Adaptive Park-and-ride Choice on Time-dependent Stochastic Multimodal Transportation Network

Abstract

In transportation networks with stochastic and dynamic travel times, park-and-ride decisions are often made adaptively considering the realized state of traffic. That is, users continue driving towards their destination if the congestion level is low, but may consider taking transit when the congestion level is high. This adaptive behavior determines whether and where people park-and-ride. We propose to use a Markov decision process to model the problem of commuters’ adaptive park-and-ride choice behavior in a transportation network with time-dependent and stochastic link travel times. The model evaluates a routing policy by minimizing the expected cost of travel that leverages the online information about the travel time on outgoing links in making park-and-ride decisions. We provide a case study of park-and-ride facilities located on freeway I-394 in Twin Cities, Minnesota. The results show a significant improvement in the travel time by the use of park-and-ride during congested conditions. It also reveals the time of departure, the state of the traffic, and the location from where park-and-ride becomes an attractive option to the commuters. Finally, we show the benefit of using online routing in comparison to an offline routing algorithm.

Appendices

Appendix A: EV Algorithm (Miller-Hooks and Mahmassani 2003)

The Expected Value (EV) algorithm generates a priori LET paths with their associated expected cost from all origins to a single destination for each possible departure time in a given time horizon. Let $S_{ij}(t) = \left \{s_{ij}^{k}\right \}_{k \ge 1}$ be the set of possible realizations of travel time on link (i, j) ∈ N_a which happens with probability $\left \{p_{ij}^{k}\right \}_{k \ge 1}$ such that ${\sum }_{k} p_{ij}^{k} = 1$. Let us denote ${\lambda _{i}^{c}}(t)$ as the expected travel cost along path c from node i to the destination d departing at time t. For each $t \in \mathcal {T}$, the minimum expected cost from each node is sought. In this label-correcting algorithm, we maintain a set of labels ${{\Lambda }_{i}^{c}}$. These labels are called pareto-optimal (or p-optimal) because each label is potentially optimal for one or more time intervals. Let pathList[i] be the set of p-optimal paths from node i to d. A scan eligible list SE is maintained whose elements (j, μ) are characterized by a node j ∈ N_a and path μ ∈ pathList[j]. A set of path pointers predNode and predPath are also maintained to trace back the optimal path after the algorithm terminates.

At each iteration, an element (j, μ) is selected from SE and for each neighboring node i of j ∈ N_a, a temporary cost label $\{\kappa _{i}(t)\}_{t \in \mathcal {T}}$ is computed (Line 14). After this, we check for the Pareto-optimality of temporary labels (Line 16). If the new path is p-optimal, we add it to the pathList[i] and update the optimal cost labels and path pointers. After the algorithm terminates, a single best path from each node and for each possible departure time is selected.

Appendix B: Proofs of Various Propositions

Proof

(Proposition 1) Lines 2-4 can be done in $\mathcal {O}(1)$ time. Assuming that the shortest path algorithm is implemented using Binary heap data structure, Line 5-13 consists of two major steps, finding the node i with minimum label γ^gc from SEL (Line 6) that can be done in $\mathcal {O}(\vert N_{t}\vert \log (\vert N_{t}\vert ))$ and updating labels of new nodes which can be done in $\mathcal {O}(\vert A_{t} \vert \log (\vert N_{t}\vert ))$ time. Therefore, the worst case computational complexity of Algorithm 1 is $\left .\mathcal {O}(\vert A_{t} \vert \log \vert N_{t} \vert + \vert N_{t}\vert \log \vert N_{t}\vert )\right )$. However, if |A_t| = Ω(|N_t|), then the worst case computational complexity of Algorithm 1 can be given as $\left .\mathcal {O}(\vert N_{t}\vert \log \vert N_{t}\vert )\right )$. □

Proof

(Proposition 2) Without the loss of generality, let us assume that there exists a cyclic path of minimum length 2. Let $\mathcal {P}\{[(i, t, \theta ) , (j, t^{\prime }, \theta ^{\prime })], [(j, t^{\prime }, \theta ^{\prime }), (i, t, \theta )]\}$ be that path. Since, $t^{\prime } = c^{\theta }_{ij} + t, \ \forall i, j \ne d$, implying that $t^{\prime } + c_{ji}^{\theta ^{\prime }} = t$, which is a contradiction unless $c_{ij}^{\theta } = c_{ji}^{\theta ^{\prime }} = 0$, in which case there does not exist such transition. Hence, there does not exist a cyclic path of length 2. One can extend this argument for a cyclic path of any length. □

Proof

(Proposition 3) To show this, let us consider various subsets of the state space created as below:

$$ \begin{array}{@{}rcl@{}} S^{\prime}_{0} &= & \{(d, t): t \in \mathcal{T}\} \end{array} $$

(13)

$$ \begin{array}{@{}rcl@{}} S^{\prime}_{k+1} & =& \left\{(i, t) : \underset{\underset{\theta^{\prime} \in {{\varTheta}}_{j}\left( t +c_{ij}^{\theta}\right)}{\theta: \pi^{*}(i, t, \theta) = j,}}{\sum} \mathbb{P}_{\pi} \left[(i, t, \theta), (j, t + c_{ij}^{\theta}, \theta^{\prime})\right] = 0, \forall j \notin \cup_{m = 0}^{k} S_{m} \right\}, k = 0, 1, ... \end{array} $$

(14)

The above construction of sets adds various states in the backward direction of the destination node. For example, S₁ will contain all the states associated to the nodes in the sets Z and Γ^− 1(d). Let $S_{\bar {k}}$ be the last of these sets that is non-empty. In view of the acyclicity and proper stationary optimal policy assumptions, we have $\bar {k} \le \vert N_{a} \vert$ and $\cup _{m = 0}^{\bar {k}} S_{m} = S^{\prime }$. After this, one can show using induction that

$$ (\hat{T}^{k} \hat{J})(i, t) = \hat{J}^{*}(i, t), \ \forall (i, t) \in \cup_{m = 0}^{k} S_{m}, k = 1, ..., \bar{k} $$

(15)

The mathematical induction part is same as the proof given in Bertsekas (2012). □

Proof

(Proposition 4) Lines 2-5 can be done in $\mathcal {O}(1)$ time. The control computation in line 11 can be performed in $\mathcal {O}(\vert A_{a} \vert )$. The previous operation should be performed for every (i, t, θ) ∈ S, repeatedly |N_a| number of times. Therefore, lines 6-11 can be performed in $\mathcal {O}(\vert S \vert \vert N_{a} \vert \vert A_{a} \vert )$. Furthermore, lines 12-15 can be performed in $\mathcal {O}(\vert S \vert \vert A_{a} \vert )$. Therefore, the overall complexity of the Algorithm 2 is equal to $\mathcal {O}(\vert S \vert \vert N_{a} \vert \vert A_{a} \vert )$. □

Proof

(Proposition 5) Assuming that the elements of SE are removed according to the FIFO rule, lines 1-15 are standard Bellman-Ford algorithm and can be performed in $\mathcal {O}(\vert S \vert \vert A_{a} \vert )$ time. The computational complexity of finding the optimal policy (lines 16-19) can be performed in $\mathcal {O}(\vert S \vert \vert A_{a} \vert )$. Therefore, the overall complexity of the Algorithm 3 is equal to $\mathcal {O}(\vert S \vert \vert A_{a} \vert )$. □