Depar- Criteria
Name ture time ic sec cost Color
in h in % ine
fast 10:07 5:28 2 57 114 blue
fast’ 11:07 5:28 2 57 114 blue
cheap 11:38 5:30 3 33 101 green
cheap’ 11:07 5:31 3 41 101 green
comb 10:07 5:52 2 84 113 cyan
direct 10:43 7:04 0 100 107 red
Figure 9.7: Fast, convenient, and cheap connections for a query from L¨ubeck (cyan) to Aschaffenburg (red). The traveled routes (left) and characteristics of the connections (right).
over W¨urzburg (blue). They save 2 minutes, are more reliable and 13emore expensive.
Besides, there is a direct IC connection via K¨oln (red) that is rather slow. An additional alternative (comb) takes the shortest path (cyan) and increases the reliability over the fastest connection by almost 30%. Only the direct connection is without ICE but has the longest detour. The cheaper connections use RE or RB for a part of the journey and thus save money.
9.15 Conclusion
In this chapter we have evaluated the search for special offers and night trains and later presented an in-depth analysis of our algorithm.
Special Offers The focus of our approach to search for special offers was to demonstrate how a large variety of different tariff classes can be incorporated into a multi-objective shortest path framework for travel information. We successively integrated a combined search for regular tariffs and contingent-based tariffs into MOTIS. In our computational experiments we observed that a multi-objective search with a mixture of tariff rules can be done almost as fast as just with one regular tariff. Sometimes the contingent-restricted versions run even faster, because the special offers apply to fast connections. In that case, less effort has to be put into searching cheap connections.
We also observed that our simple model to represent regular fares within Germany is not as accurate as desired. Hence, future work should concentrate on improved approxi-mations of regular fares. A tighter approximation would allow stricter dominance rules.
We do expect considerable savings of computational time from stricter dominance rules.
Night Trains Our computational study shows that a specialized night train search delivers many more attractive connections than an ordinary search. We have observed a trade-off between quality of the solution sets and computation time. Our implementation of a multi-criteria search with one additional criterion fails to find a good night train connection in a few cases, but it is the most efficient one. The pre-selection approach with a fast feeder computation never failed and delivers almost optimal quality. Both variants are fast enough to be applied in on-line information systems. With additional tuning the runtime can probably be reduced further, while maintaining high quality.
We see two promising perspectives for applying our algorithms in practice. The first is the scenario for which this section was written: the user explicitly asks for a night train connection. Then we would recommend using the pre-selection approach with fast feeder computation (Algorithm C), which delivers excellent quality. The second scenario is an ordinary query with a start interval in the evening. Here, it would be an option to run MOTIS with an additional criterion (Algorithm D) but without spending too much additional computation time. If this search finds attractive night train connections, these can be offered as alternatives to those computed for the query interval.
Overall We have thoroughly analyzed the speed-up techniques and heuristics intro-duced in Chapter 8. The impact of several of these techniques on the search space is visualized for an exemplary search in SectionB.3 on page201 in the appendix.
We have seen that the techniques goal-direction anddomination by labels at the ter-minal are essential for online capabilities. We are able to efficiently determine lower bounds forgoal-direction anddomination by terminal for all of our criteria and prune the search space accordingly, in less than 30ms. Selecting the right priority queue type and tuning the weights and criteria used ingoal-direction further improved processing times.
Contrary to single-criterion search the time spent on the priority queue is not among the dominant factors.
The reference version using optimal settings without heuristics achieves an average runtime of 412ms and answers 95% of the queries within 1.5 seconds.
Respecting the additional criterion reliability of interchanges produced additional re-sults. The computational cost for adding the criterion is lower than for ticket cost. This is mainly to the correlation with the number of interchanges which the latter does not exhibit.
Of all heuristics the one with best quality for performance tradeoff is shortcuts with theimportant station heuristicbased on theneighbors threshold. We may use threshold N4 to speed up searches to about a fifth of a second (210ms) on average at only little quality loss. The best choice is threshold N3 with nearly indiscernible quality loss and one third runtime improvement over our reference version.
Our baseline version without any speed-up techniques required several minutes per query. With our best setup andshortcutsN3 we reach an average runtime of 275ms and answered 95% of the queries in at most 1 second.
Chapter 10
A Time-Dependent Timetable Information System
Our fully realistic multi-criteria prototype MOTIS is based on a time-expanded graph because it appears to be easier to model all side constraints arising in practice in this framework. However, the major drawback of time-expanded graphs, in comparison to time-dependent models, is the higher space consumption, in particular if highly periodical regional mass transit has to be included. In addition, the time-dependent graph model seems easier to adapt in case of dynamic graph changes due to train delays. These reasons motivate our investigation of the time-dependent graph model in this chapter. It is based on the publication [DMS08].
Related Work and Our Contribution
To the best of our knowledge, no complete, realistic system has been built for exact multi-criteria search of all Pareto optimal solutions in the time-dependent graph model.
In [PSWZ08], Pyrga et al. consider constant transfer times and traffic days, but other aspects of real timetables, like footpaths and special transfer rules, are not considered. In this chapter, we describe a first prototype for multi-criteria search of all Pareto optima within a fully featured, real timetable. Its search results are guaranteed to be optimal.
We provide an extensive computational study showing the impact of several speed-up techniques. Even though the number of possible speed-up techniques is severely restricted, in order to guarantee the optimality of all search results, the performance of our prototype is already comparable to time-expanded systems, but consumes much less space.
Most previous research (in particular [PSWZ08]) has concentrated on the earliest arrival problem from a given point in time. But here we focus on a many-source shortest path version, because in a pre-trip search for train connections, a user usually wants to specify atime interval in which the journey should start. This implies that we have to perform a simultaneous search from multiple starting times. In a time-expanded graph model, this can be handled very easily: One simply adds a “super-source” and edges of length zero to all start events, thereby reducing the search to a single-source search.
In time-dependent graphs, however, solving the many-source shortest path problem is more subtle if travel time is used as an optimization criterion. Consider two sub-paths from the source to some intermediate node. Then, pathp1with start times1 and travel time t1 dominates another path p2 with start time s2 and travel time t2, with respect
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to travel time, only if t1 < t2 and s1 ≥ s2. Otherwise, both paths are incomparable.
This leads to weaker dominance during search than for the earliest arrival problem, and consequently to more non-dominated solutions which can be offered to customers. It is therefore remarkable that we still achieve a reasonable performance.
Our approach can easily be extended to further criteria. In order to exemplify this, the “reliability of transfers” is investigated as an additional criterion. The reliability of transfers is a property of a connection that captures the probability of catching all trains within the connection (cf. Chapter 6.3). Since possible train delays cannot be ignored, such a criterion is of great practical importance.
Overview The remainder of this chapter is organized as follows. In Section 10.1, we introduce the time-dependent graph model and describe the adaptations needed in order to make it suited for fully realistic timetables. A modification of Dijkstra’s algorithm that makes it capable of minimizing multiple criteria is introduced in Section 10.2. Several speed-up techniques that do not violate the optimality of the search results are proposed.
The results of the experimental analysis of our time-dependent search system are pre-sented in Section 10.3. We analyze the impact of the proposed speed-up techniques on performance. The prototype is then compared to our search on a time-expanded graph using MOTIS. The final aspect of our discussion covers the relationship between perfor-mance and the number of search criteria. Finally, Section 10.4 summarizes our results and gives an outlook on future work.