# Dynamic Optimal Travel Strategies in Intelligent Stochastic Transit Networks

^{*}

## Abstract

**:**

## 1. Introduction

_{OD}be the subgraph that includes the available line paths connecting the O-D pair. If more than one line is available to reach the destination at some bus stops, the network is classified as a multiservice transit network (MSTN). An example of a multiservice network is reported in Figure 1, where at node F two lines are available to reach destination D. If some path attributes X (e.g., waiting time, on-board time, on-board occupancy degree) are random variables, and MSTN is classified as a stochastic multiservice network (SMSTN). In the following, only ordinary randomness is taken into account. For example, effects due to large service disruptions are not considered.

_{t}(a decision node) to another state S

_{t}

_{+1}, depending on the action chosen and on the stochastic occurrences in S

_{t}. The transition probabilities conditional upon an action “a” are the probabilities of going from a state S

_{t}to each of the following states S

_{t}

_{+1}if action “a” is applied. Thus a trip from origin to destination on a stochastic network with diversion nodes is a stochastic decision process (SDPs). If the transition probabilities and the expected rewards depend only on state S and not on the previous transitions, the decision process is considered as decision-making in a Markov decision process (MDPs), and the Markov decision problem (MDPm) approach [17] can be used. MDPm problems can be more easily solved when the transition probabilities and the expected rewards are known through Bellman’s algorithms [18,19].

## 2. The Line-Based Search of an Optimal Travel Strategy

#### 2.1. State–Action Tree Representation of a Decision Process (DP)

_{7}); (2) to use only line 8 (action a

_{8}); (3) to consider lines 7 and 8 (action a

_{7+8}), comparing the expected travel time of each alternative action and then choosing the best one.

- (B, a
_{6}); (F, a_{7}); (G, a_{9}); (D, a_{12}); - (B, a
_{6}); (F, a_{8}); (E, a_{10}); (D, a_{12}); - (B, a
_{6}); (F, a_{7+8}); (G, a_{9}); (E, a_{10}); (D, a_{12}).

_{aj}(s

_{i}, s

_{i}

_{−1}) of state s

_{i}, conditional upon the previous state s

_{i}

_{−1}and action a

_{j}, is in our case the expected travel cost of arriving in s

_{i}from s

_{i}

_{−1}with a

_{j}.

#### 2.2. Line-Based Transition Probabilities

_{7}is applied, only node G can follow, with a probability equal to 1. The same holds for deterministic action a

_{8}and node E. If at node F action a

_{7+8}is applied, the transition probability of moving onto node G is equal to the probability that, when the traveler observes the system state and makes a decision at node F, the expected travel time of choosing line 7 is better than that of line 8, and hence line 7 is chosen. In the same way, the transition probability of moving onto node E is equal to the probability of line 8 being chosen. The transition probabilities are represented through a matrix. For example, given the policy: (O, a

_{B}); (B, a

_{6}); (F, a

_{7+8}); (G, a

_{9}); (E, a

_{10}); (D, a

_{12}), and its state–action tree (Figure 2b), the line-based transition probability matrix is reported in Table 2.

#### 2.3. Transition Probability Determination in the Line-Based Approach

_{i}and average waiting times w

_{i}are evaluated not in relation to single bus states, but directly in relation to the line frequencies (φ

_{i}): p

_{i}= φ

_{i}/(Σ

_{j}φ

_{j}); w

_{i}= 1/(Σ

_{j}φ

_{j}).

## 3. Optimal Travel Strategy Search in a Run-Based Approach

## 4. Analysis of the Forecasting Error of Bus At-Stop Arrival Times

_{Sy}:

_{y}and ${F}_{{S}_{x}}A{T}_{{S}_{y}}$ is the forecasted value when the traveller is at stop S

_{x}.

- ${\sigma}_{\epsilon}$ is the standard deviation (in seconds) of random forecasting error ε;
- FH is the forecasting horizon (in minutes).

**ε**, with a number of an element equal to the number of runs n at stop Sy, will be assumed below to be distributed according to a multivariate normal (MVN) with zero mean, dispersion matrix

_{Sy}**Σ**and density probability given by:

## 5. The Proposed Run-Based Search Method of Dynamic Optimal Strategies

#### 5.1. General Description of the Method

_{o}and therefore, the forecasts are carried out at t

_{o}, is shown in Figure 6.

- the traveller state space includes only decision nodes;
- when the traveller moves on the next decision node, a new optimal strategy is searched, considering the new forecasts of at-stop bus arrival time (FAT
_{S}) and at-destination arrival times (FAT_{D}) available in real time. Therefore, each new optimal strategy is conditional upon the forecasts available at the current decision node; - the bus state space is not considered, as better analysed in the following part of the paper, and hence the transition probabilities concern only transitions of the traveller between decision nodes;
- as an expected reward, the forecasted travel time (or, in general, the forecasted travel disutility) is used. Thus the optimal policy determines the combination of actions with minimum forecasted travel time up to destination. Therefore forecasting methods have to be used with expected value of FAT:E[FAT] = AT;
- considering the analysis results reported in Section 4, the forecasted arrival times F
_{Sx}AT_{Sx}at stop S_{y}forecasted when the traveler is at stop S_{x}can be expressed as follows:$${F}_{{S}_{x}}A{T}_{{S}_{y}}^{}=A{T}_{{S}_{y}}^{}+{\epsilon}_{{S}_{y}}$$_{y}and**ε**is the forecasting error vector with known MVN probability density functions;_{Sy} - all the buses are assumed to have unlimited capacity.

#### 5.2. Run-Based Transition Probabilities

_{Y}(Figure 7). This is a very frequent case, because, even if a higher number of interchanges could actually exist, in general a feasible path with more than one interchanging stop is rarely used.

_{o}(see Figure 7). The optimal travel strategy from a first boarding stop I up to destination D, at time to, is searched. From stop I some runs R

_{I1}… R

_{IY}… R

_{IN}lead to the decision stops J

_{1}, … J

_{Y}, … J

_{N}, where some runs R

_{J}

_{1}, R

_{J}

_{2}, …, R

_{JY}…, R

_{JM}lead to the last alighting stops S

_{J}

_{1}, S

_{J}

_{2}, …, S

_{JY}…, S

_{JM}. If a larger number of lines is available between two stops, all their runs can be considered as runs of one equivalent line connecting the two stops.

_{JYY}/I] for the time to can be obtained as:

_{Y}to stop S

_{JYY}, and p[J

_{Y}/I] is the transition probability of moving from the decision node I to stop J

_{Y}.

_{JYY}/J

_{Y}] is equal to the probability that, when the traveler will be at J

_{Y}, the run R

_{JYY}, which arrives downstream at stop S

_{JYY}, is chosen. Remember that, as reported in Section 4, the forecasted at-stop arrival times are time-dependent random variables. Therefore, the forecasted bus arrival times at destination provided at stop J

_{Y}could differ from those provided at origin O at t

_{o}and the probability of boarding each run R

_{JY}at stop J

_{Y}has to be estimated. The probability of boarding run ${R}_{{J}_{YY}}$ is equal to the probability that, when the traveler is at boarding stop J

_{Y}, the forecasted arrival time ${F}_{{J}_{Y}}A{T}_{D}^{{R}_{{J}_{YY}}}$, at destination D, using run ${R}_{{J}_{YY}}$ and forecasted at J

_{Y}, is less than or equal to the, also forecasted at J

_{Y}, arrival time ${F}_{{J}_{Y}}A{T}_{D}^{{R}_{{J}_{Y}}}$ at destination D, using one run R

_{JY}among the other runs R

_{J}

_{1}, …, ${R}_{{J}_{YM}}$.

_{Y}, considered by the policy action in question.

_{Y}have to be expressed as functions of the arrival times at destination forecasted at origin O. Let:

- $A{T}_{D}^{{R}_{{J}_{YY}}}$ be the true value of the arrival time at destination D using run ${R}_{{J}_{YY}}$;
- ${F}_{{J}_{Y}}A{T}_{D}^{{R}_{{J}_{YY}}}$ be the arrival time at destination D using ${R}_{{J}_{YY}}$, forecasted at J
_{Y}:$${F}_{{J}_{Y}}A{T}_{D}^{{R}_{{J}_{YY}}}=A{T}_{D}^{{R}_{{J}_{YY}}}+{\epsilon}_{{J}_{Y}}^{{R}_{{J}_{YY}}}$$ - ${F}_{O}A{T}_{D}^{{R}_{{J}_{YY}}}$ be the arrival time at destination D using run ${R}_{{J}_{YY}}$, forecasted at origin O:$${F}_{O}A{T}_{D}^{{R}_{{J}_{YY}}}=A{T}_{D}^{{R}_{{J}_{YY}}}+{\eta}_{O}^{{R}_{{J}_{YY}}}$$

_{Y}to ${S}_{{J}_{YY}}$ computed when the traveller is at origin O, as the probability of choosing ${R}_{{J}_{YY}}$, is:

_{j}(i.e., linear combination of ε and η) can be assumed Multivariate Normal (MVN) random variables with zero mean ($E\left[{\upsilon}_{j}\right]=0$) and n × n dispersion matrix

**Σ**, where n is the number of alternative runs. Thus, forecasted arrival times FAT

_{j}are also jointly distributed according to a multivariate normal distribution with mean vector actual arrival times (

**AT**) and variances and co-variances equal to those of the residuals υ

_{j},

**FAT**~ MVN(

**AT**,

**Σ**).

_{Y}to stop ${S}_{{J}_{YY}}$ is given by:

_{Y}/I] of moving from the decision node I to stop J

_{Y}can be computed in a very similar way, obtaining:

- $p\left[{R}_{{I}_{Y}}/I\right]$ is the probability of boarding run ${R}_{{I}_{Y}}$ at node I for reaching stop J
_{Y}; - ${F}_{O}A{T}_{{J}_{Y}}^{{R}_{{I}_{Y}}}$ is the arrival time at node J
_{Y}using run ${R}_{{I}_{Y}}$ at stop I forecasted at origin O; - ${F}_{O}A{T}_{{J}_{Y}}^{{R}_{J}}$ is the arrival time at node J
_{Y}using run ${R}_{I}$ at stop I forecasted at origin O; - ${\epsilon}_{I}^{{R}_{I}},\text{\hspace{1em}}{\eta}_{O}^{{R}_{I}},\text{\hspace{1em}}{\epsilon}_{I}^{{R}_{{I}_{Y}}},\text{\hspace{1em}}{\eta}_{O}^{{R}_{{I}_{Y}}}$ are the forecasted errors.

## 6. An Example of Dynamic Strategy Search Applying the Proposed Method

_{j}are the forecasted transfer times from a run to the interchanging run. In the case of actions including more than one line, they are obtained as forecasted waiting time, weighted with the probabilities of use of each interchanging run, in our case, given by the transition probabilities.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Optimal hyperpath for the MDPm with the state–action tree in Figure 2.

**Figure 6.**Run-based representation of transit services relative to nodes B and C, with at-origin forecasted bus arrival times at downstream nodes.

**Figure 11.**Run-based representation of transit services from node F with forecasted bus arrival times at the following node.

A | set of actions a_{i} |

A_{s,t} | the set of possible actions a that can be taken at state s at time t |

AT_{D}^{Ri} | the true value of arrival time at destination D by run R_{i} |

_{J}^{Ri} | the error of arrival time forecast computed at downstream node J with variance ${\sigma}_{{\epsilon}_{J}^{{R}_{i}}}^{2}$ using run R_{i} |

_{I}^{Ri} | the error of arrival time forecast computed at upstream node I with variance ${\sigma}_{{\eta}_{I}^{{R}_{i}}}^{2}$ using run R_{i} |

φ_{i} | frequency of line i |

F_{F}AT_{D}^{Ri} | arrival time at destination (D), forecasted in F, using run R_{i} |

Σ | dispersion matrix of dimension equal to the number of runs n available at stop S_{y} |

f(ε) | density probability of forecast error vector ε |

FH | forecasting horizon |

FTT | bus travel time forecast |

p[S_{JYY}/J_{Y}] | transition probability of moving from node J_{Y} to S_{JYY} |

MSTN | multiservice transit network |

MVN (AT, Σ) | multivariate normal random variable with mean vector AT and dispersion matrix Σ |

p[R_{JYY}/J_{Y}] | probability of using run R_{JYY} from node J_{Y} |

r_{aj}[s_{i}, s_{i−1}] | reward function of state s_{i} conditional upon the previous state s_{i−1} |

R_{i} | generic run |

SMSTN | stochastic multiservice transit network |

SN | stochastic network |

S_{i} | last alighting stop |

**Table 2.**Example of transition probability matrix of the state–action tree in Figure 2b.

B | F | G | E | D | |
---|---|---|---|---|---|

O [a_{B}] | 1 | 1 | p[G/O] = p[G/F] | p[E/O] = p[E/F] | 1 |

B [a_{6}] | 0 | 1 | p[G/B] = p[G/F] | p[E/B] = p[E/F] | 1 |

F [a_{7+8}] | 0 | 0 | p[G/F] | p[E/F] | 1 |

G [a_{9}] | 0 | 0 | 0 | 0 | 1 |

E [a_{10}] | 0 | 0 | 0 | 0 | 1 |

D [a_{12}] | 0 | 0 | 0 | 0 | 1 |

**Table 3.**Estimation of residual variance according to the forecasting horizon (refer to Figure 6).

Random Residual | Forecasted Arrival Time at Destination | When Choice Is Made | Forecasting Horizon [minutes] | Estimated Standard Deviation [seconds] | Estimated Variance [minutes ^{2}] |
---|---|---|---|---|---|

${\eta}_{O}^{{r}_{8.1}}$ | 08:08 | 07:02 | 66 | 402.60 | 45 |

${\eta}_{O}^{{r}_{7.1}}$ | 08:03 | 07:02 | 61 | 372.10 | 38 |

${\epsilon}_{F}^{{r}_{8.1}}$ | 08:08 | 07:35 | 33 | 201.30 | 11 |

${\epsilon}_{F}^{{r}_{7.1}}$ | 08:03 | 07:35 | 28 | 170.80 | 8 |

TOTAL | 103 (10.14^{2}) |

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Nuzzolo, A.; Comi, A.
Dynamic Optimal Travel Strategies in Intelligent Stochastic Transit Networks. *Information* **2021**, *12*, 281.
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Nuzzolo A, Comi A.
Dynamic Optimal Travel Strategies in Intelligent Stochastic Transit Networks. *Information*. 2021; 12(7):281.
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Nuzzolo, Agostino, and Antonio Comi.
2021. "Dynamic Optimal Travel Strategies in Intelligent Stochastic Transit Networks" *Information* 12, no. 7: 281.
https://doi.org/10.3390/info12070281