Quadratic Programming Model for Optimizing Demand-responsive Transit Timetables Huimin Niu Professor and Dean of Traffic and Transportation School Lanzhou Jiaotong University, China, Xuesong Zhou Associate Professor Arizona State University, U.S. 1

 Introduction  Quadratic integer programming model  Optimization model reformulation for high-resolution demand input data  Reformulation for low-resolution demand data and long planning horizon  Numerical Example  Conclusions Outline 2

1. Introduction 3 Public transit operations and management Passenger train timetable design problem on an urban rail transit line Input: Time-varying and stochastic passenger demand patterns Available train-unit fleet Practical scheduling regulations. Output:Determine the arrival and departure times for each train at each station.

Quick Review: Cumulative Flow Count Diagram A(t) = cumulative arrivals from time 0 to time t D(t) = cumulative departures from the system from time 0 to time t L(t) = number of customers in the system at any time t = A(t) – D(t) W(t) = total time spent by customers up to time t 4

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Illustration: Bulk Queue: Constant Arrival/Batch Departure 6 Train/Bus arrival time Area of Waiting Time  Quadratic Programming

Demand-sensitive Schedule 7 Reference: Urban Transit Scheduling: Framework, Review and Examples by Dr. Avishai Ceder

Many problems have not been solved appropriately in the existed researches, such as passenger demand, optimization objective, train skip-stop pattern, solution algorithm, and different applications. Motivation Periodic-schedule-based timetables 8 The periodic-schedule-based timetables are not fully sensitive and responsive to the time-varying passenger demands, which could result in long waiting times and reduced service reliability, particularly under irregular oversaturated conditions.

1) To derive mathematically tractable formulas for calculating passenger waiting times under time-varying demands from different OD pairs and non- cyclic timetables. 2) To design train timetables by jointly considering skip-stop patterns and spatially distributed OD passenger demands. 3) To consider the short-term dispatching and long-term timetabling. Main Challenges 9

Problem Statement: To optimize train timetables for a high-speed rail corridor for both real-time scheduling and medium-term planning applications. Input: 1) Time-dependent passenger demand patterns 2) Predetermined train skip-stop pattern 3) Practical regulations. Purpose: To minimize the total passenger waiting time at stations Output: Determine the arrival and departure times for each train at each station. 2. Quadratic integer programming model 10

Passenger waiting time The total passenger waiting time at station u toward station v boarding train j is: or : departure time of train j at station u on the line; : cumulative number of arrival passengers at station u heading to station v by time t; : cumulative number of departure passengers at station u heading to station v by time t; : number of passengers who arrive at station u travelling to station v at time t. 11

Formulation 1)Objective The objective function considered in this paper minimizes the total waiting times of passengers at stations. If the passengers arrive at station u with a uniform rate between two trains and j, under time-invariant demand conditions. : the arrival rate of passengers at station u 12 TD j-1 TD j

2) Constraints Train capacity constraints Linking constraints Safety headway constraints Feasible range constraints Departure time constraints 13

3. Optimization model reformulation for high- resolution demand input data Consider one-minute periods: for on-line scheduling :departure binary variable which is equal to 1 if train j departs from station u by time t; otherwise, it is 0. Introduce new binary variables :binary loading indicator variable which is equal to 1 if passengers from station u towards station v can board train j at time t; otherwise, it is 0. 14

Constrains For a particular train j and station OD pair (u, v), we introduce a constant binary parameter with an arbitrary train indicator j' whose value equals 1 if the train index is ; otherwise, it is 0. Calculation: Capacity constrain: Connection constrain: Calculation the train departure time: Objective Function: 15

4. Reformulation for low-resolution demand data and long planning horizon (1)a relatively long demand aggregation time interval (2) a long planning horizon T. (i.e., 60 min) Introduce new parameters and variables :the number of time periods; :demand time period index; the demand for the kth period covers time interval ; k : given demand flow rate for the station OD pair (u, v) during one-hour period k; : binary variable that indicates if a train departure time belongs to a specific period k; 16

Piecewise Linear Approximation The total waiting time at station u,, for the departure train j and station OD pair (u, v) can be represented as below. The number of passengers remaining in train j: Constrains Objective function The above model can be solved directly by standard optimization packages. 17

5. Numerical Example This paper applies GAMS to implement and solve the proposed train scheduling models. To accelerate the computation and ensure the accurate results, this study relaxes the integerality requirements for a number of time variables. This study uses the Shanghai-Hangzhou High-speed Rail Line of China, which comprises 9 stations, as a test case. Train free-flow running time between two adjacent stations (min) Adjacent station fairFree-flow running time Adjacent station pair Free-flow running time

Case with high-resolution minute-dependent demand 19

Hour-dependent demand case 20

Comparison of implemented models with different planning horizons Model with high-resolution demand data with long demand aggregation interval Planning horizon1 hour2 hours3 hours5 hours10 hours15 hours Number of trains Number of continuous variables Number of binary variables Number of constraints Computational time 2.7 minutes 10.7 minutes 138 minutes 16.9 minutes 40.6 minutes78 minutes 21

1) Mathematically rigorous and algorithmically tractable nonlinear mixed integer programming models. 2) For real-time scheduling applications, the simulation tests show that the computation time is intractable if the study horizon exceeds 3 hours. 3) For daily operational application, a number of test runs indicate that the computation time is generally acceptable from 5 hours to 15 hours planning horizon. 6. Conclusions 22

Thanks! 23