andR is the reward set. In our cases, the value of the reward depends on the next state of the system, effectively becoming anexpected reward, expressed as(1)rτs,a=∑s′∈SSτ+1rτs,a,s′·pτs′s,awhererτs,a,s′ is the relative reward when the system is next in state s′.An MDPs with a specified optimality criterion (hence forming a sextuple) is called aMarkov decision problem MDPm. Policies π are essentially functions that regulate, for each state, which actions to perform. The solution of an MDPm provides the decision maker with anoptimal policyπ∗ that associates to statesSS actionsAoptimising a predefined objective function.


### 3.2. Objective Optimal Travel Strategy as an Optimal Policy of MDPm

Given arunservice network, the optimal travel strategy ST∗ can be seen as the optimal policy π∗ of a finite and discreteMDPm, considering that(i)
the setTis the set of times (τ) when the traveller is at a diversion nodes and a diversion link has to be chosen;(ii)
the state space setSS is the set of diversion nodes among which travellers can move;(iii)
an actiona is a set of diversion links among which travellers can choose with a given diversion rule and the action set Asτ is the set of actionsa;(iv)
the change in the time of traveller location within the diversion node set consists in a Markov process;(v)
the transition probabilitiespτ[s′/s,a] are the probabilities of going from a diversion node (s) to each of the following diversion nodes (s′) if action a is applied;(vi)
the reward functionrτ[s;a] is the expected utility of applying action a at diversion node s;(vii)
the optimal policyπ∗ gives the best sequence of actions, considering the expected utility up to destination.To represent an MDPm, astate-action tree can be used. At every diversion node, each action can be represented with a set of outgoing links to the next diversion nodes. In Figure 2, in relation to the diachronic graph, the decision tree is reported. For example, at diversion nodeF three different actions are possible: (1) using run 7.1 (action a7) and hence stopG; (2) using run 8.1 (action a8), and hence stopE; (3) using both run 7.1 and run 8.1, with the diversion rule of comparing the expected utility of boarding the first arriving run and the expected utility of the next run and then choosing the best (action a7+8). With regard to transition probabilities, consider the case of diversion nodeF in Figure 2. If action a7+8 is applied, the probability of moving onto node G is equal to the probability of using line 7, and the probability of moving onto node E is equal to the probability of using line 8. If action a7 is applied, the probability of going onto node G is equal to 1. The same holds for action a8 and node E.Figure 2
Example of run hyperpath with a diversion choice at origin.


### 3.3. Objective Optimal Strategy Search Methods

As explored above, the search for anobjective optimal travel strategy in a transit network is equivalent to the solution of a Markov decision problem, MDPm. This solution, when the transition probabilities and the expected rewards are known or computable, can be found through exact linear or dynamic programming algorithms. In particular, efficient network algorithms based on the Bellman equation [19] can be used, as in Nguyen and Pallottino [15]. For example, in the case of the optimal strategy reported in Spiess and Florian [12], hypotheses of random arrivals of buses and users at stops and limited information on services allow the transition probabilities to be computed analytically, although such hypotheses are often not congruent with the case studies in question. A more recent example is the dynamic routing of Gentile [20]. Note that in this case operating conditions are assumed for the transit system in several cases very different from the real ones. In order to take into account the specific case study conditions without knowledge of transition probabilities, some authors use MDPm approximate solution approaches, such as enforcement learning methods (see, for example, the simulator MILATRANS, in [21]). However, this approach requires processes of exploration and exploitation with excessive computation times to reproduce each event. Other authors, in order to consider the actual conditions of the case study, use an adaptive routing problem in a stochastic time-dependent transit network, in which the link travel times are discrete random variables with known probability distributions [22]. Nuzzolo and Comi [11, 17] indicate a way to estimate the transition probabilities and the expected rewards for intelligent transit networks and thus apply an exactobjective optimal strategy search method.


### 3.4. Subjective Optimal Strategy Search

In order to find thesubjective optimal strategy given the actual conditions of the case study, some authors assume diversion rules which are too complex in relation to travellers’ cognitive capacity. For example, a comparison of optimal subhyperpaths is applied by Nuzzoloet al. [2] in the simulator DYBUSRT.


## 3.1. Objective Optimal Travel Strategies as Solutions to Markov Decision Problems

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 generaltheoretical framework forobjective optimal strategy search can be found in stochastic decision theory. If path choice is considered as decision making in a Markov decision process (MDPs), the Markov decision problem (MDPm) approach can be considered, as, for example, reported in Nuzzolo and Comi [17] and as summarised below for the reader's convenience.A Markov decision process (MDPs; [18]) can be defined by the quintuple (T;SSτ;Asτ;pτ[s′/s,a];rτ[s,a]), where(i)
T is aset of stagesτ at which the decision maker observes the state of the system and may make decisions,(ii)
SS is thestate space, where SSτ refers to the possible system states for a specific time τ,(iii)
Asτ is the set of possible actions that can be taken after observing states at time τ,(iv)
pτ[s′/s,a] are thetransition probabilities, determining how the system will move to the next state. In particular, pτ[s′/s,a] defines the transition to state s′ belonging to SSτ+1 at time τ+1 and, as a Markov process, only depends on states and chosen actiona at time τ.P is the set of transition probabilities,(v)
rτ[s,a] is thereward function, which determines the consequence for the decision maker’s choice of actiona while in states, andR is the reward set. In our cases, the value of the reward depends on the next state of the system, effectively becoming anexpected reward, expressed as(1)rτs,a=∑s′∈SSτ+1rτs,a,s′·pτs′s,awhererτs,a,s′ is the relative reward when the system is next in state s′.An MDPs with a specified optimality criterion (hence forming a sextuple) is called aMarkov decision problem MDPm. Policies π are essentially functions that regulate, for each state, which actions to perform. The solution of an MDPm provides the decision maker with anoptimal policyπ∗ that associates to statesSS actionsAoptimising a predefined objective function.


## 3.2. Objective Optimal Travel Strategy as an Optimal Policy of MDPm

Given arunservice network, the optimal travel strategy ST∗ can be seen as the optimal policy π∗ of a finite and discreteMDPm, considering that(i)
the setTis the set of times (τ) when the traveller is at a diversion nodes and a diversion link has to be chosen;(ii)
the state space setSS is the set of diversion nodes among which travellers can move;(iii)
an actiona is a set of diversion links among which travellers can choose with a given diversion rule and the action set Asτ is the set of actionsa;(iv)
the change in the time of traveller location within the diversion node set consists in a Markov process;(v)
the transition probabilitiespτ[s′/s,a] are the probabilities of going from a diversion node (s) to each of the following diversion nodes (s′) if action a is applied;(vi)
the reward functionrτ[s;a] is the expected utility of applying action a at diversion node s;(vii)
the optimal policyπ∗ gives the best sequence of actions, considering the expected utility up to destination.To represent an MDPm, astate-action tree can be used. At every diversion node, each action can be represented with a set of outgoing links to the next diversion nodes. In Figure 2, in relation to the diachronic graph, the decision tree is reported. For example, at diversion nodeF three different actions are possible: (1) using run 7.1 (action a7) and hence stopG; (2) using run 8.1 (action a8), and hence stopE; (3) using both run 7.1 and run 8.1, with the diversion rule of comparing the expected utility of boarding the first arriving run and the expected utility of the next run and then choosing the best (action a7+8). With regard to transition probabilities, consider the case of diversion nodeF in Figure 2. If action a7+8 is applied, the probability of moving onto node G is equal to the probability of using line 7, and the probability of moving onto node E is equal to the probability of using line 8. If action a7 is applied, the probability of going onto node G is equal to 1. The same holds for action a8 and node E.Figure 2
Example of run hyperpath with a diversion choice at origin.


## 3.3. Objective Optimal Strategy Search Methods

As explored above, the search for anobjective optimal travel strategy in a transit network is equivalent to the solution of a Markov decision problem, MDPm. This solution, when the transition probabilities and the expected rewards are known or computable, can be found through exact linear or dynamic programming algorithms. In particular, efficient network algorithms based on the Bellman equation [19] can be used, as in Nguyen and Pallottino [15]. For example, in the case of the optimal strategy reported in Spiess and Florian [12], hypotheses of random arrivals of buses and users at stops and limited information on services allow the transition probabilities to be computed analytically, although such hypotheses are often not congruent with the case studies in question. A more recent example is the dynamic routing of Gentile [20]. Note that in this case operating conditions are assumed for the transit system in several cases very different from the real ones. In order to take into account the specific case study conditions without knowledge of transition probabilities, some authors use MDPm approximate solution approaches, such as enforcement learning methods (see, for example, the simulator MILATRANS, in [21]). However, this approach requires processes of exploration and exploitation with excessive computation times to reproduce each event. Other authors, in order to consider the actual conditions of the case study, use an adaptive routing problem in a stochastic time-dependent transit network, in which the link travel times are discrete random variables with known probability distributions [22]. Nuzzolo and Comi [11, 17] indicate a way to estimate the transition probabilities and the expected rewards for intelligent transit networks and thus apply an exactobjective optimal strategy search method.


## 3.4. Subjective Optimal Strategy Search

In order to find thesubjective optimal strategy given the actual conditions of the case study, some authors assume diversion rules which are too complex in relation to travellers’ cognitive capacity. For example, a comparison of optimal subhyperpaths is applied by Nuzzoloet al. [2] in the simulator DYBUSRT.


## 4. The Proposed Behavioural Framework

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 [13], allows for service occurrences and information provided to travellers and considers some travellers’ cognitive limitations and simplifications. In the following subsections, the proposed behavioural framework is presented and examined in the MDPm perspective. Further, some application examples are reported.Traveller behavioural assumptions are defined in the context of(i)
an unreliable or stochastic and within-day dynamic transit service network with diversion nodes;(ii)
transit users who often travel on the origin-destination (O-D) pair (frequent users) and are equipped with advanced mobile route planners with real-time individual predictive information, supplying a set of suitable lines and relative path attributes (i.e., travel time components) from current position to destination;(iii)
subjective optimal strategy-based travel behaviour.


### 4.1. Traveller Behavioural Hypotheses

#### 4.1.1. Master Hyperpaths

Given an O-D pairod and its set of available paths at time τ, traveller u, as a frequent user onO-D pairod, and with the support of an advanced transit trip planner, is assumed to consider a subset of line paths feasible for the traveller. That is, paths that satisfy some logical and behavioural constraints, sometime called a mental map (see, e.g., [21]) and here calledmaster line hyperpath  MHPodu,τ (Figure 3). Due to the randomness of transit services, travellers do not refer exactly to time τ but to a time slice Δτ (e.g., Δτ=τ±5 min.), even if, for simplicity, we continue to use τ below.Figure 3
Example of a master line hyperpath.As a master line hyperpathMHP can depend on time slice τ and day t,due to within-day and day-to-day dynamicity of the transit service, it is indicated as MHPodu,τ,t. A master line hyperpath can be dynamically upgraded at each diversion node with respect to the service state at timeτof dayt. For example, information on disrupted lines allows such lines to be eliminated.


#### 4.1.2. Experienced Path Utility

Given aline service graph, a travel strategyST is defined through aline hyperpath HP from origin o to destination d, with a set of diversion nodes and adiversion ruledri, for each diversion node i, which determines the diversion link choice behaviour at that node. Hence a strategySTwill be indicated as ST[HP;dr],withdr the set of diversion rules dri. Note that on a service network, several strategies and therefore several relative hyperpaths can be used. Given a diversion rule and an objective functionOf,the strategy ST∗[HP∗,dr] which optimises this function is theoptimal strategy conditional upon the diversion ruledrand the objective function Of, with HP∗ the relative optimal hyperpath.As a result of random service occurrences and traveller’s choices according to a diversion ruledr, each feasible pathk from the origin to the destination has a certain probability of use and itsexperienced path utility  TUku,τ,t is a random variable.Therefore, it can be assumed that travellers consider the averageATU of long-period experienced values of all randomTUrelative to all paths of strategyST. Thus, thesubjective optimal strategy  ST∗ is the strategy with maximum average experienced utility ATU∗ perceived by the traveller.


#### 4.1.3. Dynamic Travel Choices and Diversion Rule

We assume that a travelleru, in order to optimise his/her travel utility, applies the following dynamic travel behaviour: “Given a master line hyperpath, at each diversion node an optimal diversion link is chosen (with the diversion rule reported below) and the relative path is used up to the next diversion node, where a new optimal diversion link is chosen and used.”The proposed diversion ruledri is composed as follows: given a master line hyperpath MHPodu,τ,t, at diversion nodeiand time τ of day t,traveller u considers all the diversion linksil, associates to each of them ananticipated utility, defined below, and chooses the diversion link il∗ with maximum anticipated utility.


#### 4.1.4. Diversion Link Anticipated Utility

Given a diversion nodei and a diversion linkil, the anticipated utility AUij is obtained by summing:(i)
the anticipated utilityAUk of the subpath from diversion node i up to the next diversion nodew, including the diversion linkil;(ii)
thenodal anticipated utility AUw of the diversion node w up to the next nodes w′.For example, the anticipated utility of linkB-F of Figure 2 is given by the anticipated utility of subpathB-F plus the nodal anticipated utility of nodeF, which in turn is a function of the anticipated utility of subpathsF-E-D andF-G-D.


#### 4.1.5. Anticipated Utility of Subpaths (AUk)

Given subpathk up to the next diversion nodew, theanticipated utilityAUk at time τ of day t is a linear function of the vector AXku,τ,t of its attributesAX,anticipated by traveller u at time τ of day t:(2)AUku,τ,t=∑jηju·AXk,ju,τ,twith ηju parameters of the utility function. In turn, the attributesAX anticipated by travellers are functions of path attributes forecasted (if any) by travellers and those forecasted by the information system:(3)AXk,ju,τ,t=ξu·FXk,jτ,t+1-ξu·PXk,ju,τ,t,where(i)
AXk,ju,τ,t is the j-thanticipated attribute value at time τ of day t;(ii)
FXk,jτ,t is the j-th attribute valueforecasted by the information system;(iii)
PXk,ju,τ,t is the value (if any) ofj-th attribute forecasted by traveller u at time τ of day t (traveller forecasting process);(iv)
ξu∈[0,1] is theweight given by traveller u to the information provided, dependent on the traveller’s compliance with the information system.


#### 4.1.6. Traveller Forecasted Attributes of a Path

Assuming that travellers use an exponential smoothing forecasting method [23], the values PXk,ju,τ,t of the j-th attributes forecasted by travelleru at time τ of day t are assumed as(4)PXk,ju,τ,t=υu·TXk,ju,τ,t-1+1-υu·PXk,ju,τ,t-1where(i)
TXk,ju,τ,t-1 is the value of thej-th attribute experienced by travelleru at time τ of dayt-1;(ii)
PXk,ju,τ,t-1 is the value of thej-th attribute forecasted by travelleru, at time τ of dayt-1;(iii)
νu∈[0,1] is the weight given to attributes experienced on dayt-1, depending on the memory process of travelleru.


#### 4.1.7. Nodal Anticipated Utility of Next Diversion Nodew (AUw)

The nodal anticipated utilityAUw at time τ of dayt, of the diversion node w with Gw subpaths k′ up to their next diversion nodes w′, is obtained by travellers as a function of the anticipated utilities of these subpaths k′:(5)AUwu,τ,t=∑k′∈Gwpu,τ,tk′·AUk′u,τ,twhere pu,τ,tk′ is theperceived share of using path k′ at time τ in the past days and AUk′u,τ,t is the anticipated utility of subpath k′ at time τ of day t. It is assumed that the values of shares pu,τ,tk′ perceived by traveller u at time τ of day t are given by(6)pu,τ,tk′=∑t′=1t-1αk′u,τ,t′∑j∈Gw∑t′=1t-1αju,τ,t′where(i)
pu,τ,tk′ is, at time τ of dayt, the perceived share of using path k′;(ii)
αk′u,τ,t′ is the weight given by the traveller to path k′ in relation to day t′;(7)αk′τ,t′=0if  pathk′was  not  used  at  dayt′ϕut-t′if  pathk′was  used  at  dayt′,withϕu the parameter of the traveller’s memory process.In the learning process, travellers search for the optimal weightsξ which maximise the average experienced utility (ATU), as simulated in the application test of Section 4.2 below.


#### 4.1.8. Example of Diversion Choices

As an example of a diversion choice, consider the choice at originO of the first boarding stop in the master hyperpath of Figures 2 and 3. Travelleruis assumed:(i)
to identify, within the master line hyperpath, the set of diversion links with the root onO, in our case the linksO-B andO-C;(ii)
to associate an anticipated utilityAUol to each diversion linkol, in our case:
(a)
for linkO-Cas the anticipated utility of pathO-C-E-D;(b)
for linkO-B as the sum of the anticipated utility of linkO-B and nodal anticipated utility of nodeB, which considers the anticipated utilities of pathB-G-D and the anticipated utility of pathB-F plus the nodal utility of diversion nodeF.(iii)
to use the diversion linkol∗ with the maximum anticipated utility AU∗.Subsequently, at timeτ′, when travelleru is at the (first boarding or interchanging) stop i and a run r of a line belonging to the run master hyperpath arrives (as depicted in Figure 2), s(he) is assumed:(i)
to consider the diversion linkilr to board runr and the diversion link ilw to wait;(ii)
to associate ananticipated utility to each of the two above diversion links;(iii)
to compare the anticipated utilities of these diversion links;(iv)
to board runr if the anticipated utility associated with the link incorporating runr is greater than the maximum anticipated utility associated with waiting link ilw;(v)
if the traveller does not board runr, the process is reapplied when the next run arrives.


#### 4.1.9. Model Parameter Estimation

The application of the presented model requires the knowledge of the following parameters:(i)
ηj are the parameters of the anticipated utility function AUk,(ii)
ξu∈[0,1] is the weight given by travellers to the information provided,(iii)
νu∈[0,1] is the weight given to attributes experienced on dayt-1,(iv)
ϕu is the parameter of traveller’s memory process of the perceived share of using path k′.Parametersηj can be obtained, for example, with standard stated-preference surveys and aggregate random utility model calibration. Parameters ν, ξ, and ϕ can be obtained applying a reverse assignment procedure [14], minimising the distance between measured alighting and boarding (or on-board) counts and those obtained through the model [24, 25].


### 4.2. An Application to a Real Network

An application of the proposed path choice modelling, with a unique subjective optimal strategy and the same parametersν, ξ, and ϕ for all travellers, within the assignment model in DYBUSRT [2], was carried out for the same network as in the authors’ other studies in the field of run-oriented transit assignment. The aim of the application was to assess how different values of parameters ξ and hence different combinations of the forecasted utilities of the traveller and information system affect expected values of average experienced utilityATU.The service network (Figure4) was obtained from the real service structure of the Fuorigrotta district in Naples (Italy), whose bus running time variation coefficients were appropriately modified for the purpose of the simulation. The study area consists of 11 traffic zones served by 11 transit lines, which supply 245 runs (with an average of 6 runs per hour on each transit line) from 7:00am to 10:00am on a typical workday (dayt).Figure 4
The application network [2].As regards the master line hyperpath, according to the literature on choice set formation and as reviewed by Bovy [26], the master set of path alternatives was generated from the set of all available paths and then considering logical constraints to avoid loops, successive boarding of the same run or the use of opposite lines, and behavioural constraints to eliminate unrealistic alternatives in terms of maximum values of attributes, such as number of transfers, transfer time, access and egress times, and schedule delay. Combining the residual paths, a master line hyperpath from each origino to each destinationd was generated. Level-of-service attributes composing path utilities were calculated by using a diachronic graph, whose service subgraph consists of about 10,400 nodes and 20,100 links. The experienced path utility function is the same as that reported by Nuzzoloet al. [2].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 parameterν equal to 0.3 [27, 28], while ϕ was hypothesized equal to 1.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 Table1) indicate that the weights used for combining the utilities in question strongly influence the average experienced utility and that the weights to use in order to minimise the experienced travel disutility strongly depend on the unreliability of the transit system. As expected, with increasing transit service unreliability and hence with increasing forecasting failures, the best overall performances are obtained with the use of a low ξ parameter, to give much more weight to personal than to system forecasted attribute values.Table 1
Application results.
Running time cv ξ Average value of T U ∗ Running time cv ξ Average value of T U ∗ 0.20 0.00 -1343.1 0.75 0.00 -504.1 0.25 -1058.0 0.25 -410.9 0.50 -992.2 0.50 -420.6 0.75 -713.7 0.75 -521.8 1.00 -978.8 1.00 -491.7  0.50 0.00 -874.5 1.00 0.00 -477.9 0.25 -704.3 0.25 -376.7 0.50 -649.4 0.50 -407.5 0.75 -938.3 0.75 -495.5 1.00 -973.8 1.00 -547.7 ∗The average experienced utility ATU is computed for 60 simulation days.


### 4.3. The Proposed Behavioural Framework from an MDPm Perspective

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(i)
the master hyperpath is found by considering quite simple logical and behavioural constraints (see Figure3);(ii)
the perceived shares of use of subpathsk′ at time τ in previous days, from the diversion nodew up to the next diversion node, are proxies for transition probabilities (see (6));(iii)
the anticipated utilities are proxies for expected rewards (see (2)). Indeed, the anticipated utilities are functions of the anticipated path attributes, given by a combination of the values forecasted by the information system and the values forecasted by travellers. The information-system forecasted attribute values, if obtained through statistical forecasting methods, are estimates of expected values. The traveller’s forecasted attribute values are obtained through exponential smoothing methods, hence proxies of expected values. Thus the anticipated utilities can be assumed to be proxies of expected utilities;(iv)
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 Figure2, the state-action trees are simplified, as reported in Figure 5, where at nodeB the only possible action is a5+6 while at nodeF it is action a7+8.Figure 5
Example of reduced action tree of Figure2.


## 4.1. Traveller Behavioural Hypotheses

### 4.1.1. Master Hyperpaths

Given an O-D pairod and its set of available paths at time τ, traveller u, as a frequent user onO-D pairod, and with the support of an advanced transit trip planner, is assumed to consider a subset of line paths feasible for the traveller. That is, paths that satisfy some logical and behavioural constraints, sometime called a mental map (see, e.g., [21]) and here calledmaster line hyperpath  MHPodu,τ (Figure 3). Due to the randomness of transit services, travellers do not refer exactly to time τ but to a time slice Δτ (e.g., Δτ=τ±5 min.), even if, for simplicity, we continue to use τ below.Figure 3
Example of a master line hyperpath.As a master line hyperpathMHP can depend on time slice τ and day t,due to within-day and day-to-day dynamicity of the transit service, it is indicated as MHPodu,τ,t. A master line hyperpath can be dynamically upgraded at each diversion node with respect to the service state at timeτof dayt. For example, information on disrupted lines allows such lines to be eliminated.


### 4.1.2. Experienced Path Utility

Given aline service graph, a travel strategyST is defined through aline hyperpath HP from origin o to destination d, with a set of diversion nodes and adiversion ruledri, for each diversion node i, which determines the diversion link choice behaviour at that node. Hence a strategySTwill be indicated as ST[HP;dr],withdr the set of diversion rules dri. Note that on a service network, several strategies and therefore several relative hyperpaths can be used. Given a diversion rule and an objective functionOf,the strategy ST∗[HP∗,dr] which optimises this function is theoptimal strategy conditional upon the diversion ruledrand the objective function Of, with HP∗ the relative optimal hyperpath.As a result of random service occurrences and traveller’s choices according to a diversion ruledr, each feasible pathk from the origin to the destination has a certain probability of use and itsexperienced path utility  TUku,τ,t is a random variable.Therefore, it can be assumed that travellers consider the averageATU of long-period experienced values of all randomTUrelative to all paths of strategyST. Thus, thesubjective optimal strategy  ST∗ is the strategy with maximum average experienced utility ATU∗ perceived by the traveller.


### 4.1.3. Dynamic Travel Choices and Diversion Rule

We assume that a travelleru, in order to optimise his/her travel utility, applies the following dynamic travel behaviour: “Given a master line hyperpath, at each diversion node an optimal diversion link is chosen (with the diversion rule reported below) and the relative path is used up to the next diversion node, where a new optimal diversion link is chosen and used.”The proposed diversion ruledri is composed as follows: given a master line hyperpath MHPodu,τ,t, at diversion nodeiand time τ of day t,traveller u considers all the diversion linksil, associates to each of them ananticipated utility, defined below, and chooses the diversion link il∗ with maximum anticipated utility.


### 4.1.4. Diversion Link Anticipated Utility

Given a diversion nodei and a diversion linkil, the anticipated utility AUij is obtained by summing:(i)
the anticipated utilityAUk of the subpath from diversion node i up to the next diversion nodew, including the diversion linkil;(ii)
thenodal anticipated utility AUw of the diversion node w up to the next nodes w′.For example, the anticipated utility of linkB-F of Figure 2 is given by the anticipated utility of subpathB-F plus the nodal anticipated utility of nodeF, which in turn is a function of the anticipated utility of subpathsF-E-D andF-G-D.


### 4.1.5. Anticipated Utility of Subpaths (AUk)

Given subpathk up to the next diversion nodew, theanticipated utilityAUk at time τ of day t is a linear function of the vector AXku,τ,t of its attributesAX,anticipated by traveller u at time τ of day t:(2)AUku,τ,t=∑jηju·AXk,ju,τ,twith ηju parameters of the utility function. In turn, the attributesAX anticipated by travellers are functions of path attributes forecasted (if any) by travellers and those forecasted by the information system:(3)AXk,ju,τ,t=ξu·FXk,jτ,t+1-ξu·PXk,ju,τ,t,where(i)
AXk,ju,τ,t is the j-thanticipated attribute value at time τ of day t;(ii)
FXk,jτ,t is the j-th attribute valueforecasted by the information system;(iii)
PXk,ju,τ,t is the value (if any) ofj-th attribute forecasted by traveller u at time τ of day t (traveller forecasting process);(iv)
ξu∈[0,1] is theweight given by traveller u to the information provided, dependent on the traveller’s compliance with the information system.


### 4.1.6. Traveller Forecasted Attributes of a Path

Assuming that travellers use an exponential smoothing forecasting method [23], the values PXk,ju,τ,t of the j-th attributes forecasted by travelleru at time τ of day t are assumed as(4)PXk,ju,τ,t=υu·TXk,ju,τ,t-1+1-υu·PXk,ju,τ,t-1where(i)
TXk,ju,τ,t-1 is the value of thej-th attribute experienced by travelleru at time τ of dayt-1;(ii)
PXk,ju,τ,t-1 is the value of thej-th attribute forecasted by travelleru, at time τ of dayt-1;(iii)
νu∈[0,1] is the weight given to attributes experienced on dayt-1, depending on the memory process of travelleru.


### 4.1.7. Nodal Anticipated Utility of Next Diversion Nodew (AUw)

The nodal anticipated utilityAUw at time τ of dayt, of the diversion node w with Gw subpaths k′ up to their next diversion nodes w′, is obtained by travellers as a function of the anticipated utilities of these subpaths k′:(5)AUwu,τ,t=∑k′∈Gwpu,τ,tk′·AUk′u,τ,twhere pu,τ,tk′ is theperceived share of using path k′ at time τ in the past days and AUk′u,τ,t is the anticipated utility of subpath k′ at time τ of day t. It is assumed that the values of shares pu,τ,tk′ perceived by traveller u at time τ of day t are given by(6)pu,τ,tk′=∑t′=1t-1αk′u,τ,t′∑j∈Gw∑t′=1t-1αju,τ,t′where(i)
pu,τ,tk′ is, at time τ of dayt, the perceived share of using path k′;(ii)
αk′u,τ,t′ is the weight given by the traveller to path k′ in relation to day t′;(7)αk′τ,t′=0if  pathk′was  not  used  at  dayt′ϕut-t′if  pathk′was  used  at  dayt′,withϕu the parameter of the traveller’s memory process.In the learning process, travellers search for the optimal weightsξ which maximise the average experienced utility (ATU), as simulated in the application test of Section 4.2 below.


### 4.1.8. Example of Diversion Choices

As an example of a diversion choice, consider the choice at originO of the first boarding stop in the master hyperpath of Figures 2 and 3. Travelleruis assumed:(i)
to identify, within the master line hyperpath, the set of diversion links with the root onO, in our case the linksO-B andO-C;(ii)
to associate an anticipated utilityAUol to each diversion linkol, in our case:
(a)
for linkO-Cas the anticipated utility of pathO-C-E-D;(b)
for linkO-B as the sum of the anticipated utility of linkO-B and nodal anticipated utility of nodeB, which considers the anticipated utilities of pathB-G-D and the anticipated utility of pathB-F plus the nodal utility of diversion nodeF.(iii)
to use the diversion linkol∗ with the maximum anticipated utility AU∗.Subsequently, at timeτ′, when travelleru is at the (first boarding or interchanging) stop i and a run r of a line belonging to the run master hyperpath arrives (as depicted in Figure 2), s(he) is assumed:(i)
to consider the diversion linkilr to board runr and the diversion link ilw to wait;(ii)
to associate ananticipated utility to each of the two above diversion links;(iii)
to compare the anticipated utilities of these diversion links;(iv)
to board runr if the anticipated utility associated with the link incorporating runr is greater than the maximum anticipated utility associated with waiting link ilw;(v)
if the traveller does not board runr, the process is reapplied when the next run arrives.


### 4.1.9. Model Parameter Estimation

The application of the presented model requires the knowledge of the following parameters:(i)
ηj are the parameters of the anticipated utility function AUk,(ii)
ξu∈[0,1] is the weight given by travellers to the information provided,(iii)
νu∈[0,1] is the weight given to attributes experienced on dayt-1,(iv)
ϕu is the parameter of traveller’s memory process of the perceived share of using path k′.Parametersηj can be obtained, for example, with standard stated-preference surveys and aggregate random utility model calibration. Parameters ν, ξ, and ϕ can be obtained applying a reverse assignment procedure [14], minimising the distance between measured alighting and boarding (or on-board) counts and those obtained through the model [24, 25].


## 4.1.1. Master Hyperpaths

Given an O-D pairod and its set of available paths at time τ, traveller u, as a frequent user onO-D pairod, and with the support of an advanced transit trip planner, is assumed to consider a subset of line paths feasible for the traveller. That is, paths that satisfy some logical and behavioural constraints, sometime called a mental map (see, e.g., [21]) and here calledmaster line hyperpath  MHPodu,τ (Figure 3). Due to the randomness of transit services, travellers do not refer exactly to time τ but to a time slice Δτ (e.g., Δτ=τ±5 min.), even if, for simplicity, we continue to use τ below.Figure 3
Example of a master line hyperpath.As a master line hyperpathMHP can depend on time slice τ and day t,due to within-day and day-to-day dynamicity of the transit service, it is indicated as MHPodu,τ,t. A master line hyperpath can be dynamically upgraded at each diversion node with respect to the service state at timeτof dayt. For example, information on disrupted lines allows such lines to be eliminated.


## 4.1.2. Experienced Path Utility

Given aline service graph, a travel strategyST is defined through aline hyperpath HP from origin o to destination d, with a set of diversion nodes and adiversion ruledri, for each diversion node i, which determines the diversion link choice behaviour at that node. Hence a strategySTwill be indicated as ST[HP;dr],withdr the set of diversion rules dri. Note that on a service network, several strategies and therefore several relative hyperpaths can be used. Given a diversion rule and an objective functionOf,the strategy ST∗[HP∗,dr] which optimises this function is theoptimal strategy conditional upon the diversion ruledrand the objective function Of, with HP∗ the relative optimal hyperpath.As a result of random service occurrences and traveller’s choices according to a diversion ruledr, each feasible pathk from the origin to the destination has a certain probability of use and itsexperienced path utility  TUku,τ,t is a random variable.Therefore, it can be assumed that travellers consider the averageATU of long-period experienced values of all randomTUrelative to all paths of strategyST. Thus, thesubjective optimal strategy  ST∗ is the strategy with maximum average experienced utility ATU∗ perceived by the traveller.


## 4.1.3. Dynamic Travel Choices and Diversion Rule

We assume that a travelleru, in order to optimise his/her travel utility, applies the following dynamic travel behaviour: “Given a master line hyperpath, at each diversion node an optimal diversion link is chosen (with the diversion rule reported below) and the relative path is used up to the next diversion node, where a new optimal diversion link is chosen and used.”The proposed diversion ruledri is composed as follows: given a master line hyperpath MHPodu,τ,t, at diversion nodeiand time τ of day t,traveller u considers all the diversion linksil, associates to each of them ananticipated utility, defined below, and chooses the diversion link il∗ with maximum anticipated utility.


## 4.1.4. Diversion Link Anticipated Utility

Given a diversion nodei and a diversion linkil, the anticipated utility AUij is obtained by summing:(i)
the anticipated utilityAUk of the subpath from diversion node i up to the next diversion nodew, including the diversion linkil;(ii)
thenodal anticipated utility AUw of the diversion node w up to the next nodes w′.For example, the anticipated utility of linkB-F of Figure 2 is given by the anticipated utility of subpathB-F plus the nodal anticipated utility of nodeF, which in turn is a function of the anticipated utility of subpathsF-E-D andF-G-D.


## 4.1.5. Anticipated Utility of Subpaths (AUk)

Given subpathk up to the next diversion nodew, theanticipated utilityAUk at time τ of day t is a linear function of the vector AXku,τ,t of its attributesAX,anticipated by traveller u at time τ of day t:(2)AUku,τ,t=∑jηju·AXk,ju,τ,twith ηju parameters of the utility function. In turn, the attributesAX anticipated by travellers are functions of path attributes forecasted (if any) by travellers and those forecasted by the information system:(3)AXk,ju,τ,t=ξu·FXk,jτ,t+1-ξu·PXk,ju,τ,t,where(i)
AXk,ju,τ,t is the j-thanticipated attribute value at time τ of day t;(ii)
FXk,jτ,t is the j-th attribute valueforecasted by the information system;(iii)
PXk,ju,τ,t is the value (if any) ofj-th attribute forecasted by traveller u at time τ of day t (traveller forecasting process);(iv)
ξu∈[0,1] is theweight given by traveller u to the information provided, dependent on the traveller’s compliance with the information system.


## 4.1.6. Traveller Forecasted Attributes of a Path

Assuming that travellers use an exponential smoothing forecasting method [23], the values PXk,ju,τ,t of the j-th attributes forecasted by travelleru at time τ of day t are assumed as(4)PXk,ju,τ,t=υu·TXk,ju,τ,t-1+1-υu·PXk,ju,τ,t-1where(i)
TXk,ju,τ,t-1 is the value of thej-th attribute experienced by travelleru at time τ of dayt-1;(ii)
PXk,ju,τ,t-1 is the value of thej-th attribute forecasted by travelleru, at time τ of dayt-1;(iii)
νu∈[0,1] is the weight given to attributes experienced on dayt-1, depending on the memory process of travelleru.


## 4.1.7. Nodal Anticipated Utility of Next Diversion Nodew (AUw)

The nodal anticipated utilityAUw at time τ of dayt, of the diversion node w with Gw subpaths k′ up to their next diversion nodes w′, is obtained by travellers as a function of the anticipated utilities of these subpaths k′:(5)AUwu,τ,t=∑k′∈Gwpu,τ,tk′·AUk′u,τ,twhere pu,τ,tk′ is theperceived share of using path k′ at time τ in the past days and AUk′u,τ,t is the anticipated utility of subpath k′ at time τ of day t. It is assumed that the values of shares pu,τ,tk′ perceived by traveller u at time τ of day t are given by(6)pu,τ,tk′=∑t′=1t-1αk′u,τ,t′∑j∈Gw∑t′=1t-1αju,τ,t′where(i)
pu,τ,tk′ is, at time τ of dayt, the perceived share of using path k′;(ii)
αk′u,τ,t′ is the weight given by the traveller to path k′ in relation to day t′;(7)αk′τ,t′=0if  pathk′was  not  used  at  dayt′ϕut-t′if  pathk′was  used  at  dayt′,withϕu the parameter of the traveller’s memory process.In the learning process, travellers search for the optimal weightsξ which maximise the average experienced utility (ATU), as simulated in the application test of Section 4.2 below.


## 4.1.8. Example of Diversion Choices

As an example of a diversion choice, consider the choice at originO of the first boarding stop in the master hyperpath of Figures 2 and 3. Travelleruis assumed:(i)
to identify, within the master line hyperpath, the set of diversion links with the root onO, in our case the linksO-B andO-C;(ii)
to associate an anticipated utilityAUol to each diversion linkol, in our case:
(a)
for linkO-Cas the anticipated utility of pathO-C-E-D;(b)
for linkO-B as the sum of the anticipated utility of linkO-B and nodal anticipated utility of nodeB, which considers the anticipated utilities of pathB-G-D and the anticipated utility of pathB-F plus the nodal utility of diversion nodeF.(iii)
to use the diversion linkol∗ with the maximum anticipated utility AU∗.Subsequently, at timeτ′, when travelleru is at the (first boarding or interchanging) stop i and a run r of a line belonging to the run master hyperpath arrives (as depicted in Figure 2), s(he) is assumed:(i)
to consider the diversion linkilr to board runr and the diversion link ilw to wait;(ii)
to associate ananticipated utility to each of the two above diversion links;(iii)
to compare the anticipated utilities of these diversion links;(iv)
to board runr if the anticipated utility associated with the link incorporating runr is greater than the maximum anticipated utility associated with waiting link ilw;(v)
if the traveller does not board runr, the process is reapplied when the next run arrives.


## 4.1.9. Model Parameter Estimation

The application of the presented model requires the knowledge of the following parameters:(i)
ηj are the parameters of the anticipated utility function AUk,(ii)
ξu∈[0,1] is the weight given by travellers to the information provided,(iii)
νu∈[0,1] is the weight given to attributes experienced on dayt-1,(iv)
ϕu is the parameter of traveller’s memory process of the perceived share of using path k′.Parametersηj can be obtained, for example, with standard stated-preference surveys and aggregate random utility model calibration. Parameters ν, ξ, and ϕ can be obtained applying a reverse assignment procedure [14], minimising the distance between measured alighting and boarding (or on-board) counts and those obtained through the model [24, 25].


## 4.2. An Application to a Real Network

An application of the proposed path choice modelling, with a unique subjective optimal strategy and the same parametersν, ξ, and ϕ for all travellers, within the assignment model in DYBUSRT [2], was carried out for the same network as in the authors’ other studies in the field of run-oriented transit assignment. The aim of the application was to assess how different values of parameters ξ and hence different combinations of the forecasted utilities of the traveller and information system affect expected values of average experienced utilityATU.The service network (Figure4) was obtained from the real service structure of the Fuorigrotta district in Naples (Italy), whose bus running time variation coefficients were appropriately modified for the purpose of the simulation. The study area consists of 11 traffic zones served by 11 transit lines, which supply 245 runs (with an average of 6 runs per hour on each transit line) from 7:00am to 10:00am on a typical workday (dayt).Figure 4
The application network [2].As regards the master line hyperpath, according to the literature on choice set formation and as reviewed by Bovy [26], the master set of path alternatives was generated from the set of all available paths and then considering logical constraints to avoid loops, successive boarding of the same run or the use of opposite lines, and behavioural constraints to eliminate unrealistic alternatives in terms of maximum values of attributes, such as number of transfers, transfer time, access and egress times, and schedule delay. Combining the residual paths, a master line hyperpath from each origino to each destinationd was generated. Level-of-service attributes composing path utilities were calculated by using a diachronic graph, whose service subgraph consists of about 10,400 nodes and 20,100 links. The experienced path utility function is the same as that reported by Nuzzoloet al. [2].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 parameterν equal to 0.3 [27, 28], while ϕ was hypothesized equal to 1.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 Table1) indicate that the weights used for combining the utilities in question strongly influence the average experienced utility and that the weights to use in order to minimise the experienced travel disutility strongly depend on the unreliability of the transit system. As expected, with increasing transit service unreliability and hence with increasing forecasting failures, the best overall performances are obtained with the use of a low ξ parameter, to give much more weight to personal than to system forecasted attribute values.Table 1
Application results.
Running time cv ξ Average value of T U ∗ Running time cv ξ Average value of T U ∗ 0.20 0.00 -1343.1 0.75 0.00 -504.1 0.25 -1058.0 0.25 -410.9 0.50 -992.2 0.50 -420.6 0.75 -713.7 0.75 -521.8 1.00 -978.8 1.00 -491.7  0.50 0.00 -874.5 1.00 0.00 -477.9 0.25 -704.3 0.25 -376.7 0.50 -649.4 0.50 -407.5 0.75 -938.3 0.75 -495.5 1.00 -973.8 1.00 -547.7 ∗The average experienced utility ATU is computed for 60 simulation days.


## 4.3. The Proposed Behavioural Framework from an MDPm Perspective

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(i)
the master hyperpath is found by considering quite simple logical and behavioural constraints (see Figure3);(ii)
the perceived shares of use of subpathsk′ at time τ in previous days, from the diversion nodew up to the next diversion node, are proxies for transition probabilities (see (6));(iii)
the anticipated utilities are proxies for expected rewards (see (2)). Indeed, the anticipated utilities are functions of the anticipated path attributes, given by a combination of the values forecasted by the information system and the values forecasted by travellers. The information-system forecasted attribute values, if obtained through statistical forecasting methods, are estimates of expected values. The traveller’s forecasted attribute values are obtained through exponential smoothing methods, hence proxies of expected values. Thus the anticipated utilities can be assumed to be proxies of expected utilities;(iv)
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 Figure2, the state-action trees are simplified, as reported in Figure 5, where at nodeB the only possible action is a5+6 while at nodeF it is action a7+8.Figure 5
Example of reduced action tree of Figure2.


## 5. Conclusions and Research Perspectives

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.



---

*Source: 8797328-2018-09-16.xml*