The grid based approach, a fast local evaluation technique for line planning

Abstract

Line planning is one of the important steps in the public transit planning process. It is a difficult combinatorial problem with an enormous search space. For networks from practice, which are typically large and have more complex constraints, it takes an unreasonable amount of time to find good solutions with the current methods. When the objective is to minimize passenger travel time, most of the calculation time is spent on solving the passenger routing sub-problem. This paper proposes a local evaluation technique to significantly speed up this most critical component. A change is assumed to have its biggest impact on passengers already travelling close to the change. This idea is implemented by putting a grid over the network and only re-evaluating the part of the network around the change, while still taking into account all passengers. This results in a smaller computation time for each passenger routing, with only a limited loss of quality, and it allows more iterations. On a number of benchmark instances, the results obtained by the grid approach are compared to the same algorithm without any local evaluation. The grid approach is significantly faster and allows to perform 5–20 times more evaluations per second, while having a minimal loss of quality per evaluation. When the available calculation time is limited, the grid based approach obtains the best solutions. On the larger benchmark instances, the grid based approach can use more complex neighborhoods resulting in even better solutions. The method is very flexible and can be integrated in many line planning algorithms and probably even in algorithms for other network problems.

Appendix

1.1 Partial origin–destination matrices

In this appendix, we explain in more detail how the calculation of the partial OD matrices is implemented. While the concept itself is quite simple, we noticed the implementation can be quite tricky and complicated.

To calculate the total travel time in a cut of the network, both the extended transit network and the OD matrix are necessary. Each cut of 3 by 3 cells has its own OD matrix, called a partial OD matrix. The calculation of the partial OD matrices is responsible for most of the overhead in the grid method. For efficiency purposes, these matrices are calculated at the same time when evaluating a newly implemented solution. The only thing needed to calculate these partial OD matrices is the original OD matrix, the routes the passengers will take with the current line plan and the network information.

Each passenger is assumed to enter and leave each cut at a fixed location, based on their route in the current solution. Only their path in between these two points can change, based on the local search moves considered. Starting or ending their trip in a node present in the cut counts as, respectively, entering or leaving the network. In Fig. 3, a passenger travelling from node 0 to node 9 will enter the highlighted cut (3, 2) in node 2b. As mentioned before, 2b is a border node representing the entrance of a bus line. From a demand perspective, this means that any demand from 0 (or 1 or 2) to 9 gets converted into demand from 2b to 9. All passengers that enter the cut while on this bus line will enter the cut in node 2b and thus will create demand starting (or ending) in 2b. It should be noted that a single passenger route might need to be split up in multiple routes. This means it is possible that a shortest path of a passenger enters (and leaves) a cut twice.

Calculating all these partial OD matrices might seem computationally expensive, but actually all partial OD matrices can be calculated at the same time. Each cut and partial OD matrix can be identified by the coordinates of its central cell. The cut in the bottom left corner of a network is then identified as cut (1, 1) and the cut next to it is cut (2, 1). Because of the overlap between the 3 by 3 cells, each node is present in multiple cuts.

When constructing all partial OD matrices, each shortest path (in the complete network) is considered one by one and only once. This is done by moving through the shortest path and determining in which cuts each node along the shortest path is located. Whenever a new cut is entered, the previous node is recorded as the entry node. Remember from Sect. 2.2 that during the construction of the extended network, the first node outside the cut is included as a border point. That is the reason why the previous node is recorded and not the current. When a cut is left, the same process is executed. Now the current node is recorded as the exit node. This process is then applied to all OD pairs in the network. This results in a list of entry and exit nodes for each cut and every OD pair. The partial OD matrix of a cut can now be constructed by adding the demand from the complete OD matrix to each entry and exit pair in the partial OD matrix. This means the partial OD matrices contain both physical bus nodes (from the original network) and line nodes (entry and exit nodes).

This might seem a complicated process, but the result is that every OD-pair of which the shortest path in the current solution is going through a certain cut, is taken into account when a move is evaluated in this cut, by ‘moving’ this OD-pair to an artificial entry and exit node at the border of the cut. This approach ensures that a local evaluation serves as an upper bound to the travel time. The real impact of a change (on the entire network) can only be better than the impact estimated by the local evaluation method in a cut. This is because the routing is completely optimized within the cut of the network. The only possible improvements, not considered in the cut, are when a change makes a new route possible, previously not travelling through the cut, or when the impact of a change is less negative because an alternative route (outside the cut) can compensate part of the disadvantage estimated. The actual routes calculated by the local evaluation are possible, hence they are a possible solution of a full evaluation and thus the result of the full evaluation cannot be worse.

In the current implementation, every partial OD matrix contains an entry for each node in the original extended network. This requires a lot of memory. Since these are all very sparse matrices its size can be heavily reduced. Only the nodes that have any demand, have to be present in the partial OD matrix. Note that a list with node ids that links this limited OD matrix to the original is probably required. Just doing this will massively limit the memory use. By cleverly constructing the extended network of a cut it might be possible to avoid having to link the partial OD nodes to the original. Since we have not run into any memory problems we did not implement this, but we mention it because it is likely necessary when computing larger instances.