Transit origin destination (OD) trip matrices are essential inputs for most problems regarding the planning, operation, and management of public transit systems. Traditionally, OD matrices were obtained statically from passenger surveys. However, due to the need for continuous updates, surveying a representative sample, and the high cost of conducting these surveys, advanced methods estimate transit OD using sensor collected data such as automatic passenger counts (APC).

The objective of this research is to formulate a dynamic transit origin estimation (DTOD) estimation model that is transferrable and applicable to scenarios where the available transit data is APC. The proposed methodology is a bi-level optimization model. In this model, each optimization is defined as a distinct level, and each level has its own objectives and constraints. The lower level (follower) seeks to optimize its outcomes, which are then used by the upper level (leader) to optimize its own outcomes.

In the bi-level model proposed, the lower level is a dynamic transit assignment model that simultaneously determines the dynamic average travel costs and optimal route choices of passengers in congested transit networks (i.e., estimated passenger flows). The upper level sums passenger route choices from the lower level to obtain transit OD, and minimizes the sum of error measurements between the obtained time-dependent OD matrices and dynamic real passenger counts (APC counts). As a result of considering asymmetric link cost interactions (i.e., the cost of traversing a link in the network is both a function of the flow on the link itself and on surrounding links), the transit assignment is formulated as a variational inequality. The upper level, in contrast, is formulated as a generalized least square estimation.

To evaluate the performance of the proposed DTOD estimation model, numerical examples are conducted using MATLAB, in which the model’s solution algorithm is coded. The model is tested on a small theoretical network and a real transit network in Calgary, Alberta, Canada. Sensitivity analyses of the bi-level model to different weighting schemes, link cost function parameters, and congestion levels are performed in which the model converges to unique solutions in minimal times and within acceptable ranges of error.