PaperlayerView exact revision on arXiv

Source-linked AI summary

Design and analysis of demand-adapted railway timetables

David Canca, Eva Barrena, Encarnación Algaba, Alejandro Zarzo

arXiv:1307.0970v1math.OC

TL;DR

Timetable models rarely represent realistic, variable passenger demand, even though timetables affect users’ service quality. This paper links continuous demand modeling with nonlinear integer timetable optimization; tests show train frequency is higher near demand peaks and the model’s measures match expected patterns.

  • Problem

    Many timetable approaches do not represent realistic full-day demand variation, and periodic timetables can be suboptimal from the user’s perspective when detailed demand is considered.

    Source: §1 · p. 2

  • Method

    Two linked models represent cumulative passenger demand continuously and use network and demand characteristics in a nonlinear integer program to determine timetables.

    Source: §1 · pp. 2–3, §2 · p. 4

  • Results

    Train frequency is higher close to demand peaks, and tests show the obtained measures correspond with expected patterns.

    Source: §5 · p. 19

  • Takeaways & Limitations

    The model permits different objective choices and post-study of timetables by parametrically varying train numbers or capacity, with measures spanning user and provider perspectives.

    Source: §5 · p. 19

    The constraint discussion assumes one interdeparture interval over the planning horizon, although different intervals may be more suitable in a general case.

    Source: §3.2 · p. 11

Abstract

from arXiv · show

Railway scheduling and timetabling are common stages in the classical hierarchical railway planning process and they perhaps represent the step with major influence on user's perception about quality of service. This aspect, in conjunction with their contribution to service profitability, makes them a widely studied topic in the literature, where nowadays many efforts are focused on improving the solving methods of the corresponding optimization problems. However, literature about models considering detailed descriptions of passenger demand is sparse. This paper tackles the problem of timetable determination by means of building and solving a non-linear integer programming model which fits the arrival and departure train times to a dynamic behavior of demand. The optimization model results are then used for computing several measures to characterize the quality of the obtained timetables considering jointly both user and company points of view. Some aspects are discussed, including the influence of train capacity and the validity of Random Incidence Theorem. An application to the C5 line of Madrid rapid transit system is presented. Different measures are analyzed in order to improve the insight into the proposed model and analyze in advance the influence of different objectives on the resulting timetable.

1. Introduction

Detailed passenger-demand behavior is rarely represented in timetable models, although timetable choices shape perceived service quality. This paper develops linked models to adapt line timetables to demand and assess service quality and profitability.

1. Introduction

  • Periodic timetables can be suboptimal for users when detailed passenger demand is considered, while most existing approaches do not model realistic full-day demand variation.The proposed timetable adapts each already-defined line to those variations; it does not modify the line plan.Source: §1 · p. 2
  • The paper constructs and studies relative measures of timetable service quality and line profitability using two consecutive linked models.The analysis considers both user and service-provider perspectives.Source: §1 · p. 2
  • The approach links a continuous demand model to a nonlinear integer program that determines timetables using network characteristics and demand behavior.The second model is an improved version of the authors’ earlier approach, revised for realism and for evaluating service-quality and profitability measures.Source: §1 · pp. 2–3
  • Train-launching and non-stopping decisions, along with demand-dependent stop times, make the timetable model more flexible but add binary variables and computational time.New ordering constraints are introduced to reduce the search space; stop times account for passenger arrivals between trains and free capacity at each stop.Source: §1 · p. 3
  • The models are tested on part of Madrid’s C5 suburban railway line, but assessing line profitability requires a broader network study.The authors note that maintenance, rolling stock, and crew management should also enter a global economic assessment.Source: §1 · p. 4

2. Passenger demand analysis

The demand analysis represents daily origin-to-destination mobility continuously and fits its cumulative form with sigmoid curves. This representation captures demand peaks and provides demand inputs for timetable determination.

2. Passenger demand analysis

  • OD(t) is a square station-by-station matrix whose entries represent the continuous daily evolution of travel demand between station pairs.The described demand curves can have local maxima, typically two or three per day, associated with rush hours.Source: §2 · p. 4
  • The model approximates cumulative demand with a linear combination of sigmoid curves, with the number of terms M representing the maximum number of daily demand peaks.The sigmoid parameters represent asymptotic value, time deviation, and slope.Source: §2 · p. 5
  • The sigmoid parameters are fitted by minimizing squared errors between the proposed cumulative-demand function and observed demand data at available time points.Inputs may include interval-based origin-destination matrices or station entrance and exit counts.Source: §2 · p. 5
  • The authors state that, to their knowledge, prior scheduling and timetabling models had not used variable demand data to adjust train arrival and departure times over a long period.The paper presents the cumulative-demand representation as an input for timetable determination.Source: §2 · pp. 4–5

3. Timetabling optimization model

This section presents a model that determines train times along a line in relation to cumulative passenger demand and evaluates timetable quality. It also provides a framework for comparing timetable quality and operating cost using passenger-demand and load-factor measures.

3. Timetabling optimization model

  • The formulation determines train departure times at the origin and arrival/departure times at each station using cumulative-demand approximations, and also produces timetable-quality measures.The approach extends a previous model and can be adapted to other demand characterizations, including discrete demand functions from mobility surveys.Source: §3 · pp. 5–6

3.2. Constraints

  • Timetable constraints enforce travel times and minimum headways while linking departures to passenger arrivals, boarding, waiting, train capacity, and station stop times.Capacity can be a decision variable or fixed; passenger demand that boards and demand left unserved are distinguished, and stop intervals reflect capacity-occupancy or passenger-flow time.Source: §3.2 · pp. 7–10
  • A periodic timetable can use one interdeparture interval over the planning horizon, but periodic schedules may be inappropriate during some dynamic-demand rush-hour intervals.The Random Incidence Theorem links expected waiting to interdeparture-time variance under assumptions of constant demand and capacity sufficient for arrivals; the model also targets over-waiting explicitly.Source: §3.2 · p. 11
  • The main objective minimizes total average waiting time across the planning horizon, with over-waiting also treated as important to service quality.The Random Incidence Theorem relates expected waiting to interdeparture-time variance under constant demand and sufficient capacity; the paper notes these conditions are often unmet in rapid transit and models waiting directly from arrivals and departures.Source: §3.2 · p. 11

3.3. Measures of timetable quality

  • For each fixed global capacity, varying the number of trains produces an optimal AWT Pareto frontier between service quality and operation cost.The model uses the ε-constraints method; the number of launched trains is fixed and stopping variables are set so each train stops at every station.Source: §3.3 · p. 12
  • Load factor measures supply-demand adequacy, while horizontal and vertical load factors distinguish occupancy homogeneity along the line from maximum train occupancy.A given load factor can result from many passengers over a short route segment or fewer passengers over a longer segment.Source: §3.3 · pp. 12–13
  • Served demand (SDk) measures the average percentage of arriving passengers who board train k and is linked to revenue, profitability, and attended demand.Non-served demand can lead to passenger loss in the medium term.Source: §3.3 · p. 13
  • The second decision phase confronts LF, SD, and resulting AWT to select an appropriate global capacity level and timetable as a function of operating cost.Load factors also indicate comfort through train occupancy and resource use through capacity.Source: §3.3 · pp. 12–13

4. An application of the model

The C5 application tests timetables against surveyed demand and examines how train schedules and capacity relate to waiting time and load measures. Departures align with demand peaks, while results vary with train count and capacity allocation.

4. An application of the model

  • The C5 tests fit cumulative demand functions to survey data for six stations and examine capacities from 2,000 to 10,000 passengers across one to fifty trains.Carriages are assumed to hold 40 passengers each; the 11,581-passenger demand and capacity allocations define the tested cases.Source: §4 · p. 14
  • When a train reaches full capacity on any route segment, its vertical load factor is one and its horizontal and total load factors coincide.This occurs for trains 5 through 29 in the fifty-train example; lower vertical load factor with higher horizontal load factor indicates capacity remains homogeneously high along the route.Source: §4 · pp. 15–16, Figure 6 · p. 15
  • Increasing the number of trains reduces AWT, and 8,000- versus 10,000-passenger global capacities produce practically null AWT differences.The paper notes that the higher-capacity option entails more rolling stock, while fewer carriages distributed across more trains can also yield shorter AWT.Source: §4 · p. 16, Figure 7 · p. 16

ASD ALF AVLF AHLF

The analysis compares average served demand and load-factor measures to assess timetable quality and capacity use. Their joint behavior, alongside waiting-time trade-offs, frames choices about train count and capacity.

ASD ALF AVLF AHLF

  • As train count increases, ASD and ALF both increase; at a fixed train count, greater capacity lowers ALF and raises ASD.The paper uses these opposing responses to motivate balancing the two measures.Source: ASD ALF AVLF AHLF · p. 17
  • The ASD and ALF surfaces and their level curves are considered alongside train-count/AWT curves to identify solutions balancing served demand and capacity use.The intersection curve represents solutions with equal ASD and ALF; the two sides correspond to better operator capacity use or better user service, respectively.Source: ASD ALF AVLF AHLF · p. 17, Figure 8 · p. 17, Figure 9 · p. 18
  • The ASD–ALF equilibrium curve lies around 80% for both measures and is close to the Pareto frontier for 150 wagons.In this experiment, capacity up to 10,000 passengers falls short of the 11,581-passenger daily demand, so good ASD and ALF values are sought above 50 trains, near the 200–250-wagon curves for best AWT.Source: ASD ALF AVLF AHLF · pp. 17–18, Figure 9 · p. 18
  • With fixed capacity per train, 50 trains of either five or six wagons yield the same AWT, so more capacity does not always improve this measure.These Pareto curves vary train count at a fixed number of wagons per train, unlike the fixed-global-capacity comparison.Source: ASD ALF AVLF AHLF · pp. 18–19, Figure 10 · p. 18

5. Conclusions

The paper presents non-cyclic timetable optimization using continuous demand and measures of service quality and profitability. Its tests show schedules responding to demand peaks and support parametric comparison of train-count and capacity choices, while broader network performance analysis remains open to additional operational factors.

5. Conclusions

  • The model jointly optimizes services, train capacity, stops, speeds, and arrival and departure times using a continuous full-day origin-destination demand representation.Cumulative demand is approximated by sigmoidal functions fitted through least squares, with improved approximation outside the planning horizon.Source: §5 · p. 19
  • Arrival and departure times are adjusted to cumulative demand using passenger-arrival behavior, with average waiting time as the objective.The model also discusses constraints for periodicity and determining the number of trains.Source: §5 · p. 19
  • Tests show higher train frequency near demand peaks and a correspondence between load factor and conventional occupancy; the model also supports parametric studies of train count and capacity.The authors report that the obtained measures match expectations and that different objectives can be selected.Source: §5 · p. 19
  • The proposed measures assess timetable quality using additional factors such as train capacity and occupancy, alongside user and service-provider perspectives.The analysis introduces ASD and ALF trade-offs with AWT to help providers consider train count and capacity for a target service-quality level.Source: §5 · pp. 19–20
  • A final analysis of network performance for each line would require maintenance, rolling-stock, and crew-management aspects.Source: §5 · p. 20
Loading 1307.0970v1…