3. Paper Summaries 17
3.5. Look-Ahead Approaches for Integrated Planning in Public
Optimizing operational costs in the sequential planning process of public transport planning is very difficult, since the costs can only be correctly evaluated after com-puting a vehicle schedule, which is done after the line concept and the timetable are already fixed. To reduce the operational costs, in [Pätzold et al., 2017], see Ap-pendix E, the authors propose several approaches to improving the approximation of the costs in the line pool generation, the line planning and the timetabling stage and therefore enhance the optimization of the costs while maintaining the sequential approach.
To evaluate a public transport plan (L, π,V), consisting of a line concept L, a timetable π and a vehicle schedule V, costs are computed using weight parameters (c1, . . . , c5), i.e.,
gcost(L, π,V) :=c1·durfull+c2·lenfull+c3·veh+c4·durempty+c5·lenempty. Here, durfull and durempty are the full and empty duration of the vehicle schedule, i.e., the time spend while serving a line or serving an empty or connecting trip, i.e., the connection between two consecutive trips, respectively, lenfull and lenempty are the full and empty distance, i.e., the distance driven to serve a line or a connecting trip, respectively, and veh is the number of vehicles necessary to operate the vehicle schedule. To evaluate the passengers convenience, the perceived travel timegtime(π) is used.
Therefore, the authors consider the following problem.
Problem. Find a feasible public transport plan (L, π,V) that minimizes the two objectives gcost(L, π,V)and gtime(π).
Since gcost(L, π,V) can only be computed after the vehicle schedule is known, the authors concentrate on approximating this objective already in earlier planning stages, allowing for public transport plans with lower overall costs. Therefore, the following three improvements to the sequential planning process are proposed.
Improvement 1: New Costs for Line Planning To approximate the operational costs in the line planning stage, the authors assume line-pure vehicle schedules to be used to provide an upper bound, i.e., every vehicle serves a single line and its
backwards direction. This reduces the empty distance lengthempty to zero. The empty duration can be easily computed in such a model with
empty duration after serving line l = T
2 −(durl mod T 2),
where durl is the duration of the line, fixed by the lower travel time bounds on the edges in the PTN. Similar, the number of vehicles needed for the operation of a line can be computed by
#vehicles needed for line l and backwards direction=⌈
2·(durl+Lturn)/T⌉ , where Lturn is the minimal turnover time between the service of two lines.
Using these two formulas, the cost of a line in the line planning stage is replaced by
costl = 2·c1·durl+2·c2·lenl+ c3 pmax·
⌈
2· durl+Lturn T
⌉
+2·c4· (T
2 −durl mod T 2
) , where lenl is the length of line l and pmax is the number of time periods covered by the vehicle schedule.
Improvement 2: A New Line Pool To include more lines which are well suited for a line-pure vehicle schedule into the line pool, the authors adapt the algorithm described in [Gattermann et al., 2017]. Here, the goal is to only construct lines which can be used efficiently in a line-pure vehicle schedule, i.e., without too much buffer time when serving the line and its backward direction consecutively. This is achieved by introducing the inequality
T
2 −Lturn−α≤durl mod T 2 ≤ T
2 −Lturn
as a feasibility constraint into the algorithm. Forward and backward direction of a line served consecutively therefore only differ at most 2α from a multiple of the period length, making a line-pure vehicle schedule very efficient, depending on the choice of α. Note that choosing α too small may lead to infeasible solutions, de-pending on the problem instance.
Improvement 3: Vehicle Scheduling Before Timetabling As a third improve-ment, the authors introduce turnaround activities into the timetabling model, re-stricting the possible departure times for lines. The goal is to restrict the time between the end of a line and the beginning of its backward direction such that there is always enough time to serve the lines consecutively by the same vehicle
(Lturn) as well as not too much time to make such a vehicle schedule inefficient (Lturn + 2α). This ensures the feasibility of the line-pure vehicle schedule and can therefore be interpreted as a vehicle scheduling step before timetabling. Hence, we call this improvementVS-first. After timetabling, the vehicle schedule is optimized again, potentially improving the operational costs even further.
To test the proposed improvements, the authors provide computational experi-ments on the datasets Grid and Germany, where all three improvements are tested separately and in combination.
50000 52000 54000 56000 58000 60000 62000 64000
Total Perceived Travel Time
16000 17000 18000 19000 20000 21000 22000 23000
Costs
normal pool new pool combined pool
normal cost new cost
TT first VS first
Figure 8: Different improvements for datasetGrid
Figure 8 depicts the different objective values for dataset Grid. Almost all combi-nations of improvements are able to reduce the costs of the original approach which is depicted by a circle, filled gray in the left half. The overall least costly solution can be found by combining all approaches, i.e., using a combined pool, consisting of new and old lines, the new cost structure for line planning and solving vehicle scheduling before timetabling. In general, solving timetabling first always leads to a faster solution for the passengers but increases the costs compared to solving the vehicle scheduling problem first.
Since the choice of α is crucial for the quality of the obtained lines, Figure 9 depicts the influence of this parameter on the quality for datasetGrid. The authors conclude that the choice of a smaller α most of the times improves the operational
2 4 6 8 10
α
16000 17000 18000 19000 20000 21000 22000 23000
Costs
normal pool new pool combined pool
normal cost new cost
TT first VS first
Figure 9: Influence for different α values for dataset Grid
costs, but it may not be chosen too small to still allow for the feasibility of the public transport plan.
In the end, the authors analyze the effects on the bigger dataset Germany, where improvements of more than 40% for the operational costs are possible when consid-ering thenewandcombinedpool. Additionally, solving the vehicle scheduling stage first can save up to 5% of the costs.