Optimization Models for Estimating Transit Network Origin–Destination Flows with Big Transit Data

Abstract

The increasing adoption of automatic vehicle location and automatic passenger count technologies by transit agencies produces passenger boarding and alighting count data on a continuous basis. This data provides new opportunities for origin–destination (O–D) flow estimation. However, the state-of-the-art methodologies generated flows within routes and barely considered linked trips. This paper proposes optimization models to identify transfers and approximate network-level O–D flows by: a quadratic integer program (QIP), a feasible rounding procedure for the quadratic convex programming (QCP) relaxation of the QIP, and an integer program (IP). A case study for Ann Arbor-Ypsilanti area in Michigan suggests that: The IP model outperforms the QCP in terms of accuracy and remains tractable from an efficiency standpoint, contrary to the QIP. Its O–D estimation achieves an R-Squared metric of $95.57\%$ at the traffic analysis zone level and $92.39\%$ at the stop level, compared to the ground-truths inferred from the state-of-the-practice trip-chaining methods.

Appendix

The appended figures visualize the spatial distribution of stop-level accuracy of the model estimation, with the IP approach on the Go!Pass data as an example. Figures 6 and 10 each illustrate the benchmark counts of trips originating from and terminating at each stop, in which the 2 transit centers are each treated as a stop. Figures 7 and 11 each depict the inferred counts of trips originating from and terminating at the each stop. Figures 8 and 12 each visualize the total deviation of the IP model estimation from the benchmark, also in terms of the trip counts at each stop as an origin and as a destination. Both figures show the absolute differences when comparing Figs. 7 and 11 against Figs. 6 and 10. Specifically, let $\text{OD}^*$ denote the benchmark matrix, $\text{OD}$ denote the matrix estimation, n denote the total number of stops, and $(i,j) \in \{1,\ldots ,n\}^2$ denote the indices (or the coordinates) of the origin and destination pairs. For each stop i as the origin, the data presented in Fig. 8 was calculated as $\big | \sum _{j=1}^{n} \text{OD}^*_{i,j} - \sum _{j=1}^{n} \text{OD}_{i,j} \big |$. For each stop j as the destination, the data presented in Fig. 12 was calculated as $\big | \sum _{i=1}^{n} \text{OD}^*_{i,j} - \sum _{i=1}^{n} \text{OD}_{i,j} \big |$. Figures 9 and 13 depict the L2-norm as the distance measure between the benchmark matrix and the estimations, also at the stop-level. Specifically, let the vector $\text{OD}^*_{i,\cdot }$ denote the ith row of the benchmark matrix, and the vector $\text{OD}_{i,\cdot }$ denote the ith row of the estimation matrix. Also, let the vector $\text{OD}^*_{\cdot ,j}$ denote the jth column of the benchmark matrix, and the vector $\text{OD}_{\cdot ,j}$ denote the jth column of the estimation matrix. For each stop i as the origin, the data presented in Fig. 9 was calculated as $\left\Vert \text{OD}^*_{i,\cdot } - \text{OD}_{i, \cdot }\right\Vert _2 = \sum _{j=1}^n (\text{OD}^*_{i,j} - \text{OD}_{i,j})^2$. For each stop j as the destination, the data presented in Fig. 13 was calculated as $\left\Vert \text{OD}^*_{\cdot ,j} - \text{OD}_{\cdot ,j}\right\Vert _2 = \sum _{i=1}^n (\text{OD}^*_{i,j} - \text{OD}_{i,j})^2$. The data in Figs. 8, 9, 12 and 13 have the same unit as those in Figs. 6, 7, 10, and 11, and were plotted in the same scale for comparison. In Figs. 6, 7, 10, and 11, the size of the red circles depicts the volume of flows originating from or terminating at each stop. In Figs. 8, 9, 12 and 13, the red circles of larger sizes correspond to larger differences between the estimation and the benchmark and poorer model performance.

Fig. 6

Benchmark counts of trips originating from each stop

Fig. 7

Inferred counts of trips originating from each stop

Fig. 8

Total difference between the inferred and the benchmark counts of trips originating from each stop

Fig. 9

L2-normed difference between the inferred and the benchmark counts of trips originating from each stop

Fig. 10

Benchmark counts of trips terminating at each stop

Fig. 11

Inferred counts of trips terminating at each stop

Fig. 12

Total difference between the inferred and the benchmark counts of trips terminating at each stop

Fig. 13

L2-normed difference between the inferred and the benchmark counts of trips terminating at each stop