An Iterative Approach for Integrated Planning in Public

As discussed in Section 2.4, integrated optimization has recently gained in impor-tance in mathematical public transport planning. Since solving integrated optimiza-tion problems exactly is often computaoptimiza-tionally not feasible for real-world instances, heuristic solutions are a topic of ongoing research.

Algorithm ReVehicleScheduling Input: line concept, timetable Output: vehicle schedule

Algorithm ReTimetabling Input: line concept, vehicle schedule Output: timetable

Algorithm ReLinePlanning Input: timetable, vehicle schedule Output: line concept

Figure 10: Overview of the algorithms

The objective of the authors in [Schiewe and Schiewe, 2018], see Appendix F, is to find a solution for the following problem:

Problem. Find a public transport plan(L, π,V), i.e., a line conceptLwith a corresponding timetableπand vehicle scheduleVsuch that the travel time of the passengers and the operational costs are minimized.

The authors introduce an iterative approach, where in each iteration two of the three planning stages line planning, timetabling and vehicle scheduling are fixed and the remaining stage is re-optimized, while guaranteeing the feasibility of the overall system. Figure 10 shows an overview of the proposed algorithmic scheme, combining the sequential optimization of the single planning stages in an arbitrary order. Since two of the resulting problems are new, completely new optimization problems need to be modeled and evaluated regarding their performance.

ReVehicleScheduling: For AlgorithmReVehicleScheduling, known algorithms from the vehicle scheduling literature, e.g. from [Bunte and Kliewer, 2009], can be used, since this is part of the standard sequential optimization problem. Here, the authors chose an aperiodic, cost-oriented model without a depot implemented in the open-source software framework LinTim, see [Schiewe et al., 2018a].

ReTimetabling: To achieve the feasibility of the vehicle schedule when re-opti-mizing the (periodic) timetable, additional constraints need to be added to the classic PESP IP formulation, see e.g. [Serafini and Ukovich, 1989], commonly used for solving the periodic timetabling problem. For the aperiodic vehicle scheduling problem, the authors assume that each line in the line concept L is covered pmax

times, resulting in the set oftrips T ={(p, l) : p ∈ {1, . . . , p_max}, l ∈ L} that each need to be served by a vehicle scheduleV exactly once. Two trips(p₁, l₁), (p₂, l₂)are compatible if there is sufficient time to get from the last station of line l₁ to the first station of linel₂. The authors denote this (aperiodic) times asendp1,l1 andstartp2,l2, respectively, and the minimal time betweenl₁ and l₂ asL_l1,l2. L_l1,l2 is assumed to be given by a fixed shortest path in the underlying infrastructure network, determined by some lower travel time bounds on the edges. A vehicle schedule contains a set of vehicle routes where each vehicle route is a list of compatible trips, i.e., all consecutive trips are compatible. The set of all such connecting trips is denoted as C. With this, the constraints

Ll1,l2 ≤startp2,l2 −endp1,l1 ((p1, l1),(p2, l2))∈ C (3.2) in addition to several auxiliary constraints to ensure the correct values for thestart andendvariables need to be added to the classical PESP IP model. Using this, a new solution still allows the current vehicle schedule to be feasible, since all connecting trips remain compatible due to (3.2).

ReLinePlanning: More work needs to be done to maintain feasibility of the time-table and the vehicle schedule when re-optimizing the line concept, since the lines are such an integral part of both fixed stages. Additionally, allowing aperiodic vehicle schedules makes finding new lines difficult, since lines have to appear periodically.

First, the authors define a public transport plan (L^′, π^′,V^′) to be consistent to another plan (L, π,V) if

• the vehicle paths on trips for V^′ are contained in the physical paths of the vehicle in V, including coinciding times and

• the duration for trips of new lines inL^′ allow for the service of the line.

Note that the last point is necessary, since connecting trips may not be converted to lines if their duration is too short, e.g. if passengers would not have enough time for boarding and alighting the line in each station. Although this definition restricts the possible lines for a new public transport plan, it is e.g. possible to connect different lines served by the same vehicle or split lines up, allowing for new optimization potential in the other stages of the iterative approach.

Algorithm 6.1 ReLinePlanning

1: Define line network:

2: One edge for each (aperiodic) service of an edge in the PTN

3: Label these edges with the vehicle and the starting time

4: Define collapsed line network:

5: Combine parallel edges from the line network with the same periodic

6: starting time

7: Label each of these edges with a tuple of vehicles using it and the

8: periodic starting time

9: Find set of longest pathsP, s.t. all edges in a path have identical labels

10: Set the line pool as the set of all subpaths ofP

11: Solve a line planning problem such that

12: all infrastructure edges are covered according to given minimal frequencies,

13: all collapsed edges are covered at most once

14: and the costs are minimized

Algorithm 6.1 provides an overview on re-optimizing the line concept of a given public transport plan, i.e., how to find a new public transport plan that is consis-tent with the current solution and minimizes the line costs. The authors proof the correctness of the algorithm with Theorem 6.1.

Theorem 6.1 ([Schiewe and Schiewe, 2018], Theorem 11). Let (L, π,V) be given.

Let the duration of the edges in connecting trips inV be uniquely determined and let for each edge e ∈ E the aperiodic departure times be unique for all trips (p, l)∈ V and connecting trips c∈ V, i.e., there is at most one departure using a specific edge in the PTN at any point in time. Then Algorithm ReLinePlanning finds a public transport plan (L^′, π^′,V^′) that is consistent with (L, π,V) such that line concept L^′ is feasible and minimizes the line costs.

Note that the assumption of unique departure times can be easily guaranteed by using headway constraints for the underlying public transport plan and is often satisfied in practice. The assumptions of uniquely determined durations for the connecting trips edges may on the other hand not be satisfied in practice. If this is the case, the algorithm still finds a feasible solution, but the optimality cannot be guaranteed.

After modeling the problems and proposing algorithms, these can now be com-bined iteratively. Since they can be comcom-bined in any order, each specific order provides an algorithm for re-optimizing a public transport plan. The authors inves-tigate several approaches and analyze the convergence behavior.

init Veh Tim Lin Veh Tim Lin Veh Tim Lin Veh Tim Lin Veh Tim Lin Veh Tim Lin 0.80

0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20

relative operational cost relative travel time

Figure 11: Re-optimizing datasetGrid

Theorem 6.2([Schiewe and Schiewe, 2018], Theorem 18). LetP₀be a feasible public transport plan with travel time t₀. LetP_i, i∈N⁺, be a public transport plan derived from Pi−1 by applying either ReTimetabling or ReVehicleScheduling and let ti

be the travel time of P_i. Then the sequence of travel time values (t_i)_i∈N decreases monotonically and converges.

Theorem 6.3([Schiewe and Schiewe, 2018], Theorem 19). Let P₀ be a feasible pub-lic transport plan with operational costs c₀ where duration based costs are neglected.

Let P_i, i ∈ N⁺, be a public transport plan derived from P_i−1 by applying either ReLinePlanning, ReTimetabling or ReVehicleScheduling and let c_i be the op-erational costs of P_i. Then the sequence of operational cost values (c_i)_i∈N decreases monotonically and converges.

Note, that for the cases where convergence is not guaranteed, i.e., the possible increase of the travel time in AlgorithmReLinePlanningand the possible increase of the costs when duration based costs are considered, the authors give examples for the non-convergence, see [Schiewe and Schiewe, 2018], Examples 14 and 15, respectively.

The computational experiments cover two different datasets, dataset Grid as a case study and datasetRegional, a close-to real-world representation of the regional train system in southern Lower Saxony, Germany. For dataset Regional, several demand scenarios are created and the average changes in the objectives are discussed.

Some results are presented in the following.

Figure 11 depicts a typical behavior of the objective values when re-optimizing a public transport plan. Algorithm ReLinePlanning may increase the travel time and does so in several cases while simultaneously reducing the costs or facilitating

v₂₁ Figure 12: Line concept for datasetGrid before and after applying the

re-optimiza-tion depicted in Figure 11

this for the vehicle scheduling step afterwards. Overall, the operational costs can be reduced by around 18% while only increasing the travel time by 8%, yielding a new competitive solution. Other iteration schemes provide different trade-offs, but the overall picture stays the same. Especially the monotonic behavior of the iteration schemes discussed in Theorem 6.2 and 6.3 is shown in Figure 14 of [Schiewe and Schiewe, 2018].

The influence of the re-optimization on the line concept is shown in Figure 12.

There are multiple lines staying the same, e.g. the dotted blue line, but other lines are either shortened to reduce costs of unneeded coverage, e.g. the dashed orange line, or new connections are formed, e.g. the dash-dotted cyan line. Here, passengers fromv₆ are allowed a direct connection to v₁₂,v₁₇ and v₂₂ after the re-optimization where at least one transfer was necessary beforehand.