Modelling joint activity-travel pattern scheduling problem in multi-modal transit networks

Abstract

Over the past decades, many activity-based travel behaviour models have been proposed based on individuals’ independent decision making. The modelling of individuals’ joint activity/travel choices, however, has received less attention. In reality, both independent and joint activities/travels form individual’s normal daily activity-travel patterns. Travel surveys have indicated that joint activity/travel constitutes an important part in individuals’ daily activity-travel patterns. On this basis, explicit modelling of joint activity/travel choices is an essential component for long-term transport planning. In this study, an activity-based network equilibrium model is proposed for scheduling two-individual joint activity-travel patterns (JATPs) in congested multi-modal transit networks. The proposed model can be used to comprehensively investigate individuals’ activity choices (e.g. activity start time and duration, activity sequence) and travel choices (e.g. departure time, route and mode) in multi-modal transit networks, including both independent ones and joint ones. The time-dependent JATP choice problem is converted into an equivalent static user equilibrium model by constructing a joint-activity-time-space (JATS) super-network platform. Joint travel benefit is modelled by incorporating a commonality factor in the JATP utility. A solution algorithm without prior JATP enumeration is proposed to solve the JATP scheduling problem on the JATS super-network. Numerical results show that individuals’ independent and joint activity/travel choices can be simultaneously investigated by the proposed model. The impacts of joint travel benefit on individuals’ independent and joint activity-travel choices are explicitly investigated.

Appendices

Appendix 1

The detailed steps of the proposed JATS super-network expansion algorithm for the afternoon period are given in this appendix. The algorithm for the morning period is similar and not given here.

Input: :

a multi-modal transit network $M$, two origin locations for individual A and B ($i_{A}$ and $i_{B}$), one destination location ($i_{AB}$), activity locations ($i_{\text{a}} \in I_{\text{a}}$), and number of time intervals $K$.

Output: :

the JATS super-network.

Step 1. :

Node augmentation.

For each node $i \in U$, expand the node into JATS node $\left( {ind,(i,l),k} \right),$ $ind = 1,2,12,$ $l = 0,1,$ $k = 1,2, \ldots ,K,K + 1$. Denote the JATS node set as $N$.

Step 2. :

Construction of JATS activity links.

Scan all nodes in set $N$. Construct JATS activity links $a_{\text{a}} \in A_{\text{a}}$ between $\left( {ind,(i_{\text{a}} ,0),k} \right)$ and $\left( {ind,(i_{\text{a}} ,0),k + 1} \right)$.

Step 3. :

Construction of JATS transfer links.

Scan all nodes in set $N$. Construct JATS transfer links $a_{\text{t}} \in A_{\text{t}}$ between $\left( {ind,(i_{\text{a}} ,0),k} \right)$ and $\left( {ind,(i_{\text{a}} ,1),k} \right)$.

Step 4. :

Construction of JATS direct in-vehicle links.

Find all in-vehicle links in network $M$ on the basis of physical travel links.

For each $i \in U$, find all $i' \in U$ which are connected to $i$ by in-vehicle links. Obtain the travel time $t_{{a_{\text{d}} }}^{0}$ of each in-vehicle link.

For each $i'$, construct JATS direct in-vehicle links between $\left( {ind,(i,1),k} \right)$ and $\left( {ind,(i',0),k + t_{{a_{\text{d}} }}^{0} } \right)$.

Step 5. :

Construction of JATS waiting links.

Scan all nodes in set $N$. Construct JATS waiting links $a_{\text{w}} \in A_{\text{w}}$ between $\left( {ind,(i,0),k} \right)$ and $\left( {ind,(i,0),k + 1} \right)$.

Step 6. :

Construction of JATS meeting links.

Scan all nodes in set $N$. Construct JATS meeting links $a_{\text{m}} \in A_{\text{m}}$ between $\left( {1,(i,0),k} \right)$ and $\left( {12,(i,0),k} \right)$, and between $\left( {2,(i,0),k} \right)$ and $\left( {12,(i,0),k} \right)$.

Step 7. :

Simplification of the super-network.

Delete the augmented nodes which are not two-way connected except for the origin nodes (i.e. $\left( {1,(i_{A} ,0),1} \right)$ and $\left( {2,(i_{B} ,0),1} \right)$) and the destination node (i.e. $\left( {12,(i_{AB} ,0),k + 1} \right)$). Delete the redundant links.

Appendix 2

In this appendix, the detailed steps of JATP searching algorithm for the afternoon period are presented. The algorithm for the morning period is similar and not shown here.

Let $P^{{dy_{A} y_{B} }}$ be a set of non-dominated routes maintained at nodes $y_{A}$ and $y_{B}$, and the non-dominated routes from destination $d$ to all node pairs are maintained in a scan eligible set, denoted as $SE$. At each iteration, one non-dominated route $p_{i}^{{dy_{A} y_{B} }}$ is selected from $SE$ in a first-in-first-out (FIFO) order for route extension. A temporary route is constructed by extending the selected route $p_{i}^{{dy_{A} y_{B} }}$ to its successor link whose end node is $y_{A}$ or $y_{B}$ ($y_{A}$ for example here, and the temporary route is denoted as $p_{j}^{{dy_{A} 'y_{B} }}$). The dominant relationship between the newly generated route $p_{j}^{{dy_{A} 'y_{B} }}$ and the set of non-dominated routes $P^{{dy_{A} 'y_{B} }}$ at nodes $y_{A} '$ and $y_{B}$ is determined based on JATP dominant condition (Definition 1). If $p_{j}^{{dy_{A} 'y_{B} }}$ is a non-dominated route at nodes $y_{A} '$ and $y_{B}$, it is then inserted into $P^{{dy_{A} 'y_{B} }}$ and $SE$. As the newly generated route $p_{j}^{{dy_{A} 'y_{B} }}$ may also dominate some routes in $P^{{dy_{A} 'y_{B} }}$, these dominated routes should be eliminated from $P^{{dy_{A} 'y_{B} }}$ and $SE$. The proposed algorithm continues the route searching process until $SE$ becomes empty. At the last step of this algorithm, the optimal JATP can be determined by choosing the route with the largest JATP utility.

The detailed steps of the proposed algorithm for finding the optimal joint route in JATS super-network for the afternoon period are listed as follows.

Inputs: :

destination node $d$

Returns: :

the optimal joint route in the JATS super-network (i.e. the optimal JATP)

Step 1. :

Initialization: Create a route $p_{i}^{ddd}$ from node $d$ to itself, and set $u_{activity}^{{p_{i}^{ddd} }} = 0,disu_{travel}^{{p_{i}^{ddd} }} = 0,disu_{joint}^{{p_{i}^{ddd} }} = 0$. Add $p_{i}^{ddd}$ into label-vector $P^{ddd}$ and the list of candidate labels $SE$.

Step 2. :

Label selection: Take label $p_{i}^{{dy_{A} y_{B} }} \in P^{{dy_{A} y_{B} }}$ from $SE$ in FIFO order. If $SE = \phi$, go to Step 4; otherwise go to Step 3.

Step 3. :

Route extension: If $y_{A} = y_{B}$(denoted as $y$ for uniformity), go to Step 3.1.; otherwise go to Step 3.2.

If the new label $p_{j}^{dxx'}$ (or $p_{j}^{dxx}$or $p_{j}^{dxy}$) is a non-dominated route under the JATP dominant condition, then insert the new label into $P^{dxx'}$ (or $P^{dxx}$or $P^{dxy}$) and $SE$, and remove all routes dominated by the new label from $P^{dxx'}$ (or $P^{dxx}$or $P^{dxy}$) and $SE$.

Go back to Step 2.

Step 3.1. :

For every link $a$ (with start node $x$) whose end node is $y$: If link $a$ is a meeting link, go to Step 3.1.1; If link $a$ is an activity/waiting link, go to Step 3.1.2; If link $a$ is an in-vehicle/transfer link, go to Step 3.1.3.

Step 3.1.1. :

Find the corresponding meeting link $a'$ (with start node $x'$) of the other individual. Generate a new label $p_{j}^{dxx'} \in P^{dxx'}$. Set $u_{activity}^{{p_{j}^{dxx'} }} = u_{activity}^{{p_{i}^{dyy} }}$, $disu_{travel}^{{p_{j}^{dxx'} }} = disu_{travel}^{{p_{i}^{dyy} }}$, and $disu_{joint}^{{p_{j}^{dxx'} }} = disu_{joint}^{{p_{i}^{dyy} }}$.

Step 3.1.2. :

Generate a new label $p_{j}^{dxx} \in P^{dxx}$. Set $u_{activity}^{{p_{j}^{dxx} }} = u_{activity}^{{p_{i}^{dyy} }} + u_{a}$, $disu_{travel}^{{p_{j}^{dxx} }} = disu_{travel}^{{p_{i}^{dyy} }}$, and $disu_{joint}^{{p_{j}^{dxx} }} = disu_{joint}^{{p_{i}^{dyy} }}$.

Step 3.1.3. :

Generate a new label $p_{j}^{dxx} \in P^{dxx}$. Set $u_{activity}^{{p_{j}^{dxx} }} = u_{activity}^{{p_{i}^{dyy} }}$, $disu_{travel}^{{p_{j}^{dxx} }} = disu_{travel}^{{p_{i}^{dyy} }} + u_{a}$, and $disu_{joint}^{{p_{j}^{dxx} }} = disu_{joint}^{{p_{i}^{dyy} }} + u_{a}$.

Step 3.2. :

For every link $a$ (with start node $x$) the end node of which is $y_{A}$ or $y_{B}$ (denoted as $y$ for simplicity): 

If link $a$ is an activity/waiting link, generate a new label $p_{j}^{dxy} \in P^{dxy}$, set $u_{activity}^{{p_{j}^{dxy} }} = u_{activity}^{{p_{i}^{dyy} }} + u_{a}$, $disu_{travel}^{{p_{j}^{dxy} }} = disu_{travel}^{{p_{i}^{dyy} }}$, and $disu_{joint}^{{p_{j}^{dxy} }} = disu_{joint}^{{p_{i}^{dyy} }}$; 

If link $a$ is an in-vehicle/transfer link, generate a new label $p_{j}^{dxy} \in P^{dxy}$, set $u_{activity}^{{p_{j}^{dxy} }} = u_{activity}^{{p_{i}^{dyy} }}$, $disu_{travel}^{{p_{j}^{dxy} }} = disu_{travel}^{{p_{i}^{dyy} }} + u_{a}$, and $disu_{joint}^{{p_{j}^{dxy} }} = disu_{joint}^{{p_{i}^{dyy} }}$.

Step 4. :

Determine the optimal JATP with the largest JATP utility. Stop.