3. Paper Summaries 17
3.2. System Headways in Line Planning
As discussed in Section 2.1, line planning is a well researched problem. There are several models in the literature with various objectives, e.g. for optimizing costs, see [Claessens et al., 1998], as well as passenger-oriented models such as direct traveler approaches, see [Bussieck, 1998], or travel time approaches, see [Schöbel and Scholl, 2006, Schmidt, 2014]. But solutions obtained by above models often fall short with respect to objectives that are hard to measure but used in practice, e.g. the memorability of the created system. A common concept to achieve memorability is a system or pulse headway, see [Vuchic, 2017], allowing for regular departures and transfers of the passengers. To incorporate this important practical aspect into mathematical line planning models, especially into cost-oriented ones, is a new approach presented in [Friedrich et al., 2018a], see Appendix B.
The authors define a system headway as a common divisor of the frequencies of all lines, i.e., for a given line conceptL with frequencies fl for line l∈ L a common divisor i ̸= 1 of all fl is called a system headway. With this, the requirement of a system headway can be included in a general line planning model, i.e., extending
(P) minobj(f, x) s.t. g(f, x)≤b
fl∈N0 l ∈ L0 x∈X
to
(P(i)) min obj(f, x) s.t. g(f, x)≤b
fl=αl·i l ∈ L0 αl∈N0 l ∈ L0 fl∈N0 l ∈ L0
x∈X
where obj(f, x) is an objective function dependent on the frequencies and some auxiliary variablesxand with general constraints g(f, x)≤b, some variable domain X, a line pool L0 and a (fixed) system headway of i. The authors first analyze the complexity of the arising formulation and derive the following theorem.
Theorem 2.1 ([Friedrich et al., 2018a], Theorem 1). Let (P) be a general line planning problem for a given instance based on a fixed planning period. Then problem P(i) is equivalent to a line planning problem (P’). The new line planning problem (P’) has the same number of variables and constraints as (P).
The authors also provide a formulation (Psys−head) to find the best system head-wayαfor a given problem instance, i.e., to find a line concept with a system headway, without fixing it beforehand. Since the provided formulation is a quadratic integer program and therefore not competitive in practice, further analysis of good sys-tem headway values is provided. The authors derive the property that for a given number i as a system headway, divisors ofi always provide better system headway values, see [Friedrich et al., 2018a], Lemma 1, resulting in the following corollary and limiting the search space for optimal system headways immensely.
Corollary 2.2([Friedrich et al., 2018a], Corollary 1).There always exists an optimal solution (α, f, x) to (Psys−head) in which the optimal system headway α is a prime number.
Apart from divisors, there are no known practical conditions on the relation be-tween different system headway values, e.g. there are cases where a smaller system headway may have worse objective value or may even be infeasible. Examples for both cases are given for common constraint types and objective functions. Addition-ally the authors provide classes of line planning problems where the feasibility of system headway solutions can be guaranteed, see [Friedrich et al., 2018a], Lemma 3.
Furthermore, it is possible to determine a priori bounds in special cases. For this, the authors consider a cost-oriented model without upper frequency bounds, i.e., the problem
min ∑
l∈L0
fl·costl s.t. femin ≤ ∑
l∈L0: e∈l
fl e∈E
fl = αl·i l ∈ L0 fl, αl ∈ N0 l ∈ L0 for given costs costl for every line l∈ L0.
For this problem, the worst case ratio of the optimal objective values opt(i) and opt(j) for system headwaysi and j can be determined beforehand.
Theorem 2.3 ([Friedrich et al., 2018a], Theorem 2). Let i, j ∈ N, i ≤ j. Then opt(j)≤ jiopt(i).
Luckily, these rather high theoretical bounds are not realized in practice, as can be seen in the experimental evaluations, see e.g. Figure 5a.
Unfortunately, the authors show that it is not possible to determine such bounds for passenger-oriented models. These often work with budget constraints to prevent
a trivial system that is optimal for every passenger but to costly for the operator.
But when such constraints are used it is not possible to guarantee feasibility for different system headways or provide bounds on the objective values beforehand.
1 2 3 4 5 6 7 8 9 10 The Costs of the Line Concepts
(a) Cost model with system headways for dataset Grid and bound from Theo-rem 2.3
(b) Direct travelers model with system head-ways for datasetGermany
Figure 5: Different solutions with system headways
To check the practical effects of system headways, the authors provide experi-mental evaluations on three different datasets, the benchmark datasetGrid created in [Friedrich et al., 2017a] and close-to real-world datasetsGoettingenandGermany, representing the bus network of Göttingen and the long-distance railway network of Germany, respectively. All experiments are done using the open-source software framework LinTim, see [Schiewe et al., 2018a]. For each dataset, solutions are cre-ated for every system headway value from 2 to 10, using a cost-oriented and a direct traveler model with a budget. Additionally, a solution without a system headway is computed as a reference value, marked with 1 in the figures presented here. In Figure 5, solutions are depicted for dataset Grid and dataset Germany. Figure 5a shows that despite increasing the costs for higher system headway values, the the-oretical bound is not reached in practice. Additionally, it is not always true that a higher system headway leads to higher costs, see e.g. the different cost values for system headways of 2 and 3 which results from the demand structure of the used dataset. For the direct traveler model, Figure 5b provides the insight that increasing the system headway results in worse objective values due to the inability to fill the budget efficiently. Again, removing the budget would solve this problem but would result in trivial solutions for all system headway values.
1 2 3 4 5 6 7 8 9 10 System Frequency
50 51 52 53 54 55
PerceivedTravelTime(min)
(a) Timetable quality for dataset Goettingen
1 2 3 4 5 6 7 8 9 10
System Frequency 174
175 176 177 178 179 180 181 182 183
PerceivedTravelTime(min)
(b) Timetable quality for dataset Germany
Figure 6: Quality of the timetable for different system headways
As discussed extensively in the literature, see e.g. [Goerigk et al., 2013, Burggraeve et al., 2017, Schöbel, 2017], line planning solutions should not be considered isolated from later planning stages. To check the influence of the computed solutions on the travel time of the passengers, a periodic timetable is computed for each line plan, using the heuristic MATCH approach, see [Pätzold and Schöbel, 2016]. Some of the results are depicted in Figure 6. Overall, a higher system frequency seems to provide a denser system, allowing faster travel and transfer times of the passengers.
But again, this is not always the case, sometimes leading to an increase in travel time when the system headway is increased.