Of course, the overall goal in practice is to not only find solutions for the single planning stages, but to find a good overall system, i.e., a public transport plan (L, π,V)with a line concept L, a periodic timetableπ and a vehicle scheduleV such both the passenger convenience in the timetable and the operational costs mainly determined by the vehicle schedule is optimized. Therefore looking into integrated planning is to be preferred over sequential planning.
This intent already proofed useful in other applications. [Lundqvist, 1973] pro-vides early insights into integrating several interdependencies into urban planning.
Especially for scheduling, several publications integrate other stages, see [Lenderink and Kals, 1993] and [Tan and Khoshnevis, 2000] for process planning and [Gross-mann et al., 2002] for integration of general planning problems. Futhermore, [Barratt and Oliveira, 2001] discuss integration in a supply chain context and [Darvish and Coelho, 2018] compare different sequential and integrated approaches for the same problem. Other applications include the location planning for distribution centers, see [Nozick and Turnquist, 2001], or multi-modal route planning, which in itself is a form of integrated planning, since a route through multiple transportation sys-tem is planned in an integrated fashion instead of sequentially. For an example, see [Dibbelt et al., 2015].
Due to the advances in other topics, integrated planning gained popularity in the public transport research community as well and remains an ongoing problem.
For recent overviews see [Borndörfer et al., 2017c] for a collection of several success
stories in practice and the recent special issue presented by [Meng et al., 2018].
Therefore, in the following some possible integration stages are shortly described and some corresponding literature is given.
First, the integration of line planning and timetabling is discussed. [Goerigk et al., 2013] present that the consideration of later planning stages when evaluating a line concept is crucial, since the chosen lines influence the quality of the result-ing timetable and may even lead to infeasibility in later stages. While [Schmidt, 2005] combines line sections into lines and sets their times integratedly, [Rittner and Nachtigall, 2009] choose a column generation approach to solve an integrated integer programming model. Other approaches often use heuristics to find solu-tions for the integrated problem, see e.g. [Kaspi, 2010, Kaspi and Raviv, 2013] for solving line planning with stopping patterns and timetabling using a cross-entropy heuristic or [Torres and Irarragorri, 2014] for the planning of multiple planning periods with possibly different passenger demand with two metaheuristics. More re-cently, [Burggraeve et al., 2017] presented an iterative approach, focussing on travel time in the line planning stage and robustness in the timetabling stage. Here, the transfer stations are restricted beforehand to reduce problem size.
Since the chosen passenger weights ca in the timetabling stage greatly influence the quality of the resulting timetable, many researchers investigate the effect of in-tegrating the routing decision into the timetabling model instead of solving it in a preprocessing step separately. [Borndörfer et al., 2017b] show that the theoretical gap between these two approaches is unbounded. [Siebert, 2011] introduces an inte-grated model to solve both stages at the same time while [Schmidt, 2014] includes the routing decision additionally into other stages such as line planning and provides several NP-hardness results for the resulting problems. To deal with the computa-tional complexity, [Gattermann et al., 2016] integrate the routing stage into the SAT model of [Großmann et al., 2012], since using SAT solvers to find solutions for periodic timetabling models is able to deliver good computational results in practice.
As another approach, [Schiewe and Schöbel, 2018] present an integer programming model, including exact preprocessing methods to reduce the problem size. For line planning, [Schmidt and Schöbel, 2015a] show that integrating the routing stage re-sults in an NP-hard problem. This is true for integrating routing into aperiodic timetabling as well, see [Schmidt and Schöbel, 2015b], even though the aperiodic timetabling problem itself is solvable in polynomial time. More recently, [Robenek et al., 2017] present a model integrating the routing into a mostly periodic plan, but with additional trips for peak hours.
One of the problems of solving periodic timetabling and vehicle scheduling sequen-tially is the underlying conflicts of objective functions. As discussed in Section 2.2 and 2.3, timetabling models often focus on passenger convenience, while most vehi-cle scheduling models try to optimize the operational costs. Solving both of these
stages independently therefore often leads to undesirable solutions w.r.t. the opera-tional costs, since good solutions for the passengers may not allow any cost-efficient solution. Therefore, there is much research focusing on an integrated approach to solving these two planning stages. One possible approach is to consider the effects on possible vehicle schedules in the timetabling step. [Lindner, 2000] integrates cost ap-proximations into timetabling, allowing for a model for periodic timetabling that op-timizes the costs while [Dutta et al., 2017] adds some vehicle scheduling constraints into the timetabling model. A similar approach is chosen in [Pätzold et al., 2017], see Appendix E and the summary in Section 3.5. Another approach is to integrate both problems into a single integer programming model. [Schiewe, 2018] presents such a model which is still able to solve medium-sized instances with commercial solvers to optimality in a reasonable time frame. [Schmid and Ehmke, 2015] present another bi-objective model for a vehicle scheduling problem with time windows. The goal is here to balance the departure times in timetabling and it is achieved using a metaheuristic and a weighted sum approach. For aperiodic timetabling, [Ibarra-Rojas and Rios-Solis, 2011] present an integrated model, but additionally include sync intervals for the timetable, resulting in nearly periodic plans. [Cadarso and Marín, 2012] solve a similar problem with extra shunting constraints. Since solving both problems simultaneously is computationally more challenging, other research focuses on heuristic approaches. [Mandl, 1980] presents a re-optimization of the ve-hicle schedule afterwards, trying to reduce the passenger travel time after a veve-hicle schedule is fixed. Similarly, [Petersen et al., 2013] present a model to modify the timetable during the vehicle scheduling stage to reduce the operational costs without decreasing the timetable quality too much. It is solved using a large neighborhood search heuristic. Other literature includes the local optimization of both solutions after they were computed, as is e.g. presented in [van den Heuvel et al., 2008] for periodic and in [Guihaire and Hao, 2010] for aperiodic timetabling. [Yue et al., 2017]
present an integrated model for aperiodic timetabling and vehicle scheduling as well, using a simulated annealing method and [Fonseca et al., 2018] present a matheuris-tic approach for a similar problem, changing some departures and arrivals in each iteration before computing a new vehicle schedule.
There is also some work on integrating vehicle scheduling and crew scheduling, see e.g. [Mesquita and Respício, 2009] for a branch&bound and branch&price approach for the multi-depot case.
There are some first results on integrating all three stages, namely line planning, periodic timetabling and vehicle scheduling, but due to the computational challeng-ing aspects of such big models, only heuristic approaches are able to solve reasonable sized instances. [Lübbecke et al., 2018] present such an integrated model, examining decomposition approaches for solvability of very small instances. [Li et al., 2018] inte-grate aspects of line planning and vehicle scheduling into timetabling for the special
case of one single track line. Other approaches are iterative, e.g. [Liebchen, 2008b]
presents an integrated model for timetabling and vehicle scheduling, which is then iterated with a line planning heuristic to compute public transport plans. [Schöbel, 2017] presents a theoretic meta-model, interpreting models for the sequential prob-lems as nodes in a graph called eigenmodel. These nodes can then be combined in different orderings, providing different heuristics for finding a public transport plan.
For more information on this model, see the discussion in Chapter 4. [Michaelis and Schöbel, 2009] present such a possible combination, starting with the vehicle scheduling in the sequential planning process.
There are also some more theoretical works on the benefit of integrating. [Lee et al., 1997] analyze the problem of not integrating in a supply chain context, while [Kidd et al., 2018] provide the value of integration for the same area. More generally, [Schiewe, 2018] defines the price of sequentiality, a measurement of the benefit of integration for general multi-stage problems and presents some theoret-ical results, e.g. under the assumption of some structures of objective functions and constraints. A related topic to integrated optimization is the consideration of interwoven problems, i.e., multiple optimization problems that are not structured hierarchical as in the cases of integrated optimization considered in this thesis but coequally with a shared set of variables and associated constraints. For a general introduction, see [Klamroth et al., 2017].