sustainability

Article Urban Rail Timetable Optimization to Improve Operational Efficiency with Flexible Routing Plans: A Nonlinear Integer Programming Model

Qiuchi Xue, Xin Yang * , Jianjun Wu, Huijun Sun, Haodong Yin and Yunchao Qu

State Key Laboratory of Rail Traffic Control and Safety, Jiaotong University, Beijing 100044, China * Correspondence: [email protected]; Tel.: +86-138-1115-1991

 Received: 4 May 2019; Accepted: 2 July 2019; Published: 5 July 2019 

Abstract: At present, most systems adopt an operation mode with a single long routing. The departure frequency is determined by the maximum section passenger flow. However, when the passenger flow varies greatly within different sections, this mode will lead to a low load factor in some sections, resulting in a waste of capacity. In view of this situation, this paper develops a nonlinear integer programming model to determine an optimal timetable with a balanced scheduling mode, where the wasted capacity at a constant departure frequency can be reduced with a slight increase in passenger waiting time. Then, we simplify the original model into a single-objective integer optimization model through normalization. A genetic algorithm is designed to find the optimal solution. Finally, a numerical example is presented based on real-world passenger and operation data from Beijing Metro . The results show that the double-routing optimization model can reduce wasted capacity by 9.5%, with a 4.5% increase in passenger waiting time, which illustrates the effectiveness of this optimization model.

Keywords: train operation plan; wasted capacity; nonlinear integer programming; balanced scheduling mode

1. Introduction With the growth of the urban population, traffic demand has gradually increased in China. Urban rail transit has become an important part of urban public transportation with the advantages of safety, high efficiency, convenience, reliability, low energy consumption and light environmental pollution [1]. Making full use of urban rail transit is an important way to relieve the pressure of modern urban traffic and to promote its sustainable development. The urban rail timetable is the core technical content for train operation organization. It is the guidance document for coordination among companies and its quality will affect passenger satisfaction and operation cost [2]. The idea of a reasonable timetable, which can command train operation, ensure traffic safety and improve transportation efficiency, is attracting more and more attention. Therefore, it is significantly important to optimize metro timetables in order to improve operational efficiency. At present, urban rail transit is mostly operated on the single long routing mode. With the city gradually extending outwards, a lot of radial lines connecting the urban central area to the suburbs and radial lines connecting suburbs at both ends of the urban area, have been formed. The passenger flow on these lines is unevenly distributed in each section. If trains are still operated on a single long routing mode and the departure frequency is determined by the maximum section passenger flow, this will cause a great capacity waste in some sections. For this phenomenon, some cities are actively exploring the operation method of mixed operation on double routings—Shanghai Metro , Metro and Tokyo Metro , for example. At present, train routing plans

Sustainability 2019, 11, 3701; doi:10.3390/su11133701 www.mdpi.com/journal/sustainability Sustainability 2019, 11, 3701 2 of 26 are mostly determined by the operator’s experience, based on the spatial distribution characteristics of passenger flow along the line, combined with the actual line condition, lacking systematic and quantitative research. Therefore, it is necessary to study train operation plans which systematically use mixed operation on long and short routings.

2. Literature Review The method of mixed operation on double routings was first used in public services. As early as 1987, coordination modes to find the schedule offset between different routings were used to balance loads and minimize overall cost, where the primary objective was to minimize fleet size and the secondary objective was to minimize waiting time for a given fleet size. It was shown that even when overall capacity exceeds volume on every link, there may still be one or more patterns of trip schedule that are not systematically overcrowded [3]. Cristián et al. [4] found that the mixed routings strategy can be justified in many cases with uneven load patterns. Since then, an extensive variety of models for mixed operation on double routings have been proposed by researchers. The researches mainly focused on operation cost and the main purpose was either to minimize the total cost or maximize the total benefit. Ji et al. [5] studied an optimization model on bus routes to reduce the total cost. The model optimized the schedule coordination of a long routing service mode and a short-turning service mode and balanced the bus load along the route and between the two service patterns. The relationship between operation cost and energy consumption is inseparable. With the widespread concern surrounding energy consumption, the issues of how to reduce energy consumption and capacity waste have become important for researchers. Yang et al. [6] developed a cooperative scheduling approach to optimize the timetable so as to improve the utilization of recovery energy. Yu et al. [7] established a two-stage model of a high-speed rail train operation plan. In the first stage, the proposed model aimed for the minimum total cost of train operation, which generated fewer train candidate sets, thereby improving the efficiency of the solution. In the second stage, the model gave an accurate flow distribution model to obtain the optimal economic efficiency and completed the design of the high-speed express train operation plan. In addition to energy consumption, service level is also important for operators. Szeto and Wu [8] proposed a model to reduce the number of transfers and the total time of passengers, thereby improving upon existing bus services. An integrated solution method was achieved to simultaneously determine the departure frequency and operating route. Cadarso and Marín [9] developed an integrated planning model to readjust departure frequencies by recalculating the railway timetable and rolling stock. The purpose of the model was to cater for increasing passenger demand and to better adjust these flows to passenger demand, thereby alleviating traffic congestion around urban and suburban areas. Canca et al. [10] described a short-turning strategy to optimize train operation; the optimization objective was to diminish the passenger waiting time and preserve a certain level of service quality. The model determined the service offset and turn-back stations. Huang et al. [11] constructed a load balancing approach to optimize metro train operation. The model was based on an improved route which generalized the time utility function, considering the penalties of both in-vehicle congestion and transfers, thereby improving the service quality and reducing safety risks in metro systems, at the same time. Many researchers considered reducing energy waste and improving service quality, at the same time. Sun et al. [12] put forward a multi-objective optimization model by studying specific methods on train routing to optimize high-speed railway network operation. The model considered energy consumption and trains’ average travel time and also took user satisfaction into account. Yang et al. [13] formulated a two-objective integer programming model to decrease the passenger waiting time and increase the utilization of regenerative energy, simultaneously. The model optimized the train timetable by controlling the train’s time and dwell time. D’Acierno and Botte [14] developed an analytical approach to deciding driving strategies. The framework could be used for properly supporting the implementation of eco-driving strategies, from a passenger-oriented Sustainability 2019, 11, 3701 3 of 26 perspective. The numerical examples showed that the proposed approach is useful in determining the optimal compromise between travel time increase and energy decrease. Wang et al. [15] investigated a multi-objective vehicle-routing optimization problem, where the objectives aimed to minimize total energy consumption and customer dissatisfaction, simultaneously. The study found a better balance between energy consumption diminishment and customer satisfaction and, finally, developed a mixed integer programming model to address the problem. Sun et al. [16] developed a multi-objective timetable optimization model to minimize total passenger waiting time and energy consumption. The model was verified by real-world smart-card automated collection data and it was shown that the developed model could improve passenger service and reduce pure energy consumption more efficiently in comparison with the timetable used currently. Many researchers considered reducing operation cost and improving service quality, at the same time. Site and Filippi [17] chose the net benefit composed of user waiting times minus operator costs as the objective function and developed an optimization framework to obtain the maximum net benefit. The framework gave thought to variable vehicle size and service patterns over different operation periods so as to obtain an intermediate-level planning of bus operations. Chang et al. [18] applied mixed operation on double routings to rail transit and a multi-objective programming model was developed to minimize the total cost, combining the passengers’ total travel time loss and the operator’s total operating cost. Based on the analysis of an urban rail transit train plan and users’ travel expenses, Deng et al. [19] developed a multi-objective optimization model to reduce passengers’ general travel expenses and increase the operator’s benefits, simultaneously. Codina et al. [20] presented a model balancing construction costs, operational costs and passengers’ benefits. The mode optimized users’ travel times and designed a public transit network system using the traditional approach of demand coverage in bimodal scenarios of operation. An and Zhang [21] constructed a mixed integer programming model to minimize the cost while improving transit service. The model composited bus-holding and stop-skipping strategies to obtain a real-time optimal strategy and was solved by a Lagrangian relaxation algorithm. The study found a better balance between the quality of the transit service and the operation cost. In some special cases, a mixed operation of double routing can also be used. Considering rotating maintenance, rolling stock circulation plans and a long-term maintenance policy, Canca et al. [22] proposed a mixed integer programming model to minimize wasted capacity caused by train empty movements and to balance the workload of the maintenance operation. Cadarso et al. [23] presented an integrated model for the recovery of the timetable and the rolling stock schedules, so that the recovery schedules could be easily implemented in practice and the operations could quickly return to the originally planned schedules after the recovery period. Cadarso et al. [24] proposed an approach synthetically considering the influence of passengers’ behavior, the timetable and rolling stock. It combined two models: one which was an integrated optimization model for the timetable and rolling stock and another which was a model for the passengers’ behavior. The proposed approach could quickly find the best solution in two steps and with a very good balance between various managerial goals. In summary, a large number of double routing operation models have been developed and the main optimization is to determine the return site and the departure frequency. On the basis of the aforementioned models, the purpose of this study is to develop a double-routing optimization model to minimize the objective function combining the passenger waiting time and the wasted capacity on the balanced scheduling mode, so as to obtain better capacity matching. Compared with previous studies, this study presents the following differences: i. Compared with Deng et al. [19], in which the turn-back station cannot be changed, the model established in this manuscript is used to first make decisions on the turn-back station. ii. Compared with Deng et al. [19] and Chang et al. [18], this manuscript determines the routing which the train should be operated on at xi time on the basis of considering the departure Sustainability 2019, 11, 3701 4 of 26

frequency and train capacity. Fleet size is not considered in this manuscript, because it changes infrequently in practice. iii. Compared with Deng et al. [19] and Chang et al. [18], the single-objective model is obtained by weighted summation with two different dimensions of passenger waiting time and wasted capacity in the original model and the nonlinear integer programming model is solved based on genetic algorithms. iv. Compared with Deng et al. [19] and Chang et al. [18], the relationship between the total passenger flow and the train’s optimum headway is obtained in this study, which makes it unnecessary to calculate the section passenger flow and find out the maximum section passenger flow specifically. In additional, this manuscript simplifies the algorithm for calculating departure frequency and analyzes the conditions for the double-routing operation.

Table1 shows the detailed feature comparison between closely related studies.

Table 1. Detailed features of closely related studies.

Turn-Back Departure Specific Train Implementation Fleet Size Station Frequency Departure Time Capacity Method Chang et al. Not Fuzzy Considered Considered Not considered Considered (2000) considered programming Deng et al. Not Three-phase Not changed Considered Not considered Considered (2013) considered solving strategy Not This paper Considered Considered Considered Considered Genetic algorithm considered

By optimizing the train operation plan to meet the demands of different sections’ passenger flow, the cost of transportation enterprises is reduced, under the premise that the service quality is little affected, and the efficiency of the trains used is improved. At the same time, this operation mode can alleviate wasted capacity and make the operation organization economically reasonable. The rest of this paper is organized as follows. In Section3, we provide the problem statement. In Section4, we formulate an integer optimization model with the objectives of passenger waiting time and wasted capacity. In Section5, we design a genetic algorithm (GA) to solve the formulated optimization model. Based on real-world passenger and operation data from Beijing Metro Line 4, in Section6 we conduct a numerical example. Section7 concludes the work.

3. Problem Statement Transit timetable construction at the route level is generally based on the maximum observed passenger flow. In current practice, this maximum passenger flow is observed in a specified time period. It is divided by a load standard (desired occupancy, e.g., number of seats) to obtain the number of departures required [25]. At present, urban rail transit is mostly operated on a single long routing mode, as shown in Figure1a and the train graph in Figure1b. With the city gradually extending outwards, there are metro lines connecting suburbs at both ends of the urban area. The passenger flow in the urban area is very large but in the suburbs the flow is relatively small at both ends of the line, so it is suitable to use the double-routing operation mode, as shown in Figure1c and the train graph in Figure1d. In addition, there are metro lines connecting the urban central area to the suburbs and similarly, the passenger flow in the urban area is much larger than the passenger flow in the suburbs, so it is suitable to use a simpler double-routing operation mode, as shown in Figure1e and the train graph in Figure1f. Sustainability 2019, 11, x FOR PEER REVIEW 4 of 25

capacity in the original model and the nonlinear integer programming model is solved based on genetic algorithms. iv. Compared with Deng et al. [19] and Chang et al. [18], the relationship between the total passenger flow and the train’s optimum headway is obtained in this study, which makes it unnecessary to calculate the section passenger flow and find out the maximum section passenger flow specifically. In additional, this manuscript simplifies the algorithm for calculating departure frequency and analyzes the conditions for the double-routing operation. Table 1 shows the detailed feature comparison between closely related studies.

Table 1. Detailed features of closely related studies.

Specific Turn-Back Departure Train Implementation Departure Fleet Size Station Frequency Capacity Method Time Chang Not Not Fuzzy et al. Considered Considered Considered considered considered programming (2000) Deng Not Not Not Three-phase et al. Considered Considered changed considered considered solving strategy (2013) This Not Genetic Considered Considered Considered Considered paper considered algorithm

By optimizing the train operation plan to meet the demands of different sections’ passenger flow, the cost of transportation enterprises is reduced, under the premise that the service quality is little affected, and the efficiency of the trains used is improved. At the same time, this operation mode can alleviate wasted capacity and make the operation organization economically reasonable. The rest of this paper is organized as follows. In Section 3, we provide the problem statement. In Section 4, we formulate an integer optimization model with the objectives of passenger waiting time and wasted capacity. In Section 5, we design a genetic algorithm (GA) to solve the formulated optimization model. Based on real-world passenger and operation data from Beijing Metro Line 4, in Section 6 we conduct a numerical example. Section 7 concludes the work.

3. Problem Statement Transit timetable construction at the route level is generally based on the maximum observed passenger flow. In current practice, this maximum passenger flow is observed in a specified time period. It is divided by a load standard (desired occupancy, e.g., number of seats) to obtain the number of departures required [25]. At present, urban rail transit is mostly operated on a single long routing mode, as shown in Figure 1a and the train graph in Figure 1b. With the city gradually extending outwards, there are metro lines connecting suburbs at both ends of the urban area. The passenger flow in the urban area is very large but in the suburbs the flow is relatively small at both ends of the line, so it is suitable to use the double-routing operation mode, as shown in Figure 1c and the train graph in Figure 1d. In addition, there are metro lines connecting the urban central area to the suburbs and similarly, the passenger flow in the urban area is much larger than the passenger flow in the suburbs, so it is suitable to use a simpler double-routing operation mode, as shown in Sustainability 2019, 11, 3701 5 of 26 Figure 1e and the train graph in Figure 1f.

B

A B

A Sustainability 2019, 11, x FOR PEER REVIEW 5 of 25 (a) Operation on long routing (b) Train graph on long routing B Sustainability 2019, 11, x FOR PEER REVIEW 5 of 25 D A C D B B C D A C D B A C (c) Operation on double routings (d) Train graph on double routings A B (c) Operation on double routings (d) Train graph on double routings C B A C B C A C B A (e) Operation on double routings (Ⅱ ) A (f) Train graph on double routings (Ⅱ ) (e) Operation on double routings (Ⅱ ) (f) Train graph on double routings (Ⅱ ) FigureFigure 1. IllustrationIllustrationss of of different different routing strategies and train graph graphs.s. Figure 1. Illustrations of different routing strategies and train graphs. A brief descriptiondescription ofof the the physical physical urban urban rail rail line line and and rolling rolling stock stock is shown is shown in Figure in Figure2. The 2 solid. The solidarrow arrow standsA brief stands for description the for long the routing long of the routing andphysical the and dottedurban the raildotted arrow line represents arrowand rolling represents the stock short is the shown routing. short in routing. TheFigure solid 2. The The lines solid are solid arrow stands for the long routing and the dotted arrow represents the short routing. The solid linesthe simple are the expression simple expression of the physical of the urbanphysical rail urban tracks. rail The tracks reversing. The reversing tracks have tracks been have built been at station built lines are the simple expression of the physical urban rail tracks. The reversing tracks have been built ata, consequently,stationat station a, consequently a, consequently, the train, operated the the train train onoperated operated the short onon routing the shortshort can routing ro changeuting can can thechange change travel the direction.the travel travel direction. direction.

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Station 1 Station 2 Station a Station n

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FigureFigure 2. 2.Description Description ofof an urban rail rail line line and and rolling rolling stock. stock. Figure 2. Description of an urban rail line and rolling stock. TheThe train train routing routing plan plan is is anan importantimportant part part of ofthe the train train operation operation plan in plan urban in rail urban transit. rail Its transit. formulation should be based on the full analysis of passenger flow, considering the operation Its formulationThe train routing should plan be basedis an important on the full part analysis of the of train passenger operation flow, plan considering in urban rail the transit. operation Its organization conditions. It will directly affect the formulation of the train timetable and the rolling formulationorganization conditions.should be basedIt will directly on the a fullffect analysis the formulation of passenger of the train flow, timetable considering and the the rolling operati stockon stock operation plan. The motivation for the double-routing operation is to promote a better match organization conditions. It will directly affect the formulation of the train timetable and the rolling operationbetween plan. capacity The motivation and demand. for Passenger the double-routing demand may operation vary at isdifferent to promote times a betteror sites. match Figure between 3 stock operation plan. The motivation for the double-routing operation is to promote a better match capacityillustrates and demand. the spatial Passenger distribution demand of passenger may vary flow at and diff theerent capacity times orusage sites. onFigure each section.3 illustrates The the betweenspatialspatial distribution cap distributionacity and of passenger ofdemand. passenger flowPassenger flow and reflects the demand capacity the difference mayusage varyin on passenger each at different section. flow Thein times different spatial or sites.directions distribution Figure of3 illustratespassengerand locations flowthe spatial reflects of the distribution theline. di Thefference usage of in passengerof passenger capacity in flow flow a single and in di long thefferent capacityrouting directions is usage shown and on in locations Figure each section 3a of and the. Tline.he spatialThe usagewe distribution can of find capacity that of there inpassenger a single is a lot long flow of waste routing reflects of is capacitythe shown difference in Figuremany in sections.3passengera and we In cansupportflow find in ofdifferent that the there idea directions isthat a lot of andwaste locationsoperators of capacity shouldof the in many useline. the sections.The double-routing usage In of support capacity operation of thein anda idea single reduce that long operators the numberrouting should ofis trainsshown use in the sectionsin double-routingFigure with 3a and weoperation canless findpassenger and that reduce thereflow, the the is number acapacity lot of of wasteusage trains on of in the capacity sections double within routing many less is passenger sections. shown in In flow,Figure support the 3b,capacity in of which the idea usagethe that on usage of capacity is improved on some sections. operatorsthe double should routing use is shownthe double in Figure-routing3b, inoperation which the and usage reduce of capacity the number is improved of trains on in some sections sections. with less passenger flow, the capacity usage on the double routing is shown in Figure 3b, in which the usage of capacity is improved on some sections. Sustainability 2019, 11, 3701 6 of 26 Sustainability 2019, 11, x FOR PEER REVIEW 6 of 25

(a) On the single long routing

(b) On the double routing

Figure 3. Capacity usage in different different routing plans.plans.

Since many passenger demands can be met by a trip following either a full-length or short-turn pattern, schedule coordination between the patternspatterns isis essentialessential [[3].3]. The critical issue in short-turnshort-turn service isis toto determinedetermine the the turn-back turn-back point point and and the th routee route schedule schedule to balanceto balance passenger passenger loads loads among among the tripsthe trips and and to minimize to minimize the total the total fleet sizefleet and size passenger and passenger waiting waiting times. times. The train The plan train of plan urban of railurban transit rail undertransit theunder multi-routing the multi-routing mode can mode be divided can be divided into three into parts: three train parts: formation, train formation, train operation train operation periods andperiods the correspondingand the corresponding train counts train of eachcounts routing of each in each routing period in [each19]. Theperiod design [19]. of The strategies design that of guaranteestrategies that reasonable guarantee user reasonable waiting times user with waiting small times increases with in small operation increases costs in is nowoperation an important costs is researchnow an important topic [10,26 research]. topic [10,26]. Through the above analysis,analysis, we constructconstruct aa double-routingdouble-routing optimizationoptimization model to determinedetermine the turn-backturn-back station of the short routing accordin accordingg to the uneven distribution of of passenger passenger flow. flow. Then, wewe decidedecide to to operate operate on on either either the the long long or shortor short routing routing at a particularat a particular time, basedtime, based on the on actual the passengeractual passenger flow. The flow. trains The ontrains the longon the routing long routin will turng will back turn at back the terminal,at the terminal, while thewhile trains the ontrains the shorton the routing short routing will turn will back turn at back the middle at the mi stationddle station obtained obtained by the modelby the decision.model decision.

4. Model Formulation This sectionsection developsdevelops a a double-routing double-routing optimization optimization model model to reduceto reduce both both passenger passenger waiting waiting time andtimewasted and wasted capacity, capacity, on a balanced on a balanced scheduling scheduli mode.ng Themode. following The following discussions discussions focus on detailingfocus on eachdetailing part ofeach the formulation,part of the thatformulation, is, notions, that model is, assumptions,notions, model objective assumptions, function, objective system constraints function, andsystem the constraints optimization and model. the optimization model.

4.1. Notations To facilitate the presentation of the model in this paper, the parameters are listed below. Sustainability 2019, 11, 3701 7 of 26

4.1. Notations To facilitate the presentation of the model in this paper, the parameters are listed below.

(1) Indices h station index, h = 1, 2, ... , n j section index, j = 1, 2, ... , n 1 − (2) Parameters the set of metro routes, I = {1, 2}, 1 on behalf of the long routing, 2 on behalf of the i short routing f the departure frequency for operating the long routing qo,d the volume of passenger flow from station O to station D Q passenger flow matrix consisting of qo,d ’ tz,tz the minimum return time of the train at the terminal and intermediate station I0 minimum tractive time fmin minimum departure frequency Cz rated passenger capacity α maximum section load factor (3) Decision variables Train operation plan, X = {x | i = 1, 2, ... , f }, x = {1, 2}, 1 on behalf of the long routing, 2 on X i i behalf of the short routing a index of return station in short routing, a {2, 3, ... , n 1} ∈ − (4) Intermediate variables

Q1 total volume of passenger flow which the origin or destination is outside the short routing total volume of passenger flow which the origin and destination are within the Q 2 short routing t1d,t2d average passenger waiting time corresponding to Q1,Q2 respectively Qm actual passenger volume of metro f1 the departure frequency of long routing f2 the departure frequency of short routing N1 actual capacity of long routing N2 actual capacity of short routing

4.2. Model Assumptions According to the actual operation characteristics of metro systems, the following assumptions are made to simplify the model formulation. (A1). Each station in the line has the conditions for building a reversing track. (A2). The rolling stocks are used independently for long routing and short routing. (A3). Trains running on long routing and short routing are both operated in the mode of stopping at every station. (A4). Trains running on long routing and short routing both have the same type and formation.

4.3. Objective Function The purpose of this study is to reduce the wasted capacity at a constant departure frequency, with little increase in the passenger waiting time. The total waiting time for passengers across the board depends on the average waiting time and the number of waiting passengers. The wasted capacity depends on the actual capacity and passenger volume of the metro.

4.3.1. The Total Waiting Time for Passengers Studies have shown that the average passenger waiting time is related to the status of passengers arriving at the station. The metro departure interval is relatively small: less than 10 min, generally. Passengers arriving at the station are in Random Normal distribution and not affected by the train Sustainability 2019, 11, 3701 8 of 26 timetable. The average waiting time of the overall passenger flow approaches half of the departure interval. Therefore, the average waiting time for passengers whose origin or destination is outside the short routing is shown in Equation (1a). The average waiting time for passengers whose origin and destination are within the short routing is shown in Equation (1b).

1 60 t1d = (1a) 2 × f1

1 60 t2d = (1b) 2 × f1 + f2 Automatic Fare Collection System (AFC) automates the ticket accounting and selling processes and it can give detailed data on system usage. We can obtain qo,d through AFC data and then obtain the metro line passenger flow matrix composed of qo,d. The expression of one direction is given in Equation (2).    0 q12 q1,n 1 q1,n   ······ −   0 0 q q q   23 2,n 1 2,n   . ··· . − .   . .. . .   . 0 . . .  Q =   (2)  . . .   . . .. q q   n 2,n 1 n 2,n   − − −   0 0 0 0 qn 1,n   ··· −  0 0 0 0 n n ······ × The number of waiting passengers equals the passenger flow. In this paper, a is defined as the index of the return station on the short routing. The section with the maximum passenger flow in [a, n] and [1, a] is selected as the short routing. When the short routing is [a, n], the total passenger flow in the short routing is shown in Equation (3a). When the short routing is [1, a], the total passenger flow in the short routing is shown in Equation (3b). The total passenger flow which the origin or destination is outside the short routing is shown in Equation (4).

dX=n OX=n Q2 = qo,d (3a) d=a O=a

Xd=a OX=a Q2 = qo,d (3b) d=1 O=1 dX=n OX=n Q = q Q (4) 1 o,d − 2 d=1 O=1 Based on the calculations of the average waiting time of passengers and the number of waiting passengers, the total waiting time for passengers across the board can be expressed as Formula (5).

PT(X, a) = Q t + Q t (5) 1• 1d 2• 2d 4.3.2. The Wasted Capacity The transport capacity of urban rail transit can be generally defined as the total number of passengers that can be transported within a unit time (usually one hour) in a certain direction of the line. The actual transport capacity of the metro is closely related to the departure frequency. In this paper, Sustainability 2019, 11, 3701 9 of 26 the departure frequency of the long routing is shown in Equation (6a) and the departure frequency of the short routing is shown in Equation (6b).

Xf f = 2 f x (6a) 1 − i i=1

Xf f = x f (6b) 2 i − i=1 Therefore, the actual capacity of the long routing is shown in Equation (7). When the short routing is [a, n], the actual capacity of the short routing is shown in Equation (8a). When the short routing is [1, a], the actual capacity of the short routing is shown in Equation (8b).

N = f Cz α (n 1) (7) 1 1 × × × −

N = f Cz α (n a) (8a) 2 2 × × × − N = f Cz α (a 1) (8b) 2 2 × × × − AFC data can be used to obtain information about passengers’ arrival and departure stations and times. To facilitate the calculation, we establish the Origin-Destination (OD) matrix of all stations based on the AFC data and then calculate the actual passenger volume from the passenger flow matrix. The actual passenger volume is shown in Equation (9).

dX=n OX=n Qm = qo,d (9) d=1 O=1

In summary, the wasted capacity can be expressed as Formula (10).

WN(X, a) = N + N Qm (10) 1 2 − 4.3.3. Weighted Summation The purpose of this study is to reduce both passenger waiting time and wasted capacity on balanced scheduling mode but the dimension of the two objectives is different. Therefore, the two functions are first normalized into a common scale by using the following two equations, that is,

PTmax PT PTˆ = − (11a) PTmax PT − min WNmax WN WNˆ = − (11b) WNmax MN − min where PTmax, PTmin, WNmax and WNmin are the maximum and minimum values of the two objectives. PTˆ and WNˆ are the normalized values of the two objectives. Then, the bi-objective model is transformed into a single objective model by using the linear weighted compromise algorithm. The final objective function can be expressed as Formula (12).

z(X, a) = ω PTˆ + ω WNˆ (12) 1• 2• where ω1 and ω2 are non-negative real numbers representing the importance of passenger waiting time and wasted capacity, respectively. Sustainability 2019, 11, 3701 10 of 26

4.4. System Constraints According to the actual operation characteristics of the metro system and considering the operation safety and service quality, we put forward the following constraints. The index of the return station in the short routing should meet the following constraint:

2 a n 1 (13) ≤ ≤ − To ensure the safe operation of adjacent trains, the tracking time and the departure frequency should meet the following constraint:

f + f 3600/I (14) 1 2 ≤ 0 where I is the minimum tracking time and f ,f N. The symbol 3600/I is calculated to obtain the 0 1 2 ∈ 0 maximum departure frequency. Considering that the return ability of the terminal or intermediate station is a major factor limiting the metro capacity, the departure frequency on each routing, consequently, cannot exceed the return ability of the return station. This can be expressed as:

f 3600/tz (15a) 1 ≤

f 3600/tz0 (15b) 2 ≤ where 3600/tz and 3600/tz0 are calculated to obtain the maximum departure frequency of the long routing and the short routing, which are constrained by the return ability, respectively. To fulfill the metro service requirements, the departure frequency on the long routing should meet the following constraint: f f (16) 1 ≥ min The number of trains used is directly related to transport enterprise interests. Excessive trains required can result in increased business costs. To ensure that the interests of transport enterprise do not decline, the departure frequency on each routing should meet the following constraint:

f1 + f2 = f (17) where f can be derived from the maximum section passenger flow when a single long routing is in operation.

4.5. Optimization Model Finally, combining the passenger waiting time and the wasted capacity, considering the constraints of the index of the return station in the short routing and the departure frequency on each routing, we give the double-routing optimization model in Equation (18).  min z(X, a)   s.t. 2 a n 1   ≤ ≤ −  f + f 3600/I  1 2 0  ≤  f1 3600/tz (18)  ≤   f2 3600/tz0  ≤   f1 fmin  ≥   f1 + f2 = f Sustainability 2019, 11, 3701 11 of 26

Due to the differences in the turnover times of the long and short routings, the departure frequency in the short routing can increase in response, so as to reduce the passenger waiting time in these sections when the same number of trains are used. This situation is suitable for morning and evening rush hour. It is worth noting that trains’ departure frequency is the same with trains operated on a single long routing in this model. It can be observed from constraint (17) that trains’ departure frequency on the short routing sections is unchanged and on the long routing sections is decreased, specifically. This will cause the passenger waiting time on non-double-routed sections to increase. The purpose of this model is to reduce the wasted capacity at constant departure frequency with little increase in the passenger waiting time. It is suggested to further optimize the method of increasing the departure frequency with a particularly large passenger volume. The model can be generalized when the metro line connects suburbs at both ends of the urban area. Passenger flow in the urban area is very large but in the suburbs is relatively small, so it is suitable to use the improved double-routing operation mode which determines two ends at once. At this point, the total passenger flow in the short routing should be re-determined according to two stations, a and b, as shown in Equation (19) and the actual capacity of the short routing should also be re-determined, as shown in Equation (20).

Xd=b OX=b Q2 = qo,d (19) d=a O=a

N = f Cz α a b (20) 2 2 × × × | − | 5. Solution Procedure The determination of efficient routes and schedules in systems is complex due to the vast search space and the multiple constraints involved [27–31]. It can be seen from the above statement that the established model consists of the passenger waiting time and the wasted capacity, where minimizing the passenger waiting time is a nonlinear integer programming problem and the minimization of wasted capacity is a general linear mixed integer programming problem. To solve the general mixed integer model, the common solutions include the branch and bound method and the cutting-plane method; at the same time, there are also some stable solvers, such as WebSphere ILOG CPLEX and mixed integer linear programming solver-IPSOLVE. Considering the actual situation of urban rail transit (e.g., multiple stations and large passenger flow), we select the CPLEX solver to try to solve the problem at first but unfortunately it takes a long time [32]. It can take up to half an hour for an optimal solution to optimize 35 stations and will be detrimental to the further development and expansion of the follow-up study. In addition, the solver does not solve the nonlinear optimization. Since with the developed double-routing optimization model it is difficult to find a good solution using the method of exhaustion or traditional optimization algorithms, we design a genetic algorithm (GA) in this study to solve the developed model, which greatly improves the solving efficiency and is more conducive to the expansion of the follow-up research. GA is a random global search algorithm and optimization method for seeking high-quality solutions. Due to its advantages of strong robustness, extensive generality and high efficiency, GA has been widely adopted for solving many urban rail optimization problems [16,33]. The basic steps of GA are coding, initial population generation, fitness evaluation, selection, crossover and mutation, as shown in Figure4. Sustainability 2019,, 11,, x 3701 FOR PEER REVIEW 1212 of of 25 26 Sustainability 2019, 11, x FOR PEER REVIEW 12 of 25

Figure 4. Basic steps of the genetic algorithm (GA). Figure 4. BasicBasic steps steps of of the the genetic genetic algorithm algorithm (GA). (GA). Step 1. Chromosome coding: The double-routing optimization model has f + 1 variables. The Step 1. ChromosomeChromosome coding:coding: The The double-routing double-routing optimization optimization model model has hasf + 1f + variables. 1 variables. The The first first f decision variables are the train operation plan, x1, x2, …, xf; when xi takes the value of 1, it f decision variables are the train operation plan, x , x , ... , x ; when x takes the value of 1, it means first f decision variables are the train operation plan,1 2 x1, x2, f…, xf; wheni xi takes the value of 1, it means the train will operate on the long routing at time xi; when the value is 2, it means the train will the train will operate on the long routing at time xi; wheni the value is 2, it means the train will operate meansoperate the ontrain the will short operate routing on at the time long xi. The routing f + 1 decisionat time x variable; when theis the value index is 2,of itthe means return the station train inwill on the short routing at time x . The f + 1 decision variable is the index of the return station in the short operatethe short on therouting. short Therefore, routing iat the time chromosome xi. The f + 1 can decision be divided variable into isf +the 1 parts index for of coding, the return that station is, one in routing. Therefore, the chromosome can be divided into f + 1 parts for coding, that is, one chromosome thechromosome short routing. has Therefore, f + 1 genes. the We chromosome convert decimal can variables be divided into into binary f + 1genes. parts In for this coding, way, a that solution is, one f chromosomehasto the+ 1 model genes. has is We encodedf + convert1 genes. asdecimal aWe solution convert variables of thedecimal GA. into Asvariables binary shown genes. ininto Figure Inbinary this 5, way,thisgenes. chromosome a solutionIn this way, to represents the a solution model is encoded as a solution of the GA. As shown in Figure5, this chromosome represents that at time x the to thatthe modelat time is x1 encodedthe train shouldas a solution operate of on the the GA. long As routing, shown at in time Figure x2 the 5, trainthis chromosome should operate represents on the1 train should operate on the long routing, at time x the train should operate on the short routing and thatshort at time routing x1 the and train at timeshould xf the operate train onshould the longoperate2 routing, on the at short time routing.x2 the train The should index ofoperate the return on the x shortatstation time routing finthe the trainand short at should routingtime x operatef isthe No.16. train on should the short operate routing. on Thethe short index routing. of the return The index station of in the the return short stationrouting in is the No.16. short routing is No.16.

Figure 5. Chromosome coding. Figure 5. ChromosomeChromosome coding. coding. SustainabilitySustainability 2019 2019, 11, 11, x, xFOR FOR PEER PEER REVIEW REVIEW 1313 of of 25 25 Sustainability 2019, 11, 3701 13 of 26 StepStep 2. 2. Initialization: Initialization: According According to to the the constraints constraints of of departure departure frequency frequency and and the the return return station station inin 18, 18, initialize initialize the the population population p 0p,0 ,generate generate n n0 initial0 initial solutions solutions randomly randomly and and ensure ensure that that each each initial initial solutionsolutionStep generated 2.generated Initialization: is is a afeasible feasible According solution. solution. to the constraints of departure frequency and the return station in 18,Step initializeStep 3. 3. Evaluation Evaluation the population function: function:p0, generateFitness Fitness evaluation nevaluation0 initial solutions is is used used to randomly to measure measure and the the ensure quality quality that of of eachindividual individual initial solutioncandidates.candidates. generated The The GA GA is is ais usually feasibleusually used solution.used to to find find the the ma maximumximum value value of of the the model; model; however, however, in in this this study study wewe need Stepneed to 3.to find Evaluationfind the the minimum minimum function: value value Fitness of of the the evaluation object objectiveive isfunction function used to of measureof the the doub doub thele-routingle-routing quality ofoptimization optimization individual candidates.model.model. Here, Here, The we we GAuse use isthe the usually function function used f(x) f(x) to = = findM M − −z(x) the z(x) maximumas as the the evaluation evaluation value offunction, function, the model; where where however, M M is is a aconstant in constant this study that that weensuresensures need that to that find the the the value value minimum of of f (fx()x value)is is positive. positive. of the objectiveIf If the the value value function of of f(x) f(x) of is is the bigger, bigger, double-routing the the corresponding corresponding optimization solution solution model. is is Here, we use the function f(x) = M z(x) as the evaluation function, where M is a constant that ensures better.better. − that theStepStep value 4. 4. Selection Selection of f (x) is process: positive.process: In IfIn this the this valuepaper, paper, of the f(x)the best isbest bigger, reserved reserved the correspondingse selectionlection method method solution is is adopted. adopted. is better. First, First, rouletterouletteStep is 4.is used Selectionused forfor selection process:selection and Inand this thenthen paper, thethe the individualsindividuals best reserved withwith selection thethe highesthighest method fitnessfitness is adopted. inin thethe currentcurrent First, roulettepopulationpopulation is used are are completely for completely selection copied andcopied then to to the the next individuals next generation. generation. with The the The highestcumulative cumulative fitness probability probability in the current that that populationindividual individual are completely copied to the next generation. The cumulative probability that individual x is inherited xix isi is inherited inherited to to the the next next generation generation is is i to the next generation is n n Xn00 0 p()xfxfx== ()/ () (21) pp()(xfxfxxiii)ii = f ( ()/xi)/ f (x ()i) i i (21)(21) i=i=1i1=1 StepStep 5. 5. Crossover CrossoverCrossover process: process:process: The TheThe crossover crossovercrossover is isis th thethe emost mostmost important important step step in in the the GA, GA, which which could could generategenerate new new chromosomes chromosomeschromosomes in inin the the population. population. This This This paper paper paper uses uses uses two two two points points points of of crossover. ofcrossover. crossover. First, First, First, two two twochromosomeschromosomes chromosomes are are arepaired paired paired randomly randomly randomly and and and serve serve serve as as asparent parent parent chromosomes, chromosomes, chromosomes, which which are areare selected selectedselected for forfor the the newnew population population according according to to the the crossover crossover probabil probability. probability.ity. If If the the new new chromoso chromosome chromosomeme meets meets all all constrains, constrains, taketake itit it toto to replacereplace replace thethe the parentparent parent chromosome.chromosome. chromosome. Otherwise,Otherw Otherwise,ise, keepkeep keep thethe the parentparent parent chromosome.chromosome. chromosome. The The crossovercrossover crossover processprocess isis is shownshown shown inin in FigureFigure Figure6 6.. 6.

FigureFigure 6. 6. Crossover Crossover process. process.

StepStep 6. 6. Mutation MutationMutation process: process:process: The The crossover crossover process process results results in in local local optimum optimum solutions. solutions. Thus, Thus, Thus, a a amutationmutation mutation process process process is is used used to to esca escapeescapepe local local optimums. optimums.optimums. For ForFor each eacheach chromo chromosomechromosomesome in inin thethe the population, population, population, we we first first first determinedetermine whether whether it it can can be be selectedselected selected forfor for thethe the mutationmutati mutationon operationoperation operation byby by mutationmutation mutation rate.rate. rate. For For the the selected selected parentparent chromosome,chromosome, chromosome, randomly randomly randomly select select select a a code acode code of of theof the the chromosome. chromosome. chromosome. If theIf If the the selected selected selected code code code is 0is (oris 0 0(or 1), (or take1), 1), take take 1 (or 1 1 0)(or(or to 0) replace0) to to replace replace it. If theit. it. newIf If the the chromosome new new chromosome chromosome meets all meets constrains,meets all all constrains, constrains, take it toreplace take take it theit to to parentreplace replace chromosome. the the parent parent Otherwise,chromosome.chromosome. keep Otherwise, Otherwise, the parent keep keep chromosome. the the parent parent Thechromosome chromosome mutation. processThe. The mutation mutation is shown process process in Figure is is shown shown7. in in Figure Figure 7. 7.

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0

0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0

FigureFigure 7. 7. Mutation Mutation process. process.

6.6. Numerical Numerical Example Example InIn this this section,section, section, the the GA GA isis is codedcoded coded inin in MatlabMatlab Matlab programmingprogramming programming language language to to verify verify the the validity validity of of the the double-routingdouble-routing optimization optimization model. model. This This work work mainly mainly studies studies one one direction. direction. Beijing Beijing Metro Metro Line Line 4 4 is is Sustainability 2019, 11, 3701 14 of 26

Sustainability 2019, 11, x FOR PEER REVIEW 14 of 25 a trunk line connecting the north and south areas of the western part of Beijing. We take Beijing metro a trunk line connecting the north and south areas of the western part of Beijing. We take Beijing line 4 as the research object, which is 50 km long and has 35 stations. First, we mark Tian’GongYuan metro line 4 as the research object, which is 50 km long and has 35 stations. First, we mark Station as No. 1 and the Biomedical Base Station as No. 2, continuing the marking up to No. 35, in turn. Tian’GongYuan Station as No. 1 and the Biomedical Base Station as No. 2, continuing the marking The routeup to No. diagram 35, in ofturn. Beijing The route Metro diagram line 4 is of shown Beijing in Metro Figure line8 and 4 is theshown daily in travelFigure demand8 and the on daily line 4 is showntravel in Figuredemand9. on line 4 is shown in Figure 9.

AnHeQiao North XiYuan BeiGongMen YuanMingYuan Park

ZhongGuanCun HaiDianHuangZhuang RenMin University WeiGongCun National Library XinJieKou Ping’an Li Ping an Li LingJingHuTong TaoRanTing Ring Road Ring Road Ring Road Ring Road Ring Road Ring Road Ring Road Ring Road th th th rd nd th rd 5

Beijing South Railway Station nd 4 5 3

Beijing South Railway Station 2 4 3 2 MaJiaPu JiaoMen West GongYiXiQiao

Xin’Gong

XiHongMen

GaoMiDian North GaoMiDian South ZaoYuan QingYuanLu 10 kilometres HuangCunXiDaJie HuangCun Railway Station YiHeZhuang

Biomedical Base

Tian’Gong Yuan

FigureFigure 8. 8.Route Route diagram of of Beijing Beijing Metro Metro line line 4. 4.

Figure 9. Daily travel demand of Beijing Metro line 4. FigureFigure 9. 9.Daily Daily traveltravel demand demand of of Beijing Beijing Metro Metro line line 4. 4.

Given the OD data for Beijing metro line 4 for the whole day of 9 July 2018, this paper selects the OD data from 8:00 to 9:00 to form the OD matrix, as shown in Figure 10. The uneven distribution of OD is visually reflected by different shades of color. Sustainability 2019, 11, x FOR PEER REVIEW 15 of 25 Sustainability 2019, 11, x FOR PEER REVIEW 15 of 25

GivenGiven the the OD OD data data for for Beijing Beijing metro metro line line 4 4for for the the whole whole day day of of 9 9July July 2018, 2018, this this paper paper selects selects the the Sustainability 2019, 11, 3701 15 of 26 ODOD data data from from 8:00 8:00 to to 9:00 9:00 to to form form the the OD OD matrix, matrix, as as shown shown in in Figure Figure 10. 10. The The uneven uneven distribution distribution of of ODOD is is visually visually reflected reflected by by different different shades shades of of color. color.

1 2 3 4 5 6 7 8 91011121314151617181920212223242526272829303132333435 1 2 3 4 5 6 7 8 91011121314151617181920212223242526272829303132333435 1 0 58 20 56 162 63 77 192 53 188 52 92 628 30 446 54 209 117 284 36 34 83 20 304 34 22 21 42 117 87 11 1 11 18 3 1 0 58 20 56 162 63 77 192 53 188 52 92 628 30 446 54 209 117 284 36 34 83 20 304 34 22 21 42 117 87 11 1 11 18 3 2 13 0 20 31 114 32 75 134 29 129 31 57 351 42 244 38 95 66 183 24 14 52 7 163 16 19 18 9 65 6281846 2 13 0 20 31 114 32 75 134 29 129 31 57 351 42 244 38 95 66 183 24 14 52 7 163 16 19 18 9 65 6281846 3 6 19 0 11 24 17 16 49 15 67 12 23 206 27 152 30 64 27 85 15 2 25 13 87 18 3 8 15 45 43 0 7 2 0 0 3 6 19 0 11 24 17 16 49 15 67 12 23 206 27 152 30 64 27 85 15 2 25 13 87 18 3 8 15 45 43 0 7 2 0 0 421532 01414191393113126563683627630120792063217511616417 6 181665 58 11 8 11 5 0 421532 01414191393113126563683627630120792063217511616417 6 181665 58 11 8 11 5 0 5 39 86 8 14 0 17 17 129 38 119 32 67 457 18 315 67 146 91 205 47 15 76 14 215 12 13 15 26 55 604 312163 5 39 86 8 14 0 17 17 129 38 119 32 67 457 18 315 67 146 91 205 47 15 76 14 215 12 13 15 26 55 604 312163 6 8 37 2 13 13 0 12 64 28 90 22 61 388 19 331 68 174 111 225 28 13 71 15 178 20 21 18 31 61 40105761 6 8 37 2 13 13 0 12 64 28 90 22 61 388 19 331 68 174 111 225 28 13 71 15 178 20 21 18 31 61 40105761 7 29 48 24 27 26 12 0 25 18 90 34 96 649 35 501 64 197 136 400 59 22 124 36 359 21 39 41 42 140 91 12 6 15 1 5 7 29 48 24 27 26 12 0 25 18 90 34 96 649 35 501 64 197 136 400 59 22 124 36 359 21 39 41 42 140 91 12 6 15 1 5 815 25 126 44 21 6 0 4 29 7 31323 9267 34137 81 211 28 13 43 7141 13 12 18 22 49 59 12 2 1321 815 25 126 44 21 6 0 4 29 7 31323 9267 34137 81 211 28 13 43 7141 13 12 18 22 49 59 12 2 1321 9 11 21 8 18 50 15 13 22 0 26 15 33 422 15 282 38 128 91 279 44 34 75 18 154 15 11 29 30 77 61 9 115110 9 11 21 8 18 50 15 13 22 0 26 15 33 422 15 282 38 128 91 279 44 34 75 18 154 15 11 29 30 77 61 9 115110 10 32 50 15 19 70 26 59 112 26 0 22 40 686 38 521 41 172 157 410 59 15 138 24 343 22 25 41 44 131 80 14 8 14 21 0 10 32 50 15 19 70 26 59 112 26 0 22 40 686 38 521 41 172 157 410 59 15 138 24 343 22 25 41 44 131 80 14 8 14 21 0 11 15 38 6 17 25 15 21 51 12 81 0 21 1213 44 979 128 502 314 798 238 58 245 64 792 53 81 101 98 267 211 30 16 25 10 5 11 15 38 6 17 25 15 21 51 12 81 0 21 1213 44 979 128 502 314 798 238 58 245 64 792 53 81 101 98 267 211 30 16 25 10 5 12 22 15 8 11 19 22 37 25 15 22 10 0 272 6 322 39 160 184 496 166 39 110 33 318 31 37 36 42 95 90 13 16 17 4 12 12 22 15 8 11 19 22 37 25 15 22 10 0 272 6 322 39 160 184 496 166 39 110 33 318 31 37 36 42 95 90 13 16 17 4 12 13 98 118 20 64 125 101 100 196 79 337 302 390 0 183 662 256 677 432 906 428 144 249 113 887 147 90 144 150 585 305 53 46 116 42 25 13 98 118 20 64 125 101 100 196 79 337 302 390 0 183 662 256 677 432 906 428 144 249 113 887 147 90 144 150 585 305 53 46 116 42 25 14 3 6 0 7 7 5 7 14 7 22 8 13 144 0 607 16 147 137 431 95 32 139 32 323 18 35 58 46 85 60 16 11 16 2 4 14 3 6 0 7 7 5 7 14 7 22 8 13 144 0 607 16 147 137 431 95 32 139 32 323 18 35 58 46 85 60 16 11 16 2 4 15 77 56 16 45 103 78 74 103 50 214 187 239 417 245 0 198 977 425 1715 369 93 534 128 1678 199 190 224 229 613 416 86 66 197 53 39 15 77 56 16 45 103 78 74 103 50 214 187 239 417 245 0 198 977 425 1715 369 93 534 128 1678 199 190 224 229 613 416 86 66 197 53 39 16 3 17 4 8 15 6 11 14 7 17 25 31 107 20 445 0 236 89 616 99 27 250 53 371 38 45 39 72 121 114222023342 16 3 17 4 8 15 6 11 14 7 17 25 31 107 20 445 0 236 89 616 99 27 250 53 371 38 45 39 72 121 114222023342 17 34 26 12 11 50 36 52 70 29 60 109 136 166 87 739 351 0 354 1384 429 183 233 144 639 110 108 128 132 197 306 76 130 77 244 17 17 34 26 12 11 50 36 52 70 29 60 109 136 166 87 739 351 0 354 1384 429 183 233 144 639 110 108 128 132 197 306 76 130 77 244 17 18 47 23 17 32 59 51 56 52 38 74 78 95 85 66 398 260 783 0 287 118 45 68 62 37 20 12 19 29 65 30 11 6 16 11 4 18 47 23 17 32 59 51 56 52 38 74 78 95 85 66 398 260 783 0 287 118 45 68 62 37 20 12 19 29 65 30 11 6 16 11 4 19 38 23 6 25 24 29 19 32 25 66 36 71 56 61 508 259 501 280 0 311 208 117 62 118 160 70 142 138 142 301 73 64 125 101 19 19 38 23 6 25 24 29 19 32 25 66 36 71 56 61 508 259 501 280 0 311 208 117 62 118 160 70 142 138 142 301 73 64 125 101 19 20 6 3 0 4 9 4 1 1 5 3 12 3 53 4 47 7 100 16 140 0 0 155 11 178 12 18 17 16 38 23 2 14 14 4 2 20 6 3 0 4 9 4 1 1 5 3 12 3 53 4 47 7 100 16 140 0 0 155 11 178 12 18 17 16 38 23 2 14 14 4 2 215301053 12710734125213872013100975123816182545211751870 215301053 12710734125213872013100975123816182545211751870 22 9 9 0 0 16 7 2 6 3 21 12 39 42 33 152 92 135 76 313 385 108 0 93 107 152 26 24 26 75 30 30 1310128 22 9 9 0 0 16 7 2 6 3 21 12 39 42 33 152 92 135 76 313 385 108 0 93 107 152 26 24 26 75 30 30 1310128 23 2 0 0 7 2 2 8 3 0 4 7 21 26 9 66 25 107 19 167 35 15 154 0 262 19 99 23 19 63 59 10 9 17 9 8 23 2 0 0 7 2 2 8 3 0 4 7 21 26 9 66 25 107 19 167 35 15 154 0 262 19 99 23 19 63 59 10 9 17 9 8 24 50 28 4 24 48 36 36 59 22 84 90 112 156 101 936 322 453 326 617 613 190 145 331 0 617 2662 380 351 587 556 223 191 295 408 54 24 50 28 4 24 48 36 36 59 22 84 90 112 156 101 936 322 453 326 617 613 190 145 331 0 617 2662 380 351 587 556 223 191 295 408 54 25 7 0 2 0 3 3 1 3 4 8 6 6 5 5 46 29 42 22 38 34 12 24 2 139 0 30 1 18 49 44 14 2 14 13 9 25 7 0 2 0 3 3 1 3 4 8 6 6 5 5 46 29 42 22 38 34 12 24 2 139 0 30 1 18 49 44 14 2 14 13 9 26 2 1 0 0 0 1 0 3 2 1 4 8 25 3 70 12 71 14 43 36 25 64 163 2113 299 0 1610 1604 4541 2303 729 231 832 274 142 26 2 1 0 0 0 1 0 3 2 1 4 8 25 3 70 12 71 14 43 36 25 64 163 2113 299 0 1610 1604 4541 2303 729 231 832 274 142 27 0 1 0 0 4 0 3 2 4 6 3 8 26 1 93 20 67 34 53 41 28 7 16 434 2 540 0 3 405 63 79 28 66 8 12 27 0 1 0 0 4 0 3 2 4 6 3 8 26 1 93 20 67 34 53 41 28 7 16 434 2 540 0 3 405 63 79 28 66 8 12 28 0 0 1 1 1 0 0 0 6 2 2 8 21 9 98 20 67 37 74 51 22 7 20 631 22 485 17 0 368 71 50 23 85 17 14 28 0 0 1 1 1 0 0 0 6 2 2 8 21 9 98 20 67 37 74 51 22 7 20 631 22 485 17 0 368 71 50 23 85 17 14 29 11 2 2 0 5 4 3 5 2 17 8 18 39 7 117 52 68 54 116 98 36 21 47 377 274 2493 1266 1243 0 4375 566 324 894 365 131 29 11 2 2 0 5 4 3 5 2 17 8 18 39 7 117 52 68 54 116 98 36 21 47 377 274 2493 1266 1243 0 4375 566 324 894 365 131 30 1 0 0 0 2 2 0 2 5 5 3 4 16 4 35 4 15 12 31 23 14 5 4 227 21 209 22 24 204 0 12 5 61 12 6 30 1 0 0 0 2 2 0 2 5 5 3 4 16 4 35 4 15 12 31 23 14 5 4 227 21 209 22 24 204 0 12 5 61 12 6 31 9 0 0 0 3 2 1 5 0 0 3 1 10 4 44 9 49 8 24 36 12 6 11 246 19 311 26 33 365 0 0 21 49 12 8 31 9 0 0 0 3 2 1 5 0 0 3 1 10 4 44 9 49 8 24 36 12 6 11 246 19 311 26 33 365 0 0 21 49 12 8 320000100 103111423584452216413214252251631332106034112 320000100 103111423584452216413214252251631332106034112 33 6 8 1 7 3 3 5 7 2 22 13 10 69 5 126 33 164 32 48 53 19 19 28 979 72 1089 357 445 3102 1549 320 72 0 33 18 33 6 8 1 7 3 3 5 7 2 22 13 10 69 5 126 33 164 32 48 53 19 19 28 979 72 1089 357 445 3102 1549 320 72 0 33 18 34 4 0 0 3 1 1 3 0 0 3 7 8 28 3 46 16 51 15 17 24 8 5 5 367 19 322 98 128 794 264 43 9 26 0 8 34 4 0 0 3 1 1 3 0 0 3 7 8 28 3 46 16 51 15 17 24 8 5 5 367 19 322 98 128 794 264 43 9 26 0 8 35 1 4 0 0 2 1 1 1 0 4 2 2 21 1 40 15 40 15 24 20 12 8 18 387 31 332 92 121 749 213 69 33 25 6 0 35 1 4 0 0 2 1 1 1 0 4 2 2 21 1 40 15 40 15 24 20 12 8 18 387 31 332 92 121 749 213 69 33 25 6 0 FigureFigureFigure 10. 10. Origin–Destination Origin–Destination (OD) (OD)(OD) matrix matrixmatrix from from 8: 008: 8:0000 to to to9:00 9:00 9:00 for for for Beijing Beijing Beijing Metro Metro Metro Line Line Line 4. 4. 4.

TheTheThe passengerpassenger passenger flowflow flow inin in each each each section section section is is shownis shown shown in in Figurein Figure Figure 11 11.. 11. During During During the the the research research research period, period, period, there there there were awere totalwere a of atotal 88,645total of of 88,645 trips88,645 fromtrips trips from Tian’GongYuan from Tian’GongYuan Tian’GongYuan Station Station Station to AnHeQiao to to AnHeQiao AnHeQiao North North Station Station and and and 49,681 49,681 49,681 trips trips trips from AnHeQiaofromfrom AnHeQiao AnHeQiao North North StationNorth Station Station to Tian’GongYuan to to Tian’GongYuan Tian’GongYuan Station. Stat Station. Theion. Theuneven The uneven uneven passenger passenger passenger flow flow flow distribution distribution distribution in in bothin directionsbothboth directions directions is caused is is caused bycaused the by large by the the number large large number number of morning of of morning morning commuters commuters commuters from the from from suburbs the the suburbs suburbs to the city.to to the Wethe city. choosecity. theWeWe direction choose choose the withthe direction direction high passenger with with high high flow passenger passenger from Tian’GongYuanflow flow from from Tian’GongYuan Tian’GongYuan Station to AnHeQiaoStation Station to to AnHeQiao NorthAnHeQiao Station North North as the Station as the optimization target. It can be seen from Figure 10 that the maximum passenger flow is optimizationStation as the target. optimization It can be target. seen fromIt can Figurebe seen 10 from that Figure the maximum 10 that the passenger maximum flow passenger is from CaiShiKouflow is from CaiShiKou Station to XuanWuMen Station—in this section there were 35,391 passengers bound Stationfrom CaiShiKou to XuanWuMen Station to Station—in XuanWuMen this Station—in section there this were section 35,391 there passengers were 35,391bound passengers for AnHeQiao bound for AnHeQiao North Station and 10,273 passengers bound for Tian’GongYuan Station. For this Northfor AnHeQiao Station and North 10,273 Station passengers and 10,273 bound passengers for Tian’GongYuan bound for Tian’GongYuan Station. For this Station. reason, For we this choose reason, we choose [a, n] as the short routing. [areason,, n] as thewe shortchoose routing. [a, n] as the short routing.

Figure 11. Passenger flow on each section. FigureFigure 11. 11. Passenger Passenger flow flow on on each each section. section. According to the code for the design of the metro and combining the actual operation characteristics of the metro systems, the equipment operating data and some parameters in the model are shown in Table2. Sustainability 2019, 11, 3701 16 of 26

Table 2. Equipment operating data and some parameters for Line 4.

Parameters I0 tz tz0 f min Cz α n f Value 120 s 150 s 180 s 12 tr/h 1460 100% 35 24

This paper implements GA with Matlab programming to optimize the double-routing optimization model. The relevant parameters of the GA are shown in Table3.

Table 3. Values of GA parameters.

Maximum Number of Initial Population Parameters Crossover Rate Mutation Rate Iterations Size Value 100 100 0.9 0.05

When the objective is to minimize passenger waiting time, the decision variables are obtained as shown in Table4, which indicates that all trains should run on the long routing; at the same time this will cause a certain wasted capacity. When the objective is to minimize the wasted capacity, the decision variables are received as shown in Table5, indicating that there are 15 trains running on the short routing and 9 trains on the long routing in an hour; this will cause the minimum wasted capacity and a greater passenger waiting time. A specific comparison of the effects on different objective functions is shown in Table6.

Table 4. Values of decision variables when taking passenger waiting time as the objective.

Decision Variables 1 2 3 4 5 6 7 8 9 10 11 12 13 Value 1111111111111 Decision Variables 14 15 16 17 18 19 20 21 22 23 24 25 Value 111111111112

Table 5. Values of decision variables when taking wasted capacity as the objective.

Decision Variables 1 2 3 4 5 6 7 8 9 10 11 12 13 Value 1221122212222 Decision Variables 14 15 16 17 18 19 20 21 22 23 24 25 Value 1 2 2 2 2 1 1 2 1 1 2 26

Table 6. Comparisons of the effects on different objective functions.

Taking Passenger Waiting Time Taking Wasted Capacity as the as the Objective Function Objective Function 24 tr/h, where 10 tr/h are on the long 24 tr/h, where 24 tr/h are on the Departure frequency routing and 14 tr/h are on the short long routing routing [28,35] Total waiting time for passengers 110,810 252,680 Wasted capacity 60,825 118 Total objective function 171,630 252,798

It can be seen that taking passenger waiting time or wasted capacity as the objective function is not enough, since this will either cause a certain wasted capacity or seriously affect the service quality. Therefore, the objective function combining passenger waiting time and wasted capacity is very necessary. We substitute the parameters into the double-routing model and find the solution. The variation tendencies of average fitness, optimal fitness, the average distance between individuals and scores in the iteration process of the genetic algorithm are shown in Figure 11 and thus we obtain the current best individual. Sustainability 2019, 11, x FOR PEER REVIEW 17 of 25 quality. Therefore, the objective function combining passenger waiting time and wasted capacity is very necessary. We substitute the parameters into the double-routing model and find the solution. The variation tendencies of average fitness, optimal fitness, the average distance between individuals Sustainability 2019, 11, 3701 17 of 26 and scores in the iteration process of the genetic algorithm are shown in Figure 11 and thus we obtain the current best individual. ItIt can can be be observed observed from from Figure Figure 12 that 12 that in the in iteration the iteration process process of the geneticof the algorithm,genetic algorithm, the average the fitnessaverage and fitness the optimal and the fitnessoptimal show fitness a downward show a downward trend and trend so do and the so scores do the and scores average and distance average betweendistance individuals.between individuals. The variation The variation tendencies tenden of averagecies of fitnessaverage and fitness optimal and fitnessoptimal show fitness that show the algorithmthat the algorithm has good has convergence. good convergence. When the When objective the objective function function takes the takes minimum the minimum value, the value, values the ofvalues the decisionof the decision variables variables are as are shown as shown in Table in Ta7.ble The 7. The index index of the of the return return station station in in the the short short routingrouting is is No. No. 10, 10, namely namely XiHongMen XiHongMen Station. Station. TheThe shortshort routingrouting is is [XiHongMen [XiHongMen Station, Station, AnHeQiao AnHeQiao NorthNorth Station]. Station].

FigureFigure 12. 12.Convergence Convergence ofof GA.GA.

Table 7. Values of decision variables with the objective function combining passenger waiting time and Table 7. Values of decision variables with the objective function combining passenger waiting time wasted capacity. and wasted capacity.

DecisionDecision Variables Variables1 21 32 3 4 4 55 66 77 88 9 10 10 11 11 12 1213 13 Value 1111112111121 Value 1 1 1 1 1 1 2 1 1 1 1 2 1 DecisionDecision Variables Variables14 1514 1615 16 17 17 18 18 1919 2020 2121 22 23 23 24 24 25 25 Value 1 1 2 1 1 1 2 1 1 1 1 10 Value 1 1 2 1 1 1 2 1 1 1 1 10

TheThe traintrain timetables timetables before before and and after after optimizati optimizationon are areshown shown in Figure in Figure 13; 13Figure; Figure 14. The14. Thex-coordinate x-coordinate indicates indicates the departure the departure times times of trains of trains and the and y-coordinate the y-coordinate indicates indicates the stations the stations along alongMetro Metro Line 4. Line It can 4. be It seen can befrom seen Figures from 13 Figures and 14 13 that and for 14 trains that foroperated trains on operated balanced on scheduling balanced schedulingmode, every mode, 2.5 min every a train 2.5 min will a be train sent will from be sentAnHeQiao from AnHeQiao North Station North and Station Tian’Gong and Tian’Gong Yuan Station. Yuan Station.The blue The line blue parallel line parallel to the to X-axis the X-axis in Figure in Figure 14 14represents represents XiHongMen XiHongMen Station, Station, which which is thethe turn-backturn-back station station of of the the short short routing. routing. InIn FigureFigure 14 14,, therethere areare fourfour trainstrains thatthat turnturn backback atat XiHongMenXiHongMen Station;Station; thethe restrest ofof trainstrains areare operatedoperated onon thethe samesame timetabletimetable asas beforebefore thethe optimization.optimization. WeWe comparedcompared thethe eeffectsffects beforebefore andand afterafter optimizationoptimization obtainedobtained byby usingusing thethe double-routingdouble-routing optimizationoptimization modelmodel andand thethe specificspecific contentscontents areare shownshown in in Table Table8 .8. It can be observed from Figure 13, Figure 14 and Table8 that the wasted capacity of Beijing Metro Line 4 decreased by 9.5% with the optimized train operation plan, while the passenger waiting time increased by 4.5%, which can verify that this optimization model is very effective. We take the passenger flow in two directions into consideration and substitute them into the model to obtain the result, as shown in Table9. In the best found train timetable, as shown in Figure 15, there is one Sustainability 2019, 11, x FOR PEER REVIEW 18 of 25

It can be observed from Figure 13, Figure 14 and Table 8 that the wasted capacity of Beijing Metro Line 4 decreased by 9.5% with the optimized train operation plan, while the passenger waiting time increased by 4.5%, which can verify that this optimization model is very effective. We Sustainability 2019, 11, 3701 18 of 26 take the passenger flow in two directions into consideration and substitute them into the model to obtain the result, as shown in Table 9. In the best found train timetable, as shown in Figure 15, there istrain one that train turns that backturns at back XiHongMen at XiHongMen Station, Station, while thewhile rest the of rest trains of aretrains operated are operated the same the as same before as beforethe optimization. the optimization.

Figure 13. The currently used train timetable.

Figure 14. TheThe best best train train timetable timetable as a result of the developed approach.

Table 8. Comparison of effects before and after Optimization.

Before Optimization After Optimization Operation Single long routing Mixed operation on long and short routings mode Departure 24 tr/h, where the long routing has 20 tr/h and the 24 tr/h frequency short routing has 4 tr/h Train operation The trains run every 150 s The trains run every 150 s from 8:00 to 9:00. At plan from 8:00 to 9:00. Each train 8:15, 8:27:30, 8:37:30 and 8:47:30, trains will be Sustainability 2019, 11, 3701 19 of 26

Table 8. Comparison of effects before and after Optimization.

Before Optimization After Optimization Mixed operation on long and short SustainabilityOperation 2019, 11, xmode FOR PEER REVIEW Single long routing 19 of 25 routings 24 tr/h, where the long routing has Departure frequencyis operated on long routing 24 tr/ operatedh on the short routing and at the rest of the 20 tr/h and the short routing has 4 tr/h time trains will be operated on the long routing The trains run every 150 s from 8:00 to Total waiting The trains run every 150 s from 9:00. At 8:15, 8:27:30, 8:37:30 and 8:47:30, timeTrain for operation plan 110,8108:00 to 9:00. Each train is operated trains115,830 will be operated on the short passengers on long routing routing and at the rest of the time trains will be operated on the long routing Wasted 60,825 55,043 Totalcapacity waiting time for passengers 110,810 115,830 Total objectiveWasted capacity 60,825 55,043 171,630 170,873 functionTotal objective function 171,630 170,873

Table 9. Values of decision variablesvariables whenwhen takingtaking twotwo directionsdirections intointo account.account.

DecisionDecision Variables Variables1 21 2 3 3 44 55 66 77 8 9 9 10 10 11 1112 1213 13 ValueValue 11112111111111 1 1 1 2 1 1 1 1 1 1 1 1 DecisionDecision Variables Variables14 1514 15 16 16 17 17 1818 1919 2020 21 22 22 23 23 24 2425 25 ValueValue 1 11 1 1 1 11 11 11 11 1 1 1 1 11 110 10

Figure 15. The best train timetable found when takingtaking two directions into account.

As can be seen from Table9 9,, when when two two directions directions are are taken taken into into consideration, consideration, thethe indexindex ofof thethe return station does not change but the departure frequencyfrequency on the short routing decreases and on the longlong routingrouting itit increasesincreases accordingly.accordingly. The departure frequency on a singlesingle long routingrouting cancan bebe calculatedcalculated byby thethe maximummaximum sectionsection passenger flow, flow, which is is our our consensus. consensus. Through Through the the in-depth in-depth study study of this of this model, model, we weobtain obtain the therelationship relationship between between the thetotal total passenger passenger flow flow and and trains’ trains’ optimum optimum headway, headway, which which makes it unnecessary toto calculatecalculate the the section section passenger passenger flow flow and and find find out out the maximumthe maximum section section passenger passenger flow specifically.flow specifically. To some To extent, some thisextent, facilitates this facilitates our calculation. our calculation. As shown inAs Figure shown 16, thein Figure horizontal 16, axisthe ishorizontal the ratio axis of the is totalthe ratio passenger of the total flow passenge to the maximumr flow to passenger the maximum flow; passenger the maximum flow; passenger the maximum flow herepassenger is calculated flow here according is calculated to the maximumaccording trainto the operated maximum on the train single operated long routing on the and single the loadlong factorrouting is and 130%. the The load vertical factor axisis 130%. is the The optimal vertical headway. axis is the optimal headway. Sustainability 2019, 11, x FOR PEER REVIEW 20 of 25 Sustainability 2019, 11, 3701 20 of 26

Figure 16. The relationship between time headway and percentage of passenger flow. Figure 16. The relationship between time headway and percentage of passenger flow. The following part will simulate some different compromise solutions and study some influencing factorsThe of thefollowing model, suchpart aswill the locationsimulate of some the return different station compromise in the short routing,solutions the and number study of trainssome oninfluencing the short factors routing, of the the ratio model, of passenger such as the flow location in the of short the routingreturn station and the in upper the short limit routing, of the load the factor,number in of the trains hope on of the reaching short recommendationsrouting, the ratio of for passenger metro trains flow operated in the short on the routing double and routings. the upper limitThe of the location load factor, of the shortin the routing’s hope of reaching return station recommendations will affect the for ratio metro of passenger trains operated flow and on thethe shortdouble routing’s routings. turnover time, which will have an impact on the optimal train routing plan and the passengers’The location total waitingof the short time. routing’s We set otherreturn variables station will as fixedaffect and the changeratio of thepassenger index offlow the and return the station.short routing’s The optimal turnover train ti routingme, which plan will and have the influence an impact of on the the return optimal station’s train location routing areplan shown and the in Tablepassengers’ 10. total waiting time. We set other variables as fixed and change the index of the return station. The optimal train routing plan and the influence of the return station’s location are shown in TableTable 10. 10. The optimal train routing plan and influence of the turn-back station’s location on optimal routing plans. Table 10. The optimal train routing plan and influence of the turn-back station’s location on optimal Departure Frequency Short Routing The Optimization Effect routing plans. (tr/h) Origin Terminal DepartureLong FrequencyShort Total Passenger Wasted Combining ShortStation RoutingStation Routing Routing Waiting TimeThe OptimizationCapacity EffectCost (tr/h) 1 35 8 21 3 113,770 57,452 171,222 Origin Terminal Long Short Total Passenger Wasted Combining 2 35 9 21 3 114,050 56,971 171,021 Station3 35 Station 10 Routing 20 Routing 4 Waiting 115,830 Time 55,043Capacity 170,865Cost 1 4 35 35 8 11 21 21 3 3 114,940 113,770 56,007 57,452 170,947 171,222 2 5 35 35 9 12 21 22 3 2 114,150 114,050 57,292 56,971 171,442 171,021 6 30 10 20 4 117,988 51,831 169,810 3 7 35 31 10 10 20 20 4 4 117,460 115,830 52,474 55,043 169,930 170,865 4 8 35 32 11 10 21 20 3 4 117,140 114,940 53,116 56,007 170,260 170,947 5 9 35 33 12 10 22 19 2 5 118,150 114,150 51,992 57,292 170,142 171,442 10 34 10 20 4 115,960 54,401 170,360 6 30 10 20 4 117,988 51,831 169,810 7 31 10 20 4 117,460 52,474 169,930 8 From32 Table 10, it can 10 be seen that 20 the departure 4 frequency on 117,140 long and short 53,116 routings changed 170,260 when the9 location33 of the turn-back 10 station moved. 19 When 5 the short routing 118,150 is [10,35], the 51,992 departure frequency 170,142 on the10 short34 routing is 4 tr/ 10h; at this point, 20 we select h = 4 35 as the origin 115,960 station and fix 54,401 it, moving the 170,360 terminal outwards or inwards and the frequency on the short routing reduces. When the short routing is [10,33], the departureFrom Table frequency 10, it can on thebe seen short that routing the isdepartur 5 tr/h; wee frequency select h = 10on aslong the and terminal short and routings fix it; when changed the originwhen stationthe location is moved of the outwards turn-back or inwards, station themoved. frequency When on the the short short routing routing is reduces. [10,35], It the can departure be found thatfrequency when the on shortthe short routing routing is [10 is,30 4] andtr/h; theat this short point, routing’s we select departure h = 35 frequency as the origin is 4 tr station/h, the combinedand fix it, costmoving is at the minimumterminal outwards but at this or time, inwards there and are twothe frequency stations that on need the short to be reconstructed.routing reduces. If weWhen intend the onshort achieving routing the is [10,33], optimization the departure purpose freque by reconstructingncy on the short only onerouting station, is 5 thetr/h; optimal we select routing h = 10 plan as the is [10,35]terminal and and the fix departure it; when frequency the origin on thestation short is routing moved is outwards 4 tr/h. or inwards, the frequency on the short routing reduces. It can be found that when the short routing is [10,30] and the short routing’s departure frequency is 4 tr/h, the combined cost is at the minimum but at this time, there are two Sustainability 2019, 11, x FOR PEER REVIEW 21 of 25 stations that need to be reconstructed. If we intend on achieving the optimization purpose by reconstructing only one station, the optimal routing plan is [10,35] and the departure frequency on the short routing is 4 tr/h. When the short routing is fixed, the proportion of trains on the short routing will directly affect Sustainability 2019, 11, 3701 21 of 26 the passenger’s total waiting time. Table 11 and Figure 16 show the impact of the different train proportions on passenger waiting time. At this time, the total number of trains is 24 and the short routingWhen is [10,35]. the short routing is fixed, the proportion of trains on the short routing will directly affect the passenger’s total waiting time. Table 11 and Figure 16 show the impact of the different train proportions on passengerTable waiting 11. Impact time. Atof the this different time, the proportion total number of trains. of trains is 24 and the short routingNumber is [10,35].of Total the Average Waiting Time for Increase in Average Waiting Trains on the Passenger Table 11. ImpactPassengers of the different on the proportion of trains.Time for Passengers on the Short Waiting Non-Double-Routed Section Non-Double-Routed Section RoutingNumber of Trains Time The Average Waiting Time Increase in Average Waiting Total Passenger on the Short for Passengers on the Time for Passengers on the 1 111,900Waiting Time 1.30 4.00% Routing Non-Double-Routed Section Non-Double-Routed Section 2 113,090 1.36 8.80% 3 1114,390 111,900 1.43 1.30 14.40% 4.00% 2 113,090 1.36 8.80% 4 3115,830 114,390 1.50 1.43 20.00% 14.40% 5 4117,410 115,830 1.58 1.50 26.40% 20.00% 6 5119,170 117,410 1.67 1.58 33.60% 26.40% 6 119,170 1.67 33.60% 7 121,140 1.76 40.80% 7 121,140 1.76 40.80% 8 8123,360 123,360 1.88 1.88 50.40% 50.40% 9 9125,867 125,867 2.00 2.00 60.00% 60.00% 10 10128,740 128,740 2.14 2.14 71.20% 71.20% 11 132,050 2.31 84.80% 11 12132,050 135,910 2.31 2.5 84.80% 100.00% 12 135,910 2.5 100.00%

The serviceservice levellevel isis closely closely related related to to passenger passenger waiting waiting time. time. From From Table Table 11 and11 and Figure Figure 17, it17, can it becan seen be seen that thethat total the passengertotal passenger waiting waiting time increases,time increases, with the with number the number of trains of on trains the shorton the routing short increased.routing increased. In order In to order ensure to the ensure service the quality, service the quality, number the of number trains onof thetrains short on routingthe short cannot routing be toocannot many. be Therefore,too many. we Therefore, can determine we can the determine maximum the number maximum of trains nu onmber the shortof trains routing on the based short on therouting service based quality on the requirements service quality of the requir metroements operation of the company. metro operation company.

(a) The trend of increase in passenger waiting time

Figure 17. Cont. Sustainability 2019, 11, 3701 22 of 26 Sustainability 2019, 11, x FOR PEER REVIEW 22 of 25 Sustainability 2019, 11, x FOR PEER REVIEW 22 of 25

(b) Total passenger waiting time (b) Total passenger waiting time Figure 17. The impact of the different proportion of trains on passenger waiting time. Figure 17. The impact of the different different proportion of trains on passenger waitingwaiting time.time. The passenger flow flow in the short routing refers to the flowflow of passengers whose origin and The passenger flow in the short routing refers to the flow of passengers whose origin and destination are are both both within within the the short short routing, routing, that that is, is, In In this this case case study, study, O O∈ [10,35][10,35] and and D ∈ D [10,35].[10,35]. In destination are both within the short routing, that is, In this case study, O ∈ [10,35]∈ and D ∈ [10,35].∈ In orderIn order to tostudy study the the influence influence of ofthe the ratio ratio of of the the sh shortort routing’s routing’s passenger passenger flow, flow, the the original original data data are order to study the influence of the ratio of the short routing’s passenger flow, the original data are processed byby increasing increasing or decreasingor decreasing the passengerthe passenger flow inflow [10,35] in by[10,35] 10% ,20%by or10% 30%, ,20% successively. or 30%, processed by increasing or decreasing the passenger flow in [10,35] by 10% ,20% or 30%, Thesuccessively. different The passenger different flow passenger distribution flow patterns distribution are shownpatterns in are Figure shown 18, whichin Figure are 18,+30%, which+20%, are successively. The different passenger flow distribution patterns are shown in Figure 18, which are ++30%,10%, +20%,10%, +10%,20% − and10%, 30%−20% di andfferent −30% from different the original from passengerthe original flow, passenger respectively. flow, respectively. The optimal +30%, −+20%, +10%,− −10%,− −20% and −30% different from the original passenger flow, respectively. Thetrain optimal routing train plans routing under diplansfferent under passenger different flow passenger conditions flow are conditions shown in are Table shown 12. in Table 12. The optimal train routing plans under different passenger flow conditions are shown in Table 12.

Figure 18. Passenger flow on each section under different ratios of the short routing’s passenger Figure 18. Passenger flow on each section under different ratios of the short routing’s passenger flow.Figure 18. Passenger flow on each section under different ratios of the short routing’s passenger flow. flow.

Table 12. The optimal train routing plans under different passenger flow conditions. Table 12. The optimal train routing plans under different passenger flow conditions. Magnitude of Change +30% +20% +10% 0 −10% −20% 30% Magnitude of Change +30% +20% +10% 0 −10% −20% 30% The Proportion of Passenger Flow in The Proportion of Passenger Flow in 81.6% 80.4% 78.9% 77.3% 75.4% 73.2% 70.5% [10,35] 81.6% 80.4% 78.9% 77.3% 75.4% 73.2% 70.5% [10,35] Deviation from the Average Section Deviation from the Average Section +11.6% +10.4% +8.9% +7.3% +5.4% +3.2% +0.5% Passenger Flow +11.6% +10.4% +8.9% +7.3% +5.4% +3.2% +0.5% Passenger Flow Short Routing [1,34] [1,34] [1,34] [10,35] [2,35] [2,35] [2,35] Short Routing [1,34] [1,34] [1,34] [10,35] [2,35] [2,35] [2,35] Departure Frequency on the Long Departure Frequency on the Long 25 24 23 20 22 20 18 Routing 25 24 23 20 22 20 18 Routing Departure Frequency on the Short 5 6 4 4 0 0 0 Departure Frequency on the Short 5 6 4 4 0 0 0 Sustainability 2019, 11, 3701 23 of 26

Table 12. The optimal train routing plans under different passenger flow conditions.

Magnitude of Change +30% +20% +10% 0 10% 20% 30% − − The Proportion of 81.6% 80.4% 78.9% 77.3% 75.4% 73.2% 70.5% Passenger Flow in [10,35] Deviation from the Average +11.6% +10.4% +8.9% +7.3% +5.4% +3.2% +0.5% Section Passenger Flow Short Routing [1,34] [1,34] [1,34] [10,35] [2,35] [2,35] [2,35] Departure Frequency on the 25 24 23 20 22 20 18 Long Routing Departure Frequency on the 5 6 4 4 0 0 0 Short Routing

By changing the passenger volume, it can be found that when the passenger flow in [10,35] is increased, the departure frequency is first determined according to the maximum section passenger flow and then the short routing with higher passenger flow is re-selected to complete optimization. When the passenger flow increases by 10%, the departure frequency increases to 27 tr/h. The departure frequency cannot keep increasing, due to the constraints of safe operation and the signal system, therefore, when the passenger flow increases by 20% or even 30%, it is determined as 30 tr/h. When the passenger flows in [10,35] are decreased by 10% and 20%, the volumes in the short routing are only 5.4% and 3.2% higher than the average section passenger volume. The optimal train routing plans obtained by the model are the same as the single long routing, indicating that in these cases, metro trains are not suitable for operating on the double routing. When the passenger flow in [10,35] decreases by 30%, the passenger flow of each section is basically the same. The wasted capacity caused by the single long routing will be relatively low and the requirement can be satisfied by trains operated on the single long routing. Therefore, we infer that when there are some sections whose passenger flow is more than 6% higher than the average section passenger flow, the double routing model can be considered. The change in the upper limit of the load factor will directly affect the value range of feasible solutions. In Table 13, the optimal train routing plans under different load factors are shown.

Table 13. The optimal train routing plans under different load factors.

The Upper Limit of Departure Frequency Departure Frequency Short Routing the Load Factor on the Long Routing on the Short Routing 80% [1,34] 24 1 90% [10,35] 22 2 100% [10,35] 20 4 110% [10,35] 19 5 120% [1,34] 20 4

The upper limit of the load factor will affect the passenger waiting time and the train departure frequency. When the load factor declines, the capacity of each train will decrease and passenger flow can be satisfied by increasing the departure frequency. With the increase of the load factor, the departure frequency on the short routing increases, while the frequency on the long routing decreases correspondingly. The reduction of the departure frequency on the long routing will reduce the waste of transportation capacity but at the same time will lead to the increase of passenger waiting time on non-double-routed sections, which is a mutual restriction process. When the upper limit of the load factor is 120%, the obtained short routing is very close to the complete long routing. The above experimental results basically agree with the theoretical analysis and the actual situation, which validates the feasibility of the optimization model. Sustainability 2019, 11, 3701 24 of 26

7. Conclusions According to the characteristics of the mixed operation on long and short routings, we develop a nonlinear integer optimization model to reduce both passenger waiting time and wasted capacity on the balanced scheduling mode. A genetic algorithm was used to solve the problem. Finally, we take the passenger data and operation data of Beijing Metro Line 4 as a numerical example. The train operation plan was optimized so that the wasted capacity decreased by 9.5%, in addition, the total objective function value decreases by 0.45% when the passenger waiting time increases by 4.5%, which can verify that this optimization model is very effective. The work presented in this paper is by no means complete and further research is needed in the following directions. First, for the convenience of modeling, the departure frequency after optimization is the same as when the trains operated on the single long routing and for this reason, the passenger waiting time increases a little. In fact, when ensuring that the number of trains used does not increase and when considering the differences in the turnover times of the double routings, the departure frequency can increase, so that it can make passenger waiting time decrease and not affect transport enterprises’ interests. Second, the model in this paper is established based on the real-time information from smart cards. The real-time performance should be explored by improving the algorithm, so as to generate more accurate schemes to guide the mixed operation on long and short routings. Faster algorithms need to be developed to solve large size problems. Lastly, further research will take the actual conditions of each station into account and consider the influence of different train types and formations on the double routing optimization model.

Author Contributions: All authors were involved in preparing the manuscript. Conceptualization, X.Y. and J.W.; Funding acquisition, J.W. and H.S.; Methodology, Q.X. and X.Y.; Project administration, H.S.; Resources, X.Y.; Supervision, J.W.; Data curation, H.Y. and Y.Q.; Writing–original draft, Q.X. and X.Y.; Writing–review & editing, H.Y. and Y.Q. Funding: This work was supported by the National Natural Science Foundation of China (Nos. 71701013, 71890972/71890970, 71525002, 71621001), the Beijing Municipal Natural Science Foundation (No. L181008), the Young Elite Scientists Sponsorship Program by CAST (No. 2018QNRC001), the Beijing Philosophy and Social Science Program Project (No. 13JGC087) and the State Key Laboratory of Rail Traffic Control and Safety (No. RCS2019ZZ001). Conflicts of Interest: The authors declare no conflict of interest.

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