Design of Urban Rail Transit Network Constrained by Urban Road Network, Trips and Land-Use Characteristics

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Article Design of Urban Rail Transit Network Constrained by Urban Road Network, Trips and Land-Use Characteristics

Shushan Chai * , Qinghuai Liang and Simin Zhong

School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China; [email protected] (Q.L.); [email protected] (S.Z.) * Correspondence: [email protected]

Received: 1 September 2019; Accepted: 1 November 2019; Published: 3 November 2019

Abstract: In the process of urban rail transit network design, the urban road network, urban trips and land use are the key factors to be considered. At present, the subjective and qualitative methods are usually used in most practices. In this paper, a quantitative model is developed to ensure the matching between the factors and the urban rail transit network. In the model, a basic network, which is used to define the roads that candidate lines will pass through, is firstly constructed based on the locations of large traffic volume and main passenger flow corridors. Two matching indexes are proposed: one indicates the matching degree between the network and the trip demand, which is calculated by the deviation value between two gravity centers of the stations’ importance distribution in network and the traffic zones’ trip intensity; the other one describes the matching degree between the network and the land use, which is calculated by the deviation value between the fractal dimensions of stations’ importance distribution and the traffic zones’ land-use intensity. The model takes the maximum traffic turnover per unit length of network and the minimum average volume of transfer passengers between lines as objectives. To solve the NP-hard problem in which the variables increase exponentially with the increase of network size, a neighborhood search algorithm is developed based on simulated annealing method. A real case study is carried out to show that the model and algorithm are effective.

Keywords: urban rail transit; network design; urban characteristics; neighborhood search

1. Introduction As the backbone of an urban transit system, urban rail transit plays an important role in urban development. A reasonable urban rail transit network can not only ensure the eﬃciency of urban passenger transport but also promote the sustainable development of a city. The network design, which will determine the structure and the overall layout of the network, is the most important stage in urban rail transit planning. How to meet the urban development characteristics in this stage is an important question. Urban rail transit network design can be aﬀected by many factors, such as urban space, structure and layout, passenger demand, society and economy, environmental impact, and policy, etc., and it is complicated to quantify the factors simultaneously, so subjective and qualitative methods are usually used in most practices of network design. With the development of computer technology and the application of mathematical methods, nowadays planners and researchers are gradually employing quantitative models in transit network design. Among them, a symmetrical urban spatial structure model was proposed by Fielbaum et al. [1]. Based on the city model, four transit network structures (direct lines, feeder-trunk, hub and spoke, and exclusive lines) were calculated by Fielbaum et al. [2] to identify the applicability to the urban structure. But the network designed by this method

Sustainability 2019, 11, 6128; doi:10.3390/su11216128 www.mdpi.com/journal/sustainability Sustainability 2019, 11, 6128 2 of 23 is too idealized and theorized to determine specific passenger corridors, line alignments, and station locations for a real city. In view of the environmental impact, a multi-objective model, which takes the minimum off-gas emissions and the minimum travel time as objectives, was formulated by Zhang [3]. With the model, the network with environment-friendly characteristics can be obtained. In view of the risk management policy, some risk-averse strategies (minimization of the maximum regret, value-at-risk measures) were introduced to the rail transit network design problem by Cadarso [4]. For each strategy, an optimization model with the objective of minimum risk impact was established. But the models considered the effects of each strategy separately rather than synthetically. For the quantitative methods of rail transit network design, mathematical programming is most commonly used. During the optimization process, the planning objectives determined by construction and service requirements include maximizing the benefits of operators [5–8], users [9] or both [10–13]. For research projects, the rail transit design can be classified as the design of a new rail transit network and the expansion of an existing rail transit network [14]. Among the influencing factors, the urban characteristics of road network, trips and land use are the most relevant, basic and important factors with the urban rail transit network, which should be given priority consideration. On one hand, as a kind of mass and punctual transit, urban rail transit can greatly improve the travel convenience for passengers and change the mode sharing of urban trips [14,15]. On the other hand, land use value along the lines and around the stations can be greatly improved with the impact of urban rail transit, which can effectively guide the rapid development of the city [16,17]. Therefore, in this paper, a quantitative model, which takes into account the urban characteristics simultaneously and is applicable to real cities is developed while the cities were generally regarded as connected graphs in most previous research. In the model, a generation method of a basic network, which takes into account the matching between the generated network and the urban road network, is proposed. Two matching indexes are introduced to the constraints: one indicates the matching degree between the network and the urban trips, the other indicates the matching degree between the network and the land use. The model takes the maximum transport efficiency of the network and the maximum travel convenience for passengers as objectives, which stands in the perspective of both managers and users. Due to the high complexity, solving this type of model has been proved to be an NP-hard problem [6,9,18], which means that as the network size increases, the variables increases exponentially. So a neighborhood search algorithm for a real-size network is developed based on a simulated annealing method in the paper. The structure of this paper is organized as follows. Section2 presents the determination of the matching indexes between the urban rail transit network and the urban characteristics. Section3 presents the multi-objective network design model, followed by the solving algorithm in Section4. Section5 presents the real-world case calculation and analysis. Section6 presents the conclusions.

2. Determination of Matching Indexes In the quantitative model, the urban road network, trips and land use are the key factors to be considered. The basic network, which connects the locations of large traﬃc volume and covers the passenger ﬂow corridors, will be determined to ensure the matching between the rail transit network and the urban road network. Two indexes will be proposed to evaluate quantitatively the matching degrees between the rail transit network and the urban trips, and similarly between the rail transit network and the land use. The determined basic network, the calculated matching indexes, the given candidate stations, the passenger demand, etc. are all used as inputs of the model. While generating the network, the matching degrees will be constrained in a suitable range, and the line alignments will be bounded in the basic network. It should be noted that only some of the candidate stations will be selected as the actual stations according to the principle of optimization. After iterations, a network that meets the constraints and has the optimal objective will be obtained. Sustainability 2019, 11, 6128 3 of 23

2.1. Structural Importance of Stations

2.1.1. Evaluation Indexes The structural importance of a station refers to the importance of its position in the topological network. In this paper, the urban rail transit network is regarded as a complex network, and the degree centrality, the closeness centrality and the betweenness centrality [19,20] of the nodes in the complex network are used to describe the structural importance of the stations from diﬀerent dimensions, respectively. The degree centrality DCi of station i reﬂects the direct relevance between the station i and other stations. The larger the value, the closer the relationship between the station and other stations in the network. Assuming that DCt, DCm and DCe are the index values of transfer station, intermediate station and terminal station respectively, they satisfy DCt > DCm > DCe. In the undirected graph network, DCi is calculated by Equation (1):

k DC = i (1) i n 1 − where ki is the degree value of station i, which is deﬁned as the number of edges connected to the station i. n is the number of stations in the network. The closeness centrality CCi of a station i reﬂects the convenience of arriving at the station i from other stations. The larger the closeness centrality value, the greater the extent to which the station i resides in the center of the network, and the shorter the average distance from other stations to the station i. CCi is calculated by Equation (2):

n 1 CCi = − (2) Pn dij j=1 where dij is the length of shortest path between station i and station j. In the urban rail transit network, the station i with high betweenness centrality BCi plays a key role in the transit of passenger ﬂow, which reﬂects the bridging function in the connection between other stations. The larger the index, the more the number of travel paths passing through the station. BCi is calculated by Equation (3):

i 2 X nst BCi = (3) (n 1)(n 2) gst − − s,t where gst is the number of shortest paths that have the same and the shortest length between stations s i and t, nst is the number of paths included in gst and passing through the station i. In order to coordinate the multi-dimensional importance of station i in the network topology, a comprehensive importance evaluation index Ii is proposed, as shown in Equation (4):

Ii = w1DCi + w2CCi + w3BCi (4) where Ii is the comprehensive importance degree of station i in the network structure, w1, w2, and w3 are the weights of the structural indexes from diﬀerent dimensions. With the index Ii, the importance of station i in the topological network can be evaluated comprehensively, reﬂecting the status of the station in the network structure. In the following section, Ii will be used to calculate the structural characteristics of an urban rail transit network. For the convenience of expression, the importance of the stations mentioned below refers to the comprehensive structural importance in the network. Sustainability 2019, 11, 6128 4 of 23 Sustainability 2019, 11, x FOR PEER REVIEW 4 of 23

2.1.2.The Gravity position Center status of Distributionof a station reflected by its structural importance in the network is the result of theThe connection position statusform of of the a station network reflected structure by itsat structuralits position importance. Therefore, in the the structural network isform the of result the ofnetwork the connection can be characterized form of the networkby the distribution structure atof itsthe position. stations Therefore,with different the importance structural form. of the networkIf the can importance be characterized degree byof a the station distribution is regarded of the as stations its ‘gravity with’, dithefferent more importance. important the station is, theIf more the importance ‘gravity’ it degree has in ofthe a stationnetwork. is regardedIn this paper, as its following ‘gravity’, the morecalculation important principle the station of the is,gravity the more center ‘gravity’ in physics, it has the in the concept network. of gravity In this paper, center following of stations’ the importance calculation in principle a network of the is gravityproposed. center An inurban physics, rail transit the concept network of gravity is usually center regarded of stations’ as a importanceplanar network. in a network If the stations is proposed. with Andifferent urban importance rail transit networkare assigned is usually to certain regarded values as and a planar locations, network. the distribution If the stations of sta withtions di ffcanerent be importanceexpressed in are the assigned form of to scatter certain in values the network. and locations, The equilibrium the distribution gravity of stations point of can the be scatter expressed in the in theplanar form topological of scatter innetwork the network. is the gravity The equilibrium center of gravitythe stations’ point importance of the scatter distribution, in the planar as topological shown in G G G networkFigure 1.is In the Figure gravity 1, centerA , ofB theand stations’ C are importance the ‘gravity distribution,’ of station A, as B shown and C, in respectively Figure1. In Figure, and t1he, O O GpointA, G B andS isG theC are gravity the ‘gravity’ center. of T stationhe calculation A, B and method C, respectively, of the coordinates and the point of OSS is is the shown gravity in center.Equation Thes (5) calculation–(6): method of the coordinates of OS is shown in Equations (5)–(6):

nn PxI  xiiiIi i=1 x1  i 1 (5) x1 = n (5) Pn I  iIi ii=1 1

n Pn yI  yiiI i1 i i y  i=1 1 = n (6) y1 n (6) PIi i1 Ii i=1

where (,)xy1 is the coordinate pair of the gravity center of stations’ importance, n is the number where (x1, y11) is the coordinate pair of the gravity center of stations’ importance, n is the number of stationsof stations in urbanin urban rail rail transit transit network, network,(xi , (,)yxyiii) is the is coordinatethe coordinate pair pair of station of stationi. i .

Figure 1. The gravity center of stations stations’’ importance distribution.distribution.

The gravitygravity centercenter reﬂects reflects the the emphasis emphasis of of rail rail transit transit development development on urbanon urban spatial spatial location. location. The locationThe location of the of gravity the gravity center center will changewill change with thewith development the development of rail of transit, rail transit, and the and focus the areafocus of area the centerof the center reﬂects reflect the importances the importance of the urban of the area urban to aarea certain to a extent. certain The extent. index The will index be used will to be calculate used to thecalculate matching the matching degree between degree thebetween network the andnetwork the urban and the trips. urban trips.

2.1.3. Fractal Dimension 2.1.3. Fractal Dimension Fractal theory originated in the late 1960s and was founded by Mandelbrot, B [[21]21].. The theory was originally used toto describedescribe highlyhighly irregularirregular graphics.graphics. Fractal dimension is the most important parameter of fractal theory. It is used to describe the dimension of complex irregular graphics graphics,, which is thethe measuremeasure of spatialspatial scale and overall complexity of the graphics. I Inn 1991 1991,, L. B Benguiguienguigui and and M. M. Daoud [[22]22] inspected the railway system in the suburb of Paris and found that the railway network showed a branched fractal structure. After that, the fractal theory waswas gradually applied in the ﬁeldfield of transportation. Sustainability 2019, 11, 6128 5 of 23

InSustainability fact, there 2019, 11 are, x FOR fractal PEER REVIEW features in a certain space–time range between the city5 of and 23 the transportation network systems, and their fractal geometry properties reﬂect the self-similar characteristicsIn fact, of there the systems. are fractal In featuresthis paper, in a the certain city is space regarded–time rangeas a circular between area, the and city theand radiusthe is transportation network systems, and their fractal geometry properties reflect the self-similar centered around the urban center. Figure2 shows the urban rail transit lines in the city with radius r, characteristics of the systems. In this paper, the city is regarded as a circular area, and the radius is wherecentered the point around O represents the urban center the urban. Figure center. 2 shows The the sum urban of therail structuraltransit lines importancein the city with degrees radius of all stationsr , Iwhere(r) within the point the radiusO representsr is calculated the urban bycenter Equation. The sum (7): of the structural importance degrees of all stations Ir() within the radius r is calculated by Equation (7): Xnr I(r) =nr Ii(r) (7) I()() r i= Ii1 r (7) i1 ( ) wherewhereIi r isIri () the is structural the structural importance importance degree degree ofof station ii within the the radius radius r r, , nnr r isis the the number number of stationsof stations within within the radius the radiusr. r .

FigureFigure 2.2. Urban Urban rail rail transittransit lines in in the the city city with with radius radius r .r .

AccordingAccording to the to the fractal fractal theory, theory,I( r)Ir()and andr satisfy r satisfy the followingthe following relationship: relationship:

I() r r DS (8) I(r) rDS (8) ∝ where DS is the fractal dimension of the stations’ importance distribution. where DSTishe the city fractal is divided dimension into several of the circles stations’ by the importance radii with different distribution. lengths, and the stations with Thedifferent city importance is divided are into scattered several in circles the circles. by the So radiithe fractal with dimensions different lengths, corresponding and the to stationsdifferent with differentradii importance can be obtained. are scattered in the circles. So the fractal dimensions corresponding to different radii can beThe obtained. fractal dimensions corresponding to different radii can reflect the change of rail transit Thesupply fractal capacity dimensions from an correspondingurban center to a to peripheral different radiiarea. In can fact, reflect the urban the change development of rail transitis biased supply towards the central area, so the supply capacity of the central area is usually larger than that of the capacity from an urban center to a peripheral area. In fact, the urban development is biased towards peripheral area. The index will be used to calculate the matching degree between the network and the central area, so the supply capacity of the central area is usually larger than that of the peripheral the urban land use. area. The index will be used to calculate the matching degree between the network and the urban land use.2.2. Matching with Urban Characteristics

2.2. Matching2.2.1. Urban with Road Urban Network Characteristics

2.2.1. UrbanUsually, Road urban Network rail transit lines are laid along the roads, and only in special cases can be constructed through land blocks. Therefore, it is necessary to limit the roads, so as to determine the Usually,range and urban direction rail transitof the searchable lines are laidsections along when the generating roads, and the only network. in special In this cases way, can not be only constructed the throughplanner’s land blocks.subjective Therefore, and groundless it is necessary decision can to limit be avoided the roads, with so mathematical as to determine model the, but range also and directionreasonable of the route searchable boundaries sections can whenbe grasped. generating We refer the to network. the topological In this network way, not used only to thelimit planner’s the subjectivesearch and range groundless and the search decision direction can be as avoidedthe ‘basic with network mathematical’. model, but also reasonable route boundaries can be grasped. We refer to the topological network used to limit the search range and the search direction as the ‘basic network’. Sustainability 2019, 11, 6128 6 of 23 Sustainability 2019, 11, x FOR PEER REVIEW 6 of 23

The locations of large traffic volume and passenger flow corridors are the main factors affecting The locations of large traffic volume and passenger flow corridors are the main factors affecting the line alignments and network structure. Among the factors, the locations with large traffic volume the line alignments and network structure. Among the factors, the locations with large traffic volume are the main source of production and attraction of passenger flow, including transportation hubs and are the main source of production and attraction of passenger flow, including transportation hubs centers of a district. The transportation hubs include a bus transfer hub, bus terminal, railway station and centers of a district. The transportation hubs include a bus transfer hub, bus terminal, railway and airport, etc. The centers of a district include commercial and cultural sites, large enterprises, large station and airport, etc. The centers of a district include commercial and cultural sites, large residential areas and tourist attractions, etc. So the basic network should be established by connecting enterprises, large residential areas and tourist attractions, etc. So the basic network should be the main locations of large traffic volume and covering the main passenger flow corridors. Before established by connecting the main locations of large traffic volume and covering the main passenger establishing the basic network, the main passenger corridors are determined by a ‘cobweb assignment’ flow corridors. Before establishing the basic network, the main passenger corridors are determined method, and the steps are as follows: by a ‘cobweb assignment’ method, and the steps are as follows: Step 1: Generate OD (origin-destination) desire lines between traffic zones according to the Step 1: Generate OD (origin-destination) desire lines between traffic zones according to the predicted passenger demand. predicted passenger demand. Step 2: Eliminate the inter-zone desire lines while the desire lines between adjacent zones are Step 2: Eliminate the inter-zone desire lines while the desire lines between adjacent zones are retained. The network formed by the retained desire lines is called ‘virtual cobweb’. retained. The network formed by the retained desire lines is called ‘virtual cobweb’. Step 3: Passenger assignment is carried out on the ‘virtual cobweb’ with all-or-nothing method. Step 3: Passenger assignment is carried out on the ‘virtual cobweb’ with all-or-nothing method. Step 4: Judging the main passenger flow corridors according to the scale of passenger Step 4: Judging the main passenger flow corridors according to the scale of passenger flow flow accumulation. accumulation. Usually, most of the research on actual transit network designs are based on the premise that the Usually, most of the research on actual transit network designs are based on the premise that basic network is known because of lacking effective determination method. Therefore, the determining the basic network is known because of lacking effective determination method. Therefore, the process of the basic network proposed in this paper is as follows: determining process of the basic network proposed in this paper is as follows: Step 1: Selection of trunk roads that meet the conditions for rail transit construction. Step 1: Selection of trunk roads that meet the conditions for rail transit construction. According to the urban traffic road network in the area requiring rail transit construction, the According to the urban traffic road network in the area requiring rail transit construction, the trunk roads along the passenger corridors and connecting the locations with large traffic volume trunk roads along the passenger corridors and connecting the locations with large traffic volume are are selected. selected. Step 2: Selection of other roads that can connect the locations of large traffic volume. Step 2: Selection of other roads that can connect the locations of large traffic volume. If the selected trunk roads cannot connect all the necessary locations of large traffic volume, other If the selected trunk roads cannot connect all the necessary locations of large traffic volume, roads around the locations that meet the construction conditions and land-use requirements should other roads around the locations that meet the construction conditions and land-use requirements be selected. should be selected. Step 3: Step 3: The mainmain locationslocationsof of large large tra trafficffic volume volume are are connected connected along along the the selected selected roads, roads, and and the topology network formed is the basic network. the topology network formed is the basic network. The locations with close distance and large traffic volume are merged. In this way, not only the The locations with close distance and large traffic volume are merged. In this way, not only the complexity of solving the model, but also the detour distance of the generated transit lines can be complexity of solving the model, but also the detour distance of the generated transit lines can be reduced. Figure3 shows the urban rail transit lines laid along the main routes in the basic network. reduced. Figure 3 shows the urban rail transit lines laid along the main routes in the basic network.

Figure 3. Urban rail transit lines laid along the main routes in the basicbasic network.network.

2.2.2. Trip Intensity Sustainability 2019, 11, x FOR PEER REVIEW 7 of 23

There is a certain difference for trip volumes in different traffic zones, and the volume in each zone includes the production and attraction of the trips. So, the trip intensity in each zone can be Sustainabilitycalculated 2019by the, 11 ,amount 6128 of the production and attraction of the zone, as shown in Equation (9)7: of 23

SPAi i i (9) 2.2.2. Trip Intensity where Si is the trip intensity in traffic zone i , Pi and Ai are the production and attraction of trip volumeThere in traffic is a certain zone dii ,ff respectivelyerence for trip. volumes in different traffic zones, and the volume in each zoneUsually, includes the the trip production intensity and of each attraction zone ofcan the be trips. considered So, the to trip be intensityconcentrated in each in the zone centroid can be positioncalculated of bythe the zone amount. So, if ofthe the centroid production position ands attractionare assigned of the to corresponding zone, as shown trip in Equation-intensity (9):values, the distribution of the trip volume can be expressed in the form of scatter in the city. Similarly, if the trip intensity of each zone is regarded as theSi = ‘gravityPi + Ai’ of the zone, the equilibrium point of the(9) ‘gravity’ is the gravity center of trip intensity, as shown in Figure 4. In Figure 4, each area represents where Si is the trip intensity in traffic zone i, Pi and Ai are the production and attraction of trip volume the traffic zone, o1 ~o9 are the centroid positions of traffic zones, GG19~ are the ‘gravity’ values, in traffic zone i, respectively. and O is the gravity center of trip intensity. The coordinates of the gravity center can be calculated Usually,P the trip intensity of each zone can be considered to be concentrated in the centroid positionby Equation of thes (10) zone.–(11) So,: if the centroid positions are assigned to corresponding trip-intensity values, the distribution of the trip volume can be expressedm in the form of scatter in the city. Similarly, if xS the trip intensity of each zone is regarded as the ‘gravity’ii of the zone, the equilibrium point of the i1 x2  ‘gravity’ is the gravity center of trip intensity, as shownm in Figure4. In Figure4, each area represents(10) Si the traffic zone, o1 o9 are the centroid positions of traffic zones, G1 G9 are the ‘gravity’ values, and ∼ i1 ∼ OP is the gravity center of trip intensity. The coordinates of the gravity center can be calculated by m Equations (10)–(11): yS  mii i1 P y2  m xiS i (11) = i=S1 x2  mi (10) i1 P Si i=1 where (,)xy2 2 is the coordinate pair of the gravity center of urban trip intensity, m is the number Pm of traffic zones divided in the city, (,)xyii is the coordinateyiSi pair of centroid position of traffic zone i=1 i . y = (11) 2 Pm The gravity center of urban trip intensity reflectsSi the bias in the distribution of urban trip = volume. If the trip intensity in an area increases, thei 1gravity center will develop toward the area. On wherethe contrary,(x2, y2 )ifis the the trip coordinate intensity pair decreases, of the gravity the development center of urban direction trip intensity,of the gravitym is thecenter number will be of reversed.traffic zones divided in the city, (xi, yi) is the coordinate pair of centroid position of traffic zone i.

Figure 4. TThehe gravity gravity center center of urban urban trip intensity intensity..

UTherban gravity rail transit center ofis urbana main trip transportation intensity reﬂects mode the for bias the in residents the distribution in the ofcity urban. The trip rail volume. transit networkIf the trip should intensity not in an only area satisfy increases, the capacity the gravity demand center will of passenger develop toward transport the, area. but On also the match contrary, the if the trip intensity decreases, the development direction of the gravity center will be reversed. Urban rail transit is a main transportation mode for the residents in the city. The rail transit network should not only satisfy the capacity demand of passenger transport, but also match the Sustainability 2019, 11, x FOR PEER REVIEW 8 of 23

distribution of trip volume in the city. So the importance distribution of rail transit stations should match the different trip volume in traffic zones. In this paper, the deviation value between the gravity

center OP of urban trip intensity and the gravity center OS of stations’ importance distribution is used to characterize the matching degree between the network and urban trips, as shown in Figure

5. The deviation value is expressed as the ratio of Euclidean distance between OP and OS to urban radius, as shown in Equation (12). Sustainability 2019, 11, 6128 8 of 23 22 ()()x12 x  y  y 12 (12) d(,) OPS O  distribution of trip volume in the city. So the importancer ' distribution of rail transit stations should matchwhere thed diis ﬀtheerent deviation trip volume value in of tra gravityﬃc zones. centers In thisbetween paper, the the stations deviation’ importance value between distribution the gravity and centerthe tripO Pintensityof urban, tripr ' intensityis the radius and the calculated gravity center from O theS of total stations’ area importance Ar( ') of distributionthe city, satisfying is used to characterize2 the matching degree between the network and urban trips, as shown in Figure5. The A( r ')  r ' . deviation value is expressed as the ratio of Euclidean distance between OP and OS to urban radius, as Furthermore, the matching degree m between urban rail transit network and urban trips can shown in Equation (12). C q be calculated by Equation (13). The value of mC is between2 0 and 1,2 and the closer the index value (x1 x2) + (y1 y2) is to 1, the higher the matching ddegree(O , O between) = the− network and− urban trips. (12) P S r 1 0 where d is the deviation value of gravitym centers between the stations’ importance distribution and(13 the1 C 1 d ( O , O ) trip intensity, r is the radius calculated from the totalPS area A(r ) of the city, satisfying A(r ) = πr 2.) 0 0 0 0

FigureFigure 5.5. DeviationDeviation of the two gravity centers.centers.

2.2.3.Furthermore, Land-Use Intensity the matching degree mC between urban rail transit network and urban trips can be calculated by Equation (13). The value of m is between 0 and 1, and the closer the index value is to 1, There is a feedback relationship betweenC urban rail transit and land use. Land-use intensity is the higher the matching degree between the network and urban trips. the root cause of passenger volume for urban rail transit while the rail transit could have a positive guiding role in the development of surrounding land1 use. Therefore, in the stage of network design, mC = (13) it is necessary to ensure the matching between1 + d the(O Purban, OS) rail transit network and the land-use intensity [23]. 2.2.3.Land Land-Use-use intensity Intensity directly affects the accumulation of urban population, so it can be calculated by the population size within the area. Similarly, according to fractal theory, the relationship between There is a feedback relationship between urban rail transit and land use. Land-use intensity is the the radius r of the urban area and the land-use intensity satisfies Equation (14): root cause of passenger volume for urban rail transit while the rail transit could have a positive guiding role in the development of surrounding land use.D Therefore, in the stage of network design,(14 it2 is P() r r P necessary to ensure the matching between the urban rail transit network and the land-use intensity [23) ]. Land-use intensity directly aﬀects the accumulation of urban population, so it can be calculated where P() is the land-use intensity calculated by the sum of resident and employed population, by the population size within the area. Similarly, according to fractal theory, the relationship between D theP radiusis the rfractalof the dimension urban area of and the the land land-use-use intensity intensity. satisﬁes Equation (14):

P(r) rDP (14) ∝ where P( ) is the land-use intensity calculated by the sum of resident and employed population, DP is · the fractal dimension of the land-use intensity. Sustainability 2019, 11, x FOR PEER REVIEW 9 of 23

SustainabilityDeriving2019 , 11r , 6128on both sides of Equation Error! Reference source not found., we can obtain9 of 23 Equation (15):

dP() r D 1 Deriving r on both sides of Equation (14), we Dr canP obtain Equation (15): (15) dr p

dP(r) 2 DP 1 The area of the circle with radius r is A() rDpr r −, satisfying dA( r ) dr 2 r , so the land-(15)use dr ∝ intensity per unit area ()r can be obtained from Equation (16): The area of the circle with radius r is A(r) = πr2, satisfying dA(r)/dr = 2πr, so the land-use dP() r ( ) DP 2 intensity per unit area ρ r can be obtained()r from Equation Dp r (16): (16) dA() r dP(r) DP 2 As can be seen from Equationρ(r )Error! = ReferenceDpr − source not found., when D (16)2 , dA(r) ∝ p DP 2 (r ) Dp (1 r ) , and as the statistical area expands with the increased radius r , ()r gradually DP 2 As can be seen from Equation (16), when Dp < 2, ρ(r) Dp(1/Dr)|=2− |, and()r as the statistical area decreases from the city center to the periphery of the city. ∝When p , remains unchanged. expands with the increased radius r, ρ(r) gradually decreases from the city center to the periphery of When Dp  2 , ()r gradually increases from the city center to the periphery of the city. Therefore, the city. When Dp = 2, ρ(r) remains unchanged. When Dp > 2, ρ(r) gradually increases from the city centerthe fractal to the dimension periphery ofPr() the reflects city. Therefore, the chang theing fractal trend dimension of urbanP land(r) reﬂects-use intensity the changing from trend the city of urbancenter land-useto the periphery intensity of from the city the. city center to the periphery of the city. However, the city does does not not develop develop evenly evenly in in all all directions. directions. Therefore, Therefore, in inthis this paper, paper, the the city city is isdivided divided into into several several fan fan-shaped-shaped area areass in in the the radial radial direction direction.. T Thehe fractal fractal dimension of land-useland-use intensity in each fan fan-shaped-shaped area area should should be be calculated calculated.. If If the the city city is is divided divided into into mm' areasareas (as(as seen in 0 Figure6 6,, the the city city is is divided divided into into four four fan-shaped fan-shaped areas area usings using four four quadrants.), quadrants. the), the fractal fractal dimension dimension of land-useof land-use intensity intensity in in di ﬀdifferenterent areas area cans can be be expressed expressed by by the the vector vector[D [,,,,]1DDDD, D12, 2D3, 3... , Dmm ' ],, wherewhere thethe 0 subscript number of the vector element represents the area number. So,So, the fractal dimension vector

DDSS of of stations’ stations’ importance importance distribution distribution and and the the fractal fractal dimension dimension vector vectorD P DofP land-use of land-use intensity intensity can becan expressed be expressed as [D asS1 , D[,,,,]DDDDS2, DS3, ... , DSm ] and and[DP 1[,,,,],DDDDDP2, DP3, ... , DPm ], respectively,, respectively, where whereDSi Dis the is S1 S 2 S 30 Sm ' P1 P 2 P 3 Pm0 ' Si fractal dimension of stations’ importance distribution in area i, DPi is the fractal dimension of land-use the fractal dimension of stations’ importance distribution in area i , DPi is the fractal dimension of intensity in area i, i = 1, 2, ... m . The deviation value between D and DP is used to describe the land-use intensity in area i , im0 1,2, ' . The deviation value betweenS and is used to matching between the network and urban land use, as shown in Equation (17): describe the matching between the network and urban land use, as shown in Equation (17): s m m' P0 2 ()DD(DSi D2 Pi) i=1Si− Pi i1 (17) d((,)D DS, D P) = (17) SP m'm 0 ( ) where ddD(,) DSSP, D DP is is the the deviationdeviation value value ofof fractal fractal dimensions dimensions between between the the stations’stations’ importance importance distribution and the land-useland-use intensity.intensity.

Figure 6. The city is divided into four fan-shaped areas using four quadrants. Sustainability 2019, 11, 6128 10 of 23

Therefore, the matching degree mF between the network and land use can be expressed by Equation (18). The value of mF is between 0 and 1, and the closer the index value is to 1, the higher the matching degree between the network and the land use.

1 mF = (18) 1 + d(DS, DP)

3. Model Formulation

3.1. Notations The station and section of urban rail transit are abstracted as node i in N and edge i, j in E, respectively. The topological network for urban rail transit is established, which is expressed as G = G(N, E). The notations in the model are shown in Tables1–3.

Table 1. Sets in the model.

Sets Meaning L The set of lines of the candidate network, satisfying l L. ∈ E The set of sections of the basic network, satisfying i, j E. ∈ N The set of stations of the basic network, satisfying i, j N. ∈ N(i) The set of adjacent points of station i. W The set of OD pairs between the locations of large traﬃc volume, satisfying w W. ∈ Np The set of eﬀective paths between OD pairs, satisfying p Np. ∈

Table 2. Known quantities in the model.

Notations Meaning dij The length of section i, j . gw The passenger demand between OD pair w. Nl l nodesmin The minimum number of stations of line in the candidate network. Nl l nodesmax The maximum number of stations of line in the candidate network. dl l min The minimum length of line in the candidate network. l dmax The maximum length of line l in the candidate network. Nlmin The minimum number of lines in the candidate network. N max The maximum number of lines in the candidate network. l Cij The capacity of section i, j . xi The abscissa of station i. yi The ordinate of station i. a The station repeatability, satisfying a > 0 and a is an integer. b The section repeatability, satisfying b > 0 and b is an integer. c The minimum angle between adjacent sections in a line. mCmin The minimum matching degree between the network and urban trips. mFmin The minimum matching degree between the network and land-use intensity.

Table 3. Variables in the model.

Variables Meaning eij If the candidate network has the section i, j , eij = 1, 0 otherwise. l l = eij If the line l in the candidate network has the section i, j , eij 1, 0 otherwise. vi If the candidate network has the station i, vi = 1, 0 otherwise. l l = vi If the line l in the candidate network has the station i, vi 1, 0 otherwise. wpl wpl = fij If the OD pair w passes the section i, j of line l along the path p, fij 1, 0 otherwise. f wpll0 If the OD pair w transfers from line l to line l along the path p, f wpll0 = 1, 0 otherwise. 0 qij The cross-sectional passenger volume of section i, j . Pwp The selection probability of path p between OD pair w. Sustainability 2019, 11, 6128 11 of 23

3.2. Objective Functions In this paper, we focus on how to consider the characteristics of urban road network, trips and land use in the quantitative network design. So other factors involved in the model have to be simplified. The model is formulated based on the assumption that passenger demand for the urban rail transit is already known. In this paper, the logit-based multi-path passenger flow assignment method is adopted. The utility function for path selection is expressed by passenger’s travel time which is the main influencing factor, as shown in Equation (19):

p p p p T = T + T + T , w W, p Np (19) w r d t ∈ ∈ p p p where Tw is the travel time for OD pair w along the path p, Tr is the travel time in sections, Td is the p dwell time at intermediate stations, Tt is the transfer time. During the trip, too many transfers will seriously aﬀect the convenience for passengers, and thus reduce the attraction of the urban rail transit. In order to describe the inﬂuence of transfers, the transfer penalty is used for transfer time calculation, as shown in Equation (20): X p = p ( + )γ Tt Ttk λk 1 (20) k

p th where Ttk is the transfer time in the k transfer station along the path p, λ is the passenger’s transfer sensitivity, γ is the adjustment coefficient which is usually larger than 1. The larger the γ, the greater the difference for sensitivity levels between different transfer times in the path. The effective paths, which are defined by Equation (21), are searched for between each OD pair in the network. So the path selection probability Pwp is calculated by Equation (22):

p p T (1 + H)T min (21) w ≤ w

pmin where H is the detour coeﬃcient and is set to 1.5 in this paper, Tw is the travel time in the shortest path.

p exp( θTw) Pwp = P − p , w W, p Np (22) exp( θTw) ∈ ∈ p Np − ∈ where θ is the calibration parameter. As a kind of public transport, urban rail transit should not only achieve the purpose of proﬁtability for the managers, but also take into account the service function to meet passengers’ demand. In this paper, the objective functions are formulated from the perspective of both managers and passengers to improve the transport eﬃciency of the network and the convenience for passengers.

Objective 1: Maximize the transport eﬃciency of the network From the managers’ point of view, the network should transport as much passengers as possible to improve the operating income. Passenger turnover quantity refers to the product of the passenger volume and the distance transported in a certain period of time, and the unit is p km. The larger the · passenger turnover quantity, the higher the transport capacity of the network. Therefore, the passenger turnover quantity per unit length of the network is used to characterize the transport eﬃciency, which is expressed by Equation (23): P qijdijeij i,j E { }∈ maxZ1 = P (23) dijeij i,j E { }∈ Sustainability 2019, 11, x FOR PEER REVIEW 12 of 23

passenger turnover quantity, the higher the transport capacity of the network. Therefore, the passenger turnover quantity per unit length of the network is used to characterize the transport efficiency, which is expressed by Equation (23): Sustainability 2019, 11, 6128 12 of 23 q d e  qij d ij e ij {,}i j E max Z1  (23) de  deij ij Objective 2: Maximize the travel convenience for{,}i jpassengers E In orderObjective to explore 2: Maximize the relationship the travel convenience between the for transfer passengers times and the travel time in paths, the eﬀective pathsIn order between to explore each the relationship OD pair in between the metro the transfer network times of Beijing and the (astravel shown time in in path Figures, the7) are calculated,effective and paths the between results areeach shown OD pair in in Figure the metro8. As network can be of seen, Beijing a path (as shown with more in Figure transfer 7) are times tendscalculated to have more, and the average results travel are shown time, in and Figure the greater8. As can the be transfer seen, a path sensitivity, with more the transfer more obvious times the phenomenon.tends to have Therefore, more average the average travel time travel, and time the greater of all passengersthe transfer sensitivity, can be reduced the more while obvious the averagethe phenomenon. Therefore, the average travel time of all passengers can be reduced while the average transfer times in paths are reduced. transfer times in paths are reduced.

FigureFigure 7. The7. The metro metro topological topological networknetwork in in Beijing Beijing 2017 2017 (288 (288 stations stations).).

Sustainability 2019, 11, x FOR PEER REVIEW 13 of 23 (a) (b)

(c) (d) FigureFigure 8. The 8. relationshipThe relationship between between the transfer the transfer times times and theand travel the travel time in time paths: in paths(a) λ:= (a)0 (insensitive  0 to transfer);(insensitive (b) λto =transfer);1, γ = 2;(b) ( c) λ 1=, 2, γ 2=; 2;(c) (d) λ2=, 3, γ2=; (d)2.   3 ,   2 .

In order to improve the travel convenience for passengers, it is necessary to improve the directness of traveling for passengers. Reducing the transfer times for passengers is a very effective method to achieve this. Therefore, the average number of transfer passengers between lines is used as the second objective, which is expressed by Equation (24):

wpll'    gw P wp f w W l,',' l  L l  l p  N min Z  p (24) 2 LL( 1) 2

Both the two objectives have the same unit, so the problem can be easily converted into a single objective problem with weighted sum approach for the convenience of solving. The transformed objective function is shown in Equation (25):

max Z=1 Z 1 2 Z 2 (25)

where 121, satisfying 011 , 012 .

3.3. Constraints Constraint 1: Network topological relationship.

ll eij v i ,,,{,} l  L i  N i j  E (26)

ll eij v j ,,,{,} l  L j  N i j  E (27)

ll eij e ji ,,{,} l  L i j  E (28)

e el ,,{,} l  L i j  E ij ij (29) lL

l eij e ij ,,{,} l  L i j  E (30)

el 2, l  L , { i , j }  E  ij (31) j N() i

1 ll eij+1 v i , l L (32) 2 {,}Ei j i N

vl  a,,{,} i  N i j  E  i (33) lL Sustainability 2019, 11, 6128 13 of 23

In order to improve the travel convenience for passengers, it is necessary to improve the directness of traveling for passengers. Reducing the transfer times for passengers is a very eﬀective method to achieve this. Therefore, the average number of transfer passengers between lines is used as the second objective, which is expressed by Equation (24):

P P P wpll gwPwp f 0 w W l,l L,l,l p Np minZ = ∈ 0∈ 0 ∈ (24) 2 L ( L 1)/2 | | | | − Both the two objectives have the same unit, so the problem can be easily converted into a single objective problem with weighted sum approach for the convenience of solving. The transformed objective function is shown in Equation (25):

maxZ =β Z β Z (25) 1 1 − 2 2 where β + β = 1, satisfying 0 β 1, 0 β 1. 1 2 ≤ 1 ≤ ≤ 2 ≤ 3.3. Constraints Constraint 1: Network topological relationship. el vl, l L, i N, i, j E (26) ij ≤ i ∈ ∈ ∈ el vl , l L, j N, i, j E (27) ij ≤ j ∈ ∈ ∈ el = el , l L, i, j E (28) ij ji ∈ ∈ X e el , l L, i, j E (29) ij ≤ ij ∈ { } ∈ l L ∈ el e , l L, i, j E (30) ij ≤ ij ∈ ∈ X el 2, l L, i, j E (31) ij ≤ ∈ { } ∈ j N(i) ∈ 1 X X el +1 = vl, l L (32) 2 ij i ∈ i,j E i N { }∈ ∈ X vl a, i N, i, j E (33) i ≤ ∈ { } ∈ l L ∈ X el b, i, j E (34) ij ≤ { } ∈ l L ∈ Equations (26) and (27) indicate that if the station i or station j is not on the line l, the section i, j will not on the line l. Equation (28) indicates that the network is regarded as undirected graph. Equations (29) and (30) indicate that if the network has the section i, j E, multiple lines in the ∈ network are allowed to pass through the section i, j . Equations (31) and (32) indicate that the line l has neither branches nor loops. Equation (33) indicates that there are up to a lines have the station i in the network. Similarly, Equation (34) indicates that there are up to b lines have the section i, j . Constraint 2: Passenger ﬂow balance.

 w X X X X  1, k O wpl wpl  − ∈ w f f = 1, k D , w W, p Np (35) ik − kj  ∈ ∈ ∈ l L i: i,k E l L j: k,j E  else ∈ { }∈ ∈ { }∈ 0, Sustainability 2019, 11, x FOR PEER REVIEW 14 of 23

el  b,{,} i j E  ij (34) lL

Equations (26) and (27) indicate that if the station i or station j is not on the line l , the section {,}ij will not on the line . Equation (28) indicates that the network is regarded as undirected graph. Equations (29) and (30) indicate that if the network has the section {,}i j E , multiple lines in the network are allowed to pass through the section {,}ij. Equations (31) and (32) indicate that the line l has neither branches nor loops. Equation (33) indicates that there are up to a lines have the station in the network. Similarly, Equation (34) indicates that there are up to b lines have the section {,}ij. Constraint 2: Passenger flow balance.

w 1,kO

 w ffwpl wpl ,k DwW , p  N  ik   kj 1 , p (35) lLiikE:{ , }  lLjkjE  :{ , }   0,else 

wpl wpl l fijfe ji ij ,,{,},, l L i j  E w  W p  N p (36)

Equation (35) indicates that for OD pair w , if station k is the departure station Ow , there is only the inflow at the station; if station k is the intermediate station, the inflow is equal to the outflow at Sustainability 2019, 11, 6128 14 of 23 the station; if station k is the terminal station Dw , there is only outflow at the station. Equation (36) indicates that there is no backflow in the line l . wpl wpl l Constraint 3: Matching fdegrees+ f betweene , l networkL, i, j andE, wurbanW, characteristicsp Np . (36) ij ji ≤ ij ∈ ∈ ∈ ∈ w Equation (35) indicates that for OD pairmmCCminw, if station1 k is the departure station O , there is( only37) the inflow at the station; if station k is the intermediate station, the inflow is equal to the outflow at mmFFmin 1 (38) the station; if station k is the terminal station Dw, there is only outflow at the station. Equation (36) indicatesEquation that theres (37) isand no (38) backflow indicate in thethat line the lmatching. degrees mC and mF should meet the upper and lowerConstraint limits 3:, respectively Matching degrees. between network and urban characteristics. Constraint 4: Line alignments. m m 1 (37) In Figure 9, i , j and k represent threeCmin adjacent≤ C ≤ stations in line l .  is the angle between the vectors ji and jk . In order to ensurem the line malignmentF 1 goes smoothly, the angle  between (38) Fmin ≤ ≤ the adjacent sections should not be too small, as seen in Equation (39). Equations (37) and (38) indicate that the matching degrees mC and mF should meet the upper and lower limits, respectively. c  (39) Constraint 4: Line alignments. The cosine of the angle between the vectors and can be easily calculated by Equation In Figure9, i, j and k represent three adjacent stations in line l. α is the angle between the vectors →ji (40). So Equation (39) can be expressed by Equation (41). and →jk. In order to ensure the line alignment goes smoothly, the angle α between the adjacent sections (x x )( x  x )  ( y  y )( y  y ) should not be too small,cos as seen in Equationi j (39). k j i j k j 2 2 2 2 (40) ()()()()xi x j  y i  y j x k  x j  y k  y j c α π (39) ≤ ≤ where (,)xyii, (,)xyjj and (,)xykk are the coordinate pairs of station i , j and k , respectively.

Figure 9. The angle between adjacent sections in a line.line.

The cosine of the angle between the vectors →ji and →jk can be easily calculated by Equation (40). So c arccos (41) Equation (39) can be expressed by Equation (41).

(xi xj)(xk xj) + (yi yj)(yk yj) cos α = q − − q − − (40) (x x )2 + (y y )2 (x x )2 + (y y )2 i − j i − j k − j k − j where (xi, yi), (xj, yj) and (xk, yk) are the coordinate pairs of station i, j and k, respectively.

c arccosα π (41) ≤ ≤ Constraint 5: Network size.

Nl Nl Nl , l L (42) nodesmin ≤ nodes ≤ nodesmax ∈ X dl d el dl , l L (43) min ≤ ij ij ≤ max ∈ i,j E { }∈ N L N (44) lmin ≤ | | ≤ lmax Equations (42)–(44) indicate that the number of stations in line l, the length of line l and the l number of lines in the network should meet the upper and lower limits, respectively, where Nnodes is the number of stations in the line l. Sustainability 2019, 11, x FOR PEER REVIEW 15 of 23

Constraint 5: Network size.

l l l NnodesminN nodes  N nodes max， l  L (42)

dl d e l  d l ， l  L min ij ij max (43) {,}i j E

NLNllmin max (3)

Equations (42)–(44) indicate that the number of stations in line l , the length of line l and the Sustainability 2019, 11, 6128 15 ofl 23 number of lines in the network should meet the upper and lower limits, respectively, where N nodes is the number of stations in the line l . Constraint 6: Capacity of sections.sections. g P fwpl  C,,{,} l  L i j  E X X w wp ijwpl ij (45) w W p N p gwPwp f C , l L, i, j E (45) ij ≤ ij ∈ { } ∈ w W p Np Equation (45) indicates that∈ cross∈ -sectional passenger volume of sections in each line is less than

Cij . Equation (45) indicates that cross-sectional passenger volume of sections in each line is less than Cij. 4. Neighborhood Search Algorithm 4. Neighborhood Search Algorithm The problem dealt with in this paper is NP-hard. It has high structural complexity, multiple constraintsThe problem and large dealt search with space, in this etc. paper So a is neighborhood NP-hard. It has search high algorithm structural [7] complexity, is developed multiple based constraintson a simulated and large annealing search algorithm space, etc.. SoA a simulated neighborhood annealing search algori algorithmthm [is7 ] a is random developed optimization based on a simulatedalgorithm annealing based on algorithm.a Monte Carlo A simulated iterative annealing solution algorithm strategy, isand a random has strong optimization robustness, algorithm global basedconvergence on a Monte and wide Carlo adaptability, iterative solution which strategy, is suitable and for has solving strong robustness,the problem. global convergence and wideIn adaptability, order to obtain which a variety is suitable of perturbation for solving the solution problem.s in the iteration process, four neighborhood transformIn order operators to obtain are a variety designed. of perturbation The functions solutions of these in operatorsthe iteration are process, as follows: four neighborhoodadding a line transformrandomly, operatorsdeleting a areline designed. randomly, The adding functions a section of thesein a line operators randomly are and as follows: deleting addinga section a linein a randomly,line randomly deleting, as shown a line in randomly, Figures 10 adding–13. Different a section neighborhood in a line randomly transform and operators deleting aare section executed in a lineaccording randomly, to a as certain shown probability. in Figures 10Different–13. Diﬀ erent types neighborhood of perturbations transform are randomly operators applied are executed to the according to a certain probability. Diﬀerent types of perturbations are randomly applied to the current solution Gcurr to obtain new solution Gnew . Each operator can be executed independently or G G currentsimultaneously, solution forcurr example,to obtain one new line solution can be addednew. Each while operator another can line be can executed be deleted independently at the same or simultaneously, for example, one line can be added while another line can be deleted at the same time. time.

Sustainability 2019, 11, x FOR PEER REVIEWFigure 10. Add a l lineine r randomly.andomly. 16 of 23

FigureFigure 11.11. DeleteDelete aa lineline randomly.randomly.

Figure 12. Add a section in a line randomly.

Figure 13. Delete a section in a line randomly.

Let fG()curr , fG()new and fG()best be the objective function values of the current solution

Gcurr , the new solution Gnew (perturbation solution) and the optimal solution Gbest in the existing

iteration process, respectively. If f()() Gnew f G curr , Gnew is accepted as the new current solution.

Otherwise, Gnew is accepted according to the Metropolis criterion. Metropolis criterion is that the algorithm allows the inferior solution to be accepted with a certain probability, so that the algorithm can jump out of the trap of the local optimum. The acceptance probability in the criterion is calculated by Equation (46). As the temperature decreases, the probability of accepting an inferior solution decreases gradually.

[f ( Gnew ) f ( G curr )]/ per  (46) Sustainability 2019, 11, x FOR PEER REVIEW 16 of 23

Sustainability 2019, 11, 6128 16 of 23 Figure 11. Delete a line randomly.

Figure 12. Add a s sectionection in a lineline randomly.randomly.

Figure 13. Delete a s sectionection in a lineline randomly.randomly.

fG()fG() fG() Let f (fGG()currcurr), ,f (GfG()newnew) and andf ( GfGbest())bestbe thebe objective the objective function function values values of the of current the current solution solutionGcurr, G G theGcurr new, the solution new solutionnew (perturbation Gnew (perturbation solution) solution) and the and optimal the optimal solution solutionbest in theGbest existing in the iteration existing process, respectively. If f (Gnew) > f (Gcurr), Gnew is accepted as the new current solution. Otherwise, iteration process, respectively. If f()() Gnew f G curr , Gnew is accepted as the new current solution. Gnew is accepted according to the Metropolis criterion. Metropolis criterion is that the algorithm allows Otherwise, Gnew is accepted according to the Metropolis criterion. Metropolis criterion is that the the inferior solutionnew to be accepted with a certain probability, so that the algorithm can jump out of the algorithm allows the inferior solution to be accepted with a certain probability, so that the algorithm trapalgorithm of the allows local optimum. the inferior The solution acceptance to be probabilityaccepted with in the a certain criterion probability, is calculated so that by Equationthe algorithm (46). can jump out of the trap of the local optimum. The acceptance probability in the criterion is calculated Ascan the jump temperature out of the trap decreases, of the local the probability optimum. ofThe accepting acceptance an inferiorprobability solution in the decreases criterion gradually.is calculated by Equation (46). As the temperature decreases, the probability of accepting an inferior solution decreases gradually. [ f (Gnew) f (Gcurr)]/τ decreases gradually. pr = e − (46) [f ( G ) f ( G )]/ [f ( Gnew ) f ( G curr )]/ per  (46) where τ is the current temperature. r The algorithm is designed with the memory function of the optimal solution Gbest in the existing iteration process, that is, to remember the ‘best so far’ state. As the iteration proceeds, the solution Gbest in the memory base is continuously updated until the algorithm converges. Therefore, the objective value will approach the global optimum gradually in the iterative process. A solution has many kinds of changes after perturbation, so even if the current solution is perturbed into the solution that has appeared in previous iterations, it will change in other directions when perturbed again, which will not have a negative impact on the results. The overall ﬂow of the algorithm is shown in Figure 14. Sustainability 2019, 11, x FOR PEER REVIEW 17 of 23

where  is the current temperature.

The algorithm is designed with the memory function of the optimal solution Gbest in the existing iteration process, that is, to remember the ‘best so far’ state. As the iteration proceeds, the solution in the memory base is continuously updated until the algorithm converges. Therefore, the objective value will approach the global optimum gradually in the iterative process. A solution has many kinds of changes after perturbation, so even if the current solution is perturbed into the solution that has appeared in previous iterations, it will change in other directions when perturbed Sustainabilityagain, which2019 will, 11, not 6128 have a negative impact on the results. The overall flow of the algorithm is shown17 of 23 in Figure 14.

Figure 14. Overall ﬂowflow chart of the algorithm.algorithm.

5. R Real-Worldeal-World Case Application

5.1. Basic Network 5.1. Basic Network Baotou, which is a prefecture-level city in Inner Mongolia, has the total area of over 2000 square kilometers. The urban area was divided into 433 traﬃc zones, and 64 locations of large traﬃc volume were selected in the urban central area, as shown in Figure 15. The urban passenger corridors determined using the ‘cobweb assignment method’ are shown in Figure 16. As can be seen, the urban passenger corridors are mainly located in the east–west axis. According to the passenger corridors and the main axis of urban development, the basic network is formed using the method presented in Section 2.2.1, as shown in Figure 17. Sustainability 2019, 11, x FOR PEER REVIEW 18 of 23 Sustainability 2019, 11, x FOR PEER REVIEW 18 of 23 Sustainability 2019, 11, x FOR PEER REVIEW 18 of 23 Baotou, which is a prefecture-level city in Inner Mongolia, has the total area of over 2000 square Baotou, which is a prefecture-level city in Inner Mongolia, has the total area of over 2000 square kilometers.Baotou, T whichhe urban is a area prefecture was divided-level cityinto in433 Inner traffic Mongolia, zones, and has 64 the locations total area of largeof over traffic 2000 volume square kilometers. The urban area was divided into 433 traffic zones, and 64 locations of large traffic volume kilometers.were selected The in urban the urbanarea was central divided area into, as 433 shown traffic in zones, Figure and 15 64. Thelocations urban of passengerlarge traffic corridors volume were selected in the urban central area, as shown in Figure 15. The urban passenger corridors weredetermined selected using in thethe ‘cobweburban central assignment area, method’ as shown are in shown Figure in 15 Figure. The 16 urban. As can passenger be seen, t corridorshe urban determined using the ‘cobweb assignment method’ are shown in Figure 16. As can be seen, the urban determinedpassenger corridors using the are ‘cobweb mainly assignment located in method’the east– arewest shown axis. inAccording Figure 16 to. As the can passenger be seen, tcorridorshe urban passenger corridors are mainly located in the east–west axis. According to the passenger corridors passengerand the main corridors axis of areurban mainly development, located in the the basic east –networkwest axis. is formedAccording using to thethe methodpassenger presented corridors in and the main axis of urban development, the basic network is formed using the method presented in andSustainabilitySection the 2.2main.12019, asaxis, 11 shown, 6128of urban in Figure development, 17. the basic network is formed using the method presented18 of in 23 Section 2.2.1.1,, asas shownshown inin Figure 17..

Figure 15. Selected locations of large traffic volume in the urban central area. Figure 15. Selected locations of large traffic volume in the urban central area. Figure 15. SelectedSelected locations locations of of large large traffic traﬃc vo volumelume in the urban central area area..

Figure 16. Urban passenger corridors in the urban central area. Figure 16. UUrbanrban passenger corridors in the urban central area.area. Figure 16. Urban passenger corridors in the urban central area.

Figure 17. BBasicasic network in the urban central area.area. Figure 17. Basic network in the urban central area. Figure 17. Basic network in the urban central area. 5.2. Parameters 5.2. Parameters 5.2. Parameters 5.2. PInarameters the algorithm, the annealing rate φ is set to 0.9, and the maximum number of iterations is set to 500. In this case, the passenger’s transfer sensitivity λ is set to 0. The values of other parameters in the model are shown in Table4. Sustainability 2019, 11, x FOR PEER REVIEW 19 of 23 Sustainability 2019, 11, x FOR PEER REVIEW 19 of 23

In the algorithm, the annealing rate  is set to 0.9, and the maximum number of iterations is In the algorithm, the annealing rate  is set to 0.9, and the maximum number of iterations is set to 500. In this case, the passenger’s transfer sensitivity  is set to 0. The values of other set to 500. In this case, the passenger’s transfer sensitivity  is set to 0. The values of other parameters in the model are shown in Table 4. parameters in the model are shown in Table 4.

Table 4. Parameter values in the model. Sustainability 2019, 11, 6128 Table 4. Parameter values in the model. 19 of 23 Parameter Value Parameter Value Parameter Value Parameter Value Table1 4. Parameter0.65 valuesNl inmax the model.6 1 0.65 Nl max 6 2 0.35 Cij 45000 Parameter2 Value0.35 ParameterCij 45000 Value l Nnodesl min 15 a 3 β1 Nnodes min 0.6515 aN lmax 3 6 l β2 Nnodesl max 0.35 8 b C i j 2 45000 Nnodes max 8 b 2 Nl 15 a 3 nodesmin l 5 dl 5 Nl min 815 km c b  2 nodesmax dmin 15 km c 12 l 12 5 d l 15 km c π min d l km m 12 l d max 35 mC min 0.7 dmax max 35 km35 km CmminCmin 0.7 0.7 N m Nlmin l min 3 3 FmminFmin 0.65 0.65 Nl min 3 mF min 0.65

5.3. Urban Characteristic Indexes 5.3. Urban Characteristic Indexes The trip intensity in the long-term planning year in the traffic zones is shown in Figure 18. The trip intensity in the long-termlong-term planning year year in in the traffic traffic zones is is shown in in Figure 1818.. According to the centroid coordinates of each traffic zone, the gravity center of urban trip intensity According toto thethe centroidcentroid coordinatescoordinates of of each each tra trafficffic zone, zone, the the gravity gravity center center of of urban urban trip trip intensity intensity is is calculated. The land-use intensity of the traffic zones in the long-term planning year is predicted, calculated.is calculated The. The land-use land-use intensity intensity of of the the tra trafficffic zones zones in in the the long-term long-term planning planning year year is predicted,is predicted, as as shown in Figure 19. As we can see from Figures 18 and 19, there is a strong correlation between shownas shown in Figurein Figure 19. 19 As. As we we can can see see from from Figures Figure 18s and18 and 19, there19, there is a is strong a strong correlation correlation between between the the trip intensity and the land-use intensity in the traffic zones. tripthe trip intensity intensity and and the land-usethe land-use intensity intensity in the in tratheffi trafficc zones. zones.

Figure 18. The trip intensity in traffic zones. Figure 18. The trip intensity in tratrafficﬃc zones.zones.

Figure 19.19. The land-useland-use intensity in tratrafficﬃc zones.zones. Figure 19. The land-use intensity in traffic zones.

The city is divided into multiple circles centered at the gravity center OP by the radii with different lengths, and the difference in length between adjacent circles is 500 m. From Figure 19, we can see that the city does not develop evenly in all directions. Therefore, it is necessary to divide the city into different areas and calculate the fractal dimension of land-use intensity in each area separately. In this paper, the city is divided into four areas using four quadrants, as shown in Figure 20. Sustainability 2019, 11, x FOR PEER REVIEW 20 of 23

The city is divided into multiple circles centered at the gravity center OP by the radii with different lengths, and the difference in length between adjacent circles is 500 m . From Figure 19, we Sustainability 2019, 11, x FOR PEER REVIEW 20 of 23 can see that the city does not develop evenly in all directions. Therefore, it is necessary to divide the city into different areas and calculate the fractal dimension of land-use O intensity in each area Sustainability 2019The, 11 city, 6128 is divided into multiple circles centered at the gravity center P by the radii with 20 of 23 separately. In this paper, the city is divided into four areas using four quadrants, as shown in Figure different lengths, and the difference in length between adjacent circles is 500 m . From Figure 19, we 20can. see that the city does not develop evenly in all directions. Therefore, it is necessary to divide the city into different areas and calculate the fractal dimension of land-use intensity in each area separately. In this paper, the city is divided into four areas using four quadrants, as shown in Figure 20.

FigureFigure 20. 20.Division Division of of urbanurban circle circle layer layerss and and area areas.s.

TakingTaking logarithms logarithms on both on both sides sides of of Equation Equation (14),(14), E Equationquation (47) (47) can can be obtained be obtained.. We can We see can thatsee that Figure 20. Division of urban circle layers and areas. the fractal dimension Dp can be got by linear regression. the fractal dimension Dp can be got by linear regression.

Taking logarithms on both sides ofln EquationP ( r ) D p(14), ln r Equation a0 (47) can be obtained. We can see that(47) the fractal dimension Dp can be got byln linearP(r )regression. = Dp ln r + a0 (47) where a0 is a constant. lnP (rDra )=+ ln The double logarithmic relationships betweenp land0-use intensity and radii in the areas of(47) four where a0 is a constant. quadrants are recorded, and the fractal dimension of land-use intensity in each quadrant is obtained Thewhere double a0 logarithmic is a constant. relationships between land-use intensity and radii in the areas of four by regression fitting, as shown in Figure 21. As we can see from the fitting coefficients, the calculated quadrants areThe recorded, double logarithmic and the fractal relationships dimension between of land-use intensity intensity and in radii each in quadrantthe areas of is four obtained by fractal dimension vector of land-use intensity is DP  (1.506,1.509,1.549,1.131) . The fractal regressionquadrants ﬁtting, asare shownrecorded, in and Figure the fractal 21. As dimension we can seeof land-use from the intensity fitting in coefficients, each quadrant the is obtained calculated fractal dimensionsby regression are fitting, less than as shown2. So a cincording Figure 21.to E Asquation we can Error! see from Reference the fitting source coefficients, not found. the calculated, we can see dimensionthat vector urban of land land-use-use intensity intensity gradually is DP = decreases (1.506, 1.509, from 1.549,t=he city 1.131 center). The to the fractal outside, dimensions and the are less fractal dimension vector of land-use intensity is DP (1.506,1.509,1.549,1.131) . The fractal than 2. Sodecreasingdimensions according trend are to lessof Equation the than area 2. Soin (16), accordingthe wefourth can to quadrant seeEquation that is urbanError! the fastest Reference land-use. source intensity not graduallyfound., we can decreases see from the city centerthat urban to the land-use outside, intensity and the gr decreasingadually decreases trend from of the the area city in center the fourth to the quadrantoutside, and is thethe fastest. decreasing trend of the area in the fourth quadrant is the fastest.

(a) (b) Sustainability 2019, 11, x FOR PEER REVIEW 21 of 23 (a) (b)

(c) (d) Figure 21.FigureFractal 21. Fractal ﬁtting fitting of land-use of land-use intensity intensity inin the areas areas within within different diﬀerent quadrants: quadrants: (a) within (a ) within quadrantquadrant 1; (b) within 1; (b) within quadrant quadrant 2; ( c2;) (c) within within quadrant quadrant 3; 3; (d) (d within) within quadrant quadrant 4. 4.

5.4. Results Based on the model and algorithm, the real-world case is calculated and the optimal solution in the existing iteration process (called optimal solution for short) is obtained. The optimal objective

value Z is about 8664.2. The first objective value Z1 is about 32,287 while the second objective value

Z2 is about 35,207. The iterative process of the algorithm is shown in Figure 22. As can be seen, the algorithm can converge quickly and has strong stability.

Figure 22. Process of iteration.

The optimal network is shown in Figure 23. The optimal network has 5 lines with a total length of 110.6 km , and the details of the lines is shown in Table 5. The network presents the radial structure as a whole, and the central area is supplemented by a grid-like structure. The distance between the

gravity center OS of stations’ importance distribution and the gravity center OP of urban trip

intensity is about 973.9 m , as shown in Figure 23. The matching degree mC between the optimal

network and the urban trips is about 0.947, and the matching degree mF between the optimal network and the urban land use is about 0.893. In general, the network takes the east–west direction as the main axis, and the line alignments basically follow the routes covered by the main passenger flow corridors, which conforms to the banded development requirement of the city. The results showed that the layout of the network has a good match with the structure of the city. Sustainability 2019, 11, x FOR PEER REVIEW 21 of 23

Figure 21. Fractal fitting of land-use intensity in the areas within different quadrants: (a) within Sustainability 2019, 11, 6128 21 of 23 quadrant 1; (b) within quadrant 2; (c) within quadrant 3; (d) within quadrant 4.

5.4. R Resultsesults Based on the model and algorithm, the real-worldreal-world case is calculated and the optimal solution in thethe existingexisting iterationiteration processprocess (called(called optimaloptimal solutionsolution forfor short)short) isis obtained.obtained. TheThe optimaloptimal objectiveobjective Z Z valuevalue Z is is about about 8664.2. 8664.2. The The ﬁrst first objective objective valuevalueZ 1 1is is about about 32,287 32,287 while while the the second second objectiveobjective value

ZZ22 is is about about 35,207. 35,207. The The iterativeiterative processprocess ofof thethe algorithmalgorithm isis shownshown inin FigureFigure 2222.. AsAs cancan bebe seen,seen, the algorithm can converge quicklyquickly andand hashas strongstrong stability.stability.

Figure 22.22. Process of iteration.iteration.

The optimaloptimal networknetwork isis shown shown in in Figure Figure 23 2.3 The. The optimal optimal network network has has 5 lines5 lines with with a totala total length length of 110.6of 110.6 km,km and, and the t detailshe details of theof the lines lines is shown is shown in Table in Table5. The 5. The network network presents presents the radial the radial structure structure as a whole,as a whole, and theand central the central area isarea supplemented is supplemented by a grid-like by a grid structure.-like structure. The distance The distance between between the gravity the center OS of stations’ importance distribution and the gravity center OP of urban trip intensity is about gravity center OS of stations’ importance distribution and the gravity center OP of urban trip 973.9 m, as shown in Figure 23. The matching degree mC between the optimal network and the urban intensity is about 973.9 m , as shown in Figure 23. The matching degree mC between the optimal trips is about 0.947, and the matching degree mF between the optimal network and the urban land m usenetwork is about and 0.893. the urban In general, tripsthe is networkabout 0.947, takes and the theeast–west matching direction degree as theF main between axis, theand optimal the line alignmentsnetwork and basically the urban follow land the use routes is about covered 0.893. by In the general, main passengerthe network ﬂow takes corridors, the east which–west conformsdirection toas thethe bandedmain axis, development and the line requirement alignments of basically the city. follow The results the routes showed covered that the by layout the main of the passenger network hasflowSustainability a corridors,good match 2019 , which 11 with, x FOR the conforms PEER structure REVIEW to of the the banded city. development requirement of the city. The22 resultsof 23 showed that the layout of the network has a good match with the structure of the city.

FigureFigure 23.23. The optimaloptimal network network..

Table 5. Details of the lines in the optimal network.

Line Sequence of stations in the line Length 1 8–7–6–11–15–23–31–41–46–53–58–61–62–63 31.1 km 2 13–18–19–25–32–43–41–40–39 18.8 km 3 23–24–42–47–48–52–54–55–56–60 23.1 km 4 4–9–14–15–23–22–27–29 16.8 km 5 2–3–5–10–21–30–29–40 20.8 km

6. Conclusions In this paper, a quantitative model is developed to ensure that the generated urban rail transit network has a good match with urban road network, trips and land-use characteristics. The main contributions of the model are as follows: • The basic network, which is determined according to the main locations of large traffic volume and main passenger flow corridors, is constructed to ensure matching between the network and the urban road network. With the basic network, the search range and the search direction of the line alignments can be bounded while generating the transit network. • The matching index, which is calculated by the deviation value between two gravity centers of the stations’ importance distribution in the network and the traffic zones’ trip intensity, is proposed to constrain the matching degree between the network and the urban trip demand. • Similarly, the matching index, which is calculated by the deviation value between the fractal dimensions of stations’ importance distribution and the traffic zones’ land-use intensity, is proposed to constrain the matching degree between the network and the land use. To solve the model, a neighborhood search algorithm is developed based on the simulated annealing method. A real-size network is calculated with the model and algorithm. The generated network conforms to the development requirement of the city, which shows a good match between the network and the urban characteristics.

Author Contributions: Shushan Chai: Conceptualization, methodology and study design, writing – original draft. Qinghuai Liang: Literature review, critical revision and editing. Simin Zhong: Calculation and formal analysis.

Funding: This research received no external funding.

Conflicts of Interest: The authors declare no conflict of interest.

References Sustainability 2019, 11, 6128 22 of 23

Table 5. Details of the lines in the optimal network.

The basic network, which is determined according to the main locations of large traffic volume • and main passenger flow corridors, is constructed to ensure matching between the network and the urban road network. With the basic network, the search range and the search direction of the line alignments can be bounded while generating the transit network. The matching index, which is calculated by the deviation value between two gravity centers of the • stations’ importance distribution in the network and the traffic zones’ trip intensity, is proposed to constrain the matching degree between the network and the urban trip demand. Similarly, the matching index, which is calculated by the deviation value between the fractal • dimensions of stations’ importance distribution and the traffic zones’ land-use intensity, is proposed to constrain the matching degree between the network and the land use.

To solve the model, a neighborhood search algorithm is developed based on the simulated annealing method. A real-size network is calculated with the model and algorithm. The generated network conforms to the development requirement of the city, which shows a good match between the network and the urban characteristics.

Author Contributions: S.C.: Conceptualization, methodology and study design, writing – original draft. Q.L.: Literature review, critical revision and editing. S.Z.: Calculation and formal analysis. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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