Cooperative Passenger Inflow Control in Urban Mass Transit Network with Constraint on Capacity of Station

In urban mass transit network, when passengers’ trip demands exceed capacity of transport, the numbers of passengers accumulating in the original or transfer stations always exceed the safety limitation of those stations. It is necessary to control passenger inflow of stations to assure the safety of stations and the efficiency of passengers. We define time of delay (TD) to evaluate inflow control solutions, which is the sum of waiting time outside of stations caused by inflow control and extra waiting time on platform waiting for next coming train because of insufficient capacity of first coming train. We build a model about cooperative passenger inflow control in the whole network (CPICN) with constraint on capacity of station.The objective of CPICN is to minimize the average time of delay (ATD) and maximum time of delay (MTD). Particle swarm optimization for constrained optimization problem is used to find the optimal solution. The numeral experiments are carried out to prove the feasibility and efficiency of the model proposed in this paper.


Introduction
More and more passengers travel by urban mass transit (UMT) because UMT is rapid, punctual, and green.With the increasing number of passengers, the capacity of urban mass transit network (UMTN) cannot satisfy the trip demands of passengers.There are lots of passengers who queue and wait because of insufficient capacity of trains [1].Dangers are brought while there are too many waiting passengers in stations.In the cities in China, such as, Beijing, Guangzhou and Shanghai, the UMT corporations usually control passenger inflow in stations where there are too many waiting passengers to assure safety [2,3].
The approaches of passenger inflow control include controlling passenger inflow in a special station [4], in two stations [5], in stations on a special line [6], or in stations on different lines [7].In the studies above, the model of controlling passenger inflow in a special station, two stations, and stations on a special line has been described accurately and solved by some methods.Controlling passenger inflow of stations in different lines is described in a qualitative way and there is no model and no effective method.
In UMTN, many trips involve at least one transfer [8].Transferring passenger takes a great proportion of waiting passengers in transfer stations.The number of waiting passengers cannot be controlled under safety limitation only by controlling inflow of this transfer station.It is necessary to limit the inflow of original station which constitutes the transferring passengers and waiting passengers in this transfer station.So the cooperative inflow control involving stations in the whole network is needed to reduce the pressure of transfer station and improve the safety and efficiency of passengers' trips.This will be described in detail in Section 2.1.
Thus we build model to describe cooperative passenger inflow control in the whole network (CPICN).This model and the solving method will provide theoretic support for planning control measure in real operation.
The rest of this paper is organized as follows.Section 2 analyzes the necessity and evaluation criteria of CPICN.Section 3 builds the model, including assumption, objective, and constraints.Section 4 solves the model by PSO algorithm.Section 5 carries out numerical experiments to prove the feasibility and effectiveness of this model and solving method.Section 6 summarizes the contributions in this paper and puts forward the further research.

The Necessity of CPICN.
A simple network is constructed as shown in Figure 1 to illustrate the composition of waiting passengers in transfer station.Figure 1(b) describes some detail in Figure 1(a).In the simple network there are 5 stations which are  0 ,

How to Evaluate Solution of CPICN.
Passenger inflow control can change the number of passengers waiting outside or inside of stations.The waiting time of passengers can be changed along with it.The waiting time of a passenger may consist of several parts shown in Figure 2.These waiting parts can be classified into three types: 1, 2, and 3.1 represents the waiting time outside of station caused by inflow control, 2 represents the waiting time used to wait for the first coming train, and 3 represents the waiting time used to wait for the next coming train which can be boarded because of insufficient capacity of the first coming train.2 is inevitable for a trip.1 and 3 are extra delay for passengers' trip, which can be changed and controlled by inflow control.Here we name the sum of 1 and 3 as time of delay (TD).
Denote TDOC  as the time used by passenger  to wait outside of station because of inflow control, which corresponds to 1.Denote TDIO  as the extra time used to wait for next train that can be boarded, which corresponds to 3.TD for a special passenger  can be described as follows: Inflow control leads to less TD for some passengers while more TD for some other passengers.So we use ATD to evaluate the solution which represents average TD.It is computed according to (2), where PA is the set of passengers in whole network and PAC is the number of passengers in PA.For the fixed transport capacity of network, if ATD is less, more passengers are served and the control solution is better Moreover, it exists that some passengers have longer TD than others.If TD  is much longer than TD   (  ∈ PA−{}), it is unfair for passenger .So we define MTD to represent the longest TD  , which can be described as (3).The solution is better while MTD is less MTD = max (TD  ) ,  ∈ PA. (3) Thus, the general efficiency and individual interest are both considered.The solution of CPICN is needed to satisfy minimum ATD and MTD.

Assumptions of Model.
To simplify the problem and highlight the key conceptions, some assumptions are given as follows.
(1) The time-dependent O-D trip demand is known and discretized into time segments.It is a uniform random passenger arrival distribution in a time segment Δts.
(2) The control solution is time varying for every station.
(3) Passengers queue and board trains according to firstin-first-out discipline.
(4) All passengers who enter transit network will reach their planned destination and will not give up the trip at midway.
(5) All transit trains have fixed capacity and operate precisely according to specified timetables.
(6) Capacity of platform is used as the capacity of station supplied for waiting passengers.
(7) In a transfer station, different lines use different platforms.Equation (4c) represents that the number of passengers aboard train tr should not exceed the capacity of tr, where tp tr is denoted as number of passengers aboard tr, tc tr as capacity of tr.

Model of
In (4d),  ts  is the maximum number of passengers allowed to enter station   under control in time segment ts, and  ts  is the number of passengers who want to enter station   in this time segment. ts  can be expressed as (5), where  ts  is the number of arrival passengers who arrive at   in time segment ts, and ts − 1 is the time segment before ts Equation (4e) corresponds to assumption (4a), (4b), (4c), (4d), and (4e), where tp  tr is the number of passengers on tr when tr arrives at station   .al  tr and ab  tr represent the number of passengers alighting from tr and boarding tr at   , respectively.tp +1 tr represents the number of passengers on tr when tr arrives at next station  +1 .

Solution Procedure
4.1.Method Used to Solve Model.According to assumption (2), the solution of model can be expressed as in (6), where  is the number of stations and  is the number of time segments (the solution of model): The maximum numbers of passengers allowed to enter station in different time segments of different stations influence each other for the constraints of model.Because of the complexity, particle swarm optimization (PSO) for constrained optimization problem is chosen to solve the model.
Particle swarm optimization (PSO) originally was developed by Eberhart and Kennedy in 1995 [9], which belongs to a class of methods known as evolutionary computation.It hypothesizes that there are  particles in the D dimensions' space.Fitness function used to evaluate particles in space.For the particle  of iterative  + 1 generation, the position and velocity of particles are adjusted according to the position and velocity in the previous  generation, as follows: where   and V  are position and velocity of particle  in dimensionality . 1 and  2 are weight coefficients of the particle individual and  1 and  2 are two random functions whose values are between 0 and 1.   represents the previous best local position of the particle .  represents the best global position found up by the whole swarm.
An important improvement in the basic PSO is the use of an inertia factor  [10] that multiplies the velocity term in (7), as shown in (9).In general,  starts large and is decremented from  ini to  end during the evolution of the algorithm as shown in (10), where Gk is the largest generation of algorithm Clerc and Kennedy [11] proposed an alternative version of PSO in which the convergence factor is used, and the velocity adjustment is given as where  is the convergent factor, which is computed as (12), where PSO for constrained optimization problem finds optimal location of particle which has best fitness and satisfies the constraints.The approaches focus on several ways, such as adding penalty function to fitness function [12], valuing the infeasible solution by constraints violation [13,14], adapting PSO parameters according to constraints violation [15,16], generating new feasible solution [17], and adjusting parameters to make infeasible solution become feasible [18].
Referencing [13,18], when adjusting infeasible solution to feasible, the rule updating best solution is described as follows.
Rule 1. (1) Any feasible solution is preferred to any infeasible solution.
(2) Between two feasible solutions, the one having smaller ATD and smaller MTD is preferred.
(3) Between two infeasible solutions, the one having smaller constraint violation is preferred.
Moreover, infeasible solution is valuable for finding optimal solution.Some researchers proposed "Do not kill unfeasible individuals" [19] and "Unfeasible individuals may well contain information valuable enough for them to be preserved" [20].So we divide particles into two subpopulations.The best solution in population 1 must be feasible, and the best solution in population 2 is not constrained by feasible space.Two populations cross obeying fix rate .
The rules updating best solution of population 1 (po 1 ) and population 2 (po 2 ) are described as Rules 2 and 3, respectively.

Rule 2. (1)
Any feasible solution is preferred to any infeasible solution.
(2) Between two feasible solutions, the one having smaller ADT and smaller MDT is preferred.Rule 3. The one having smaller ADT and smaller MDT is preferred.
When crossing, velocity of particle  is computed according to (13), where  is the index of population, such as  1  () is the best solution in po 1 and  2  () is the best solution in po 2 Adjust infeasible solution to feasible solution referencing [18] when the rate of infeasible solution exceeds up limit of rIF.rIF of population po is computed as (14), where IF po is the number of infeasible solutions in po and  po is the size of po.The up limit of rIF is  in this paper 4.2.Solving Procedure.Based on the above method, the flow for solving procedure is shown in Figure 3, where network loading includes load lines, stations, transfer relationship, and schedule.
In the procedure, fitness function is computed by simulation referencing [21][22][23].During simulating, set the limit of inflow according to location of particle, update the number of passengers outside of stations, in stations, and on the trains with the train arriving and departing, and compute MTD and ATD at the end of simulating.

Background of Experiments.
In order to demonstrate the key features of CPICN, experiments are carried out on a simple network with many-to-one OD flow shown in Figure 1.The inflows of  0 ,  1 ,  2 ,  3 are the same, which are time varying as shown in Table 1.
The safe limit of platform is 1800.The capacity of train is 1856.The headway of trains is 180 seconds and the time holding on station of trains is 30 seconds.The running time between two stations of trains is 300 seconds.The transfer time in transfer station is 120 seconds.The distribution of optimal solutions under Algorithm A ( po = 80) is shown in Figure 4.

Effectiveness of
Choose a running; the procedure of convergence of ADT and MDT are shown in Figures 5 and 6, respectively.

Analyzing the Results of
Experiments.An optimal solution of model A is chosen, in which ATD is 194 seconds   A is better than that of model B. The maximum number of waiting passengers under control of model A is less than the safety limitation of station.The comparison of maximum number of passengers delayed under control against the one without control is shown in Table 6.It can be observed that the large number of passengers waiting next coming trains in  0 is shared by other stations through inflow control.

Conclusion
In this paper, CPICN is proposed considering the safety of stations.The model of CPICN is built with the objective of minimizing ADT and MDT under constraint of capacity of station and train, O-D demand, schedule, and flow conservation.The PSO method for constrained problem is used to solving the model, which integrates the approaches about adjusting infeasible solution to feasible solution and crossing infeasible solution to feasible solution.The numerical experiments are carried out to prove the feasibility and effectiveness of the model and solving algorithm.
For the further research, the solving algorithm will be experimented and applied to the real network.Moreover the route choice about passengers under control measure will be the focus.

Figure 1 :
Figure 1: Compositions of waiting passengers in transfer station.

Figure 2 :
Figure 2: Example of compositions of waiting time.

Table 1 :
Time varying O-D demand.
[18]rithm.Algorithms are programed by Java and compiled by JDK 7.25 for Windows x64 in Eclipse 10.Experiments are tested on a personal computer with an Intel Core i5, 2.50 GHz CPU, and 4 GB RAM.The parameters of algorithms are set as shown in Table2.The algorithm in this paper is named as Algorithm A. The algorithm proposed by Sun et al.[18]is named as

Table 2 :
Parameters of algorithm.
[6] key indicators of the solutions are compared with the past research which is proposed by Zhao et al.[6].The model in this paper is named as model A. The model proposed by Zhao et al.[6]is named as model B. Comparison of key indicators under different conditions is shown in Table5.The key indicators become better under control than the one without control.The value of MDT under control of model t Figure 5: Convergence curve of ATD.

Table 3 :
Comparison of solution with different algorithms.

Table 4 :
Example of optimal solution.

Table 6 :
Comparison of maximum number of passenger delayed.