Research ArticleTraffic Network Modeling and Extended Max-Pressure TrafficControl Strategy Based on Granular Computing Theory

ShenXue Hao 12 and LiCai Yang 1

1School of Control Science and Engineering Shandong University Jinan 250061 China2School of Information and Engineering Shandong Yingcai University Jinan 250104 China

Correspondence should be addressed to LiCai Yang yanglcsdueducn

Received 3 December 2018 Accepted 27 February 2019 Published 18 March 2019

Academic Editor Roberta Di Pace

Copyright copy 2019 ShenXueHao and LiCai YangThis is an open access article distributed under theCreative Commons AttributionLicensewhichpermits unrestricteduse distribution and reproduction in anymedium provided the original work is properly cited

Reasonable traffic network model and flexible traffic control strategy play important roles in improving the urban traffic controlefficiency Introducing granular computing theory into traffic network modeling and traffic control is a useful attempt sincegranular computing is closer to the human thinking in solving problems In this paper the traffic elements are depicted usingS-rough set to achieve the granulation partition of traffic network Four layers are partitioned in the proposed hierarchicalmultigranularity traffic network model such as vehicle layer platoon layer segment and intersection layer and subregion layerEach traffic granule is represented in rough representation form and the dynamic characteristics are described using the elementarytransfer operations based on S-rough set theory As an application on the proposed traffic networkmodel an extendedmax-pressuretraffic control strategy is applied on the platoon and segment and intersection layer Simulation results illustrate that the proposedtraffic network model and traffic control strategy achieve better performance

1 Introduction

Congestion has become a major problem in urban trafficsystems Reasonable traffic data management and optimaltraffic control strategy play an important role in traffic controlsystem [1]With the development of data collection technolo-gies more and more traffic data can be collected using fordesign the traffic control strategy How tomake full use of themassive traffic data is an essential issue in traffic control andmanagement system including vehicle information platooninformation road segment information and subregion infor-mation Furthermore based on the reasonable traffic dataprocess and management model how to design an optimaltraffic control strategy is the other important problem Theresolutions of the two essential questions can improve theefficiency of urban traffic control system In this paper wetry to build a hierarchical multigranularity traffic networkmodel to depict and manage the traffic elements in urbantraffic system using granular computing theory and thendesign a max-pressure traffic light control strategy based on

the proposed model to improve the efficiency of urban trafficcontrol system

On the one hand the idea of granular computing isconsistent with the human thinking and problem solvingway [2 3] it has attracted more attention of researchersfor intelligent transportation system Granulation involvesdecomposition of whole into parts and complex problem canbe decomposed into several subtasks for efficiency improvingand problems simplification [4] From this view if the urbantraffic network is viewed as multilayer and multigranularmodel using granular computing theory itmay be an effectiveway to solve the traffic control and management problemsAs discussion in [5] traffic information processing andrepresentation are the basis of urban traffic control andmanagement The proper construction of traffic informationgranules and granularity selection enable us to obtain moreintuitive and accurate traffic information In order to achievethe granulation in traffic data processing Seng et al ana-lyzed the basic theory of granular computing and discussedthe applications of granular computing in the intelligent

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 2752763 11 pageshttpsdoiorg10115520192752763

2 Mathematical Problems in Engineering

transportation system In their paper the granulationmethodwas presented based on rough set theory [6] Yao et aldetected the traffic congestion on the city road networkusing granular computing in their paper They constructedthe hierarchical congestion recognition model and optimizedthe granular level and granularity of congestion throughspace conversion [7] Li et al introduced rough set theoryand granular computing theory into road safety indicatoranalysis In their paper the individual road safety indicatorswere combined into a composite indicator to reduce thedimensions of selected risk factors [8] He et al combinedthe social network and community structure to build amodelwith multiple granularities and levels and decomposed thecomplex structure of large network into different granularityspaces to search the fine-grained or coarse-grained shortestpath [9] These proposed granulation ideas and methodsin traffic data processing can really reduce the complexityof problem solving in traffic control and management sys-tem

On the other hand the traffic data collected fromdifferentdevices contains measure errors these data may be incom-plete inaccurate and inconsistent [10 11] Rough set theoryas one important part of granular computing theory [12] candeal with the uncertain features of traffic data well for thetraffic problem solving [13] Researchers have applied roughset theory in intelligent transportation system area such astraffic data processing traffic events recognition and trafficflow prediction since the rough representation of traffic datacan express the actual situations of traffic network improvethe accuracy of traffic information and obtain the hidinginformation In our previous research work the rough settheory was utilized for processing the field traffic data withvagueness to extract car-following behavior decision rulesand construct a data-driven car-following model [11] Xionget al integrated rough set and Bayesian network for roadwaytraffic accident analysis [14] Deshpande et al combinedsupport vector machine and rough set theory to achieve thetraffic flow prediction [15]These applications based on roughset theory show better performance in traffic data processing

Inspired by the ideas of granular computing theory inthis paper we granulate the traffic elements in traffic networkusing singular rough set theory and build a hierarchicalmultigranularity model to represent the urban traffic net-work And based on the traffic network model an extendedmax-pressure traffic control strategy is proposed to improvethe efficiency of urban traffic system The contributionsof this paper lie in two aspects (1) The traffic networkis modeled as a multilayer and multigranularity modelincluding vehicle granule platoon granule and subregiongranule (2) The proposed extended max-pressure trafficcontrol strategy computes the light phase pressure using therough representation of platoon and segment granules whichcan reflect the actual situations of traffic network

The remainder of this paper is organized as follows Theproposed hierarchical multigranularity model is presented inSection 2 The extended max-pressure traffic control strategyis given in Section 3 In Section 4 the simulations are carriedout to analyze the performance of the proposed strategy Thispaper is concluded in Section 5

2 Traffic Network Modeling Based onGranular Computing Theory

On the view of multigranularity the world is made up ofgranules with different size attributes and states In the realworldmany complex systems can be decomposed asmultiplelayers and multiple granules [16] The common granules canbe extracted from the real system to simplify the complexityof problems On this view the urban traffic network is alsoa kind of hierarchical multigranularity model in reality withdifferent types of traffic elements such as vehicle platoonroad segment intersection and subregion

As shown in Figure 1 according to the concepts ofelements in traffic network there are four layers vehicle layerplatoon layer intersection and segments layer and subregionlayer Based on the states of vehicles travelling on a roadsegment they may construct platoons in which vehicles havesimilar speed headway and the platoon state can be com-puted according to these states of vehicles in it Similarly thetraffic state of one road segment can be calculated accordingto the states of platoons travelling on it The road segmentswith the connected intersections having similar traffic statealso can be divided into a subregion in which similar trafficcontrol strategy can be applied From this view the trafficnetwork can be viewed as a hierarchical multigranularitymodel in which the lower level traffic elements compose anupper level traffic element However these traffic elementscannot be represented by the common set concept since theremay be lower level elements that are not in an upper level setcompletely That is some lower level elements maybe belongto an upper level set partly Furthermore the state of trafficelement changes in time the dynamic of element state shouldbe described in suchmodelTherefore in this paper we try tomodel the traffic network as a hierarchical multigranularitymodel using the singular rough set theory in the followingsections

21 S-Rough Set eory According to the discussion in theprevious section there are four layers in the modeled trafficnetwork such as vehicle layer platoon layer segment andintersection layer and subregion layer Based on the roughset theory proposed by Professor Z Pawlak in 1982 [17] thesetraffic elements can be represented using the upper and lowerapproximation set The basic concepts of rough set theory areintroduced as follows

To the information system there is a nonempty finiteset of objects called universe U For 119883 sube 119880 let R be anequivalence relation and [119909]119877 denote the equivalence classof R defined by x Then the approximation set of X is definedas

119877 (119883) = cup 119909 | 119909 isin 119880 [119909]119877 sube 119883 (1)

119877 (119883) = cup 119909 | 119909 isin 119880 [119909]119877 cap 119883 = (2)

As shown in Figure 2 the area surrounded by black dottedline is the crisp set X the green blocks area is the lowerapproximation set of X and the grey together with greenblocks area is the upper approximation set of X The upperand lower approximation sets of X can describe the rough

Mathematical Problems in Engineering 3

PlatoonVehicle driving into a platoon

Vehicle driving out of a platoon

(a) Vehicle dynamic on platoon layer

IntersectionPlatoon

Upstream segment

Platoon

Downstream segmentPlatoon driving into downstream segment

(b) Platoon dynamic on segment layer

Sub-region

Boundary of sub-region

Segment-intersection is transferred into sub-region due to the state changing

Segment-intersection is transferred out of sub-region

due to the state changing

(c) Segment-intersection dynamic on subregion layer

Figure 1 The dynamic of traffic elements in traffic network

U

R(X)

Boundary (X) = R (X) minus R(X)

Figure 2The rough representation of set X

representation of the crisp set X Boundary(X) is the bound-ary ofX computed using the difference of the upper and lowerapproximation set This rough representation is consistentwith the realities in world that objects maybe belong to a setpartly This kind of description can express the real worldaccurately

However the rough set defined above cannot describethe dynamic of X That is when the element state has been

U

f (u )

x1u1

u2

x2

1

f (u2)

(x1)

(x2)f

f

Figure 3The dynamic characteristics of S-rough set

changed the element may be transferred into or out of X Asshown in Figure 3 for u1notin X u2notin X 1199091 isin 119883 1199092 isin 119883 thereare some element operations f (u) and 1198911015840(119909) which make u1

become f (u1) with f (u1) belonging to X completely make u2

become f (u2) with f (u2) entering X partly make x1 become1198911015840(1199091) with 1198911015840(1199091) not belonging to X completely and makex2 become 1198911015840(1199092) with 1198911015840(1199092) leaving X partly This type ofset X with dynamic elements change cannot be describedaccurately using rough set theory Therefore the singularrough set theory (simplified as S-rough set) proposed byProfessor Shi is introduced to solve this problem [18 19] in

4 Mathematical Problems in Engineering

which the dynamic of set X is described using the elementarytransfer operations

Let 119865 cup 1198651015840 be the elementary transfer on U there are119865 = 1198911 1198912 119891119898 and1198651015840 = 11989110158401 1198911015840

2 1198911015840119898The effect of119891 isin 119865 is defined as follows for element uisinU and u notin X there

exists f (u)=x xisinX That is f makes X occur with singularityThe existence of f makes the boundary of X extend outwardsSimilarly the effect of 1198911015840 isin 1198651015840 is defined as follows forelement xisinX there exists 1198911015840(119909) = 119906 119906 notin 119883 1198911015840 makes theboundary of X shrink inwards The existence of both f and1198911015840 makes X occur with singularity in two directions and Xbecomes 119883lowast that is X has two-direction dynamic characterLetX119891 denote the extension of X1198831198911015840 denote the shrink of Xand the two-direction singular set119883lowast can be defined formallyas follows

119883119891 = 119906 | 119906 isin 119880 119906 notin 119883119891 (119906) = 119909 isin 119883 (3)

1198831198911015840 = 119909 | 119909 isin 1198831198911015840 (119909) = 119906 notin 119883 (4)

119883lowast = 119883 1198831198911015840 cup 119883119891 (5)

where f (u) is transferred into X and 1198911015840(119906) is transferred outof X completely According to the rough set definition ie(1)-(2) the approximation of119883lowast can be described as

(119877 119865 cup 1198651015840)∘(119883lowast)

= cup 119909 | [119891 (119909)]119877 sube 119883lowast cup [119909]119877 sube 119883lowast (6)

(119877 119865 cup 1198651015840)∘ (119883lowast)= cup 119909 | [119891 (119909)]119877 cap 119883lowast = and [119909]119877 cap 119883lowast = (7)

Therefore the existence of 119865 cup 1198651015840 makes X become 119883lowastthe dynamic characteristics of X with element transfer-in ortransfer-out completely can be described using (6)-(7) Inreality there exist the cases of elements transfer partly That isforuisinUunotinX there is f (u) =x and x belongs toX partly (ieuis transferred into X partly) or for xisinX there is 1198911015840(119909)=u andu belongs to X partly (ie x is transferred out of X partly)For this case the upper assistant set As+(119883lowast) and the lowerassistant set Asminus(119883lowast) are defined as (8)-(9) where As+(119883lowast) ismade up of the elements transferred into X partly Asminus(119883lowast)is made up of the elements transferred out of X partly [20]

119860119904+ (119883lowast)= 119909 | 119906 isin 119880 119906 notin 119883119891 (119906) = 119909 119909 enters 119883 partly (8)

119860119904minus (119883lowast)= 119909 | 119909 isin 1198831198911015840 (119909) = 119906 119906 leaves 119883 partly (9)

To sum up the dynamic and the rough representation of Xcan be described using (6)-(9) According to the definitionsof S-rough set the elements in traffic network can be defined

in similar ways In the next section these traffic elements ineach layer are defined respectively

22 Traffic Granules Definition Based on S-Rough Set Basedon the discussion in previous sections the traffic elementssuch as vehicle platoon intersection and segments andsubregion can be represented using S-rough set theirdynamic characteristics and rough representation can also bedepicted

As shown in Figure 4 there are four layers in the hierar-chical multigranularity traffic network model Vehicle is thesmallest granule in this model Vehicle granules with similarstates form a platoon granule There may be several platoongranules on a road segment Segments with the connectedintersections with similar traffic states construct a subregionin which the intersection and segment is considered as abasic granule The whole traffic network may be partitionedinto several subregions On the same layer the lower levelgranules may be transferred from one upper level granule toanother For example as shown in Figure 4 vehicle granulesin platoon 1 may be transferred into platoon 2 when its stateschanged Similarly one intersection and segment granulemay be transferred into or out of a subregion granule dueto the changing traffic state of road segment Furthermorethere exists the partly elementary transfer case the vehicle atthe head of a platoon will drive out from the platoon whenit accelerates and the vehicle at the end of a platoon willdrive out from the platoon when it decelerates In this casethe vehicle belongs to the platoon in position view whileits acceleration is higher than the other vehicles it can besaid that the vehicle granule belongs to the platoon partlyThere are the similar cases to the other granules This isconsistent with S-rough set definition Therefore using S-rough set to represent the granules in traffic network candescribe the traffic network dynamic more accurately Inthe following context the singular rough representations ofthese traffic elements are listed except for the smallest vehiclegranule

(1) Platoon granule Let these vehicles travelling on thesame segment be the universe U119881 R119875 is the equivalencerelation about the attributes of speed acceleration andposition on U119881 Plt denotes a platoon 119875119897119905lowast denotes the two-direction singular set of Plt 119865119881 cup 1198651015840

119881 is the elementarytransfer and there are F119881=1198911198811 1198911198812 119891119881119898 and 1198651015840

119881 =1198911015840

1198811 11989110158401198812 1198911015840

119881119898 where f119881(veh) denotes that granuleveh is transferred into Plt due to the changing of speedposition and acceleration 119891119881

1015840(veh) denotes that granule vehis transferred out of Plt due to the changing of speed positionand acceleration Then the singular rough set of a platoon isdefined as

119875119897119905lowast = 119875119897119905 V119890ℎ | V119890ℎ isin 119875119897119905 1198911015840119881 (V119890ℎ) notin 119875119897119905

cup V119890ℎ | V119890ℎ isin 119880119881 V119890ℎ notin 119875119897119905 119891119881 (V119890ℎ) isin 119875119897119905 (10)

(119877119875 119865119881 cup 1198651015840119881)∘ (119875119897119905lowast) = cup V119890ℎ | [119891119881 (V119890ℎ)]119877119875 sube 119875119897119905lowast

cup [V119890ℎ]119877119875 sube 119875119897119905lowast (11)

Mathematical Problems in Engineering 5

Traffic NetworkSub-Region

Intersection amp Segments

Intersection amp Segments

helliphellip

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

helliphellip

Sub-Region

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

helliphellip

Element transfer at platoon level

Element transfer at segment level

Element transfer at region level

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

helliphellip

helliphellip

Figure 4 Hierarchical multigranularity traffic network model

(119877119875 119865119881 cup 1198651015840119881)∘ (119875119897119905lowast) = cup V119890ℎ | [119891119881 (V119890ℎ)]119877119875 cap 119875119897119905lowast

= and [V119890ℎ]119877119875 cap 119875119897119905lowast = (12)

119860119904+ (119875119897119905lowast) = 119891119881 (V119890ℎ) | V119890ℎ isin 119880119881 V119890ℎnotin 119875119897119905 119891119881 (V119890ℎ) enters 119875119897119905 partly (13)

119860119904minus (119875119897119905lowast) = V119890ℎ | V119890ℎisin 119875119897119905 1198911015840

119881 (V119890ℎ) leaves 119875119897119905 partly (14)

(2) Segment granule Let these platoons travelling on the samesegment be the universe U119875 R119878119890119892 denotes the equivalencerelation about the attributes of average speed and positionon U119875 Seg denotes the segment with multiple platoonsand 119878119890119892lowast denotes the two-direction singular set of SegF119875cup1198651015840

119875 is the elementary transfer on U119875 where 119865119875 =1198911198751 1198911198752 119891119875119898 and 1198651015840

119875 = 11989110158401198751 1198911015840

1198752 1198911015840119875119898 119891119875(119901119897119905)

denotes that platoon plt is driving into Seg 1198911198751015840(119901119897119905) denotes

that platoon plt is driving out of SegThen the singular roughset representation of segment is defined as

119878119890119892lowast = 119878119890119892 119901119897119905 | 119901119897119905 isin 119878119890119892 1198911015840119875 (119901119897119905) notin 119878119890119892

cup 119901119897119905 | 119901119897119905 isin 119880119875 119901119897119905 notin 119878119890119892 119891119875 (119901119897119905) isin 119878119890119892 (15)

(119877119878119890119892 119865119875 cup 1198651015840119875)∘ (119878119890119892lowast) = cup 119901119897119905 | [119891119875 (119901119897119905)]119877119878119890119892

cup [119901119897119905]119877119878119890119892 sube 119878119890119892lowast(16)

(119877119878119890119892 119865119875 cup 1198651015840119875)∘ (119878119890119892lowast) = cup 119901119897119905 | [119891119875 (119901119897119905)]119877119878119890119892

cap 119878119890119892lowast = and [119901119897119905]119877119878119890119892 cap 119878119890119892lowast = (17)

119860119904+ (119878119890119892lowast) = 119891119875 (119901119897119905) | 119901119897119905 isin 119880119875 119901119897119905notin 119878119890119892 119891119875 (119901119897119905) enters 119878119890119892 partly (18)

119860119904minus (119878119890119892lowast) = 119901119897119905 | 119901119897119905isin 119878 1198911015840

119875 (119901119897119905) leaves 119878119890119892 partly (19)

Actually for each road segment of traffic network theremay be several lanes corresponding to several vehicle flowscontrolled by different traffic light phases of intersection Tofacilitate the control strategy design the segment granule ispartitioned into subgranules corresponding to different lanesFor example the granule seg119886119887 denotes the platoon rough seton segment a where these platoons are leaving for segment bThe rough set definition of seg119886119887 can also be described using(15)-(19) with attribute lane id added In this case the stateseg119886 can be calculated by all the granules seg119886119887

(3) Subregion granule Since there are both segmentsand intersections in the subregion granule and the states ofsegments with the connected intersections affect the trafficcontrol strategy there are correlations between upstream anddownstream segments the ldquoUpstreamSegment-Intersection-DownstreamSegmentrdquo (simplified as uid) is viewed as thebasic granule in subregion granule definition These uidscompose the universe U119878 Let SR denote a subregion in

6 Mathematical Problems in Engineering

which the segments of uids have similar traffic state andthere are correlations between the upstream and downstreamsegment of each uid Since the uid is made up of upstreamsegment useg downstream segment dseg and the connectedintersection intersc if the useg and dseg are both transferredinto or out of SR it is to say that the uid is transferred into orout of SR completely If only useg or dseg is transferred intoor out of SR it is to say that the uid is transferred into or outof SR partly Let 119878119877lowast denote the two-direction singular setof SR and 119865119878 cup 1198651015840

119878 is the elementary transfer on U119878 where119865119880 = 1198911198801 1198911198802 119891119880119898 1198651015840119880 = 1198911015840

1198801 11989110158401198802 1198911015840

119880119898f119880(uid) indicates that the basic granule uid is transferred intoSR due to the state changing 1198911015840

119880(uid) indicates that uid istransferred out of SR due to the state changing The roughrepresentation of 119878119877lowast can be defined as

119878119877lowast = 119878119877 119906119894119889 | 119906119894119889 isin 119878119877 1198911015840119880 (119906119894119889) notin 119878119877

cup 119906119894119889 | 119906119894119889 isin 119880119880 119906119894119889 notin 119878119877 119891119880 (119906119894119889) isin 119878119877 (20)

(119877119878119877 119865119880 cup 1198651015840119880)∘ (119878119877lowast) = cup 119906119894119889 | [119891119880 (119906119894119889)]119877119878119877

cup [119906119894119889]119877119878119877 sube 119878119877lowast (21)

(119877119878119877 119865119880 cup 1198651015840119880)∘ (119878119877lowast) = cup 119906119894119889 | [119891119880 (119906119894119889)]119877119878119877

cap 119878119877lowast = and [119906119894119889]119877119878119877 cap 119878119877lowast = (22)

119860119904+ (119878119877lowast) = 119891119880 (119906119894119889) | 119906119894119889 isin 119880119878119877 119906119894119889notin 119878119877 119891119880 (119906119894119889) enters 119878119877 partly (23)

119860119904minus (119878119877lowast) = 119906119894119889 | 119906119894119889isin 119878119877 1198911015840

119880 (119906119894119889) leaves 119878119877 partly (24)

23Maintenance and Construction of Traffic Granules Basedon the definitions of the hierarchical multigranularity trafficnetwork model traffic control strategy can be designedaiming at different traffic demand to improve the efficiency ofurban traffic control systemThe key issue that this model canbe utilized for traffic control and management is to maintainthe integrity of these granules in real time For the traffic dataprocessing efficiently the distributed traffic datamanagementframework has been discussed in our previous research work[21] In this section it is assumed that the basic vehicle stateinformation can be collected by the communication deviceequipped in vehicles and managed by the distributed trafficdata centers

For the granules in traffic network model U119881 U119875and U119878 are the vehicle universe platoon universe andsegment universe respectively The smallest granule in thismodel is vehicle granule the related attributes are listedas veh id veh speed veh acc veh pos segment id lane idpreceding id following id pre dist back dist veh class pas-sengers veh weight etc These attributes represent vehicle idspeed acceleration position road segment id that vehicletravels on lane id that vehicle travels on the distance to theprevious vehicle the distance to the vehicle in back vehicle

class the number of passengers in vehicle and vehicle weightin mergence respectively The attributes such as veh classpassengers and veh weight can be utilized for light phasepressure computation in the traffic light control strategydesign Inwhat follows the constructions of threemain trafficgranules are listed in detail

(1) Maintenance and construction of platoon granule Theequivalence relation R119875 on U119881 is used for constructinga platoon on each road segment it is defined as followsvehicles with neighboring position and same segment idsame lane id similar speed and acceleration The elementarytransfer is defined as follows For veh notin p ifMinspeed119901119897119886119905119900119900119899 leveh speed le Maxspeed119901119897119886119905119900119900119899 |V119890ℎ 119886119888119888| le acc119905ℎ119903119890119904ℎ119900119897119889 pre distle dist119905ℎ119903119890119904ℎ119900119897119889 or back dist le dist119905ℎ119903119890119904ℎ119900119897119889 then veh is trans-ferred into platoon p completely if (pre dist gt dist119905ℎ119903119890119904ℎ119900119897119889 andveh accgt acc119905ℎ119903119890119904ℎ119900119897119889) or (back dist gt dist119905ℎ119903119890119904ℎ119900119897119889 and veh accltacc119905ℎ119903119890119904ℎ119900119897119889) then veh is driving into platoon p since veh isaccelerating from the back of p or decelerating in front of pafter a period of time veh will drive into platoon p In thiscase it is said that veh is transferred into p partly For visinpif there is |V119890ℎ 119886119888119888| gt acc119905ℎ119903119890119904ℎ119900119897119889 veh in p is accelerating ordecelerating after a period of time veh will leave platoon pIn this case it is said that veh is transferred out of p partly Ifveh is the head or rear of p when back dist gt dist119905ℎ119903119890119904ℎ119900119897119889 orpre dist gt dist119905ℎ119903119890119904ℎ119900119897119889 it is said that veh has been transferredout of p completely

(2) Maintenance and construction of road segment Forthe universe 119880119875 the basic granule is platoon the relatedattributes including platoon id ave speed max speed minspeed ave headwaymax headwaymin headway veh countseg id lane id platoon weight rightofway flag etc Theseattributes represent the vehicle id average speed maximumspeed of platoon minimum speed of platoon average head-way maximum headway minimum headway number ofvehicles in platoon segment id that the platoon is travellingon lane id the weight of platoon and whether the platoonobtains the right of way to pass the downstream intersectionrespectively The equivalence relation R119878 on U119875 is definedas follows platoons with same segment id and lane id Theelementary transfer of platoon granule on segment granuleis defined as follows For 119901 isin 119904 if rightofway flag is true andveh count lt veh count119905ℎ119903119890119904ℎ119900119897119889 it is said that platoon119901 canpassthrough the downstream intersection in green time then 119901 istransferred out of segment 119904 completely if rightofway flag istrue and veh count gt veh count119905ℎ119903119890119904ℎ119900119897119889 it is said that part ofplatoon 119901 will pass through the downstream intersection ingreen time then 119901 is transferred out of segment 119904 partly For119901 notin 119904 there are similar situations

(3) Maintenance and construction of subregion granuleFor constructing the subregion granule uid is the basicgranule which constitutes the universe U119880 The relatedattributes include uid id upseg state downseg state andcorrelation degree representing uid id state of upstream seg-ment state of downstream segment and correlation degreeof the upstream and downstream segments respectively Thestate of segment can be extracted from platoon granules on itusing the attributes seg id veh count ave speed queuelengthand seg cap where seg cap is the capacity of a road segmentThe correlation degree can be calculated according to the state

Mathematical Problems in Engineering 7

qab

a bi

qghfgh

fab

gh

d

d

qgd

qgd

rbc

rbc

qbc

qbc

c

c

Figure 5 Vehicle flows on the upstream and downstream segments

and capacity of upstreamanddownstream segments [22]Theequivalence relation R119878119877 onU119878 is defined as follows segmentshave similar state and the correlation degree is more than agiven thresholdThe elementary transfer is defined as followsFor uid isinSR if the number of vehicles and the average speedof upstream and downstream segments are both exceedingthe given range of subregion SR that is the upstream anddownstream segment states of uid are not close to the stateof SR it is said that uid is transferred out of SR completely Ifonly the downstream or upstream segment state is not closeto the state of SR it is said that uid is transferred out ofSR partly For uid notin SR if uid is adjacent with SR and theupstream and downstream segment states of uid are similarto the state of SR it is said that uid is transferred into SRcompletely If only upstream or downstream segment state issimilar to the SR state it is said that uid is transferred into SRpartly

Based on the construction and maintenance of granulesin the proposed traffic network model discussed above thesetraffic granules can be maintained in a distributed modeunder a distributed traffic data management frameworkThen the traffic information of different layers and granular-ities can be extracted Based on the multigranularity trafficinformation including vehicle information platoon infor-mation segment information and subregion informationthe traffic control and management system may adopt moresuitable strategies to improve the efficiency However stillsome issues have not been presented in detail in this proposedmodel such as the parameters selection of traffic statetraffic state computation of road segment and the correlationdegree calculation between downstream and upstream roadsegmentsThese problems will be researched in further workIn this paper to illustrate the validity and feasibility ofthe proposed hierarchical multigranularity traffic networkmodel we propose a max-pressure based traffic controlstrategy in the next section

3 Traffic Control Strategy Based onthe Proposed Traffic Model

In this section an application based on the proposed trafficnetwork model is discussed The proposed traffic networkmodel is combined with the max-pressure traffic controlalgorithm to design an extended max-pressure traffic controlstrategy in which the light phase pressure is computed usingthe rough representations of platoon and segment granules

31 Max-Pressure Traffic Control Strategy The max-pressuretraffic control strategy is proposed by Varaiya [23] based on

the backpressure traffic control algorithm [24 25] Accordingto the strategy the urban traffic network is viewed as a queu-ing network and at each time slot the active phase of eachintersection is selected depending on the light phase pressurecomputed using local queue length of each connected roadsegment

As shown in Figure 5 on segments a and g there arequeued vehicles waiting for passing through intersection ithe vehicle flows f 119886119887 and f 119892ℎ are controlled by light phase pof intersection i For the vehicle flow f 119886119887 when it enters thedownstream segment b it will drive into the queued vehicles1199021198871198881015840 and 11990211988711988810158401015840 according to the proportions 1199031198871198881015840 and 11990311988711988810158401015840 where1198881015840 and 11988810158401015840 are the downstream segments of bThere are similarsituations for the vehicle flow 119891119892ℎ The pressure of light phasep of intersection i is computed using (25) and the active lightphase in next time slot is selected using (26)

119901119903119890119904119904119906119903119890119901 = sum119891119886119887isin119901

120583119886119887 (119901) (119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)) (25)

where119891119886119887 isin 119901means that the vehicle flow f 119886119887 is controlled bylight phase p120583119886119887(p) is the number of vehicles passing throughintersection per unit time when p is activated

119901lowast = argmax119901isin119875119894

(119901119903119890119904119904119906119903119890119901 (119905)) (26)

where pisinP119894 means that p is one of the light phases of lightsignal group 119875119894

In this method the light phase pressure is concernedwith the number of queued vehicles in front of the stop lineon the upstream and downstream segments The detailedvehicle information the arriving vehicles (for the upstreamsegment) and the departing vehicles (for the downstreamsegment) are not considered Based on the proposed trafficnetwork model there may be several platoons with differentstates on the upstream and downstream segments On theupstream segment there are stopped platoons in front ofthe stop line and arriving platoons heading for the stop lineOn the downstream segment there may be the departingplatoon when the corresponding downstream light phaseis activated These situations are considered in the trafficnetwork rough representation discussed in previous sectionsTherefore in the next section the light phase pressurecomputation method is extended based on the proposedhierarchical multigranularity traffic network model

32 Extended Max-Pressure Traffic Control Algorithm Actu-ally the light phase pressure is affected by four types of

8 Mathematical Problems in Engineering

platoons on the segment For the platoons on upstreamsegment the stopped platoons waiting for passing throughthe intersection the platoons that are driving towards thestop line and the platoons on the neighboring lanes that aregoing to drive into the corresponding lanes will affect thelight phase pressure For the platoons on downstream seg-ment the departing platoon affects the light phase pressurecomputation too if the downstream light phase is activatedSince the rough representation of traffic granules includesdetailed traffic information the vehicle weight the numberof passengers vehicle class etc also can be considered Thenbased on the rough representation of platoon and segment inthe proposed traffic network model the light phase pressurecan be computed more accurately

To simplify the description the rough set representationof 119883lowast with elementary transfer in two directions is denotedby the lower approximation set appr (119883lowast 119865 cup 1198651015840) and theupper approximation set 119886119901119901119903minus(119883lowast 119865 cup 1198651015840) appr (119883lowast F)represents the one-direction lower approximation set withelement transfer-in operations and appr (119883lowast 1198651015840) representsthe one direction lower approximation set with elementtransfer-out operations The upper assistant set As+(119883lowast F)represents the element set that elements enter 119883lowast partlyunder transfer operations F The lower assistant set Asminus(119883lowast1198651015840) represents the element set that elements leave 119883lowast partlyunder transfer operations 1198651015840 Then the pressure from eachplatoon can be computed using

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) = sum

V119890ℎisin119886119901119901119903 (119875lowast119886119887119865119881)

120572119908V119890ℎ

+ sumV119890ℎisin119860119904+(119875lowast

119886119887119865119881)

120573119908V119890ℎ

(27)

where 120572 120573 is the coefficient of the vehicle weight and 119908V119890ℎ isthe vehicle weight determined by the vehicle type it meansthat the different types of vehicles generate different pressuredue to the different number of passengers and emergence ofthese vehicles

The pressure of the upstream segment is computed using

119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

= sum119901119897119905isin119886119901119901119903 (119904119890119892lowast

119886119887119865119875)cup119860119904+(119904119890119892lowast

119886119887119865119875)

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) (28)

where useg represents an upstream segment and seg119886119887 is asubgranule on segment a in which platoons on segment aare waiting for driving into segment b

The pressure of the downstream segment is computedusing

119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887 119905)

= sum119888isin1198881015840 11988810158401015840

119901119897119905isin119886119901119901119903 (119904119890119892lowast1198871198881198651015840119875)cup119860119904minus(119904119890119892lowast119887119888119865

1015840119875)

119903119887119888119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119887119888 119905) (29)

According to (28) (29) the vehicles entering upstreamsegment and departing from downstream segment are con-sidered for the coordination with neighboring intersections

The pressure of light phase 119901119894119895 is computed using

119901119903119890119904119904119906119903119890119901ℎ119886119904119890 (119901119894119895 119905)

= sum119891119886119887isin119901

119894119895

119891119887119888isin119901119894+1119896

(119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

minus 119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887119888 119905)) 120583119886119887 (119905)

(30)

where 119901119894119895 is the jth light phase of intersection i 119901119894+1

119896 is thecooperative light phase of the downstream intersection i+1

Based on the extended pressure computation methoddescribed above the vehicle type vehicle state and platoonstate upstream light phase state and downstream light phasestate are considered If the element transfer operations Fand 1198651015840 are both empty the extended pressure computationmethod degenerates into max-pressure traffic control algo-rithm If 119865119875 = Φ and 1198651015840

119875 = Φ the arriving vehiclesfrom upstream segment are considered in the pressurecomputation in this case the coordination with the upstreamintersection light phase can be achieved If 119865119875 = Φ and1198651015840

119875 = Φ the departing vehicles on the downstream segmentare considered to achieve the coordination with downstreamintersection light phaseThe light phase with max-pressure isselected to be activated in the next time slot using (26)

Furthermore the vehicle class the number of passengersand vehicle emergency also can be considered using thevehicle weight Therefore based on the extended light phasecomputation method the real situations of traffic controlsystem can be reflected The strategy may be more suitablefor the actual situation of urban traffic network To illustratethe effectiveness and feasibility of the proposed model andalgorithm simulations are carried out in the next section

4 Simulations

In this section the simulation is carried out to analyze theeffectiveness and feasibility of the proposed method Thesimulation traffic network consists of 15 intersections and 76links constructed in Vissim The network includes 16 ingresssegments and 16 exit segments as shown in Figure 6(a)

The vehicle input of each ingress link is 800 vehh Thereare 15 signal controllers and 4 light phases for each controllerie north-south straight phase north-south left-turn phasewest-east straight phase and west-east left-turn phase Thereare 3 lanes on each link The left-turn vehicle flow drives onLane 3 and the straight vehicle flow drives on Lane 2The twovehicle flows are controlled by the traffic lightThe right-turnvehicle flow drives on Lane 1 and is free of the traffic lightTo obtain the traffic parameters during simulation 60 queuecounters 60 travel time sections and 60 routing decisions arelaid on the simulation traffic network as shown in Figures6(b) 6(c) and 6(d) During the simulation the green lighttime of each light phase is assumed to be 21 s which is thepedestrian clearance time (the road width is about 21 m andthe pedestrians speed is about 1 ms) [26] The yellow lighttime is assumed to be 3 s

Mathematical Problems in Engineering 9

(a)

(b) (c) (d)

Figure 6 The simulation traffic network and settings

Table 1 Traffic network performance comparison

Parameters FTMethod MPMethod EMP MethodTotal travel time (h) 633947 588139 568009Total delay time (h) 350255 298110 277457Total stopped delay (h) 253507 225692 205465Average delay time (s) 130584 111420 104057Average stopped delay (s) 94514 84353 77057Average speed (kmh) 23527 25916 26883Number of vehicles in network 448 368 334Number of vehicles that left network 9208 9264 9265Total Number of vehicles 9656 9632 9599

To illustrate the efficiency and feasibility of the pro-posed traffic network model and traffic control algorithmthree traffic light control methods are implemented in thissection including fixed-time control (FT method) max-pressure control (MP method) and the extended max-pressure control (EMP method) To simplify the simulationthe extended pressure computation only considers the caseof 119865119875 = Φ and 1198651015840

119875 = Φ in this paper the other cases will beconsidered in further research works The three traffic lightcontrol methods are implemented in Visual Studio 2010Thesimulation programs communicate with Vissim through theVissim COM programming interface to obtain the vehiclestate and traffic parameters to decide the traffic light signalat each time slot The simulation runs for 3600 s and thetraffic network performance is evaluated from 1000 s to 3600s usingVissimThis is because in the first 1000 s of simulationtime there are not enough vehicles entering the simulationtraffic network

The simulation results of the traffic network are listed inTable 1 According to the simulation results EMP methodbased on the proposed hierarchical multigranularity trafficnetwork model achieves better performance because the

light phase pressure computed using (27)-(30) is closer tothe actual traffic situation in which the vehicles on a roadsegment are represented using rough set form The depart-ing vehicles of downstream segments considered in thissimulation can achieve the coordination with downstreamintersections

As shown in Figure 7 the average queue length underEMP method is shorter than the other traffic light controlmethods The shorter queue length in traffic network meansit can effectively reduce the occurrence of spill-over ondownstream road segment so as to avoid the traffic collapsedue to vehicle queue overflow EMP method is based onthe proposed hierarchical multigranularity traffic networkmodel in which the traffic elements dynamic characteris-tics are depicted accurately and the light phase states ofdownstream intersections are also considered to computethe light phase pressure therefore EMP method achievesbetter performance As shown in Figure 8 the averagetravel time under EMP control is lower than the other twomethods

Furthermore using the extended pressure computationmethod based on the proposed traffic network the priority

10 Mathematical Problems in Engineering

MPFTEMP

05

101520253035404550

Que

ue L

engt

h (v

eh)

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000Time (s)

Figure 7 Average queue length of the three-traffic-light controlmethod

MPFTEMP

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000

Time (s)

0102030405060708090

100

Trav

el Ti

me (

s)

Figure 8 Average travel time of the three-traffic-light controlmethod

control for some particular vehicles can be achieved con-veniently since different classes of vehicles can be set withdifferent vehicle weight in the EMPmethod Different weightvalues can result in different traffic efficiency for differenttypes of vehicles For example when the type of bus is addedto the simulation traffic network the higher weight valuecan make buses have higher average speed and the otherclasses of vehicles maybe achieve lower average speed Forthis issue the vehicle weight selection is important As shownin Figure 9 the different bus weight values are simulatedand they result in different average speed According to themultiple simulation results and the real traffic demand thevehicle weight can be determined

To sum up since the hierarchical multigranularity trafficmodel can represent the actual traffic network situationsaccurately and the EMP method can achieve traffic controlflexibly this paper may be a useful attempt in solving urbantraffic control problem

5 Conclusions

In this paper a hierarchical multigranularity traffic networkis proposed and on the basis of the proposed traffic networkan extended max-pressure traffic control method is imple-mented The simulation results illustrate that the proposedtraffic network model and EMP method achieve better

Average speed [Car]Average speed [Bus]Average speed [All vehicles]

24245

25255

26265

27

Aver

age s

peed

15 2 25 3 35 4 45 5 551Vehicle weight of Bus

Figure 9 Simulation results with different bus weight values

performance In the proposed model the traffic elementsin urban traffic network are represented using S-rough setin which the dynamic characteristics of traffic elementsare depicted This type of traffic element description canrepresent the traffic network situation accurately so as todesign the more efficient traffic control strategy

However the hierarchical multigranularity traffic net-work model is only a preliminary exploration and it needsto be further improved in the future research work such asthe traffic subregion formulation and the boundary controlof traffic subregion For the traffic subregion formulationthe correlation of downstream and upstream road segmentsshould be computed considering traffic states and the geo-graphical characteristics of each pair of upstream and down-stream segments The control strategy of subregion boundarycan compute the traffic pressure considering the remainingcapacity of downstream segment to speed up the vehicleflow release towards to the segment with larger remainingcapacity These issues will be studied in our further researchwork to improve the hierarchical multigranularity traffic net-workmodel Furthermore in the future work the completelymultigranularity traffic control strategies should be furtherresearched using granular computing theory to achieve thetraffic system multigranularity control completely

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial NaturalScience Foundation China [grant number ZR2018FM027]Project of Shandong Province Higher Educational Scienceand Technology Program [grant number J17KB181] KeyResearch and Development Project in Shandong ProvinceChina [grant numbers 2015GGX101048 2016GSF120009]

Mathematical Problems in Engineering 11

and National Natural Science Foundation of Youth FundProject [grant number 61501284]

References

[1] Q He K L Head and J Ding ldquoMulti-modal traffic signalcontrol with priority signal actuation and coordinationrdquoTrans-portation Research Part C Emerging Technologies vol 46 pp65ndash82 2014

[2] Y Yao ldquoInterpreting concept learning in cognitive informaticsand granular computingrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 39 no 4 pp 855ndash8662009

[3] J T Yao A V Vasilakos andW Pedrycz ldquoGranular computingperspectives and challengesrdquo IEEE Transactions on Cyberneticsvol 43 no 6 pp 1977ndash1989 2013

[4] H Liu andM Cocea ldquoGranular computing-based approach forclassification towards reduction of bias in ensemble learningrdquoGranular Computing vol 2 no 3 pp 131ndash139 2017

[5] X Ji L Liu and Q Wu ldquoTraffic management and decision-making support based on granular computingrdquo Journal ofHighway and Transportation Research and Development vol2008 no 10 pp 129ndash133 2008

[6] S Dewen C Xinhong C Jing and F Xujian ldquoGranularcomputing in intelligent transportation An exploratory studyrdquoCybernetics and Information Technologies vol 15 no 6 pp 122ndash134 2015

[7] C Yao J-C Gu and Q Yang ldquoTraffic congestion identificationand analysis of urban road network based on granular comput-ingrdquo Applied Mechanics and Materials vol 641-642 pp 916ndash922 2014

[8] T Li D Ruan Y Shen E Hermans and G Wets ldquoA newweighting approach based on rough set theory and granularcomputing for road safety indicator analysisrdquo InternationalJournal of Computational Intelligence vol 32 no 4 pp 517ndash5342016

[9] F He R Liu Y Zhang et al ldquoBGrR large-scale network routingspeedup techniques based on granular computingrdquo ComputerScience vol 41 no 11 pp 265ndash281 2014

[10] R Deb and A W-C Liew ldquoMissing value imputation forthe analysis of incomplete traffic accident datardquo InformationSciences vol 339 pp 274ndash289 2016

[11] S Hao L Yang and Y Shi ldquoData-driven car-following modelbased on rough set theoryrdquo IET Intelligent Transport Systemsvol 12 no 1 pp 49ndash57 2018

[12] L Livi and A Sadeghian ldquoGranular computing computationalintelligence and the analysis of non-geometric input spacesrdquoGranular Computing vol 1 no 1 pp 13ndash20 2016

[13] X Yang H Da-Wei S Bing and Z Duo-Jia ldquoCity traffic flowbreakdown prediction based on fuzzy rough setrdquo Open Physicsvol 15 no 1 pp 292ndash299 2017

[14] X Xiong L Chen and J Liang ldquoAnalysis of roadway trafficaccidents based on rough sets and Bayesian networksrdquo Promet- Traffic - Traffico vol 30 no 1 pp 71ndash81 2018

[15] M Deshpande and P Bajaj ldquoPerformance improvement of traf-fic flow prediction model using combination of support vectormachine and rough setrdquo International Journal of ComputerApplications vol 163 no 2 pp 31ndash35 2017

[16] Y Y Yao ldquoGranular computing basic issues and possible solu-tionsrdquo in Proceedings of the Fi13h International Conference onComputing and Information vol 1 pp 186ndash189 2000

[17] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Science vol 11 no 5 pp 341ndash356 1982

[18] K Shi ldquoS-rough sets and its applications in diagnosis-rec-ognition for diseaserdquo in Proceedings of the 1st InternationalConference on Machine Learning and Cybernetics pp 50ndash54IEEE Beijing China 2002

[19] K Q Shi and Y Q Cui ldquoS-Rough sets and their general struc-turesrdquo Journal of Shandong University Natural Science Shan-dong Daxue Xuebao Lixue Ban vol 37 no 6 pp 471ndash474 2002

[20] H Haiqing X Peijun and F Haiyan ldquoS-rough recognitionof knowledge and general threshold encryption authenticationscheme of recognition conclusionrdquo in Proceedings of the SixthInternational Conference on Intelligent Systems Design andApplications pp 816ndash824 Jian China October 2006

[21] S Hao L Yang and Y Shi ldquoReal-time data-driven trajectoryreconstruction based on rough set theoryrdquo Advances in Trans-portation Studies vol 44 pp 119ndash132 2018

[22] Y Yang L Jia Y Qin S Han and H Dong ldquoUnderstandingstructure of urban traffic network based on spatial-temporalcorrelation analysisrdquo Modern Physics Letters B CondensedMatter Physics Statistical Physics AtomicMolecular andOpticalPhysics vol 31 no 22 Article ID 1750230 19 pages 2017

[23] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[24] TWongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 USA September 2012

[25] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo IFAC Proceedings Volumes vol 47 no 3 pp11332ndash11337 2014

[26] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

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Mathematical Problems in Engineering 3

PlatoonVehicle driving into a platoon

Vehicle driving out of a platoon

(a) Vehicle dynamic on platoon layer

IntersectionPlatoon

Upstream segment

Platoon

Downstream segmentPlatoon driving into downstream segment

(b) Platoon dynamic on segment layer

Sub-region

Boundary of sub-region

Segment-intersection is transferred into sub-region due to the state changing

Segment-intersection is transferred out of sub-region

due to the state changing

(c) Segment-intersection dynamic on subregion layer

Figure 1 The dynamic of traffic elements in traffic network

U

R(X)

Boundary (X) = R (X) minus R(X)

Figure 2The rough representation of set X

representation of the crisp set X Boundary(X) is the bound-ary ofX computed using the difference of the upper and lowerapproximation set This rough representation is consistentwith the realities in world that objects maybe belong to a setpartly This kind of description can express the real worldaccurately

However the rough set defined above cannot describethe dynamic of X That is when the element state has been

U

f (u )

x1u1

u2

x2

1

f (u2)

(x1)

(x2)f

f

Figure 3The dynamic characteristics of S-rough set

changed the element may be transferred into or out of X Asshown in Figure 3 for u1notin X u2notin X 1199091 isin 119883 1199092 isin 119883 thereare some element operations f (u) and 1198911015840(119909) which make u1

become f (u1) with f (u1) belonging to X completely make u2

become f (u2) with f (u2) entering X partly make x1 become1198911015840(1199091) with 1198911015840(1199091) not belonging to X completely and makex2 become 1198911015840(1199092) with 1198911015840(1199092) leaving X partly This type ofset X with dynamic elements change cannot be describedaccurately using rough set theory Therefore the singularrough set theory (simplified as S-rough set) proposed byProfessor Shi is introduced to solve this problem [18 19] in

4 Mathematical Problems in Engineering

which the dynamic of set X is described using the elementarytransfer operations

Let 119865 cup 1198651015840 be the elementary transfer on U there are119865 = 1198911 1198912 119891119898 and1198651015840 = 11989110158401 1198911015840

2 1198911015840119898The effect of119891 isin 119865 is defined as follows for element uisinU and u notin X there

exists f (u)=x xisinX That is f makes X occur with singularityThe existence of f makes the boundary of X extend outwardsSimilarly the effect of 1198911015840 isin 1198651015840 is defined as follows forelement xisinX there exists 1198911015840(119909) = 119906 119906 notin 119883 1198911015840 makes theboundary of X shrink inwards The existence of both f and1198911015840 makes X occur with singularity in two directions and Xbecomes 119883lowast that is X has two-direction dynamic characterLetX119891 denote the extension of X1198831198911015840 denote the shrink of Xand the two-direction singular set119883lowast can be defined formallyas follows

119883119891 = 119906 | 119906 isin 119880 119906 notin 119883119891 (119906) = 119909 isin 119883 (3)

1198831198911015840 = 119909 | 119909 isin 1198831198911015840 (119909) = 119906 notin 119883 (4)

119883lowast = 119883 1198831198911015840 cup 119883119891 (5)

where f (u) is transferred into X and 1198911015840(119906) is transferred outof X completely According to the rough set definition ie(1)-(2) the approximation of119883lowast can be described as

(119877 119865 cup 1198651015840)∘(119883lowast)

= cup 119909 | [119891 (119909)]119877 sube 119883lowast cup [119909]119877 sube 119883lowast (6)

(119877 119865 cup 1198651015840)∘ (119883lowast)= cup 119909 | [119891 (119909)]119877 cap 119883lowast = and [119909]119877 cap 119883lowast = (7)

Therefore the existence of 119865 cup 1198651015840 makes X become 119883lowastthe dynamic characteristics of X with element transfer-in ortransfer-out completely can be described using (6)-(7) Inreality there exist the cases of elements transfer partly That isforuisinUunotinX there is f (u) =x and x belongs toX partly (ieuis transferred into X partly) or for xisinX there is 1198911015840(119909)=u andu belongs to X partly (ie x is transferred out of X partly)For this case the upper assistant set As+(119883lowast) and the lowerassistant set Asminus(119883lowast) are defined as (8)-(9) where As+(119883lowast) ismade up of the elements transferred into X partly Asminus(119883lowast)is made up of the elements transferred out of X partly [20]

119860119904+ (119883lowast)= 119909 | 119906 isin 119880 119906 notin 119883119891 (119906) = 119909 119909 enters 119883 partly (8)

119860119904minus (119883lowast)= 119909 | 119909 isin 1198831198911015840 (119909) = 119906 119906 leaves 119883 partly (9)

To sum up the dynamic and the rough representation of Xcan be described using (6)-(9) According to the definitionsof S-rough set the elements in traffic network can be defined

in similar ways In the next section these traffic elements ineach layer are defined respectively

22 Traffic Granules Definition Based on S-Rough Set Basedon the discussion in previous sections the traffic elementssuch as vehicle platoon intersection and segments andsubregion can be represented using S-rough set theirdynamic characteristics and rough representation can also bedepicted

As shown in Figure 4 there are four layers in the hierar-chical multigranularity traffic network model Vehicle is thesmallest granule in this model Vehicle granules with similarstates form a platoon granule There may be several platoongranules on a road segment Segments with the connectedintersections with similar traffic states construct a subregionin which the intersection and segment is considered as abasic granule The whole traffic network may be partitionedinto several subregions On the same layer the lower levelgranules may be transferred from one upper level granule toanother For example as shown in Figure 4 vehicle granulesin platoon 1 may be transferred into platoon 2 when its stateschanged Similarly one intersection and segment granulemay be transferred into or out of a subregion granule dueto the changing traffic state of road segment Furthermorethere exists the partly elementary transfer case the vehicle atthe head of a platoon will drive out from the platoon whenit accelerates and the vehicle at the end of a platoon willdrive out from the platoon when it decelerates In this casethe vehicle belongs to the platoon in position view whileits acceleration is higher than the other vehicles it can besaid that the vehicle granule belongs to the platoon partlyThere are the similar cases to the other granules This isconsistent with S-rough set definition Therefore using S-rough set to represent the granules in traffic network candescribe the traffic network dynamic more accurately Inthe following context the singular rough representations ofthese traffic elements are listed except for the smallest vehiclegranule

(1) Platoon granule Let these vehicles travelling on thesame segment be the universe U119881 R119875 is the equivalencerelation about the attributes of speed acceleration andposition on U119881 Plt denotes a platoon 119875119897119905lowast denotes the two-direction singular set of Plt 119865119881 cup 1198651015840

119881 is the elementarytransfer and there are F119881=1198911198811 1198911198812 119891119881119898 and 1198651015840

119881 =1198911015840

1198811 11989110158401198812 1198911015840

119881119898 where f119881(veh) denotes that granuleveh is transferred into Plt due to the changing of speedposition and acceleration 119891119881

1015840(veh) denotes that granule vehis transferred out of Plt due to the changing of speed positionand acceleration Then the singular rough set of a platoon isdefined as

119875119897119905lowast = 119875119897119905 V119890ℎ | V119890ℎ isin 119875119897119905 1198911015840119881 (V119890ℎ) notin 119875119897119905

cup V119890ℎ | V119890ℎ isin 119880119881 V119890ℎ notin 119875119897119905 119891119881 (V119890ℎ) isin 119875119897119905 (10)

(119877119875 119865119881 cup 1198651015840119881)∘ (119875119897119905lowast) = cup V119890ℎ | [119891119881 (V119890ℎ)]119877119875 sube 119875119897119905lowast

cup [V119890ℎ]119877119875 sube 119875119897119905lowast (11)

Mathematical Problems in Engineering 5

Traffic NetworkSub-Region

Intersection amp Segments

Intersection amp Segments

helliphellip

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

helliphellip

Sub-Region

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

helliphellip

Element transfer at platoon level

Element transfer at segment level

Element transfer at region level

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

helliphellip

helliphellip

Figure 4 Hierarchical multigranularity traffic network model

(119877119875 119865119881 cup 1198651015840119881)∘ (119875119897119905lowast) = cup V119890ℎ | [119891119881 (V119890ℎ)]119877119875 cap 119875119897119905lowast

= and [V119890ℎ]119877119875 cap 119875119897119905lowast = (12)

119860119904+ (119875119897119905lowast) = 119891119881 (V119890ℎ) | V119890ℎ isin 119880119881 V119890ℎnotin 119875119897119905 119891119881 (V119890ℎ) enters 119875119897119905 partly (13)

119860119904minus (119875119897119905lowast) = V119890ℎ | V119890ℎisin 119875119897119905 1198911015840

119881 (V119890ℎ) leaves 119875119897119905 partly (14)

(2) Segment granule Let these platoons travelling on the samesegment be the universe U119875 R119878119890119892 denotes the equivalencerelation about the attributes of average speed and positionon U119875 Seg denotes the segment with multiple platoonsand 119878119890119892lowast denotes the two-direction singular set of SegF119875cup1198651015840

119875 is the elementary transfer on U119875 where 119865119875 =1198911198751 1198911198752 119891119875119898 and 1198651015840

119875 = 11989110158401198751 1198911015840

1198752 1198911015840119875119898 119891119875(119901119897119905)

denotes that platoon plt is driving into Seg 1198911198751015840(119901119897119905) denotes

that platoon plt is driving out of SegThen the singular roughset representation of segment is defined as

119878119890119892lowast = 119878119890119892 119901119897119905 | 119901119897119905 isin 119878119890119892 1198911015840119875 (119901119897119905) notin 119878119890119892

cup 119901119897119905 | 119901119897119905 isin 119880119875 119901119897119905 notin 119878119890119892 119891119875 (119901119897119905) isin 119878119890119892 (15)

(119877119878119890119892 119865119875 cup 1198651015840119875)∘ (119878119890119892lowast) = cup 119901119897119905 | [119891119875 (119901119897119905)]119877119878119890119892

cup [119901119897119905]119877119878119890119892 sube 119878119890119892lowast(16)

(119877119878119890119892 119865119875 cup 1198651015840119875)∘ (119878119890119892lowast) = cup 119901119897119905 | [119891119875 (119901119897119905)]119877119878119890119892

cap 119878119890119892lowast = and [119901119897119905]119877119878119890119892 cap 119878119890119892lowast = (17)

119860119904+ (119878119890119892lowast) = 119891119875 (119901119897119905) | 119901119897119905 isin 119880119875 119901119897119905notin 119878119890119892 119891119875 (119901119897119905) enters 119878119890119892 partly (18)

119860119904minus (119878119890119892lowast) = 119901119897119905 | 119901119897119905isin 119878 1198911015840

119875 (119901119897119905) leaves 119878119890119892 partly (19)

Actually for each road segment of traffic network theremay be several lanes corresponding to several vehicle flowscontrolled by different traffic light phases of intersection Tofacilitate the control strategy design the segment granule ispartitioned into subgranules corresponding to different lanesFor example the granule seg119886119887 denotes the platoon rough seton segment a where these platoons are leaving for segment bThe rough set definition of seg119886119887 can also be described using(15)-(19) with attribute lane id added In this case the stateseg119886 can be calculated by all the granules seg119886119887

(3) Subregion granule Since there are both segmentsand intersections in the subregion granule and the states ofsegments with the connected intersections affect the trafficcontrol strategy there are correlations between upstream anddownstream segments the ldquoUpstreamSegment-Intersection-DownstreamSegmentrdquo (simplified as uid) is viewed as thebasic granule in subregion granule definition These uidscompose the universe U119878 Let SR denote a subregion in

6 Mathematical Problems in Engineering

which the segments of uids have similar traffic state andthere are correlations between the upstream and downstreamsegment of each uid Since the uid is made up of upstreamsegment useg downstream segment dseg and the connectedintersection intersc if the useg and dseg are both transferredinto or out of SR it is to say that the uid is transferred into orout of SR completely If only useg or dseg is transferred intoor out of SR it is to say that the uid is transferred into or outof SR partly Let 119878119877lowast denote the two-direction singular setof SR and 119865119878 cup 1198651015840

119878 is the elementary transfer on U119878 where119865119880 = 1198911198801 1198911198802 119891119880119898 1198651015840119880 = 1198911015840

1198801 11989110158401198802 1198911015840

119880119898f119880(uid) indicates that the basic granule uid is transferred intoSR due to the state changing 1198911015840

119880(uid) indicates that uid istransferred out of SR due to the state changing The roughrepresentation of 119878119877lowast can be defined as

119878119877lowast = 119878119877 119906119894119889 | 119906119894119889 isin 119878119877 1198911015840119880 (119906119894119889) notin 119878119877

cup 119906119894119889 | 119906119894119889 isin 119880119880 119906119894119889 notin 119878119877 119891119880 (119906119894119889) isin 119878119877 (20)

(119877119878119877 119865119880 cup 1198651015840119880)∘ (119878119877lowast) = cup 119906119894119889 | [119891119880 (119906119894119889)]119877119878119877

cup [119906119894119889]119877119878119877 sube 119878119877lowast (21)

(119877119878119877 119865119880 cup 1198651015840119880)∘ (119878119877lowast) = cup 119906119894119889 | [119891119880 (119906119894119889)]119877119878119877

cap 119878119877lowast = and [119906119894119889]119877119878119877 cap 119878119877lowast = (22)

119860119904+ (119878119877lowast) = 119891119880 (119906119894119889) | 119906119894119889 isin 119880119878119877 119906119894119889notin 119878119877 119891119880 (119906119894119889) enters 119878119877 partly (23)

119860119904minus (119878119877lowast) = 119906119894119889 | 119906119894119889isin 119878119877 1198911015840

119880 (119906119894119889) leaves 119878119877 partly (24)

23Maintenance and Construction of Traffic Granules Basedon the definitions of the hierarchical multigranularity trafficnetwork model traffic control strategy can be designedaiming at different traffic demand to improve the efficiency ofurban traffic control systemThe key issue that this model canbe utilized for traffic control and management is to maintainthe integrity of these granules in real time For the traffic dataprocessing efficiently the distributed traffic datamanagementframework has been discussed in our previous research work[21] In this section it is assumed that the basic vehicle stateinformation can be collected by the communication deviceequipped in vehicles and managed by the distributed trafficdata centers

For the granules in traffic network model U119881 U119875and U119878 are the vehicle universe platoon universe andsegment universe respectively The smallest granule in thismodel is vehicle granule the related attributes are listedas veh id veh speed veh acc veh pos segment id lane idpreceding id following id pre dist back dist veh class pas-sengers veh weight etc These attributes represent vehicle idspeed acceleration position road segment id that vehicletravels on lane id that vehicle travels on the distance to theprevious vehicle the distance to the vehicle in back vehicle

class the number of passengers in vehicle and vehicle weightin mergence respectively The attributes such as veh classpassengers and veh weight can be utilized for light phasepressure computation in the traffic light control strategydesign Inwhat follows the constructions of threemain trafficgranules are listed in detail

(1) Maintenance and construction of platoon granule Theequivalence relation R119875 on U119881 is used for constructinga platoon on each road segment it is defined as followsvehicles with neighboring position and same segment idsame lane id similar speed and acceleration The elementarytransfer is defined as follows For veh notin p ifMinspeed119901119897119886119905119900119900119899 leveh speed le Maxspeed119901119897119886119905119900119900119899 |V119890ℎ 119886119888119888| le acc119905ℎ119903119890119904ℎ119900119897119889 pre distle dist119905ℎ119903119890119904ℎ119900119897119889 or back dist le dist119905ℎ119903119890119904ℎ119900119897119889 then veh is trans-ferred into platoon p completely if (pre dist gt dist119905ℎ119903119890119904ℎ119900119897119889 andveh accgt acc119905ℎ119903119890119904ℎ119900119897119889) or (back dist gt dist119905ℎ119903119890119904ℎ119900119897119889 and veh accltacc119905ℎ119903119890119904ℎ119900119897119889) then veh is driving into platoon p since veh isaccelerating from the back of p or decelerating in front of pafter a period of time veh will drive into platoon p In thiscase it is said that veh is transferred into p partly For visinpif there is |V119890ℎ 119886119888119888| gt acc119905ℎ119903119890119904ℎ119900119897119889 veh in p is accelerating ordecelerating after a period of time veh will leave platoon pIn this case it is said that veh is transferred out of p partly Ifveh is the head or rear of p when back dist gt dist119905ℎ119903119890119904ℎ119900119897119889 orpre dist gt dist119905ℎ119903119890119904ℎ119900119897119889 it is said that veh has been transferredout of p completely

(2) Maintenance and construction of road segment Forthe universe 119880119875 the basic granule is platoon the relatedattributes including platoon id ave speed max speed minspeed ave headwaymax headwaymin headway veh countseg id lane id platoon weight rightofway flag etc Theseattributes represent the vehicle id average speed maximumspeed of platoon minimum speed of platoon average head-way maximum headway minimum headway number ofvehicles in platoon segment id that the platoon is travellingon lane id the weight of platoon and whether the platoonobtains the right of way to pass the downstream intersectionrespectively The equivalence relation R119878 on U119875 is definedas follows platoons with same segment id and lane id Theelementary transfer of platoon granule on segment granuleis defined as follows For 119901 isin 119904 if rightofway flag is true andveh count lt veh count119905ℎ119903119890119904ℎ119900119897119889 it is said that platoon119901 canpassthrough the downstream intersection in green time then 119901 istransferred out of segment 119904 completely if rightofway flag istrue and veh count gt veh count119905ℎ119903119890119904ℎ119900119897119889 it is said that part ofplatoon 119901 will pass through the downstream intersection ingreen time then 119901 is transferred out of segment 119904 partly For119901 notin 119904 there are similar situations

(3) Maintenance and construction of subregion granuleFor constructing the subregion granule uid is the basicgranule which constitutes the universe U119880 The relatedattributes include uid id upseg state downseg state andcorrelation degree representing uid id state of upstream seg-ment state of downstream segment and correlation degreeof the upstream and downstream segments respectively Thestate of segment can be extracted from platoon granules on itusing the attributes seg id veh count ave speed queuelengthand seg cap where seg cap is the capacity of a road segmentThe correlation degree can be calculated according to the state

Mathematical Problems in Engineering 7

qab

a bi

qghfgh

fab

gh

d

d

qgd

qgd

rbc

rbc

qbc

qbc

c

c

Figure 5 Vehicle flows on the upstream and downstream segments

and capacity of upstreamanddownstream segments [22]Theequivalence relation R119878119877 onU119878 is defined as follows segmentshave similar state and the correlation degree is more than agiven thresholdThe elementary transfer is defined as followsFor uid isinSR if the number of vehicles and the average speedof upstream and downstream segments are both exceedingthe given range of subregion SR that is the upstream anddownstream segment states of uid are not close to the stateof SR it is said that uid is transferred out of SR completely Ifonly the downstream or upstream segment state is not closeto the state of SR it is said that uid is transferred out ofSR partly For uid notin SR if uid is adjacent with SR and theupstream and downstream segment states of uid are similarto the state of SR it is said that uid is transferred into SRcompletely If only upstream or downstream segment state issimilar to the SR state it is said that uid is transferred into SRpartly

Based on the construction and maintenance of granulesin the proposed traffic network model discussed above thesetraffic granules can be maintained in a distributed modeunder a distributed traffic data management frameworkThen the traffic information of different layers and granular-ities can be extracted Based on the multigranularity trafficinformation including vehicle information platoon infor-mation segment information and subregion informationthe traffic control and management system may adopt moresuitable strategies to improve the efficiency However stillsome issues have not been presented in detail in this proposedmodel such as the parameters selection of traffic statetraffic state computation of road segment and the correlationdegree calculation between downstream and upstream roadsegmentsThese problems will be researched in further workIn this paper to illustrate the validity and feasibility ofthe proposed hierarchical multigranularity traffic networkmodel we propose a max-pressure based traffic controlstrategy in the next section

3 Traffic Control Strategy Based onthe Proposed Traffic Model

In this section an application based on the proposed trafficnetwork model is discussed The proposed traffic networkmodel is combined with the max-pressure traffic controlalgorithm to design an extended max-pressure traffic controlstrategy in which the light phase pressure is computed usingthe rough representations of platoon and segment granules

31 Max-Pressure Traffic Control Strategy The max-pressuretraffic control strategy is proposed by Varaiya [23] based on

the backpressure traffic control algorithm [24 25] Accordingto the strategy the urban traffic network is viewed as a queu-ing network and at each time slot the active phase of eachintersection is selected depending on the light phase pressurecomputed using local queue length of each connected roadsegment

As shown in Figure 5 on segments a and g there arequeued vehicles waiting for passing through intersection ithe vehicle flows f 119886119887 and f 119892ℎ are controlled by light phase pof intersection i For the vehicle flow f 119886119887 when it enters thedownstream segment b it will drive into the queued vehicles1199021198871198881015840 and 11990211988711988810158401015840 according to the proportions 1199031198871198881015840 and 11990311988711988810158401015840 where1198881015840 and 11988810158401015840 are the downstream segments of bThere are similarsituations for the vehicle flow 119891119892ℎ The pressure of light phasep of intersection i is computed using (25) and the active lightphase in next time slot is selected using (26)

119901119903119890119904119904119906119903119890119901 = sum119891119886119887isin119901

120583119886119887 (119901) (119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)) (25)

where119891119886119887 isin 119901means that the vehicle flow f 119886119887 is controlled bylight phase p120583119886119887(p) is the number of vehicles passing throughintersection per unit time when p is activated

119901lowast = argmax119901isin119875119894

(119901119903119890119904119904119906119903119890119901 (119905)) (26)

where pisinP119894 means that p is one of the light phases of lightsignal group 119875119894

In this method the light phase pressure is concernedwith the number of queued vehicles in front of the stop lineon the upstream and downstream segments The detailedvehicle information the arriving vehicles (for the upstreamsegment) and the departing vehicles (for the downstreamsegment) are not considered Based on the proposed trafficnetwork model there may be several platoons with differentstates on the upstream and downstream segments On theupstream segment there are stopped platoons in front ofthe stop line and arriving platoons heading for the stop lineOn the downstream segment there may be the departingplatoon when the corresponding downstream light phaseis activated These situations are considered in the trafficnetwork rough representation discussed in previous sectionsTherefore in the next section the light phase pressurecomputation method is extended based on the proposedhierarchical multigranularity traffic network model

32 Extended Max-Pressure Traffic Control Algorithm Actu-ally the light phase pressure is affected by four types of

8 Mathematical Problems in Engineering

platoons on the segment For the platoons on upstreamsegment the stopped platoons waiting for passing throughthe intersection the platoons that are driving towards thestop line and the platoons on the neighboring lanes that aregoing to drive into the corresponding lanes will affect thelight phase pressure For the platoons on downstream seg-ment the departing platoon affects the light phase pressurecomputation too if the downstream light phase is activatedSince the rough representation of traffic granules includesdetailed traffic information the vehicle weight the numberof passengers vehicle class etc also can be considered Thenbased on the rough representation of platoon and segment inthe proposed traffic network model the light phase pressurecan be computed more accurately

To simplify the description the rough set representationof 119883lowast with elementary transfer in two directions is denotedby the lower approximation set appr (119883lowast 119865 cup 1198651015840) and theupper approximation set 119886119901119901119903minus(119883lowast 119865 cup 1198651015840) appr (119883lowast F)represents the one-direction lower approximation set withelement transfer-in operations and appr (119883lowast 1198651015840) representsthe one direction lower approximation set with elementtransfer-out operations The upper assistant set As+(119883lowast F)represents the element set that elements enter 119883lowast partlyunder transfer operations F The lower assistant set Asminus(119883lowast1198651015840) represents the element set that elements leave 119883lowast partlyunder transfer operations 1198651015840 Then the pressure from eachplatoon can be computed using

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) = sum

V119890ℎisin119886119901119901119903 (119875lowast119886119887119865119881)

120572119908V119890ℎ

+ sumV119890ℎisin119860119904+(119875lowast

119886119887119865119881)

120573119908V119890ℎ

(27)

where 120572 120573 is the coefficient of the vehicle weight and 119908V119890ℎ isthe vehicle weight determined by the vehicle type it meansthat the different types of vehicles generate different pressuredue to the different number of passengers and emergence ofthese vehicles

The pressure of the upstream segment is computed using

119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

= sum119901119897119905isin119886119901119901119903 (119904119890119892lowast

119886119887119865119875)cup119860119904+(119904119890119892lowast

119886119887119865119875)

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) (28)

where useg represents an upstream segment and seg119886119887 is asubgranule on segment a in which platoons on segment aare waiting for driving into segment b

The pressure of the downstream segment is computedusing

119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887 119905)

= sum119888isin1198881015840 11988810158401015840

119901119897119905isin119886119901119901119903 (119904119890119892lowast1198871198881198651015840119875)cup119860119904minus(119904119890119892lowast119887119888119865

1015840119875)

119903119887119888119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119887119888 119905) (29)

According to (28) (29) the vehicles entering upstreamsegment and departing from downstream segment are con-sidered for the coordination with neighboring intersections

The pressure of light phase 119901119894119895 is computed using

119901119903119890119904119904119906119903119890119901ℎ119886119904119890 (119901119894119895 119905)

= sum119891119886119887isin119901

119894119895

119891119887119888isin119901119894+1119896

(119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

minus 119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887119888 119905)) 120583119886119887 (119905)

(30)

where 119901119894119895 is the jth light phase of intersection i 119901119894+1

119896 is thecooperative light phase of the downstream intersection i+1

Based on the extended pressure computation methoddescribed above the vehicle type vehicle state and platoonstate upstream light phase state and downstream light phasestate are considered If the element transfer operations Fand 1198651015840 are both empty the extended pressure computationmethod degenerates into max-pressure traffic control algo-rithm If 119865119875 = Φ and 1198651015840

119875 = Φ the arriving vehiclesfrom upstream segment are considered in the pressurecomputation in this case the coordination with the upstreamintersection light phase can be achieved If 119865119875 = Φ and1198651015840

119875 = Φ the departing vehicles on the downstream segmentare considered to achieve the coordination with downstreamintersection light phaseThe light phase with max-pressure isselected to be activated in the next time slot using (26)

Furthermore the vehicle class the number of passengersand vehicle emergency also can be considered using thevehicle weight Therefore based on the extended light phasecomputation method the real situations of traffic controlsystem can be reflected The strategy may be more suitablefor the actual situation of urban traffic network To illustratethe effectiveness and feasibility of the proposed model andalgorithm simulations are carried out in the next section

4 Simulations

In this section the simulation is carried out to analyze theeffectiveness and feasibility of the proposed method Thesimulation traffic network consists of 15 intersections and 76links constructed in Vissim The network includes 16 ingresssegments and 16 exit segments as shown in Figure 6(a)

The vehicle input of each ingress link is 800 vehh Thereare 15 signal controllers and 4 light phases for each controllerie north-south straight phase north-south left-turn phasewest-east straight phase and west-east left-turn phase Thereare 3 lanes on each link The left-turn vehicle flow drives onLane 3 and the straight vehicle flow drives on Lane 2The twovehicle flows are controlled by the traffic lightThe right-turnvehicle flow drives on Lane 1 and is free of the traffic lightTo obtain the traffic parameters during simulation 60 queuecounters 60 travel time sections and 60 routing decisions arelaid on the simulation traffic network as shown in Figures6(b) 6(c) and 6(d) During the simulation the green lighttime of each light phase is assumed to be 21 s which is thepedestrian clearance time (the road width is about 21 m andthe pedestrians speed is about 1 ms) [26] The yellow lighttime is assumed to be 3 s

Mathematical Problems in Engineering 9

(a)

(b) (c) (d)

Figure 6 The simulation traffic network and settings

Table 1 Traffic network performance comparison

Parameters FTMethod MPMethod EMP MethodTotal travel time (h) 633947 588139 568009Total delay time (h) 350255 298110 277457Total stopped delay (h) 253507 225692 205465Average delay time (s) 130584 111420 104057Average stopped delay (s) 94514 84353 77057Average speed (kmh) 23527 25916 26883Number of vehicles in network 448 368 334Number of vehicles that left network 9208 9264 9265Total Number of vehicles 9656 9632 9599

To illustrate the efficiency and feasibility of the pro-posed traffic network model and traffic control algorithmthree traffic light control methods are implemented in thissection including fixed-time control (FT method) max-pressure control (MP method) and the extended max-pressure control (EMP method) To simplify the simulationthe extended pressure computation only considers the caseof 119865119875 = Φ and 1198651015840

119875 = Φ in this paper the other cases will beconsidered in further research works The three traffic lightcontrol methods are implemented in Visual Studio 2010Thesimulation programs communicate with Vissim through theVissim COM programming interface to obtain the vehiclestate and traffic parameters to decide the traffic light signalat each time slot The simulation runs for 3600 s and thetraffic network performance is evaluated from 1000 s to 3600s usingVissimThis is because in the first 1000 s of simulationtime there are not enough vehicles entering the simulationtraffic network

The simulation results of the traffic network are listed inTable 1 According to the simulation results EMP methodbased on the proposed hierarchical multigranularity trafficnetwork model achieves better performance because the

light phase pressure computed using (27)-(30) is closer tothe actual traffic situation in which the vehicles on a roadsegment are represented using rough set form The depart-ing vehicles of downstream segments considered in thissimulation can achieve the coordination with downstreamintersections

As shown in Figure 7 the average queue length underEMP method is shorter than the other traffic light controlmethods The shorter queue length in traffic network meansit can effectively reduce the occurrence of spill-over ondownstream road segment so as to avoid the traffic collapsedue to vehicle queue overflow EMP method is based onthe proposed hierarchical multigranularity traffic networkmodel in which the traffic elements dynamic characteris-tics are depicted accurately and the light phase states ofdownstream intersections are also considered to computethe light phase pressure therefore EMP method achievesbetter performance As shown in Figure 8 the averagetravel time under EMP control is lower than the other twomethods

Furthermore using the extended pressure computationmethod based on the proposed traffic network the priority

10 Mathematical Problems in Engineering

MPFTEMP

05

101520253035404550

Que

ue L

engt

h (v

eh)

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000Time (s)

Figure 7 Average queue length of the three-traffic-light controlmethod

MPFTEMP

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000

Time (s)

0102030405060708090

100

Trav

el Ti

me (

s)

Figure 8 Average travel time of the three-traffic-light controlmethod

control for some particular vehicles can be achieved con-veniently since different classes of vehicles can be set withdifferent vehicle weight in the EMPmethod Different weightvalues can result in different traffic efficiency for differenttypes of vehicles For example when the type of bus is addedto the simulation traffic network the higher weight valuecan make buses have higher average speed and the otherclasses of vehicles maybe achieve lower average speed Forthis issue the vehicle weight selection is important As shownin Figure 9 the different bus weight values are simulatedand they result in different average speed According to themultiple simulation results and the real traffic demand thevehicle weight can be determined

To sum up since the hierarchical multigranularity trafficmodel can represent the actual traffic network situationsaccurately and the EMP method can achieve traffic controlflexibly this paper may be a useful attempt in solving urbantraffic control problem

5 Conclusions

In this paper a hierarchical multigranularity traffic networkis proposed and on the basis of the proposed traffic networkan extended max-pressure traffic control method is imple-mented The simulation results illustrate that the proposedtraffic network model and EMP method achieve better

Average speed [Car]Average speed [Bus]Average speed [All vehicles]

24245

25255

26265

27

Aver

age s

peed

15 2 25 3 35 4 45 5 551Vehicle weight of Bus

Figure 9 Simulation results with different bus weight values

performance In the proposed model the traffic elementsin urban traffic network are represented using S-rough setin which the dynamic characteristics of traffic elementsare depicted This type of traffic element description canrepresent the traffic network situation accurately so as todesign the more efficient traffic control strategy

However the hierarchical multigranularity traffic net-work model is only a preliminary exploration and it needsto be further improved in the future research work such asthe traffic subregion formulation and the boundary controlof traffic subregion For the traffic subregion formulationthe correlation of downstream and upstream road segmentsshould be computed considering traffic states and the geo-graphical characteristics of each pair of upstream and down-stream segments The control strategy of subregion boundarycan compute the traffic pressure considering the remainingcapacity of downstream segment to speed up the vehicleflow release towards to the segment with larger remainingcapacity These issues will be studied in our further researchwork to improve the hierarchical multigranularity traffic net-workmodel Furthermore in the future work the completelymultigranularity traffic control strategies should be furtherresearched using granular computing theory to achieve thetraffic system multigranularity control completely

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial NaturalScience Foundation China [grant number ZR2018FM027]Project of Shandong Province Higher Educational Scienceand Technology Program [grant number J17KB181] KeyResearch and Development Project in Shandong ProvinceChina [grant numbers 2015GGX101048 2016GSF120009]

Mathematical Problems in Engineering 11

and National Natural Science Foundation of Youth FundProject [grant number 61501284]

References

[1] Q He K L Head and J Ding ldquoMulti-modal traffic signalcontrol with priority signal actuation and coordinationrdquoTrans-portation Research Part C Emerging Technologies vol 46 pp65ndash82 2014

[2] Y Yao ldquoInterpreting concept learning in cognitive informaticsand granular computingrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 39 no 4 pp 855ndash8662009

[3] J T Yao A V Vasilakos andW Pedrycz ldquoGranular computingperspectives and challengesrdquo IEEE Transactions on Cyberneticsvol 43 no 6 pp 1977ndash1989 2013

[4] H Liu andM Cocea ldquoGranular computing-based approach forclassification towards reduction of bias in ensemble learningrdquoGranular Computing vol 2 no 3 pp 131ndash139 2017

[5] X Ji L Liu and Q Wu ldquoTraffic management and decision-making support based on granular computingrdquo Journal ofHighway and Transportation Research and Development vol2008 no 10 pp 129ndash133 2008

[6] S Dewen C Xinhong C Jing and F Xujian ldquoGranularcomputing in intelligent transportation An exploratory studyrdquoCybernetics and Information Technologies vol 15 no 6 pp 122ndash134 2015

[7] C Yao J-C Gu and Q Yang ldquoTraffic congestion identificationand analysis of urban road network based on granular comput-ingrdquo Applied Mechanics and Materials vol 641-642 pp 916ndash922 2014

[8] T Li D Ruan Y Shen E Hermans and G Wets ldquoA newweighting approach based on rough set theory and granularcomputing for road safety indicator analysisrdquo InternationalJournal of Computational Intelligence vol 32 no 4 pp 517ndash5342016

[9] F He R Liu Y Zhang et al ldquoBGrR large-scale network routingspeedup techniques based on granular computingrdquo ComputerScience vol 41 no 11 pp 265ndash281 2014

[10] R Deb and A W-C Liew ldquoMissing value imputation forthe analysis of incomplete traffic accident datardquo InformationSciences vol 339 pp 274ndash289 2016

[11] S Hao L Yang and Y Shi ldquoData-driven car-following modelbased on rough set theoryrdquo IET Intelligent Transport Systemsvol 12 no 1 pp 49ndash57 2018

[12] L Livi and A Sadeghian ldquoGranular computing computationalintelligence and the analysis of non-geometric input spacesrdquoGranular Computing vol 1 no 1 pp 13ndash20 2016

[13] X Yang H Da-Wei S Bing and Z Duo-Jia ldquoCity traffic flowbreakdown prediction based on fuzzy rough setrdquo Open Physicsvol 15 no 1 pp 292ndash299 2017

[14] X Xiong L Chen and J Liang ldquoAnalysis of roadway trafficaccidents based on rough sets and Bayesian networksrdquo Promet- Traffic - Traffico vol 30 no 1 pp 71ndash81 2018

[15] M Deshpande and P Bajaj ldquoPerformance improvement of traf-fic flow prediction model using combination of support vectormachine and rough setrdquo International Journal of ComputerApplications vol 163 no 2 pp 31ndash35 2017

[16] Y Y Yao ldquoGranular computing basic issues and possible solu-tionsrdquo in Proceedings of the Fi13h International Conference onComputing and Information vol 1 pp 186ndash189 2000

[17] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Science vol 11 no 5 pp 341ndash356 1982

[18] K Shi ldquoS-rough sets and its applications in diagnosis-rec-ognition for diseaserdquo in Proceedings of the 1st InternationalConference on Machine Learning and Cybernetics pp 50ndash54IEEE Beijing China 2002

[19] K Q Shi and Y Q Cui ldquoS-Rough sets and their general struc-turesrdquo Journal of Shandong University Natural Science Shan-dong Daxue Xuebao Lixue Ban vol 37 no 6 pp 471ndash474 2002

[20] H Haiqing X Peijun and F Haiyan ldquoS-rough recognitionof knowledge and general threshold encryption authenticationscheme of recognition conclusionrdquo in Proceedings of the SixthInternational Conference on Intelligent Systems Design andApplications pp 816ndash824 Jian China October 2006

[21] S Hao L Yang and Y Shi ldquoReal-time data-driven trajectoryreconstruction based on rough set theoryrdquo Advances in Trans-portation Studies vol 44 pp 119ndash132 2018

[22] Y Yang L Jia Y Qin S Han and H Dong ldquoUnderstandingstructure of urban traffic network based on spatial-temporalcorrelation analysisrdquo Modern Physics Letters B CondensedMatter Physics Statistical Physics AtomicMolecular andOpticalPhysics vol 31 no 22 Article ID 1750230 19 pages 2017

[23] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[24] TWongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 USA September 2012

[25] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo IFAC Proceedings Volumes vol 47 no 3 pp11332ndash11337 2014

[26] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

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4 Mathematical Problems in Engineering

which the dynamic of set X is described using the elementarytransfer operations

Let 119865 cup 1198651015840 be the elementary transfer on U there are119865 = 1198911 1198912 119891119898 and1198651015840 = 11989110158401 1198911015840

2 1198911015840119898The effect of119891 isin 119865 is defined as follows for element uisinU and u notin X there

exists f (u)=x xisinX That is f makes X occur with singularityThe existence of f makes the boundary of X extend outwardsSimilarly the effect of 1198911015840 isin 1198651015840 is defined as follows forelement xisinX there exists 1198911015840(119909) = 119906 119906 notin 119883 1198911015840 makes theboundary of X shrink inwards The existence of both f and1198911015840 makes X occur with singularity in two directions and Xbecomes 119883lowast that is X has two-direction dynamic characterLetX119891 denote the extension of X1198831198911015840 denote the shrink of Xand the two-direction singular set119883lowast can be defined formallyas follows

119883119891 = 119906 | 119906 isin 119880 119906 notin 119883119891 (119906) = 119909 isin 119883 (3)

1198831198911015840 = 119909 | 119909 isin 1198831198911015840 (119909) = 119906 notin 119883 (4)

119883lowast = 119883 1198831198911015840 cup 119883119891 (5)

where f (u) is transferred into X and 1198911015840(119906) is transferred outof X completely According to the rough set definition ie(1)-(2) the approximation of119883lowast can be described as

(119877 119865 cup 1198651015840)∘(119883lowast)

= cup 119909 | [119891 (119909)]119877 sube 119883lowast cup [119909]119877 sube 119883lowast (6)

(119877 119865 cup 1198651015840)∘ (119883lowast)= cup 119909 | [119891 (119909)]119877 cap 119883lowast = and [119909]119877 cap 119883lowast = (7)

Therefore the existence of 119865 cup 1198651015840 makes X become 119883lowastthe dynamic characteristics of X with element transfer-in ortransfer-out completely can be described using (6)-(7) Inreality there exist the cases of elements transfer partly That isforuisinUunotinX there is f (u) =x and x belongs toX partly (ieuis transferred into X partly) or for xisinX there is 1198911015840(119909)=u andu belongs to X partly (ie x is transferred out of X partly)For this case the upper assistant set As+(119883lowast) and the lowerassistant set Asminus(119883lowast) are defined as (8)-(9) where As+(119883lowast) ismade up of the elements transferred into X partly Asminus(119883lowast)is made up of the elements transferred out of X partly [20]

119860119904+ (119883lowast)= 119909 | 119906 isin 119880 119906 notin 119883119891 (119906) = 119909 119909 enters 119883 partly (8)

119860119904minus (119883lowast)= 119909 | 119909 isin 1198831198911015840 (119909) = 119906 119906 leaves 119883 partly (9)

To sum up the dynamic and the rough representation of Xcan be described using (6)-(9) According to the definitionsof S-rough set the elements in traffic network can be defined

in similar ways In the next section these traffic elements ineach layer are defined respectively

22 Traffic Granules Definition Based on S-Rough Set Basedon the discussion in previous sections the traffic elementssuch as vehicle platoon intersection and segments andsubregion can be represented using S-rough set theirdynamic characteristics and rough representation can also bedepicted

As shown in Figure 4 there are four layers in the hierar-chical multigranularity traffic network model Vehicle is thesmallest granule in this model Vehicle granules with similarstates form a platoon granule There may be several platoongranules on a road segment Segments with the connectedintersections with similar traffic states construct a subregionin which the intersection and segment is considered as abasic granule The whole traffic network may be partitionedinto several subregions On the same layer the lower levelgranules may be transferred from one upper level granule toanother For example as shown in Figure 4 vehicle granulesin platoon 1 may be transferred into platoon 2 when its stateschanged Similarly one intersection and segment granulemay be transferred into or out of a subregion granule dueto the changing traffic state of road segment Furthermorethere exists the partly elementary transfer case the vehicle atthe head of a platoon will drive out from the platoon whenit accelerates and the vehicle at the end of a platoon willdrive out from the platoon when it decelerates In this casethe vehicle belongs to the platoon in position view whileits acceleration is higher than the other vehicles it can besaid that the vehicle granule belongs to the platoon partlyThere are the similar cases to the other granules This isconsistent with S-rough set definition Therefore using S-rough set to represent the granules in traffic network candescribe the traffic network dynamic more accurately Inthe following context the singular rough representations ofthese traffic elements are listed except for the smallest vehiclegranule

(1) Platoon granule Let these vehicles travelling on thesame segment be the universe U119881 R119875 is the equivalencerelation about the attributes of speed acceleration andposition on U119881 Plt denotes a platoon 119875119897119905lowast denotes the two-direction singular set of Plt 119865119881 cup 1198651015840

119881 is the elementarytransfer and there are F119881=1198911198811 1198911198812 119891119881119898 and 1198651015840

119881 =1198911015840

1198811 11989110158401198812 1198911015840

119881119898 where f119881(veh) denotes that granuleveh is transferred into Plt due to the changing of speedposition and acceleration 119891119881

1015840(veh) denotes that granule vehis transferred out of Plt due to the changing of speed positionand acceleration Then the singular rough set of a platoon isdefined as

119875119897119905lowast = 119875119897119905 V119890ℎ | V119890ℎ isin 119875119897119905 1198911015840119881 (V119890ℎ) notin 119875119897119905

cup V119890ℎ | V119890ℎ isin 119880119881 V119890ℎ notin 119875119897119905 119891119881 (V119890ℎ) isin 119875119897119905 (10)

(119877119875 119865119881 cup 1198651015840119881)∘ (119875119897119905lowast) = cup V119890ℎ | [119891119881 (V119890ℎ)]119877119875 sube 119875119897119905lowast

cup [V119890ℎ]119877119875 sube 119875119897119905lowast (11)

Mathematical Problems in Engineering 5

Traffic NetworkSub-Region

Intersection amp Segments

Intersection amp Segments

helliphellip

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

helliphellip

Sub-Region

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

helliphellip

Element transfer at platoon level

Element transfer at segment level

Element transfer at region level

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

helliphellip

helliphellip

Figure 4 Hierarchical multigranularity traffic network model

(119877119875 119865119881 cup 1198651015840119881)∘ (119875119897119905lowast) = cup V119890ℎ | [119891119881 (V119890ℎ)]119877119875 cap 119875119897119905lowast

= and [V119890ℎ]119877119875 cap 119875119897119905lowast = (12)

119860119904+ (119875119897119905lowast) = 119891119881 (V119890ℎ) | V119890ℎ isin 119880119881 V119890ℎnotin 119875119897119905 119891119881 (V119890ℎ) enters 119875119897119905 partly (13)

119860119904minus (119875119897119905lowast) = V119890ℎ | V119890ℎisin 119875119897119905 1198911015840

119881 (V119890ℎ) leaves 119875119897119905 partly (14)

(2) Segment granule Let these platoons travelling on the samesegment be the universe U119875 R119878119890119892 denotes the equivalencerelation about the attributes of average speed and positionon U119875 Seg denotes the segment with multiple platoonsand 119878119890119892lowast denotes the two-direction singular set of SegF119875cup1198651015840

119875 is the elementary transfer on U119875 where 119865119875 =1198911198751 1198911198752 119891119875119898 and 1198651015840

119875 = 11989110158401198751 1198911015840

1198752 1198911015840119875119898 119891119875(119901119897119905)

denotes that platoon plt is driving into Seg 1198911198751015840(119901119897119905) denotes

that platoon plt is driving out of SegThen the singular roughset representation of segment is defined as

119878119890119892lowast = 119878119890119892 119901119897119905 | 119901119897119905 isin 119878119890119892 1198911015840119875 (119901119897119905) notin 119878119890119892

cup 119901119897119905 | 119901119897119905 isin 119880119875 119901119897119905 notin 119878119890119892 119891119875 (119901119897119905) isin 119878119890119892 (15)

(119877119878119890119892 119865119875 cup 1198651015840119875)∘ (119878119890119892lowast) = cup 119901119897119905 | [119891119875 (119901119897119905)]119877119878119890119892

cup [119901119897119905]119877119878119890119892 sube 119878119890119892lowast(16)

(119877119878119890119892 119865119875 cup 1198651015840119875)∘ (119878119890119892lowast) = cup 119901119897119905 | [119891119875 (119901119897119905)]119877119878119890119892

cap 119878119890119892lowast = and [119901119897119905]119877119878119890119892 cap 119878119890119892lowast = (17)

119860119904+ (119878119890119892lowast) = 119891119875 (119901119897119905) | 119901119897119905 isin 119880119875 119901119897119905notin 119878119890119892 119891119875 (119901119897119905) enters 119878119890119892 partly (18)

119860119904minus (119878119890119892lowast) = 119901119897119905 | 119901119897119905isin 119878 1198911015840

119875 (119901119897119905) leaves 119878119890119892 partly (19)

Actually for each road segment of traffic network theremay be several lanes corresponding to several vehicle flowscontrolled by different traffic light phases of intersection Tofacilitate the control strategy design the segment granule ispartitioned into subgranules corresponding to different lanesFor example the granule seg119886119887 denotes the platoon rough seton segment a where these platoons are leaving for segment bThe rough set definition of seg119886119887 can also be described using(15)-(19) with attribute lane id added In this case the stateseg119886 can be calculated by all the granules seg119886119887

(3) Subregion granule Since there are both segmentsand intersections in the subregion granule and the states ofsegments with the connected intersections affect the trafficcontrol strategy there are correlations between upstream anddownstream segments the ldquoUpstreamSegment-Intersection-DownstreamSegmentrdquo (simplified as uid) is viewed as thebasic granule in subregion granule definition These uidscompose the universe U119878 Let SR denote a subregion in

6 Mathematical Problems in Engineering

which the segments of uids have similar traffic state andthere are correlations between the upstream and downstreamsegment of each uid Since the uid is made up of upstreamsegment useg downstream segment dseg and the connectedintersection intersc if the useg and dseg are both transferredinto or out of SR it is to say that the uid is transferred into orout of SR completely If only useg or dseg is transferred intoor out of SR it is to say that the uid is transferred into or outof SR partly Let 119878119877lowast denote the two-direction singular setof SR and 119865119878 cup 1198651015840

119878 is the elementary transfer on U119878 where119865119880 = 1198911198801 1198911198802 119891119880119898 1198651015840119880 = 1198911015840

1198801 11989110158401198802 1198911015840

119880119898f119880(uid) indicates that the basic granule uid is transferred intoSR due to the state changing 1198911015840

119880(uid) indicates that uid istransferred out of SR due to the state changing The roughrepresentation of 119878119877lowast can be defined as

119878119877lowast = 119878119877 119906119894119889 | 119906119894119889 isin 119878119877 1198911015840119880 (119906119894119889) notin 119878119877

cup 119906119894119889 | 119906119894119889 isin 119880119880 119906119894119889 notin 119878119877 119891119880 (119906119894119889) isin 119878119877 (20)

(119877119878119877 119865119880 cup 1198651015840119880)∘ (119878119877lowast) = cup 119906119894119889 | [119891119880 (119906119894119889)]119877119878119877

cup [119906119894119889]119877119878119877 sube 119878119877lowast (21)

(119877119878119877 119865119880 cup 1198651015840119880)∘ (119878119877lowast) = cup 119906119894119889 | [119891119880 (119906119894119889)]119877119878119877

cap 119878119877lowast = and [119906119894119889]119877119878119877 cap 119878119877lowast = (22)

119860119904+ (119878119877lowast) = 119891119880 (119906119894119889) | 119906119894119889 isin 119880119878119877 119906119894119889notin 119878119877 119891119880 (119906119894119889) enters 119878119877 partly (23)

119860119904minus (119878119877lowast) = 119906119894119889 | 119906119894119889isin 119878119877 1198911015840

119880 (119906119894119889) leaves 119878119877 partly (24)

23Maintenance and Construction of Traffic Granules Basedon the definitions of the hierarchical multigranularity trafficnetwork model traffic control strategy can be designedaiming at different traffic demand to improve the efficiency ofurban traffic control systemThe key issue that this model canbe utilized for traffic control and management is to maintainthe integrity of these granules in real time For the traffic dataprocessing efficiently the distributed traffic datamanagementframework has been discussed in our previous research work[21] In this section it is assumed that the basic vehicle stateinformation can be collected by the communication deviceequipped in vehicles and managed by the distributed trafficdata centers

For the granules in traffic network model U119881 U119875and U119878 are the vehicle universe platoon universe andsegment universe respectively The smallest granule in thismodel is vehicle granule the related attributes are listedas veh id veh speed veh acc veh pos segment id lane idpreceding id following id pre dist back dist veh class pas-sengers veh weight etc These attributes represent vehicle idspeed acceleration position road segment id that vehicletravels on lane id that vehicle travels on the distance to theprevious vehicle the distance to the vehicle in back vehicle

class the number of passengers in vehicle and vehicle weightin mergence respectively The attributes such as veh classpassengers and veh weight can be utilized for light phasepressure computation in the traffic light control strategydesign Inwhat follows the constructions of threemain trafficgranules are listed in detail

(1) Maintenance and construction of platoon granule Theequivalence relation R119875 on U119881 is used for constructinga platoon on each road segment it is defined as followsvehicles with neighboring position and same segment idsame lane id similar speed and acceleration The elementarytransfer is defined as follows For veh notin p ifMinspeed119901119897119886119905119900119900119899 leveh speed le Maxspeed119901119897119886119905119900119900119899 |V119890ℎ 119886119888119888| le acc119905ℎ119903119890119904ℎ119900119897119889 pre distle dist119905ℎ119903119890119904ℎ119900119897119889 or back dist le dist119905ℎ119903119890119904ℎ119900119897119889 then veh is trans-ferred into platoon p completely if (pre dist gt dist119905ℎ119903119890119904ℎ119900119897119889 andveh accgt acc119905ℎ119903119890119904ℎ119900119897119889) or (back dist gt dist119905ℎ119903119890119904ℎ119900119897119889 and veh accltacc119905ℎ119903119890119904ℎ119900119897119889) then veh is driving into platoon p since veh isaccelerating from the back of p or decelerating in front of pafter a period of time veh will drive into platoon p In thiscase it is said that veh is transferred into p partly For visinpif there is |V119890ℎ 119886119888119888| gt acc119905ℎ119903119890119904ℎ119900119897119889 veh in p is accelerating ordecelerating after a period of time veh will leave platoon pIn this case it is said that veh is transferred out of p partly Ifveh is the head or rear of p when back dist gt dist119905ℎ119903119890119904ℎ119900119897119889 orpre dist gt dist119905ℎ119903119890119904ℎ119900119897119889 it is said that veh has been transferredout of p completely

(2) Maintenance and construction of road segment Forthe universe 119880119875 the basic granule is platoon the relatedattributes including platoon id ave speed max speed minspeed ave headwaymax headwaymin headway veh countseg id lane id platoon weight rightofway flag etc Theseattributes represent the vehicle id average speed maximumspeed of platoon minimum speed of platoon average head-way maximum headway minimum headway number ofvehicles in platoon segment id that the platoon is travellingon lane id the weight of platoon and whether the platoonobtains the right of way to pass the downstream intersectionrespectively The equivalence relation R119878 on U119875 is definedas follows platoons with same segment id and lane id Theelementary transfer of platoon granule on segment granuleis defined as follows For 119901 isin 119904 if rightofway flag is true andveh count lt veh count119905ℎ119903119890119904ℎ119900119897119889 it is said that platoon119901 canpassthrough the downstream intersection in green time then 119901 istransferred out of segment 119904 completely if rightofway flag istrue and veh count gt veh count119905ℎ119903119890119904ℎ119900119897119889 it is said that part ofplatoon 119901 will pass through the downstream intersection ingreen time then 119901 is transferred out of segment 119904 partly For119901 notin 119904 there are similar situations

(3) Maintenance and construction of subregion granuleFor constructing the subregion granule uid is the basicgranule which constitutes the universe U119880 The relatedattributes include uid id upseg state downseg state andcorrelation degree representing uid id state of upstream seg-ment state of downstream segment and correlation degreeof the upstream and downstream segments respectively Thestate of segment can be extracted from platoon granules on itusing the attributes seg id veh count ave speed queuelengthand seg cap where seg cap is the capacity of a road segmentThe correlation degree can be calculated according to the state

Mathematical Problems in Engineering 7

qab

a bi

qghfgh

fab

gh

d

d

qgd

qgd

rbc

rbc

qbc

qbc

c

c

Figure 5 Vehicle flows on the upstream and downstream segments

and capacity of upstreamanddownstream segments [22]Theequivalence relation R119878119877 onU119878 is defined as follows segmentshave similar state and the correlation degree is more than agiven thresholdThe elementary transfer is defined as followsFor uid isinSR if the number of vehicles and the average speedof upstream and downstream segments are both exceedingthe given range of subregion SR that is the upstream anddownstream segment states of uid are not close to the stateof SR it is said that uid is transferred out of SR completely Ifonly the downstream or upstream segment state is not closeto the state of SR it is said that uid is transferred out ofSR partly For uid notin SR if uid is adjacent with SR and theupstream and downstream segment states of uid are similarto the state of SR it is said that uid is transferred into SRcompletely If only upstream or downstream segment state issimilar to the SR state it is said that uid is transferred into SRpartly

Based on the construction and maintenance of granulesin the proposed traffic network model discussed above thesetraffic granules can be maintained in a distributed modeunder a distributed traffic data management frameworkThen the traffic information of different layers and granular-ities can be extracted Based on the multigranularity trafficinformation including vehicle information platoon infor-mation segment information and subregion informationthe traffic control and management system may adopt moresuitable strategies to improve the efficiency However stillsome issues have not been presented in detail in this proposedmodel such as the parameters selection of traffic statetraffic state computation of road segment and the correlationdegree calculation between downstream and upstream roadsegmentsThese problems will be researched in further workIn this paper to illustrate the validity and feasibility ofthe proposed hierarchical multigranularity traffic networkmodel we propose a max-pressure based traffic controlstrategy in the next section

3 Traffic Control Strategy Based onthe Proposed Traffic Model

In this section an application based on the proposed trafficnetwork model is discussed The proposed traffic networkmodel is combined with the max-pressure traffic controlalgorithm to design an extended max-pressure traffic controlstrategy in which the light phase pressure is computed usingthe rough representations of platoon and segment granules

31 Max-Pressure Traffic Control Strategy The max-pressuretraffic control strategy is proposed by Varaiya [23] based on

the backpressure traffic control algorithm [24 25] Accordingto the strategy the urban traffic network is viewed as a queu-ing network and at each time slot the active phase of eachintersection is selected depending on the light phase pressurecomputed using local queue length of each connected roadsegment

As shown in Figure 5 on segments a and g there arequeued vehicles waiting for passing through intersection ithe vehicle flows f 119886119887 and f 119892ℎ are controlled by light phase pof intersection i For the vehicle flow f 119886119887 when it enters thedownstream segment b it will drive into the queued vehicles1199021198871198881015840 and 11990211988711988810158401015840 according to the proportions 1199031198871198881015840 and 11990311988711988810158401015840 where1198881015840 and 11988810158401015840 are the downstream segments of bThere are similarsituations for the vehicle flow 119891119892ℎ The pressure of light phasep of intersection i is computed using (25) and the active lightphase in next time slot is selected using (26)

119901119903119890119904119904119906119903119890119901 = sum119891119886119887isin119901

120583119886119887 (119901) (119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)) (25)

where119891119886119887 isin 119901means that the vehicle flow f 119886119887 is controlled bylight phase p120583119886119887(p) is the number of vehicles passing throughintersection per unit time when p is activated

119901lowast = argmax119901isin119875119894

(119901119903119890119904119904119906119903119890119901 (119905)) (26)

where pisinP119894 means that p is one of the light phases of lightsignal group 119875119894

In this method the light phase pressure is concernedwith the number of queued vehicles in front of the stop lineon the upstream and downstream segments The detailedvehicle information the arriving vehicles (for the upstreamsegment) and the departing vehicles (for the downstreamsegment) are not considered Based on the proposed trafficnetwork model there may be several platoons with differentstates on the upstream and downstream segments On theupstream segment there are stopped platoons in front ofthe stop line and arriving platoons heading for the stop lineOn the downstream segment there may be the departingplatoon when the corresponding downstream light phaseis activated These situations are considered in the trafficnetwork rough representation discussed in previous sectionsTherefore in the next section the light phase pressurecomputation method is extended based on the proposedhierarchical multigranularity traffic network model

32 Extended Max-Pressure Traffic Control Algorithm Actu-ally the light phase pressure is affected by four types of

8 Mathematical Problems in Engineering

platoons on the segment For the platoons on upstreamsegment the stopped platoons waiting for passing throughthe intersection the platoons that are driving towards thestop line and the platoons on the neighboring lanes that aregoing to drive into the corresponding lanes will affect thelight phase pressure For the platoons on downstream seg-ment the departing platoon affects the light phase pressurecomputation too if the downstream light phase is activatedSince the rough representation of traffic granules includesdetailed traffic information the vehicle weight the numberof passengers vehicle class etc also can be considered Thenbased on the rough representation of platoon and segment inthe proposed traffic network model the light phase pressurecan be computed more accurately

To simplify the description the rough set representationof 119883lowast with elementary transfer in two directions is denotedby the lower approximation set appr (119883lowast 119865 cup 1198651015840) and theupper approximation set 119886119901119901119903minus(119883lowast 119865 cup 1198651015840) appr (119883lowast F)represents the one-direction lower approximation set withelement transfer-in operations and appr (119883lowast 1198651015840) representsthe one direction lower approximation set with elementtransfer-out operations The upper assistant set As+(119883lowast F)represents the element set that elements enter 119883lowast partlyunder transfer operations F The lower assistant set Asminus(119883lowast1198651015840) represents the element set that elements leave 119883lowast partlyunder transfer operations 1198651015840 Then the pressure from eachplatoon can be computed using

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) = sum

V119890ℎisin119886119901119901119903 (119875lowast119886119887119865119881)

120572119908V119890ℎ

+ sumV119890ℎisin119860119904+(119875lowast

119886119887119865119881)

120573119908V119890ℎ

(27)

where 120572 120573 is the coefficient of the vehicle weight and 119908V119890ℎ isthe vehicle weight determined by the vehicle type it meansthat the different types of vehicles generate different pressuredue to the different number of passengers and emergence ofthese vehicles

The pressure of the upstream segment is computed using

119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

= sum119901119897119905isin119886119901119901119903 (119904119890119892lowast

119886119887119865119875)cup119860119904+(119904119890119892lowast

119886119887119865119875)

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) (28)

where useg represents an upstream segment and seg119886119887 is asubgranule on segment a in which platoons on segment aare waiting for driving into segment b

The pressure of the downstream segment is computedusing

119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887 119905)

= sum119888isin1198881015840 11988810158401015840

119901119897119905isin119886119901119901119903 (119904119890119892lowast1198871198881198651015840119875)cup119860119904minus(119904119890119892lowast119887119888119865

1015840119875)

119903119887119888119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119887119888 119905) (29)

According to (28) (29) the vehicles entering upstreamsegment and departing from downstream segment are con-sidered for the coordination with neighboring intersections

The pressure of light phase 119901119894119895 is computed using

119901119903119890119904119904119906119903119890119901ℎ119886119904119890 (119901119894119895 119905)

= sum119891119886119887isin119901

119894119895

119891119887119888isin119901119894+1119896

(119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

minus 119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887119888 119905)) 120583119886119887 (119905)

(30)

where 119901119894119895 is the jth light phase of intersection i 119901119894+1

119896 is thecooperative light phase of the downstream intersection i+1

Based on the extended pressure computation methoddescribed above the vehicle type vehicle state and platoonstate upstream light phase state and downstream light phasestate are considered If the element transfer operations Fand 1198651015840 are both empty the extended pressure computationmethod degenerates into max-pressure traffic control algo-rithm If 119865119875 = Φ and 1198651015840

119875 = Φ the arriving vehiclesfrom upstream segment are considered in the pressurecomputation in this case the coordination with the upstreamintersection light phase can be achieved If 119865119875 = Φ and1198651015840

119875 = Φ the departing vehicles on the downstream segmentare considered to achieve the coordination with downstreamintersection light phaseThe light phase with max-pressure isselected to be activated in the next time slot using (26)

Furthermore the vehicle class the number of passengersand vehicle emergency also can be considered using thevehicle weight Therefore based on the extended light phasecomputation method the real situations of traffic controlsystem can be reflected The strategy may be more suitablefor the actual situation of urban traffic network To illustratethe effectiveness and feasibility of the proposed model andalgorithm simulations are carried out in the next section

4 Simulations

In this section the simulation is carried out to analyze theeffectiveness and feasibility of the proposed method Thesimulation traffic network consists of 15 intersections and 76links constructed in Vissim The network includes 16 ingresssegments and 16 exit segments as shown in Figure 6(a)

The vehicle input of each ingress link is 800 vehh Thereare 15 signal controllers and 4 light phases for each controllerie north-south straight phase north-south left-turn phasewest-east straight phase and west-east left-turn phase Thereare 3 lanes on each link The left-turn vehicle flow drives onLane 3 and the straight vehicle flow drives on Lane 2The twovehicle flows are controlled by the traffic lightThe right-turnvehicle flow drives on Lane 1 and is free of the traffic lightTo obtain the traffic parameters during simulation 60 queuecounters 60 travel time sections and 60 routing decisions arelaid on the simulation traffic network as shown in Figures6(b) 6(c) and 6(d) During the simulation the green lighttime of each light phase is assumed to be 21 s which is thepedestrian clearance time (the road width is about 21 m andthe pedestrians speed is about 1 ms) [26] The yellow lighttime is assumed to be 3 s

Mathematical Problems in Engineering 9

(a)

(b) (c) (d)

Figure 6 The simulation traffic network and settings

Table 1 Traffic network performance comparison

Parameters FTMethod MPMethod EMP MethodTotal travel time (h) 633947 588139 568009Total delay time (h) 350255 298110 277457Total stopped delay (h) 253507 225692 205465Average delay time (s) 130584 111420 104057Average stopped delay (s) 94514 84353 77057Average speed (kmh) 23527 25916 26883Number of vehicles in network 448 368 334Number of vehicles that left network 9208 9264 9265Total Number of vehicles 9656 9632 9599

To illustrate the efficiency and feasibility of the pro-posed traffic network model and traffic control algorithmthree traffic light control methods are implemented in thissection including fixed-time control (FT method) max-pressure control (MP method) and the extended max-pressure control (EMP method) To simplify the simulationthe extended pressure computation only considers the caseof 119865119875 = Φ and 1198651015840

119875 = Φ in this paper the other cases will beconsidered in further research works The three traffic lightcontrol methods are implemented in Visual Studio 2010Thesimulation programs communicate with Vissim through theVissim COM programming interface to obtain the vehiclestate and traffic parameters to decide the traffic light signalat each time slot The simulation runs for 3600 s and thetraffic network performance is evaluated from 1000 s to 3600s usingVissimThis is because in the first 1000 s of simulationtime there are not enough vehicles entering the simulationtraffic network

The simulation results of the traffic network are listed inTable 1 According to the simulation results EMP methodbased on the proposed hierarchical multigranularity trafficnetwork model achieves better performance because the

light phase pressure computed using (27)-(30) is closer tothe actual traffic situation in which the vehicles on a roadsegment are represented using rough set form The depart-ing vehicles of downstream segments considered in thissimulation can achieve the coordination with downstreamintersections

As shown in Figure 7 the average queue length underEMP method is shorter than the other traffic light controlmethods The shorter queue length in traffic network meansit can effectively reduce the occurrence of spill-over ondownstream road segment so as to avoid the traffic collapsedue to vehicle queue overflow EMP method is based onthe proposed hierarchical multigranularity traffic networkmodel in which the traffic elements dynamic characteris-tics are depicted accurately and the light phase states ofdownstream intersections are also considered to computethe light phase pressure therefore EMP method achievesbetter performance As shown in Figure 8 the averagetravel time under EMP control is lower than the other twomethods

Furthermore using the extended pressure computationmethod based on the proposed traffic network the priority

10 Mathematical Problems in Engineering

MPFTEMP

05

101520253035404550

Que

ue L

engt

h (v

eh)

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000Time (s)

Figure 7 Average queue length of the three-traffic-light controlmethod

MPFTEMP

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000

Time (s)

0102030405060708090

100

Trav

el Ti

me (

s)

Figure 8 Average travel time of the three-traffic-light controlmethod

control for some particular vehicles can be achieved con-veniently since different classes of vehicles can be set withdifferent vehicle weight in the EMPmethod Different weightvalues can result in different traffic efficiency for differenttypes of vehicles For example when the type of bus is addedto the simulation traffic network the higher weight valuecan make buses have higher average speed and the otherclasses of vehicles maybe achieve lower average speed Forthis issue the vehicle weight selection is important As shownin Figure 9 the different bus weight values are simulatedand they result in different average speed According to themultiple simulation results and the real traffic demand thevehicle weight can be determined

To sum up since the hierarchical multigranularity trafficmodel can represent the actual traffic network situationsaccurately and the EMP method can achieve traffic controlflexibly this paper may be a useful attempt in solving urbantraffic control problem

5 Conclusions

In this paper a hierarchical multigranularity traffic networkis proposed and on the basis of the proposed traffic networkan extended max-pressure traffic control method is imple-mented The simulation results illustrate that the proposedtraffic network model and EMP method achieve better

Average speed [Car]Average speed [Bus]Average speed [All vehicles]

24245

25255

26265

27

Aver

age s

peed

15 2 25 3 35 4 45 5 551Vehicle weight of Bus

Figure 9 Simulation results with different bus weight values

performance In the proposed model the traffic elementsin urban traffic network are represented using S-rough setin which the dynamic characteristics of traffic elementsare depicted This type of traffic element description canrepresent the traffic network situation accurately so as todesign the more efficient traffic control strategy

However the hierarchical multigranularity traffic net-work model is only a preliminary exploration and it needsto be further improved in the future research work such asthe traffic subregion formulation and the boundary controlof traffic subregion For the traffic subregion formulationthe correlation of downstream and upstream road segmentsshould be computed considering traffic states and the geo-graphical characteristics of each pair of upstream and down-stream segments The control strategy of subregion boundarycan compute the traffic pressure considering the remainingcapacity of downstream segment to speed up the vehicleflow release towards to the segment with larger remainingcapacity These issues will be studied in our further researchwork to improve the hierarchical multigranularity traffic net-workmodel Furthermore in the future work the completelymultigranularity traffic control strategies should be furtherresearched using granular computing theory to achieve thetraffic system multigranularity control completely

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial NaturalScience Foundation China [grant number ZR2018FM027]Project of Shandong Province Higher Educational Scienceand Technology Program [grant number J17KB181] KeyResearch and Development Project in Shandong ProvinceChina [grant numbers 2015GGX101048 2016GSF120009]

Mathematical Problems in Engineering 11

and National Natural Science Foundation of Youth FundProject [grant number 61501284]

References

[1] Q He K L Head and J Ding ldquoMulti-modal traffic signalcontrol with priority signal actuation and coordinationrdquoTrans-portation Research Part C Emerging Technologies vol 46 pp65ndash82 2014

[2] Y Yao ldquoInterpreting concept learning in cognitive informaticsand granular computingrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 39 no 4 pp 855ndash8662009

[3] J T Yao A V Vasilakos andW Pedrycz ldquoGranular computingperspectives and challengesrdquo IEEE Transactions on Cyberneticsvol 43 no 6 pp 1977ndash1989 2013

[4] H Liu andM Cocea ldquoGranular computing-based approach forclassification towards reduction of bias in ensemble learningrdquoGranular Computing vol 2 no 3 pp 131ndash139 2017

[5] X Ji L Liu and Q Wu ldquoTraffic management and decision-making support based on granular computingrdquo Journal ofHighway and Transportation Research and Development vol2008 no 10 pp 129ndash133 2008

[6] S Dewen C Xinhong C Jing and F Xujian ldquoGranularcomputing in intelligent transportation An exploratory studyrdquoCybernetics and Information Technologies vol 15 no 6 pp 122ndash134 2015

[7] C Yao J-C Gu and Q Yang ldquoTraffic congestion identificationand analysis of urban road network based on granular comput-ingrdquo Applied Mechanics and Materials vol 641-642 pp 916ndash922 2014

[8] T Li D Ruan Y Shen E Hermans and G Wets ldquoA newweighting approach based on rough set theory and granularcomputing for road safety indicator analysisrdquo InternationalJournal of Computational Intelligence vol 32 no 4 pp 517ndash5342016

[9] F He R Liu Y Zhang et al ldquoBGrR large-scale network routingspeedup techniques based on granular computingrdquo ComputerScience vol 41 no 11 pp 265ndash281 2014

[10] R Deb and A W-C Liew ldquoMissing value imputation forthe analysis of incomplete traffic accident datardquo InformationSciences vol 339 pp 274ndash289 2016

[11] S Hao L Yang and Y Shi ldquoData-driven car-following modelbased on rough set theoryrdquo IET Intelligent Transport Systemsvol 12 no 1 pp 49ndash57 2018

[12] L Livi and A Sadeghian ldquoGranular computing computationalintelligence and the analysis of non-geometric input spacesrdquoGranular Computing vol 1 no 1 pp 13ndash20 2016

[13] X Yang H Da-Wei S Bing and Z Duo-Jia ldquoCity traffic flowbreakdown prediction based on fuzzy rough setrdquo Open Physicsvol 15 no 1 pp 292ndash299 2017

[14] X Xiong L Chen and J Liang ldquoAnalysis of roadway trafficaccidents based on rough sets and Bayesian networksrdquo Promet- Traffic - Traffico vol 30 no 1 pp 71ndash81 2018

[15] M Deshpande and P Bajaj ldquoPerformance improvement of traf-fic flow prediction model using combination of support vectormachine and rough setrdquo International Journal of ComputerApplications vol 163 no 2 pp 31ndash35 2017

[16] Y Y Yao ldquoGranular computing basic issues and possible solu-tionsrdquo in Proceedings of the Fi13h International Conference onComputing and Information vol 1 pp 186ndash189 2000

[17] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Science vol 11 no 5 pp 341ndash356 1982

[18] K Shi ldquoS-rough sets and its applications in diagnosis-rec-ognition for diseaserdquo in Proceedings of the 1st InternationalConference on Machine Learning and Cybernetics pp 50ndash54IEEE Beijing China 2002

[19] K Q Shi and Y Q Cui ldquoS-Rough sets and their general struc-turesrdquo Journal of Shandong University Natural Science Shan-dong Daxue Xuebao Lixue Ban vol 37 no 6 pp 471ndash474 2002

[20] H Haiqing X Peijun and F Haiyan ldquoS-rough recognitionof knowledge and general threshold encryption authenticationscheme of recognition conclusionrdquo in Proceedings of the SixthInternational Conference on Intelligent Systems Design andApplications pp 816ndash824 Jian China October 2006

[21] S Hao L Yang and Y Shi ldquoReal-time data-driven trajectoryreconstruction based on rough set theoryrdquo Advances in Trans-portation Studies vol 44 pp 119ndash132 2018

[22] Y Yang L Jia Y Qin S Han and H Dong ldquoUnderstandingstructure of urban traffic network based on spatial-temporalcorrelation analysisrdquo Modern Physics Letters B CondensedMatter Physics Statistical Physics AtomicMolecular andOpticalPhysics vol 31 no 22 Article ID 1750230 19 pages 2017

[23] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[24] TWongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 USA September 2012

[25] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo IFAC Proceedings Volumes vol 47 no 3 pp11332ndash11337 2014

[26] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

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Mathematical Problems in Engineering 5

Traffic NetworkSub-Region

Intersection amp Segments

Intersection amp Segments

helliphellip

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

helliphellip

Sub-Region

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

helliphellip

Element transfer at platoon level

Element transfer at segment level

Element transfer at region level

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

PlatoonVehicle Vehicle Vehicle Vehicle Vehicle Vehicle

helliphellip

helliphellip

Figure 4 Hierarchical multigranularity traffic network model

(119877119875 119865119881 cup 1198651015840119881)∘ (119875119897119905lowast) = cup V119890ℎ | [119891119881 (V119890ℎ)]119877119875 cap 119875119897119905lowast

= and [V119890ℎ]119877119875 cap 119875119897119905lowast = (12)

119860119904+ (119875119897119905lowast) = 119891119881 (V119890ℎ) | V119890ℎ isin 119880119881 V119890ℎnotin 119875119897119905 119891119881 (V119890ℎ) enters 119875119897119905 partly (13)

119860119904minus (119875119897119905lowast) = V119890ℎ | V119890ℎisin 119875119897119905 1198911015840

119881 (V119890ℎ) leaves 119875119897119905 partly (14)

(2) Segment granule Let these platoons travelling on the samesegment be the universe U119875 R119878119890119892 denotes the equivalencerelation about the attributes of average speed and positionon U119875 Seg denotes the segment with multiple platoonsand 119878119890119892lowast denotes the two-direction singular set of SegF119875cup1198651015840

119875 is the elementary transfer on U119875 where 119865119875 =1198911198751 1198911198752 119891119875119898 and 1198651015840

119875 = 11989110158401198751 1198911015840

1198752 1198911015840119875119898 119891119875(119901119897119905)

denotes that platoon plt is driving into Seg 1198911198751015840(119901119897119905) denotes

that platoon plt is driving out of SegThen the singular roughset representation of segment is defined as

119878119890119892lowast = 119878119890119892 119901119897119905 | 119901119897119905 isin 119878119890119892 1198911015840119875 (119901119897119905) notin 119878119890119892

cup 119901119897119905 | 119901119897119905 isin 119880119875 119901119897119905 notin 119878119890119892 119891119875 (119901119897119905) isin 119878119890119892 (15)

(119877119878119890119892 119865119875 cup 1198651015840119875)∘ (119878119890119892lowast) = cup 119901119897119905 | [119891119875 (119901119897119905)]119877119878119890119892

cup [119901119897119905]119877119878119890119892 sube 119878119890119892lowast(16)

(119877119878119890119892 119865119875 cup 1198651015840119875)∘ (119878119890119892lowast) = cup 119901119897119905 | [119891119875 (119901119897119905)]119877119878119890119892

cap 119878119890119892lowast = and [119901119897119905]119877119878119890119892 cap 119878119890119892lowast = (17)

119860119904+ (119878119890119892lowast) = 119891119875 (119901119897119905) | 119901119897119905 isin 119880119875 119901119897119905notin 119878119890119892 119891119875 (119901119897119905) enters 119878119890119892 partly (18)

119860119904minus (119878119890119892lowast) = 119901119897119905 | 119901119897119905isin 119878 1198911015840

119875 (119901119897119905) leaves 119878119890119892 partly (19)

Actually for each road segment of traffic network theremay be several lanes corresponding to several vehicle flowscontrolled by different traffic light phases of intersection Tofacilitate the control strategy design the segment granule ispartitioned into subgranules corresponding to different lanesFor example the granule seg119886119887 denotes the platoon rough seton segment a where these platoons are leaving for segment bThe rough set definition of seg119886119887 can also be described using(15)-(19) with attribute lane id added In this case the stateseg119886 can be calculated by all the granules seg119886119887

(3) Subregion granule Since there are both segmentsand intersections in the subregion granule and the states ofsegments with the connected intersections affect the trafficcontrol strategy there are correlations between upstream anddownstream segments the ldquoUpstreamSegment-Intersection-DownstreamSegmentrdquo (simplified as uid) is viewed as thebasic granule in subregion granule definition These uidscompose the universe U119878 Let SR denote a subregion in

6 Mathematical Problems in Engineering

which the segments of uids have similar traffic state andthere are correlations between the upstream and downstreamsegment of each uid Since the uid is made up of upstreamsegment useg downstream segment dseg and the connectedintersection intersc if the useg and dseg are both transferredinto or out of SR it is to say that the uid is transferred into orout of SR completely If only useg or dseg is transferred intoor out of SR it is to say that the uid is transferred into or outof SR partly Let 119878119877lowast denote the two-direction singular setof SR and 119865119878 cup 1198651015840

119878 is the elementary transfer on U119878 where119865119880 = 1198911198801 1198911198802 119891119880119898 1198651015840119880 = 1198911015840

1198801 11989110158401198802 1198911015840

119880119898f119880(uid) indicates that the basic granule uid is transferred intoSR due to the state changing 1198911015840

119880(uid) indicates that uid istransferred out of SR due to the state changing The roughrepresentation of 119878119877lowast can be defined as

119878119877lowast = 119878119877 119906119894119889 | 119906119894119889 isin 119878119877 1198911015840119880 (119906119894119889) notin 119878119877

cup 119906119894119889 | 119906119894119889 isin 119880119880 119906119894119889 notin 119878119877 119891119880 (119906119894119889) isin 119878119877 (20)

(119877119878119877 119865119880 cup 1198651015840119880)∘ (119878119877lowast) = cup 119906119894119889 | [119891119880 (119906119894119889)]119877119878119877

cup [119906119894119889]119877119878119877 sube 119878119877lowast (21)

(119877119878119877 119865119880 cup 1198651015840119880)∘ (119878119877lowast) = cup 119906119894119889 | [119891119880 (119906119894119889)]119877119878119877

cap 119878119877lowast = and [119906119894119889]119877119878119877 cap 119878119877lowast = (22)

119860119904+ (119878119877lowast) = 119891119880 (119906119894119889) | 119906119894119889 isin 119880119878119877 119906119894119889notin 119878119877 119891119880 (119906119894119889) enters 119878119877 partly (23)

119860119904minus (119878119877lowast) = 119906119894119889 | 119906119894119889isin 119878119877 1198911015840

119880 (119906119894119889) leaves 119878119877 partly (24)

23Maintenance and Construction of Traffic Granules Basedon the definitions of the hierarchical multigranularity trafficnetwork model traffic control strategy can be designedaiming at different traffic demand to improve the efficiency ofurban traffic control systemThe key issue that this model canbe utilized for traffic control and management is to maintainthe integrity of these granules in real time For the traffic dataprocessing efficiently the distributed traffic datamanagementframework has been discussed in our previous research work[21] In this section it is assumed that the basic vehicle stateinformation can be collected by the communication deviceequipped in vehicles and managed by the distributed trafficdata centers

For the granules in traffic network model U119881 U119875and U119878 are the vehicle universe platoon universe andsegment universe respectively The smallest granule in thismodel is vehicle granule the related attributes are listedas veh id veh speed veh acc veh pos segment id lane idpreceding id following id pre dist back dist veh class pas-sengers veh weight etc These attributes represent vehicle idspeed acceleration position road segment id that vehicletravels on lane id that vehicle travels on the distance to theprevious vehicle the distance to the vehicle in back vehicle

class the number of passengers in vehicle and vehicle weightin mergence respectively The attributes such as veh classpassengers and veh weight can be utilized for light phasepressure computation in the traffic light control strategydesign Inwhat follows the constructions of threemain trafficgranules are listed in detail

(1) Maintenance and construction of platoon granule Theequivalence relation R119875 on U119881 is used for constructinga platoon on each road segment it is defined as followsvehicles with neighboring position and same segment idsame lane id similar speed and acceleration The elementarytransfer is defined as follows For veh notin p ifMinspeed119901119897119886119905119900119900119899 leveh speed le Maxspeed119901119897119886119905119900119900119899 |V119890ℎ 119886119888119888| le acc119905ℎ119903119890119904ℎ119900119897119889 pre distle dist119905ℎ119903119890119904ℎ119900119897119889 or back dist le dist119905ℎ119903119890119904ℎ119900119897119889 then veh is trans-ferred into platoon p completely if (pre dist gt dist119905ℎ119903119890119904ℎ119900119897119889 andveh accgt acc119905ℎ119903119890119904ℎ119900119897119889) or (back dist gt dist119905ℎ119903119890119904ℎ119900119897119889 and veh accltacc119905ℎ119903119890119904ℎ119900119897119889) then veh is driving into platoon p since veh isaccelerating from the back of p or decelerating in front of pafter a period of time veh will drive into platoon p In thiscase it is said that veh is transferred into p partly For visinpif there is |V119890ℎ 119886119888119888| gt acc119905ℎ119903119890119904ℎ119900119897119889 veh in p is accelerating ordecelerating after a period of time veh will leave platoon pIn this case it is said that veh is transferred out of p partly Ifveh is the head or rear of p when back dist gt dist119905ℎ119903119890119904ℎ119900119897119889 orpre dist gt dist119905ℎ119903119890119904ℎ119900119897119889 it is said that veh has been transferredout of p completely

(2) Maintenance and construction of road segment Forthe universe 119880119875 the basic granule is platoon the relatedattributes including platoon id ave speed max speed minspeed ave headwaymax headwaymin headway veh countseg id lane id platoon weight rightofway flag etc Theseattributes represent the vehicle id average speed maximumspeed of platoon minimum speed of platoon average head-way maximum headway minimum headway number ofvehicles in platoon segment id that the platoon is travellingon lane id the weight of platoon and whether the platoonobtains the right of way to pass the downstream intersectionrespectively The equivalence relation R119878 on U119875 is definedas follows platoons with same segment id and lane id Theelementary transfer of platoon granule on segment granuleis defined as follows For 119901 isin 119904 if rightofway flag is true andveh count lt veh count119905ℎ119903119890119904ℎ119900119897119889 it is said that platoon119901 canpassthrough the downstream intersection in green time then 119901 istransferred out of segment 119904 completely if rightofway flag istrue and veh count gt veh count119905ℎ119903119890119904ℎ119900119897119889 it is said that part ofplatoon 119901 will pass through the downstream intersection ingreen time then 119901 is transferred out of segment 119904 partly For119901 notin 119904 there are similar situations

(3) Maintenance and construction of subregion granuleFor constructing the subregion granule uid is the basicgranule which constitutes the universe U119880 The relatedattributes include uid id upseg state downseg state andcorrelation degree representing uid id state of upstream seg-ment state of downstream segment and correlation degreeof the upstream and downstream segments respectively Thestate of segment can be extracted from platoon granules on itusing the attributes seg id veh count ave speed queuelengthand seg cap where seg cap is the capacity of a road segmentThe correlation degree can be calculated according to the state

Mathematical Problems in Engineering 7

qab

a bi

qghfgh

fab

gh

d

d

qgd

qgd

rbc

rbc

qbc

qbc

c

c

Figure 5 Vehicle flows on the upstream and downstream segments

and capacity of upstreamanddownstream segments [22]Theequivalence relation R119878119877 onU119878 is defined as follows segmentshave similar state and the correlation degree is more than agiven thresholdThe elementary transfer is defined as followsFor uid isinSR if the number of vehicles and the average speedof upstream and downstream segments are both exceedingthe given range of subregion SR that is the upstream anddownstream segment states of uid are not close to the stateof SR it is said that uid is transferred out of SR completely Ifonly the downstream or upstream segment state is not closeto the state of SR it is said that uid is transferred out ofSR partly For uid notin SR if uid is adjacent with SR and theupstream and downstream segment states of uid are similarto the state of SR it is said that uid is transferred into SRcompletely If only upstream or downstream segment state issimilar to the SR state it is said that uid is transferred into SRpartly

Based on the construction and maintenance of granulesin the proposed traffic network model discussed above thesetraffic granules can be maintained in a distributed modeunder a distributed traffic data management frameworkThen the traffic information of different layers and granular-ities can be extracted Based on the multigranularity trafficinformation including vehicle information platoon infor-mation segment information and subregion informationthe traffic control and management system may adopt moresuitable strategies to improve the efficiency However stillsome issues have not been presented in detail in this proposedmodel such as the parameters selection of traffic statetraffic state computation of road segment and the correlationdegree calculation between downstream and upstream roadsegmentsThese problems will be researched in further workIn this paper to illustrate the validity and feasibility ofthe proposed hierarchical multigranularity traffic networkmodel we propose a max-pressure based traffic controlstrategy in the next section

3 Traffic Control Strategy Based onthe Proposed Traffic Model

In this section an application based on the proposed trafficnetwork model is discussed The proposed traffic networkmodel is combined with the max-pressure traffic controlalgorithm to design an extended max-pressure traffic controlstrategy in which the light phase pressure is computed usingthe rough representations of platoon and segment granules

31 Max-Pressure Traffic Control Strategy The max-pressuretraffic control strategy is proposed by Varaiya [23] based on

the backpressure traffic control algorithm [24 25] Accordingto the strategy the urban traffic network is viewed as a queu-ing network and at each time slot the active phase of eachintersection is selected depending on the light phase pressurecomputed using local queue length of each connected roadsegment

As shown in Figure 5 on segments a and g there arequeued vehicles waiting for passing through intersection ithe vehicle flows f 119886119887 and f 119892ℎ are controlled by light phase pof intersection i For the vehicle flow f 119886119887 when it enters thedownstream segment b it will drive into the queued vehicles1199021198871198881015840 and 11990211988711988810158401015840 according to the proportions 1199031198871198881015840 and 11990311988711988810158401015840 where1198881015840 and 11988810158401015840 are the downstream segments of bThere are similarsituations for the vehicle flow 119891119892ℎ The pressure of light phasep of intersection i is computed using (25) and the active lightphase in next time slot is selected using (26)

119901119903119890119904119904119906119903119890119901 = sum119891119886119887isin119901

120583119886119887 (119901) (119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)) (25)

where119891119886119887 isin 119901means that the vehicle flow f 119886119887 is controlled bylight phase p120583119886119887(p) is the number of vehicles passing throughintersection per unit time when p is activated

119901lowast = argmax119901isin119875119894

(119901119903119890119904119904119906119903119890119901 (119905)) (26)

where pisinP119894 means that p is one of the light phases of lightsignal group 119875119894

In this method the light phase pressure is concernedwith the number of queued vehicles in front of the stop lineon the upstream and downstream segments The detailedvehicle information the arriving vehicles (for the upstreamsegment) and the departing vehicles (for the downstreamsegment) are not considered Based on the proposed trafficnetwork model there may be several platoons with differentstates on the upstream and downstream segments On theupstream segment there are stopped platoons in front ofthe stop line and arriving platoons heading for the stop lineOn the downstream segment there may be the departingplatoon when the corresponding downstream light phaseis activated These situations are considered in the trafficnetwork rough representation discussed in previous sectionsTherefore in the next section the light phase pressurecomputation method is extended based on the proposedhierarchical multigranularity traffic network model

32 Extended Max-Pressure Traffic Control Algorithm Actu-ally the light phase pressure is affected by four types of

8 Mathematical Problems in Engineering

platoons on the segment For the platoons on upstreamsegment the stopped platoons waiting for passing throughthe intersection the platoons that are driving towards thestop line and the platoons on the neighboring lanes that aregoing to drive into the corresponding lanes will affect thelight phase pressure For the platoons on downstream seg-ment the departing platoon affects the light phase pressurecomputation too if the downstream light phase is activatedSince the rough representation of traffic granules includesdetailed traffic information the vehicle weight the numberof passengers vehicle class etc also can be considered Thenbased on the rough representation of platoon and segment inthe proposed traffic network model the light phase pressurecan be computed more accurately

To simplify the description the rough set representationof 119883lowast with elementary transfer in two directions is denotedby the lower approximation set appr (119883lowast 119865 cup 1198651015840) and theupper approximation set 119886119901119901119903minus(119883lowast 119865 cup 1198651015840) appr (119883lowast F)represents the one-direction lower approximation set withelement transfer-in operations and appr (119883lowast 1198651015840) representsthe one direction lower approximation set with elementtransfer-out operations The upper assistant set As+(119883lowast F)represents the element set that elements enter 119883lowast partlyunder transfer operations F The lower assistant set Asminus(119883lowast1198651015840) represents the element set that elements leave 119883lowast partlyunder transfer operations 1198651015840 Then the pressure from eachplatoon can be computed using

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) = sum

V119890ℎisin119886119901119901119903 (119875lowast119886119887119865119881)

120572119908V119890ℎ

+ sumV119890ℎisin119860119904+(119875lowast

119886119887119865119881)

120573119908V119890ℎ

(27)

where 120572 120573 is the coefficient of the vehicle weight and 119908V119890ℎ isthe vehicle weight determined by the vehicle type it meansthat the different types of vehicles generate different pressuredue to the different number of passengers and emergence ofthese vehicles

The pressure of the upstream segment is computed using

119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

= sum119901119897119905isin119886119901119901119903 (119904119890119892lowast

119886119887119865119875)cup119860119904+(119904119890119892lowast

119886119887119865119875)

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) (28)

where useg represents an upstream segment and seg119886119887 is asubgranule on segment a in which platoons on segment aare waiting for driving into segment b

The pressure of the downstream segment is computedusing

119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887 119905)

= sum119888isin1198881015840 11988810158401015840

119901119897119905isin119886119901119901119903 (119904119890119892lowast1198871198881198651015840119875)cup119860119904minus(119904119890119892lowast119887119888119865

1015840119875)

119903119887119888119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119887119888 119905) (29)

According to (28) (29) the vehicles entering upstreamsegment and departing from downstream segment are con-sidered for the coordination with neighboring intersections

The pressure of light phase 119901119894119895 is computed using

119901119903119890119904119904119906119903119890119901ℎ119886119904119890 (119901119894119895 119905)

= sum119891119886119887isin119901

119894119895

119891119887119888isin119901119894+1119896

(119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

minus 119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887119888 119905)) 120583119886119887 (119905)

(30)

where 119901119894119895 is the jth light phase of intersection i 119901119894+1

119896 is thecooperative light phase of the downstream intersection i+1

Based on the extended pressure computation methoddescribed above the vehicle type vehicle state and platoonstate upstream light phase state and downstream light phasestate are considered If the element transfer operations Fand 1198651015840 are both empty the extended pressure computationmethod degenerates into max-pressure traffic control algo-rithm If 119865119875 = Φ and 1198651015840

119875 = Φ the arriving vehiclesfrom upstream segment are considered in the pressurecomputation in this case the coordination with the upstreamintersection light phase can be achieved If 119865119875 = Φ and1198651015840

119875 = Φ the departing vehicles on the downstream segmentare considered to achieve the coordination with downstreamintersection light phaseThe light phase with max-pressure isselected to be activated in the next time slot using (26)

Furthermore the vehicle class the number of passengersand vehicle emergency also can be considered using thevehicle weight Therefore based on the extended light phasecomputation method the real situations of traffic controlsystem can be reflected The strategy may be more suitablefor the actual situation of urban traffic network To illustratethe effectiveness and feasibility of the proposed model andalgorithm simulations are carried out in the next section

4 Simulations

In this section the simulation is carried out to analyze theeffectiveness and feasibility of the proposed method Thesimulation traffic network consists of 15 intersections and 76links constructed in Vissim The network includes 16 ingresssegments and 16 exit segments as shown in Figure 6(a)

The vehicle input of each ingress link is 800 vehh Thereare 15 signal controllers and 4 light phases for each controllerie north-south straight phase north-south left-turn phasewest-east straight phase and west-east left-turn phase Thereare 3 lanes on each link The left-turn vehicle flow drives onLane 3 and the straight vehicle flow drives on Lane 2The twovehicle flows are controlled by the traffic lightThe right-turnvehicle flow drives on Lane 1 and is free of the traffic lightTo obtain the traffic parameters during simulation 60 queuecounters 60 travel time sections and 60 routing decisions arelaid on the simulation traffic network as shown in Figures6(b) 6(c) and 6(d) During the simulation the green lighttime of each light phase is assumed to be 21 s which is thepedestrian clearance time (the road width is about 21 m andthe pedestrians speed is about 1 ms) [26] The yellow lighttime is assumed to be 3 s

Mathematical Problems in Engineering 9

(a)

(b) (c) (d)

Figure 6 The simulation traffic network and settings

Table 1 Traffic network performance comparison

Parameters FTMethod MPMethod EMP MethodTotal travel time (h) 633947 588139 568009Total delay time (h) 350255 298110 277457Total stopped delay (h) 253507 225692 205465Average delay time (s) 130584 111420 104057Average stopped delay (s) 94514 84353 77057Average speed (kmh) 23527 25916 26883Number of vehicles in network 448 368 334Number of vehicles that left network 9208 9264 9265Total Number of vehicles 9656 9632 9599

To illustrate the efficiency and feasibility of the pro-posed traffic network model and traffic control algorithmthree traffic light control methods are implemented in thissection including fixed-time control (FT method) max-pressure control (MP method) and the extended max-pressure control (EMP method) To simplify the simulationthe extended pressure computation only considers the caseof 119865119875 = Φ and 1198651015840

119875 = Φ in this paper the other cases will beconsidered in further research works The three traffic lightcontrol methods are implemented in Visual Studio 2010Thesimulation programs communicate with Vissim through theVissim COM programming interface to obtain the vehiclestate and traffic parameters to decide the traffic light signalat each time slot The simulation runs for 3600 s and thetraffic network performance is evaluated from 1000 s to 3600s usingVissimThis is because in the first 1000 s of simulationtime there are not enough vehicles entering the simulationtraffic network

The simulation results of the traffic network are listed inTable 1 According to the simulation results EMP methodbased on the proposed hierarchical multigranularity trafficnetwork model achieves better performance because the

light phase pressure computed using (27)-(30) is closer tothe actual traffic situation in which the vehicles on a roadsegment are represented using rough set form The depart-ing vehicles of downstream segments considered in thissimulation can achieve the coordination with downstreamintersections

As shown in Figure 7 the average queue length underEMP method is shorter than the other traffic light controlmethods The shorter queue length in traffic network meansit can effectively reduce the occurrence of spill-over ondownstream road segment so as to avoid the traffic collapsedue to vehicle queue overflow EMP method is based onthe proposed hierarchical multigranularity traffic networkmodel in which the traffic elements dynamic characteris-tics are depicted accurately and the light phase states ofdownstream intersections are also considered to computethe light phase pressure therefore EMP method achievesbetter performance As shown in Figure 8 the averagetravel time under EMP control is lower than the other twomethods

Furthermore using the extended pressure computationmethod based on the proposed traffic network the priority

10 Mathematical Problems in Engineering

MPFTEMP

05

101520253035404550

Que

ue L

engt

h (v

eh)

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000Time (s)

Figure 7 Average queue length of the three-traffic-light controlmethod

MPFTEMP

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000

Time (s)

0102030405060708090

100

Trav

el Ti

me (

s)

Figure 8 Average travel time of the three-traffic-light controlmethod

control for some particular vehicles can be achieved con-veniently since different classes of vehicles can be set withdifferent vehicle weight in the EMPmethod Different weightvalues can result in different traffic efficiency for differenttypes of vehicles For example when the type of bus is addedto the simulation traffic network the higher weight valuecan make buses have higher average speed and the otherclasses of vehicles maybe achieve lower average speed Forthis issue the vehicle weight selection is important As shownin Figure 9 the different bus weight values are simulatedand they result in different average speed According to themultiple simulation results and the real traffic demand thevehicle weight can be determined

To sum up since the hierarchical multigranularity trafficmodel can represent the actual traffic network situationsaccurately and the EMP method can achieve traffic controlflexibly this paper may be a useful attempt in solving urbantraffic control problem

5 Conclusions

In this paper a hierarchical multigranularity traffic networkis proposed and on the basis of the proposed traffic networkan extended max-pressure traffic control method is imple-mented The simulation results illustrate that the proposedtraffic network model and EMP method achieve better

Average speed [Car]Average speed [Bus]Average speed [All vehicles]

24245

25255

26265

27

Aver

age s

peed

15 2 25 3 35 4 45 5 551Vehicle weight of Bus

Figure 9 Simulation results with different bus weight values

performance In the proposed model the traffic elementsin urban traffic network are represented using S-rough setin which the dynamic characteristics of traffic elementsare depicted This type of traffic element description canrepresent the traffic network situation accurately so as todesign the more efficient traffic control strategy

However the hierarchical multigranularity traffic net-work model is only a preliminary exploration and it needsto be further improved in the future research work such asthe traffic subregion formulation and the boundary controlof traffic subregion For the traffic subregion formulationthe correlation of downstream and upstream road segmentsshould be computed considering traffic states and the geo-graphical characteristics of each pair of upstream and down-stream segments The control strategy of subregion boundarycan compute the traffic pressure considering the remainingcapacity of downstream segment to speed up the vehicleflow release towards to the segment with larger remainingcapacity These issues will be studied in our further researchwork to improve the hierarchical multigranularity traffic net-workmodel Furthermore in the future work the completelymultigranularity traffic control strategies should be furtherresearched using granular computing theory to achieve thetraffic system multigranularity control completely

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial NaturalScience Foundation China [grant number ZR2018FM027]Project of Shandong Province Higher Educational Scienceand Technology Program [grant number J17KB181] KeyResearch and Development Project in Shandong ProvinceChina [grant numbers 2015GGX101048 2016GSF120009]

Mathematical Problems in Engineering 11

and National Natural Science Foundation of Youth FundProject [grant number 61501284]

References

[1] Q He K L Head and J Ding ldquoMulti-modal traffic signalcontrol with priority signal actuation and coordinationrdquoTrans-portation Research Part C Emerging Technologies vol 46 pp65ndash82 2014

[2] Y Yao ldquoInterpreting concept learning in cognitive informaticsand granular computingrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 39 no 4 pp 855ndash8662009

[3] J T Yao A V Vasilakos andW Pedrycz ldquoGranular computingperspectives and challengesrdquo IEEE Transactions on Cyberneticsvol 43 no 6 pp 1977ndash1989 2013

[4] H Liu andM Cocea ldquoGranular computing-based approach forclassification towards reduction of bias in ensemble learningrdquoGranular Computing vol 2 no 3 pp 131ndash139 2017

[5] X Ji L Liu and Q Wu ldquoTraffic management and decision-making support based on granular computingrdquo Journal ofHighway and Transportation Research and Development vol2008 no 10 pp 129ndash133 2008

[6] S Dewen C Xinhong C Jing and F Xujian ldquoGranularcomputing in intelligent transportation An exploratory studyrdquoCybernetics and Information Technologies vol 15 no 6 pp 122ndash134 2015

[7] C Yao J-C Gu and Q Yang ldquoTraffic congestion identificationand analysis of urban road network based on granular comput-ingrdquo Applied Mechanics and Materials vol 641-642 pp 916ndash922 2014

[8] T Li D Ruan Y Shen E Hermans and G Wets ldquoA newweighting approach based on rough set theory and granularcomputing for road safety indicator analysisrdquo InternationalJournal of Computational Intelligence vol 32 no 4 pp 517ndash5342016

[9] F He R Liu Y Zhang et al ldquoBGrR large-scale network routingspeedup techniques based on granular computingrdquo ComputerScience vol 41 no 11 pp 265ndash281 2014

[10] R Deb and A W-C Liew ldquoMissing value imputation forthe analysis of incomplete traffic accident datardquo InformationSciences vol 339 pp 274ndash289 2016

[11] S Hao L Yang and Y Shi ldquoData-driven car-following modelbased on rough set theoryrdquo IET Intelligent Transport Systemsvol 12 no 1 pp 49ndash57 2018

[12] L Livi and A Sadeghian ldquoGranular computing computationalintelligence and the analysis of non-geometric input spacesrdquoGranular Computing vol 1 no 1 pp 13ndash20 2016

[13] X Yang H Da-Wei S Bing and Z Duo-Jia ldquoCity traffic flowbreakdown prediction based on fuzzy rough setrdquo Open Physicsvol 15 no 1 pp 292ndash299 2017

[14] X Xiong L Chen and J Liang ldquoAnalysis of roadway trafficaccidents based on rough sets and Bayesian networksrdquo Promet- Traffic - Traffico vol 30 no 1 pp 71ndash81 2018

[15] M Deshpande and P Bajaj ldquoPerformance improvement of traf-fic flow prediction model using combination of support vectormachine and rough setrdquo International Journal of ComputerApplications vol 163 no 2 pp 31ndash35 2017

[16] Y Y Yao ldquoGranular computing basic issues and possible solu-tionsrdquo in Proceedings of the Fi13h International Conference onComputing and Information vol 1 pp 186ndash189 2000

[17] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Science vol 11 no 5 pp 341ndash356 1982

[18] K Shi ldquoS-rough sets and its applications in diagnosis-rec-ognition for diseaserdquo in Proceedings of the 1st InternationalConference on Machine Learning and Cybernetics pp 50ndash54IEEE Beijing China 2002

[19] K Q Shi and Y Q Cui ldquoS-Rough sets and their general struc-turesrdquo Journal of Shandong University Natural Science Shan-dong Daxue Xuebao Lixue Ban vol 37 no 6 pp 471ndash474 2002

[20] H Haiqing X Peijun and F Haiyan ldquoS-rough recognitionof knowledge and general threshold encryption authenticationscheme of recognition conclusionrdquo in Proceedings of the SixthInternational Conference on Intelligent Systems Design andApplications pp 816ndash824 Jian China October 2006

[21] S Hao L Yang and Y Shi ldquoReal-time data-driven trajectoryreconstruction based on rough set theoryrdquo Advances in Trans-portation Studies vol 44 pp 119ndash132 2018

[22] Y Yang L Jia Y Qin S Han and H Dong ldquoUnderstandingstructure of urban traffic network based on spatial-temporalcorrelation analysisrdquo Modern Physics Letters B CondensedMatter Physics Statistical Physics AtomicMolecular andOpticalPhysics vol 31 no 22 Article ID 1750230 19 pages 2017

[23] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[24] TWongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 USA September 2012

[25] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo IFAC Proceedings Volumes vol 47 no 3 pp11332ndash11337 2014

[26] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

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6 Mathematical Problems in Engineering

which the segments of uids have similar traffic state andthere are correlations between the upstream and downstreamsegment of each uid Since the uid is made up of upstreamsegment useg downstream segment dseg and the connectedintersection intersc if the useg and dseg are both transferredinto or out of SR it is to say that the uid is transferred into orout of SR completely If only useg or dseg is transferred intoor out of SR it is to say that the uid is transferred into or outof SR partly Let 119878119877lowast denote the two-direction singular setof SR and 119865119878 cup 1198651015840

119878 is the elementary transfer on U119878 where119865119880 = 1198911198801 1198911198802 119891119880119898 1198651015840119880 = 1198911015840

1198801 11989110158401198802 1198911015840

119880119898f119880(uid) indicates that the basic granule uid is transferred intoSR due to the state changing 1198911015840

119880(uid) indicates that uid istransferred out of SR due to the state changing The roughrepresentation of 119878119877lowast can be defined as

119878119877lowast = 119878119877 119906119894119889 | 119906119894119889 isin 119878119877 1198911015840119880 (119906119894119889) notin 119878119877

cup 119906119894119889 | 119906119894119889 isin 119880119880 119906119894119889 notin 119878119877 119891119880 (119906119894119889) isin 119878119877 (20)

(119877119878119877 119865119880 cup 1198651015840119880)∘ (119878119877lowast) = cup 119906119894119889 | [119891119880 (119906119894119889)]119877119878119877

cup [119906119894119889]119877119878119877 sube 119878119877lowast (21)

(119877119878119877 119865119880 cup 1198651015840119880)∘ (119878119877lowast) = cup 119906119894119889 | [119891119880 (119906119894119889)]119877119878119877

cap 119878119877lowast = and [119906119894119889]119877119878119877 cap 119878119877lowast = (22)

119860119904+ (119878119877lowast) = 119891119880 (119906119894119889) | 119906119894119889 isin 119880119878119877 119906119894119889notin 119878119877 119891119880 (119906119894119889) enters 119878119877 partly (23)

119860119904minus (119878119877lowast) = 119906119894119889 | 119906119894119889isin 119878119877 1198911015840

119880 (119906119894119889) leaves 119878119877 partly (24)

23Maintenance and Construction of Traffic Granules Basedon the definitions of the hierarchical multigranularity trafficnetwork model traffic control strategy can be designedaiming at different traffic demand to improve the efficiency ofurban traffic control systemThe key issue that this model canbe utilized for traffic control and management is to maintainthe integrity of these granules in real time For the traffic dataprocessing efficiently the distributed traffic datamanagementframework has been discussed in our previous research work[21] In this section it is assumed that the basic vehicle stateinformation can be collected by the communication deviceequipped in vehicles and managed by the distributed trafficdata centers

For the granules in traffic network model U119881 U119875and U119878 are the vehicle universe platoon universe andsegment universe respectively The smallest granule in thismodel is vehicle granule the related attributes are listedas veh id veh speed veh acc veh pos segment id lane idpreceding id following id pre dist back dist veh class pas-sengers veh weight etc These attributes represent vehicle idspeed acceleration position road segment id that vehicletravels on lane id that vehicle travels on the distance to theprevious vehicle the distance to the vehicle in back vehicle

class the number of passengers in vehicle and vehicle weightin mergence respectively The attributes such as veh classpassengers and veh weight can be utilized for light phasepressure computation in the traffic light control strategydesign Inwhat follows the constructions of threemain trafficgranules are listed in detail

(1) Maintenance and construction of platoon granule Theequivalence relation R119875 on U119881 is used for constructinga platoon on each road segment it is defined as followsvehicles with neighboring position and same segment idsame lane id similar speed and acceleration The elementarytransfer is defined as follows For veh notin p ifMinspeed119901119897119886119905119900119900119899 leveh speed le Maxspeed119901119897119886119905119900119900119899 |V119890ℎ 119886119888119888| le acc119905ℎ119903119890119904ℎ119900119897119889 pre distle dist119905ℎ119903119890119904ℎ119900119897119889 or back dist le dist119905ℎ119903119890119904ℎ119900119897119889 then veh is trans-ferred into platoon p completely if (pre dist gt dist119905ℎ119903119890119904ℎ119900119897119889 andveh accgt acc119905ℎ119903119890119904ℎ119900119897119889) or (back dist gt dist119905ℎ119903119890119904ℎ119900119897119889 and veh accltacc119905ℎ119903119890119904ℎ119900119897119889) then veh is driving into platoon p since veh isaccelerating from the back of p or decelerating in front of pafter a period of time veh will drive into platoon p In thiscase it is said that veh is transferred into p partly For visinpif there is |V119890ℎ 119886119888119888| gt acc119905ℎ119903119890119904ℎ119900119897119889 veh in p is accelerating ordecelerating after a period of time veh will leave platoon pIn this case it is said that veh is transferred out of p partly Ifveh is the head or rear of p when back dist gt dist119905ℎ119903119890119904ℎ119900119897119889 orpre dist gt dist119905ℎ119903119890119904ℎ119900119897119889 it is said that veh has been transferredout of p completely

(2) Maintenance and construction of road segment Forthe universe 119880119875 the basic granule is platoon the relatedattributes including platoon id ave speed max speed minspeed ave headwaymax headwaymin headway veh countseg id lane id platoon weight rightofway flag etc Theseattributes represent the vehicle id average speed maximumspeed of platoon minimum speed of platoon average head-way maximum headway minimum headway number ofvehicles in platoon segment id that the platoon is travellingon lane id the weight of platoon and whether the platoonobtains the right of way to pass the downstream intersectionrespectively The equivalence relation R119878 on U119875 is definedas follows platoons with same segment id and lane id Theelementary transfer of platoon granule on segment granuleis defined as follows For 119901 isin 119904 if rightofway flag is true andveh count lt veh count119905ℎ119903119890119904ℎ119900119897119889 it is said that platoon119901 canpassthrough the downstream intersection in green time then 119901 istransferred out of segment 119904 completely if rightofway flag istrue and veh count gt veh count119905ℎ119903119890119904ℎ119900119897119889 it is said that part ofplatoon 119901 will pass through the downstream intersection ingreen time then 119901 is transferred out of segment 119904 partly For119901 notin 119904 there are similar situations

(3) Maintenance and construction of subregion granuleFor constructing the subregion granule uid is the basicgranule which constitutes the universe U119880 The relatedattributes include uid id upseg state downseg state andcorrelation degree representing uid id state of upstream seg-ment state of downstream segment and correlation degreeof the upstream and downstream segments respectively Thestate of segment can be extracted from platoon granules on itusing the attributes seg id veh count ave speed queuelengthand seg cap where seg cap is the capacity of a road segmentThe correlation degree can be calculated according to the state

Mathematical Problems in Engineering 7

qab

a bi

qghfgh

fab

gh

d

d

qgd

qgd

rbc

rbc

qbc

qbc

c

c

Figure 5 Vehicle flows on the upstream and downstream segments

and capacity of upstreamanddownstream segments [22]Theequivalence relation R119878119877 onU119878 is defined as follows segmentshave similar state and the correlation degree is more than agiven thresholdThe elementary transfer is defined as followsFor uid isinSR if the number of vehicles and the average speedof upstream and downstream segments are both exceedingthe given range of subregion SR that is the upstream anddownstream segment states of uid are not close to the stateof SR it is said that uid is transferred out of SR completely Ifonly the downstream or upstream segment state is not closeto the state of SR it is said that uid is transferred out ofSR partly For uid notin SR if uid is adjacent with SR and theupstream and downstream segment states of uid are similarto the state of SR it is said that uid is transferred into SRcompletely If only upstream or downstream segment state issimilar to the SR state it is said that uid is transferred into SRpartly

Based on the construction and maintenance of granulesin the proposed traffic network model discussed above thesetraffic granules can be maintained in a distributed modeunder a distributed traffic data management frameworkThen the traffic information of different layers and granular-ities can be extracted Based on the multigranularity trafficinformation including vehicle information platoon infor-mation segment information and subregion informationthe traffic control and management system may adopt moresuitable strategies to improve the efficiency However stillsome issues have not been presented in detail in this proposedmodel such as the parameters selection of traffic statetraffic state computation of road segment and the correlationdegree calculation between downstream and upstream roadsegmentsThese problems will be researched in further workIn this paper to illustrate the validity and feasibility ofthe proposed hierarchical multigranularity traffic networkmodel we propose a max-pressure based traffic controlstrategy in the next section

3 Traffic Control Strategy Based onthe Proposed Traffic Model

In this section an application based on the proposed trafficnetwork model is discussed The proposed traffic networkmodel is combined with the max-pressure traffic controlalgorithm to design an extended max-pressure traffic controlstrategy in which the light phase pressure is computed usingthe rough representations of platoon and segment granules

31 Max-Pressure Traffic Control Strategy The max-pressuretraffic control strategy is proposed by Varaiya [23] based on

the backpressure traffic control algorithm [24 25] Accordingto the strategy the urban traffic network is viewed as a queu-ing network and at each time slot the active phase of eachintersection is selected depending on the light phase pressurecomputed using local queue length of each connected roadsegment

As shown in Figure 5 on segments a and g there arequeued vehicles waiting for passing through intersection ithe vehicle flows f 119886119887 and f 119892ℎ are controlled by light phase pof intersection i For the vehicle flow f 119886119887 when it enters thedownstream segment b it will drive into the queued vehicles1199021198871198881015840 and 11990211988711988810158401015840 according to the proportions 1199031198871198881015840 and 11990311988711988810158401015840 where1198881015840 and 11988810158401015840 are the downstream segments of bThere are similarsituations for the vehicle flow 119891119892ℎ The pressure of light phasep of intersection i is computed using (25) and the active lightphase in next time slot is selected using (26)

119901119903119890119904119904119906119903119890119901 = sum119891119886119887isin119901

120583119886119887 (119901) (119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)) (25)

where119891119886119887 isin 119901means that the vehicle flow f 119886119887 is controlled bylight phase p120583119886119887(p) is the number of vehicles passing throughintersection per unit time when p is activated

119901lowast = argmax119901isin119875119894

(119901119903119890119904119904119906119903119890119901 (119905)) (26)

where pisinP119894 means that p is one of the light phases of lightsignal group 119875119894

In this method the light phase pressure is concernedwith the number of queued vehicles in front of the stop lineon the upstream and downstream segments The detailedvehicle information the arriving vehicles (for the upstreamsegment) and the departing vehicles (for the downstreamsegment) are not considered Based on the proposed trafficnetwork model there may be several platoons with differentstates on the upstream and downstream segments On theupstream segment there are stopped platoons in front ofthe stop line and arriving platoons heading for the stop lineOn the downstream segment there may be the departingplatoon when the corresponding downstream light phaseis activated These situations are considered in the trafficnetwork rough representation discussed in previous sectionsTherefore in the next section the light phase pressurecomputation method is extended based on the proposedhierarchical multigranularity traffic network model

32 Extended Max-Pressure Traffic Control Algorithm Actu-ally the light phase pressure is affected by four types of

8 Mathematical Problems in Engineering

platoons on the segment For the platoons on upstreamsegment the stopped platoons waiting for passing throughthe intersection the platoons that are driving towards thestop line and the platoons on the neighboring lanes that aregoing to drive into the corresponding lanes will affect thelight phase pressure For the platoons on downstream seg-ment the departing platoon affects the light phase pressurecomputation too if the downstream light phase is activatedSince the rough representation of traffic granules includesdetailed traffic information the vehicle weight the numberof passengers vehicle class etc also can be considered Thenbased on the rough representation of platoon and segment inthe proposed traffic network model the light phase pressurecan be computed more accurately

To simplify the description the rough set representationof 119883lowast with elementary transfer in two directions is denotedby the lower approximation set appr (119883lowast 119865 cup 1198651015840) and theupper approximation set 119886119901119901119903minus(119883lowast 119865 cup 1198651015840) appr (119883lowast F)represents the one-direction lower approximation set withelement transfer-in operations and appr (119883lowast 1198651015840) representsthe one direction lower approximation set with elementtransfer-out operations The upper assistant set As+(119883lowast F)represents the element set that elements enter 119883lowast partlyunder transfer operations F The lower assistant set Asminus(119883lowast1198651015840) represents the element set that elements leave 119883lowast partlyunder transfer operations 1198651015840 Then the pressure from eachplatoon can be computed using

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) = sum

V119890ℎisin119886119901119901119903 (119875lowast119886119887119865119881)

120572119908V119890ℎ

+ sumV119890ℎisin119860119904+(119875lowast

119886119887119865119881)

120573119908V119890ℎ

(27)

where 120572 120573 is the coefficient of the vehicle weight and 119908V119890ℎ isthe vehicle weight determined by the vehicle type it meansthat the different types of vehicles generate different pressuredue to the different number of passengers and emergence ofthese vehicles

The pressure of the upstream segment is computed using

119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

= sum119901119897119905isin119886119901119901119903 (119904119890119892lowast

119886119887119865119875)cup119860119904+(119904119890119892lowast

119886119887119865119875)

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) (28)

where useg represents an upstream segment and seg119886119887 is asubgranule on segment a in which platoons on segment aare waiting for driving into segment b

The pressure of the downstream segment is computedusing

119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887 119905)

= sum119888isin1198881015840 11988810158401015840

119901119897119905isin119886119901119901119903 (119904119890119892lowast1198871198881198651015840119875)cup119860119904minus(119904119890119892lowast119887119888119865

1015840119875)

119903119887119888119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119887119888 119905) (29)

According to (28) (29) the vehicles entering upstreamsegment and departing from downstream segment are con-sidered for the coordination with neighboring intersections

The pressure of light phase 119901119894119895 is computed using

119901119903119890119904119904119906119903119890119901ℎ119886119904119890 (119901119894119895 119905)

= sum119891119886119887isin119901

119894119895

119891119887119888isin119901119894+1119896

(119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

minus 119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887119888 119905)) 120583119886119887 (119905)

(30)

where 119901119894119895 is the jth light phase of intersection i 119901119894+1

119896 is thecooperative light phase of the downstream intersection i+1

Based on the extended pressure computation methoddescribed above the vehicle type vehicle state and platoonstate upstream light phase state and downstream light phasestate are considered If the element transfer operations Fand 1198651015840 are both empty the extended pressure computationmethod degenerates into max-pressure traffic control algo-rithm If 119865119875 = Φ and 1198651015840

119875 = Φ the arriving vehiclesfrom upstream segment are considered in the pressurecomputation in this case the coordination with the upstreamintersection light phase can be achieved If 119865119875 = Φ and1198651015840

119875 = Φ the departing vehicles on the downstream segmentare considered to achieve the coordination with downstreamintersection light phaseThe light phase with max-pressure isselected to be activated in the next time slot using (26)

Furthermore the vehicle class the number of passengersand vehicle emergency also can be considered using thevehicle weight Therefore based on the extended light phasecomputation method the real situations of traffic controlsystem can be reflected The strategy may be more suitablefor the actual situation of urban traffic network To illustratethe effectiveness and feasibility of the proposed model andalgorithm simulations are carried out in the next section

4 Simulations

In this section the simulation is carried out to analyze theeffectiveness and feasibility of the proposed method Thesimulation traffic network consists of 15 intersections and 76links constructed in Vissim The network includes 16 ingresssegments and 16 exit segments as shown in Figure 6(a)

The vehicle input of each ingress link is 800 vehh Thereare 15 signal controllers and 4 light phases for each controllerie north-south straight phase north-south left-turn phasewest-east straight phase and west-east left-turn phase Thereare 3 lanes on each link The left-turn vehicle flow drives onLane 3 and the straight vehicle flow drives on Lane 2The twovehicle flows are controlled by the traffic lightThe right-turnvehicle flow drives on Lane 1 and is free of the traffic lightTo obtain the traffic parameters during simulation 60 queuecounters 60 travel time sections and 60 routing decisions arelaid on the simulation traffic network as shown in Figures6(b) 6(c) and 6(d) During the simulation the green lighttime of each light phase is assumed to be 21 s which is thepedestrian clearance time (the road width is about 21 m andthe pedestrians speed is about 1 ms) [26] The yellow lighttime is assumed to be 3 s

Mathematical Problems in Engineering 9

(a)

(b) (c) (d)

Figure 6 The simulation traffic network and settings

Table 1 Traffic network performance comparison

Parameters FTMethod MPMethod EMP MethodTotal travel time (h) 633947 588139 568009Total delay time (h) 350255 298110 277457Total stopped delay (h) 253507 225692 205465Average delay time (s) 130584 111420 104057Average stopped delay (s) 94514 84353 77057Average speed (kmh) 23527 25916 26883Number of vehicles in network 448 368 334Number of vehicles that left network 9208 9264 9265Total Number of vehicles 9656 9632 9599

To illustrate the efficiency and feasibility of the pro-posed traffic network model and traffic control algorithmthree traffic light control methods are implemented in thissection including fixed-time control (FT method) max-pressure control (MP method) and the extended max-pressure control (EMP method) To simplify the simulationthe extended pressure computation only considers the caseof 119865119875 = Φ and 1198651015840

119875 = Φ in this paper the other cases will beconsidered in further research works The three traffic lightcontrol methods are implemented in Visual Studio 2010Thesimulation programs communicate with Vissim through theVissim COM programming interface to obtain the vehiclestate and traffic parameters to decide the traffic light signalat each time slot The simulation runs for 3600 s and thetraffic network performance is evaluated from 1000 s to 3600s usingVissimThis is because in the first 1000 s of simulationtime there are not enough vehicles entering the simulationtraffic network

The simulation results of the traffic network are listed inTable 1 According to the simulation results EMP methodbased on the proposed hierarchical multigranularity trafficnetwork model achieves better performance because the

light phase pressure computed using (27)-(30) is closer tothe actual traffic situation in which the vehicles on a roadsegment are represented using rough set form The depart-ing vehicles of downstream segments considered in thissimulation can achieve the coordination with downstreamintersections

As shown in Figure 7 the average queue length underEMP method is shorter than the other traffic light controlmethods The shorter queue length in traffic network meansit can effectively reduce the occurrence of spill-over ondownstream road segment so as to avoid the traffic collapsedue to vehicle queue overflow EMP method is based onthe proposed hierarchical multigranularity traffic networkmodel in which the traffic elements dynamic characteris-tics are depicted accurately and the light phase states ofdownstream intersections are also considered to computethe light phase pressure therefore EMP method achievesbetter performance As shown in Figure 8 the averagetravel time under EMP control is lower than the other twomethods

Furthermore using the extended pressure computationmethod based on the proposed traffic network the priority

10 Mathematical Problems in Engineering

MPFTEMP

05

101520253035404550

Que

ue L

engt

h (v

eh)

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000Time (s)

Figure 7 Average queue length of the three-traffic-light controlmethod

MPFTEMP

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000

Time (s)

0102030405060708090

100

Trav

el Ti

me (

s)

Figure 8 Average travel time of the three-traffic-light controlmethod

control for some particular vehicles can be achieved con-veniently since different classes of vehicles can be set withdifferent vehicle weight in the EMPmethod Different weightvalues can result in different traffic efficiency for differenttypes of vehicles For example when the type of bus is addedto the simulation traffic network the higher weight valuecan make buses have higher average speed and the otherclasses of vehicles maybe achieve lower average speed Forthis issue the vehicle weight selection is important As shownin Figure 9 the different bus weight values are simulatedand they result in different average speed According to themultiple simulation results and the real traffic demand thevehicle weight can be determined

To sum up since the hierarchical multigranularity trafficmodel can represent the actual traffic network situationsaccurately and the EMP method can achieve traffic controlflexibly this paper may be a useful attempt in solving urbantraffic control problem

5 Conclusions

In this paper a hierarchical multigranularity traffic networkis proposed and on the basis of the proposed traffic networkan extended max-pressure traffic control method is imple-mented The simulation results illustrate that the proposedtraffic network model and EMP method achieve better

Average speed [Car]Average speed [Bus]Average speed [All vehicles]

24245

25255

26265

27

Aver

age s

peed

15 2 25 3 35 4 45 5 551Vehicle weight of Bus

Figure 9 Simulation results with different bus weight values

performance In the proposed model the traffic elementsin urban traffic network are represented using S-rough setin which the dynamic characteristics of traffic elementsare depicted This type of traffic element description canrepresent the traffic network situation accurately so as todesign the more efficient traffic control strategy

However the hierarchical multigranularity traffic net-work model is only a preliminary exploration and it needsto be further improved in the future research work such asthe traffic subregion formulation and the boundary controlof traffic subregion For the traffic subregion formulationthe correlation of downstream and upstream road segmentsshould be computed considering traffic states and the geo-graphical characteristics of each pair of upstream and down-stream segments The control strategy of subregion boundarycan compute the traffic pressure considering the remainingcapacity of downstream segment to speed up the vehicleflow release towards to the segment with larger remainingcapacity These issues will be studied in our further researchwork to improve the hierarchical multigranularity traffic net-workmodel Furthermore in the future work the completelymultigranularity traffic control strategies should be furtherresearched using granular computing theory to achieve thetraffic system multigranularity control completely

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial NaturalScience Foundation China [grant number ZR2018FM027]Project of Shandong Province Higher Educational Scienceand Technology Program [grant number J17KB181] KeyResearch and Development Project in Shandong ProvinceChina [grant numbers 2015GGX101048 2016GSF120009]

Mathematical Problems in Engineering 11

and National Natural Science Foundation of Youth FundProject [grant number 61501284]

References

[1] Q He K L Head and J Ding ldquoMulti-modal traffic signalcontrol with priority signal actuation and coordinationrdquoTrans-portation Research Part C Emerging Technologies vol 46 pp65ndash82 2014

[2] Y Yao ldquoInterpreting concept learning in cognitive informaticsand granular computingrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 39 no 4 pp 855ndash8662009

[3] J T Yao A V Vasilakos andW Pedrycz ldquoGranular computingperspectives and challengesrdquo IEEE Transactions on Cyberneticsvol 43 no 6 pp 1977ndash1989 2013

[4] H Liu andM Cocea ldquoGranular computing-based approach forclassification towards reduction of bias in ensemble learningrdquoGranular Computing vol 2 no 3 pp 131ndash139 2017

[5] X Ji L Liu and Q Wu ldquoTraffic management and decision-making support based on granular computingrdquo Journal ofHighway and Transportation Research and Development vol2008 no 10 pp 129ndash133 2008

[6] S Dewen C Xinhong C Jing and F Xujian ldquoGranularcomputing in intelligent transportation An exploratory studyrdquoCybernetics and Information Technologies vol 15 no 6 pp 122ndash134 2015

[7] C Yao J-C Gu and Q Yang ldquoTraffic congestion identificationand analysis of urban road network based on granular comput-ingrdquo Applied Mechanics and Materials vol 641-642 pp 916ndash922 2014

[8] T Li D Ruan Y Shen E Hermans and G Wets ldquoA newweighting approach based on rough set theory and granularcomputing for road safety indicator analysisrdquo InternationalJournal of Computational Intelligence vol 32 no 4 pp 517ndash5342016

[9] F He R Liu Y Zhang et al ldquoBGrR large-scale network routingspeedup techniques based on granular computingrdquo ComputerScience vol 41 no 11 pp 265ndash281 2014

[10] R Deb and A W-C Liew ldquoMissing value imputation forthe analysis of incomplete traffic accident datardquo InformationSciences vol 339 pp 274ndash289 2016

[11] S Hao L Yang and Y Shi ldquoData-driven car-following modelbased on rough set theoryrdquo IET Intelligent Transport Systemsvol 12 no 1 pp 49ndash57 2018

[12] L Livi and A Sadeghian ldquoGranular computing computationalintelligence and the analysis of non-geometric input spacesrdquoGranular Computing vol 1 no 1 pp 13ndash20 2016

[13] X Yang H Da-Wei S Bing and Z Duo-Jia ldquoCity traffic flowbreakdown prediction based on fuzzy rough setrdquo Open Physicsvol 15 no 1 pp 292ndash299 2017

[14] X Xiong L Chen and J Liang ldquoAnalysis of roadway trafficaccidents based on rough sets and Bayesian networksrdquo Promet- Traffic - Traffico vol 30 no 1 pp 71ndash81 2018

[15] M Deshpande and P Bajaj ldquoPerformance improvement of traf-fic flow prediction model using combination of support vectormachine and rough setrdquo International Journal of ComputerApplications vol 163 no 2 pp 31ndash35 2017

[16] Y Y Yao ldquoGranular computing basic issues and possible solu-tionsrdquo in Proceedings of the Fi13h International Conference onComputing and Information vol 1 pp 186ndash189 2000

[17] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Science vol 11 no 5 pp 341ndash356 1982

[18] K Shi ldquoS-rough sets and its applications in diagnosis-rec-ognition for diseaserdquo in Proceedings of the 1st InternationalConference on Machine Learning and Cybernetics pp 50ndash54IEEE Beijing China 2002

[19] K Q Shi and Y Q Cui ldquoS-Rough sets and their general struc-turesrdquo Journal of Shandong University Natural Science Shan-dong Daxue Xuebao Lixue Ban vol 37 no 6 pp 471ndash474 2002

[20] H Haiqing X Peijun and F Haiyan ldquoS-rough recognitionof knowledge and general threshold encryption authenticationscheme of recognition conclusionrdquo in Proceedings of the SixthInternational Conference on Intelligent Systems Design andApplications pp 816ndash824 Jian China October 2006

[21] S Hao L Yang and Y Shi ldquoReal-time data-driven trajectoryreconstruction based on rough set theoryrdquo Advances in Trans-portation Studies vol 44 pp 119ndash132 2018

[22] Y Yang L Jia Y Qin S Han and H Dong ldquoUnderstandingstructure of urban traffic network based on spatial-temporalcorrelation analysisrdquo Modern Physics Letters B CondensedMatter Physics Statistical Physics AtomicMolecular andOpticalPhysics vol 31 no 22 Article ID 1750230 19 pages 2017

[23] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[24] TWongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 USA September 2012

[25] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo IFAC Proceedings Volumes vol 47 no 3 pp11332ndash11337 2014

[26] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

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Mathematical Problems in Engineering 7

qab

a bi

qghfgh

fab

gh

d

d

qgd

qgd

rbc

rbc

qbc

qbc

c

c

Figure 5 Vehicle flows on the upstream and downstream segments

and capacity of upstreamanddownstream segments [22]Theequivalence relation R119878119877 onU119878 is defined as follows segmentshave similar state and the correlation degree is more than agiven thresholdThe elementary transfer is defined as followsFor uid isinSR if the number of vehicles and the average speedof upstream and downstream segments are both exceedingthe given range of subregion SR that is the upstream anddownstream segment states of uid are not close to the stateof SR it is said that uid is transferred out of SR completely Ifonly the downstream or upstream segment state is not closeto the state of SR it is said that uid is transferred out ofSR partly For uid notin SR if uid is adjacent with SR and theupstream and downstream segment states of uid are similarto the state of SR it is said that uid is transferred into SRcompletely If only upstream or downstream segment state issimilar to the SR state it is said that uid is transferred into SRpartly

Based on the construction and maintenance of granulesin the proposed traffic network model discussed above thesetraffic granules can be maintained in a distributed modeunder a distributed traffic data management frameworkThen the traffic information of different layers and granular-ities can be extracted Based on the multigranularity trafficinformation including vehicle information platoon infor-mation segment information and subregion informationthe traffic control and management system may adopt moresuitable strategies to improve the efficiency However stillsome issues have not been presented in detail in this proposedmodel such as the parameters selection of traffic statetraffic state computation of road segment and the correlationdegree calculation between downstream and upstream roadsegmentsThese problems will be researched in further workIn this paper to illustrate the validity and feasibility ofthe proposed hierarchical multigranularity traffic networkmodel we propose a max-pressure based traffic controlstrategy in the next section

3 Traffic Control Strategy Based onthe Proposed Traffic Model

In this section an application based on the proposed trafficnetwork model is discussed The proposed traffic networkmodel is combined with the max-pressure traffic controlalgorithm to design an extended max-pressure traffic controlstrategy in which the light phase pressure is computed usingthe rough representations of platoon and segment granules

31 Max-Pressure Traffic Control Strategy The max-pressuretraffic control strategy is proposed by Varaiya [23] based on

the backpressure traffic control algorithm [24 25] Accordingto the strategy the urban traffic network is viewed as a queu-ing network and at each time slot the active phase of eachintersection is selected depending on the light phase pressurecomputed using local queue length of each connected roadsegment

As shown in Figure 5 on segments a and g there arequeued vehicles waiting for passing through intersection ithe vehicle flows f 119886119887 and f 119892ℎ are controlled by light phase pof intersection i For the vehicle flow f 119886119887 when it enters thedownstream segment b it will drive into the queued vehicles1199021198871198881015840 and 11990211988711988810158401015840 according to the proportions 1199031198871198881015840 and 11990311988711988810158401015840 where1198881015840 and 11988810158401015840 are the downstream segments of bThere are similarsituations for the vehicle flow 119891119892ℎ The pressure of light phasep of intersection i is computed using (25) and the active lightphase in next time slot is selected using (26)

119901119903119890119904119904119906119903119890119901 = sum119891119886119887isin119901

120583119886119887 (119901) (119902119886119887 (119905) minus sum119888

119903119887119888119902119887119888 (119905)) (25)

where119891119886119887 isin 119901means that the vehicle flow f 119886119887 is controlled bylight phase p120583119886119887(p) is the number of vehicles passing throughintersection per unit time when p is activated

119901lowast = argmax119901isin119875119894

(119901119903119890119904119904119906119903119890119901 (119905)) (26)

where pisinP119894 means that p is one of the light phases of lightsignal group 119875119894

In this method the light phase pressure is concernedwith the number of queued vehicles in front of the stop lineon the upstream and downstream segments The detailedvehicle information the arriving vehicles (for the upstreamsegment) and the departing vehicles (for the downstreamsegment) are not considered Based on the proposed trafficnetwork model there may be several platoons with differentstates on the upstream and downstream segments On theupstream segment there are stopped platoons in front ofthe stop line and arriving platoons heading for the stop lineOn the downstream segment there may be the departingplatoon when the corresponding downstream light phaseis activated These situations are considered in the trafficnetwork rough representation discussed in previous sectionsTherefore in the next section the light phase pressurecomputation method is extended based on the proposedhierarchical multigranularity traffic network model

32 Extended Max-Pressure Traffic Control Algorithm Actu-ally the light phase pressure is affected by four types of

8 Mathematical Problems in Engineering

platoons on the segment For the platoons on upstreamsegment the stopped platoons waiting for passing throughthe intersection the platoons that are driving towards thestop line and the platoons on the neighboring lanes that aregoing to drive into the corresponding lanes will affect thelight phase pressure For the platoons on downstream seg-ment the departing platoon affects the light phase pressurecomputation too if the downstream light phase is activatedSince the rough representation of traffic granules includesdetailed traffic information the vehicle weight the numberof passengers vehicle class etc also can be considered Thenbased on the rough representation of platoon and segment inthe proposed traffic network model the light phase pressurecan be computed more accurately

To simplify the description the rough set representationof 119883lowast with elementary transfer in two directions is denotedby the lower approximation set appr (119883lowast 119865 cup 1198651015840) and theupper approximation set 119886119901119901119903minus(119883lowast 119865 cup 1198651015840) appr (119883lowast F)represents the one-direction lower approximation set withelement transfer-in operations and appr (119883lowast 1198651015840) representsthe one direction lower approximation set with elementtransfer-out operations The upper assistant set As+(119883lowast F)represents the element set that elements enter 119883lowast partlyunder transfer operations F The lower assistant set Asminus(119883lowast1198651015840) represents the element set that elements leave 119883lowast partlyunder transfer operations 1198651015840 Then the pressure from eachplatoon can be computed using

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) = sum

V119890ℎisin119886119901119901119903 (119875lowast119886119887119865119881)

120572119908V119890ℎ

+ sumV119890ℎisin119860119904+(119875lowast

119886119887119865119881)

120573119908V119890ℎ

(27)

where 120572 120573 is the coefficient of the vehicle weight and 119908V119890ℎ isthe vehicle weight determined by the vehicle type it meansthat the different types of vehicles generate different pressuredue to the different number of passengers and emergence ofthese vehicles

The pressure of the upstream segment is computed using

119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

= sum119901119897119905isin119886119901119901119903 (119904119890119892lowast

119886119887119865119875)cup119860119904+(119904119890119892lowast

119886119887119865119875)

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) (28)

where useg represents an upstream segment and seg119886119887 is asubgranule on segment a in which platoons on segment aare waiting for driving into segment b

The pressure of the downstream segment is computedusing

119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887 119905)

= sum119888isin1198881015840 11988810158401015840

119901119897119905isin119886119901119901119903 (119904119890119892lowast1198871198881198651015840119875)cup119860119904minus(119904119890119892lowast119887119888119865

1015840119875)

119903119887119888119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119887119888 119905) (29)

According to (28) (29) the vehicles entering upstreamsegment and departing from downstream segment are con-sidered for the coordination with neighboring intersections

The pressure of light phase 119901119894119895 is computed using

119901119903119890119904119904119906119903119890119901ℎ119886119904119890 (119901119894119895 119905)

= sum119891119886119887isin119901

119894119895

119891119887119888isin119901119894+1119896

(119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

minus 119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887119888 119905)) 120583119886119887 (119905)

(30)

where 119901119894119895 is the jth light phase of intersection i 119901119894+1

119896 is thecooperative light phase of the downstream intersection i+1

Based on the extended pressure computation methoddescribed above the vehicle type vehicle state and platoonstate upstream light phase state and downstream light phasestate are considered If the element transfer operations Fand 1198651015840 are both empty the extended pressure computationmethod degenerates into max-pressure traffic control algo-rithm If 119865119875 = Φ and 1198651015840

119875 = Φ the arriving vehiclesfrom upstream segment are considered in the pressurecomputation in this case the coordination with the upstreamintersection light phase can be achieved If 119865119875 = Φ and1198651015840

119875 = Φ the departing vehicles on the downstream segmentare considered to achieve the coordination with downstreamintersection light phaseThe light phase with max-pressure isselected to be activated in the next time slot using (26)

Furthermore the vehicle class the number of passengersand vehicle emergency also can be considered using thevehicle weight Therefore based on the extended light phasecomputation method the real situations of traffic controlsystem can be reflected The strategy may be more suitablefor the actual situation of urban traffic network To illustratethe effectiveness and feasibility of the proposed model andalgorithm simulations are carried out in the next section

4 Simulations

In this section the simulation is carried out to analyze theeffectiveness and feasibility of the proposed method Thesimulation traffic network consists of 15 intersections and 76links constructed in Vissim The network includes 16 ingresssegments and 16 exit segments as shown in Figure 6(a)

The vehicle input of each ingress link is 800 vehh Thereare 15 signal controllers and 4 light phases for each controllerie north-south straight phase north-south left-turn phasewest-east straight phase and west-east left-turn phase Thereare 3 lanes on each link The left-turn vehicle flow drives onLane 3 and the straight vehicle flow drives on Lane 2The twovehicle flows are controlled by the traffic lightThe right-turnvehicle flow drives on Lane 1 and is free of the traffic lightTo obtain the traffic parameters during simulation 60 queuecounters 60 travel time sections and 60 routing decisions arelaid on the simulation traffic network as shown in Figures6(b) 6(c) and 6(d) During the simulation the green lighttime of each light phase is assumed to be 21 s which is thepedestrian clearance time (the road width is about 21 m andthe pedestrians speed is about 1 ms) [26] The yellow lighttime is assumed to be 3 s

Mathematical Problems in Engineering 9

(a)

(b) (c) (d)

Figure 6 The simulation traffic network and settings

Table 1 Traffic network performance comparison

Parameters FTMethod MPMethod EMP MethodTotal travel time (h) 633947 588139 568009Total delay time (h) 350255 298110 277457Total stopped delay (h) 253507 225692 205465Average delay time (s) 130584 111420 104057Average stopped delay (s) 94514 84353 77057Average speed (kmh) 23527 25916 26883Number of vehicles in network 448 368 334Number of vehicles that left network 9208 9264 9265Total Number of vehicles 9656 9632 9599

To illustrate the efficiency and feasibility of the pro-posed traffic network model and traffic control algorithmthree traffic light control methods are implemented in thissection including fixed-time control (FT method) max-pressure control (MP method) and the extended max-pressure control (EMP method) To simplify the simulationthe extended pressure computation only considers the caseof 119865119875 = Φ and 1198651015840

119875 = Φ in this paper the other cases will beconsidered in further research works The three traffic lightcontrol methods are implemented in Visual Studio 2010Thesimulation programs communicate with Vissim through theVissim COM programming interface to obtain the vehiclestate and traffic parameters to decide the traffic light signalat each time slot The simulation runs for 3600 s and thetraffic network performance is evaluated from 1000 s to 3600s usingVissimThis is because in the first 1000 s of simulationtime there are not enough vehicles entering the simulationtraffic network

The simulation results of the traffic network are listed inTable 1 According to the simulation results EMP methodbased on the proposed hierarchical multigranularity trafficnetwork model achieves better performance because the

light phase pressure computed using (27)-(30) is closer tothe actual traffic situation in which the vehicles on a roadsegment are represented using rough set form The depart-ing vehicles of downstream segments considered in thissimulation can achieve the coordination with downstreamintersections

As shown in Figure 7 the average queue length underEMP method is shorter than the other traffic light controlmethods The shorter queue length in traffic network meansit can effectively reduce the occurrence of spill-over ondownstream road segment so as to avoid the traffic collapsedue to vehicle queue overflow EMP method is based onthe proposed hierarchical multigranularity traffic networkmodel in which the traffic elements dynamic characteris-tics are depicted accurately and the light phase states ofdownstream intersections are also considered to computethe light phase pressure therefore EMP method achievesbetter performance As shown in Figure 8 the averagetravel time under EMP control is lower than the other twomethods

Furthermore using the extended pressure computationmethod based on the proposed traffic network the priority

10 Mathematical Problems in Engineering

MPFTEMP

05

101520253035404550

Que

ue L

engt

h (v

eh)

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000Time (s)

Figure 7 Average queue length of the three-traffic-light controlmethod

MPFTEMP

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000

Time (s)

0102030405060708090

100

Trav

el Ti

me (

s)

Figure 8 Average travel time of the three-traffic-light controlmethod

control for some particular vehicles can be achieved con-veniently since different classes of vehicles can be set withdifferent vehicle weight in the EMPmethod Different weightvalues can result in different traffic efficiency for differenttypes of vehicles For example when the type of bus is addedto the simulation traffic network the higher weight valuecan make buses have higher average speed and the otherclasses of vehicles maybe achieve lower average speed Forthis issue the vehicle weight selection is important As shownin Figure 9 the different bus weight values are simulatedand they result in different average speed According to themultiple simulation results and the real traffic demand thevehicle weight can be determined

To sum up since the hierarchical multigranularity trafficmodel can represent the actual traffic network situationsaccurately and the EMP method can achieve traffic controlflexibly this paper may be a useful attempt in solving urbantraffic control problem

5 Conclusions

In this paper a hierarchical multigranularity traffic networkis proposed and on the basis of the proposed traffic networkan extended max-pressure traffic control method is imple-mented The simulation results illustrate that the proposedtraffic network model and EMP method achieve better

Average speed [Car]Average speed [Bus]Average speed [All vehicles]

24245

25255

26265

27

Aver

age s

peed

15 2 25 3 35 4 45 5 551Vehicle weight of Bus

Figure 9 Simulation results with different bus weight values

performance In the proposed model the traffic elementsin urban traffic network are represented using S-rough setin which the dynamic characteristics of traffic elementsare depicted This type of traffic element description canrepresent the traffic network situation accurately so as todesign the more efficient traffic control strategy

However the hierarchical multigranularity traffic net-work model is only a preliminary exploration and it needsto be further improved in the future research work such asthe traffic subregion formulation and the boundary controlof traffic subregion For the traffic subregion formulationthe correlation of downstream and upstream road segmentsshould be computed considering traffic states and the geo-graphical characteristics of each pair of upstream and down-stream segments The control strategy of subregion boundarycan compute the traffic pressure considering the remainingcapacity of downstream segment to speed up the vehicleflow release towards to the segment with larger remainingcapacity These issues will be studied in our further researchwork to improve the hierarchical multigranularity traffic net-workmodel Furthermore in the future work the completelymultigranularity traffic control strategies should be furtherresearched using granular computing theory to achieve thetraffic system multigranularity control completely

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial NaturalScience Foundation China [grant number ZR2018FM027]Project of Shandong Province Higher Educational Scienceand Technology Program [grant number J17KB181] KeyResearch and Development Project in Shandong ProvinceChina [grant numbers 2015GGX101048 2016GSF120009]

Mathematical Problems in Engineering 11

and National Natural Science Foundation of Youth FundProject [grant number 61501284]

References

[1] Q He K L Head and J Ding ldquoMulti-modal traffic signalcontrol with priority signal actuation and coordinationrdquoTrans-portation Research Part C Emerging Technologies vol 46 pp65ndash82 2014

[2] Y Yao ldquoInterpreting concept learning in cognitive informaticsand granular computingrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 39 no 4 pp 855ndash8662009

[3] J T Yao A V Vasilakos andW Pedrycz ldquoGranular computingperspectives and challengesrdquo IEEE Transactions on Cyberneticsvol 43 no 6 pp 1977ndash1989 2013

[4] H Liu andM Cocea ldquoGranular computing-based approach forclassification towards reduction of bias in ensemble learningrdquoGranular Computing vol 2 no 3 pp 131ndash139 2017

[5] X Ji L Liu and Q Wu ldquoTraffic management and decision-making support based on granular computingrdquo Journal ofHighway and Transportation Research and Development vol2008 no 10 pp 129ndash133 2008

[6] S Dewen C Xinhong C Jing and F Xujian ldquoGranularcomputing in intelligent transportation An exploratory studyrdquoCybernetics and Information Technologies vol 15 no 6 pp 122ndash134 2015

[7] C Yao J-C Gu and Q Yang ldquoTraffic congestion identificationand analysis of urban road network based on granular comput-ingrdquo Applied Mechanics and Materials vol 641-642 pp 916ndash922 2014

[8] T Li D Ruan Y Shen E Hermans and G Wets ldquoA newweighting approach based on rough set theory and granularcomputing for road safety indicator analysisrdquo InternationalJournal of Computational Intelligence vol 32 no 4 pp 517ndash5342016

[9] F He R Liu Y Zhang et al ldquoBGrR large-scale network routingspeedup techniques based on granular computingrdquo ComputerScience vol 41 no 11 pp 265ndash281 2014

[10] R Deb and A W-C Liew ldquoMissing value imputation forthe analysis of incomplete traffic accident datardquo InformationSciences vol 339 pp 274ndash289 2016

[11] S Hao L Yang and Y Shi ldquoData-driven car-following modelbased on rough set theoryrdquo IET Intelligent Transport Systemsvol 12 no 1 pp 49ndash57 2018

[12] L Livi and A Sadeghian ldquoGranular computing computationalintelligence and the analysis of non-geometric input spacesrdquoGranular Computing vol 1 no 1 pp 13ndash20 2016

[13] X Yang H Da-Wei S Bing and Z Duo-Jia ldquoCity traffic flowbreakdown prediction based on fuzzy rough setrdquo Open Physicsvol 15 no 1 pp 292ndash299 2017

[14] X Xiong L Chen and J Liang ldquoAnalysis of roadway trafficaccidents based on rough sets and Bayesian networksrdquo Promet- Traffic - Traffico vol 30 no 1 pp 71ndash81 2018

[15] M Deshpande and P Bajaj ldquoPerformance improvement of traf-fic flow prediction model using combination of support vectormachine and rough setrdquo International Journal of ComputerApplications vol 163 no 2 pp 31ndash35 2017

[16] Y Y Yao ldquoGranular computing basic issues and possible solu-tionsrdquo in Proceedings of the Fi13h International Conference onComputing and Information vol 1 pp 186ndash189 2000

[17] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Science vol 11 no 5 pp 341ndash356 1982

[18] K Shi ldquoS-rough sets and its applications in diagnosis-rec-ognition for diseaserdquo in Proceedings of the 1st InternationalConference on Machine Learning and Cybernetics pp 50ndash54IEEE Beijing China 2002

[19] K Q Shi and Y Q Cui ldquoS-Rough sets and their general struc-turesrdquo Journal of Shandong University Natural Science Shan-dong Daxue Xuebao Lixue Ban vol 37 no 6 pp 471ndash474 2002

[20] H Haiqing X Peijun and F Haiyan ldquoS-rough recognitionof knowledge and general threshold encryption authenticationscheme of recognition conclusionrdquo in Proceedings of the SixthInternational Conference on Intelligent Systems Design andApplications pp 816ndash824 Jian China October 2006

[21] S Hao L Yang and Y Shi ldquoReal-time data-driven trajectoryreconstruction based on rough set theoryrdquo Advances in Trans-portation Studies vol 44 pp 119ndash132 2018

[22] Y Yang L Jia Y Qin S Han and H Dong ldquoUnderstandingstructure of urban traffic network based on spatial-temporalcorrelation analysisrdquo Modern Physics Letters B CondensedMatter Physics Statistical Physics AtomicMolecular andOpticalPhysics vol 31 no 22 Article ID 1750230 19 pages 2017

[23] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[24] TWongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 USA September 2012

[25] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo IFAC Proceedings Volumes vol 47 no 3 pp11332ndash11337 2014

[26] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

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8 Mathematical Problems in Engineering

platoons on the segment For the platoons on upstreamsegment the stopped platoons waiting for passing throughthe intersection the platoons that are driving towards thestop line and the platoons on the neighboring lanes that aregoing to drive into the corresponding lanes will affect thelight phase pressure For the platoons on downstream seg-ment the departing platoon affects the light phase pressurecomputation too if the downstream light phase is activatedSince the rough representation of traffic granules includesdetailed traffic information the vehicle weight the numberof passengers vehicle class etc also can be considered Thenbased on the rough representation of platoon and segment inthe proposed traffic network model the light phase pressurecan be computed more accurately

To simplify the description the rough set representationof 119883lowast with elementary transfer in two directions is denotedby the lower approximation set appr (119883lowast 119865 cup 1198651015840) and theupper approximation set 119886119901119901119903minus(119883lowast 119865 cup 1198651015840) appr (119883lowast F)represents the one-direction lower approximation set withelement transfer-in operations and appr (119883lowast 1198651015840) representsthe one direction lower approximation set with elementtransfer-out operations The upper assistant set As+(119883lowast F)represents the element set that elements enter 119883lowast partlyunder transfer operations F The lower assistant set Asminus(119883lowast1198651015840) represents the element set that elements leave 119883lowast partlyunder transfer operations 1198651015840 Then the pressure from eachplatoon can be computed using

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) = sum

V119890ℎisin119886119901119901119903 (119875lowast119886119887119865119881)

120572119908V119890ℎ

+ sumV119890ℎisin119860119904+(119875lowast

119886119887119865119881)

120573119908V119890ℎ

(27)

where 120572 120573 is the coefficient of the vehicle weight and 119908V119890ℎ isthe vehicle weight determined by the vehicle type it meansthat the different types of vehicles generate different pressuredue to the different number of passengers and emergence ofthese vehicles

The pressure of the upstream segment is computed using

119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

= sum119901119897119905isin119886119901119901119903 (119904119890119892lowast

119886119887119865119875)cup119860119904+(119904119890119892lowast

119886119887119865119875)

119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119886119887 119905) (28)

where useg represents an upstream segment and seg119886119887 is asubgranule on segment a in which platoons on segment aare waiting for driving into segment b

The pressure of the downstream segment is computedusing

119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887 119905)

= sum119888isin1198881015840 11988810158401015840

119901119897119905isin119886119901119901119903 (119904119890119892lowast1198871198881198651015840119875)cup119860119904minus(119904119890119892lowast119887119888119865

1015840119875)

119903119887119888119901119903119890119904119904119906119903119890119901119897119905 (119875lowast119887119888 119905) (29)

According to (28) (29) the vehicles entering upstreamsegment and departing from downstream segment are con-sidered for the coordination with neighboring intersections

The pressure of light phase 119901119894119895 is computed using

119901119903119890119904119904119906119903119890119901ℎ119886119904119890 (119901119894119895 119905)

= sum119891119886119887isin119901

119894119895

119891119887119888isin119901119894+1119896

(119901119903119890119904119904119906119903119890119906119904119890119892 (119904119890119892lowast119886119887 119905)

minus 119901119903119890119904119904119906119903119890119889119904119890119892 (119904119890119892lowast119887119888 119905)) 120583119886119887 (119905)

(30)

where 119901119894119895 is the jth light phase of intersection i 119901119894+1

119896 is thecooperative light phase of the downstream intersection i+1

Based on the extended pressure computation methoddescribed above the vehicle type vehicle state and platoonstate upstream light phase state and downstream light phasestate are considered If the element transfer operations Fand 1198651015840 are both empty the extended pressure computationmethod degenerates into max-pressure traffic control algo-rithm If 119865119875 = Φ and 1198651015840

119875 = Φ the arriving vehiclesfrom upstream segment are considered in the pressurecomputation in this case the coordination with the upstreamintersection light phase can be achieved If 119865119875 = Φ and1198651015840

119875 = Φ the departing vehicles on the downstream segmentare considered to achieve the coordination with downstreamintersection light phaseThe light phase with max-pressure isselected to be activated in the next time slot using (26)

Furthermore the vehicle class the number of passengersand vehicle emergency also can be considered using thevehicle weight Therefore based on the extended light phasecomputation method the real situations of traffic controlsystem can be reflected The strategy may be more suitablefor the actual situation of urban traffic network To illustratethe effectiveness and feasibility of the proposed model andalgorithm simulations are carried out in the next section

4 Simulations

In this section the simulation is carried out to analyze theeffectiveness and feasibility of the proposed method Thesimulation traffic network consists of 15 intersections and 76links constructed in Vissim The network includes 16 ingresssegments and 16 exit segments as shown in Figure 6(a)

The vehicle input of each ingress link is 800 vehh Thereare 15 signal controllers and 4 light phases for each controllerie north-south straight phase north-south left-turn phasewest-east straight phase and west-east left-turn phase Thereare 3 lanes on each link The left-turn vehicle flow drives onLane 3 and the straight vehicle flow drives on Lane 2The twovehicle flows are controlled by the traffic lightThe right-turnvehicle flow drives on Lane 1 and is free of the traffic lightTo obtain the traffic parameters during simulation 60 queuecounters 60 travel time sections and 60 routing decisions arelaid on the simulation traffic network as shown in Figures6(b) 6(c) and 6(d) During the simulation the green lighttime of each light phase is assumed to be 21 s which is thepedestrian clearance time (the road width is about 21 m andthe pedestrians speed is about 1 ms) [26] The yellow lighttime is assumed to be 3 s

Mathematical Problems in Engineering 9

(a)

(b) (c) (d)

Figure 6 The simulation traffic network and settings

Table 1 Traffic network performance comparison

Parameters FTMethod MPMethod EMP MethodTotal travel time (h) 633947 588139 568009Total delay time (h) 350255 298110 277457Total stopped delay (h) 253507 225692 205465Average delay time (s) 130584 111420 104057Average stopped delay (s) 94514 84353 77057Average speed (kmh) 23527 25916 26883Number of vehicles in network 448 368 334Number of vehicles that left network 9208 9264 9265Total Number of vehicles 9656 9632 9599

To illustrate the efficiency and feasibility of the pro-posed traffic network model and traffic control algorithmthree traffic light control methods are implemented in thissection including fixed-time control (FT method) max-pressure control (MP method) and the extended max-pressure control (EMP method) To simplify the simulationthe extended pressure computation only considers the caseof 119865119875 = Φ and 1198651015840

119875 = Φ in this paper the other cases will beconsidered in further research works The three traffic lightcontrol methods are implemented in Visual Studio 2010Thesimulation programs communicate with Vissim through theVissim COM programming interface to obtain the vehiclestate and traffic parameters to decide the traffic light signalat each time slot The simulation runs for 3600 s and thetraffic network performance is evaluated from 1000 s to 3600s usingVissimThis is because in the first 1000 s of simulationtime there are not enough vehicles entering the simulationtraffic network

The simulation results of the traffic network are listed inTable 1 According to the simulation results EMP methodbased on the proposed hierarchical multigranularity trafficnetwork model achieves better performance because the

light phase pressure computed using (27)-(30) is closer tothe actual traffic situation in which the vehicles on a roadsegment are represented using rough set form The depart-ing vehicles of downstream segments considered in thissimulation can achieve the coordination with downstreamintersections

As shown in Figure 7 the average queue length underEMP method is shorter than the other traffic light controlmethods The shorter queue length in traffic network meansit can effectively reduce the occurrence of spill-over ondownstream road segment so as to avoid the traffic collapsedue to vehicle queue overflow EMP method is based onthe proposed hierarchical multigranularity traffic networkmodel in which the traffic elements dynamic characteris-tics are depicted accurately and the light phase states ofdownstream intersections are also considered to computethe light phase pressure therefore EMP method achievesbetter performance As shown in Figure 8 the averagetravel time under EMP control is lower than the other twomethods

Furthermore using the extended pressure computationmethod based on the proposed traffic network the priority

10 Mathematical Problems in Engineering

MPFTEMP

05

101520253035404550

Que

ue L

engt

h (v

eh)

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000Time (s)

Figure 7 Average queue length of the three-traffic-light controlmethod

MPFTEMP

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000

Time (s)

0102030405060708090

100

Trav

el Ti

me (

s)

Figure 8 Average travel time of the three-traffic-light controlmethod

control for some particular vehicles can be achieved con-veniently since different classes of vehicles can be set withdifferent vehicle weight in the EMPmethod Different weightvalues can result in different traffic efficiency for differenttypes of vehicles For example when the type of bus is addedto the simulation traffic network the higher weight valuecan make buses have higher average speed and the otherclasses of vehicles maybe achieve lower average speed Forthis issue the vehicle weight selection is important As shownin Figure 9 the different bus weight values are simulatedand they result in different average speed According to themultiple simulation results and the real traffic demand thevehicle weight can be determined

To sum up since the hierarchical multigranularity trafficmodel can represent the actual traffic network situationsaccurately and the EMP method can achieve traffic controlflexibly this paper may be a useful attempt in solving urbantraffic control problem

5 Conclusions

In this paper a hierarchical multigranularity traffic networkis proposed and on the basis of the proposed traffic networkan extended max-pressure traffic control method is imple-mented The simulation results illustrate that the proposedtraffic network model and EMP method achieve better

Average speed [Car]Average speed [Bus]Average speed [All vehicles]

24245

25255

26265

27

Aver

age s

peed

15 2 25 3 35 4 45 5 551Vehicle weight of Bus

Figure 9 Simulation results with different bus weight values

performance In the proposed model the traffic elementsin urban traffic network are represented using S-rough setin which the dynamic characteristics of traffic elementsare depicted This type of traffic element description canrepresent the traffic network situation accurately so as todesign the more efficient traffic control strategy

However the hierarchical multigranularity traffic net-work model is only a preliminary exploration and it needsto be further improved in the future research work such asthe traffic subregion formulation and the boundary controlof traffic subregion For the traffic subregion formulationthe correlation of downstream and upstream road segmentsshould be computed considering traffic states and the geo-graphical characteristics of each pair of upstream and down-stream segments The control strategy of subregion boundarycan compute the traffic pressure considering the remainingcapacity of downstream segment to speed up the vehicleflow release towards to the segment with larger remainingcapacity These issues will be studied in our further researchwork to improve the hierarchical multigranularity traffic net-workmodel Furthermore in the future work the completelymultigranularity traffic control strategies should be furtherresearched using granular computing theory to achieve thetraffic system multigranularity control completely

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial NaturalScience Foundation China [grant number ZR2018FM027]Project of Shandong Province Higher Educational Scienceand Technology Program [grant number J17KB181] KeyResearch and Development Project in Shandong ProvinceChina [grant numbers 2015GGX101048 2016GSF120009]

Mathematical Problems in Engineering 11

and National Natural Science Foundation of Youth FundProject [grant number 61501284]

References

[1] Q He K L Head and J Ding ldquoMulti-modal traffic signalcontrol with priority signal actuation and coordinationrdquoTrans-portation Research Part C Emerging Technologies vol 46 pp65ndash82 2014

[2] Y Yao ldquoInterpreting concept learning in cognitive informaticsand granular computingrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 39 no 4 pp 855ndash8662009

[3] J T Yao A V Vasilakos andW Pedrycz ldquoGranular computingperspectives and challengesrdquo IEEE Transactions on Cyberneticsvol 43 no 6 pp 1977ndash1989 2013

[4] H Liu andM Cocea ldquoGranular computing-based approach forclassification towards reduction of bias in ensemble learningrdquoGranular Computing vol 2 no 3 pp 131ndash139 2017

[5] X Ji L Liu and Q Wu ldquoTraffic management and decision-making support based on granular computingrdquo Journal ofHighway and Transportation Research and Development vol2008 no 10 pp 129ndash133 2008

[6] S Dewen C Xinhong C Jing and F Xujian ldquoGranularcomputing in intelligent transportation An exploratory studyrdquoCybernetics and Information Technologies vol 15 no 6 pp 122ndash134 2015

[7] C Yao J-C Gu and Q Yang ldquoTraffic congestion identificationand analysis of urban road network based on granular comput-ingrdquo Applied Mechanics and Materials vol 641-642 pp 916ndash922 2014

[8] T Li D Ruan Y Shen E Hermans and G Wets ldquoA newweighting approach based on rough set theory and granularcomputing for road safety indicator analysisrdquo InternationalJournal of Computational Intelligence vol 32 no 4 pp 517ndash5342016

[9] F He R Liu Y Zhang et al ldquoBGrR large-scale network routingspeedup techniques based on granular computingrdquo ComputerScience vol 41 no 11 pp 265ndash281 2014

[10] R Deb and A W-C Liew ldquoMissing value imputation forthe analysis of incomplete traffic accident datardquo InformationSciences vol 339 pp 274ndash289 2016

[11] S Hao L Yang and Y Shi ldquoData-driven car-following modelbased on rough set theoryrdquo IET Intelligent Transport Systemsvol 12 no 1 pp 49ndash57 2018

[12] L Livi and A Sadeghian ldquoGranular computing computationalintelligence and the analysis of non-geometric input spacesrdquoGranular Computing vol 1 no 1 pp 13ndash20 2016

[13] X Yang H Da-Wei S Bing and Z Duo-Jia ldquoCity traffic flowbreakdown prediction based on fuzzy rough setrdquo Open Physicsvol 15 no 1 pp 292ndash299 2017

[14] X Xiong L Chen and J Liang ldquoAnalysis of roadway trafficaccidents based on rough sets and Bayesian networksrdquo Promet- Traffic - Traffico vol 30 no 1 pp 71ndash81 2018

[15] M Deshpande and P Bajaj ldquoPerformance improvement of traf-fic flow prediction model using combination of support vectormachine and rough setrdquo International Journal of ComputerApplications vol 163 no 2 pp 31ndash35 2017

[16] Y Y Yao ldquoGranular computing basic issues and possible solu-tionsrdquo in Proceedings of the Fi13h International Conference onComputing and Information vol 1 pp 186ndash189 2000

[17] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Science vol 11 no 5 pp 341ndash356 1982

[18] K Shi ldquoS-rough sets and its applications in diagnosis-rec-ognition for diseaserdquo in Proceedings of the 1st InternationalConference on Machine Learning and Cybernetics pp 50ndash54IEEE Beijing China 2002

[19] K Q Shi and Y Q Cui ldquoS-Rough sets and their general struc-turesrdquo Journal of Shandong University Natural Science Shan-dong Daxue Xuebao Lixue Ban vol 37 no 6 pp 471ndash474 2002

[20] H Haiqing X Peijun and F Haiyan ldquoS-rough recognitionof knowledge and general threshold encryption authenticationscheme of recognition conclusionrdquo in Proceedings of the SixthInternational Conference on Intelligent Systems Design andApplications pp 816ndash824 Jian China October 2006

[21] S Hao L Yang and Y Shi ldquoReal-time data-driven trajectoryreconstruction based on rough set theoryrdquo Advances in Trans-portation Studies vol 44 pp 119ndash132 2018

[22] Y Yang L Jia Y Qin S Han and H Dong ldquoUnderstandingstructure of urban traffic network based on spatial-temporalcorrelation analysisrdquo Modern Physics Letters B CondensedMatter Physics Statistical Physics AtomicMolecular andOpticalPhysics vol 31 no 22 Article ID 1750230 19 pages 2017

[23] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[24] TWongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 USA September 2012

[25] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo IFAC Proceedings Volumes vol 47 no 3 pp11332ndash11337 2014

[26] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

(a)

(b) (c) (d)

Figure 6 The simulation traffic network and settings

Table 1 Traffic network performance comparison

Parameters FTMethod MPMethod EMP MethodTotal travel time (h) 633947 588139 568009Total delay time (h) 350255 298110 277457Total stopped delay (h) 253507 225692 205465Average delay time (s) 130584 111420 104057Average stopped delay (s) 94514 84353 77057Average speed (kmh) 23527 25916 26883Number of vehicles in network 448 368 334Number of vehicles that left network 9208 9264 9265Total Number of vehicles 9656 9632 9599

To illustrate the efficiency and feasibility of the pro-posed traffic network model and traffic control algorithmthree traffic light control methods are implemented in thissection including fixed-time control (FT method) max-pressure control (MP method) and the extended max-pressure control (EMP method) To simplify the simulationthe extended pressure computation only considers the caseof 119865119875 = Φ and 1198651015840

119875 = Φ in this paper the other cases will beconsidered in further research works The three traffic lightcontrol methods are implemented in Visual Studio 2010Thesimulation programs communicate with Vissim through theVissim COM programming interface to obtain the vehiclestate and traffic parameters to decide the traffic light signalat each time slot The simulation runs for 3600 s and thetraffic network performance is evaluated from 1000 s to 3600s usingVissimThis is because in the first 1000 s of simulationtime there are not enough vehicles entering the simulationtraffic network

The simulation results of the traffic network are listed inTable 1 According to the simulation results EMP methodbased on the proposed hierarchical multigranularity trafficnetwork model achieves better performance because the

light phase pressure computed using (27)-(30) is closer tothe actual traffic situation in which the vehicles on a roadsegment are represented using rough set form The depart-ing vehicles of downstream segments considered in thissimulation can achieve the coordination with downstreamintersections

As shown in Figure 7 the average queue length underEMP method is shorter than the other traffic light controlmethods The shorter queue length in traffic network meansit can effectively reduce the occurrence of spill-over ondownstream road segment so as to avoid the traffic collapsedue to vehicle queue overflow EMP method is based onthe proposed hierarchical multigranularity traffic networkmodel in which the traffic elements dynamic characteris-tics are depicted accurately and the light phase states ofdownstream intersections are also considered to computethe light phase pressure therefore EMP method achievesbetter performance As shown in Figure 8 the averagetravel time under EMP control is lower than the other twomethods

Furthermore using the extended pressure computationmethod based on the proposed traffic network the priority

10 Mathematical Problems in Engineering

MPFTEMP

05

101520253035404550

Que

ue L

engt

h (v

eh)

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000Time (s)

Figure 7 Average queue length of the three-traffic-light controlmethod

MPFTEMP

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000

Time (s)

0102030405060708090

100

Trav

el Ti

me (

s)

Figure 8 Average travel time of the three-traffic-light controlmethod

control for some particular vehicles can be achieved con-veniently since different classes of vehicles can be set withdifferent vehicle weight in the EMPmethod Different weightvalues can result in different traffic efficiency for differenttypes of vehicles For example when the type of bus is addedto the simulation traffic network the higher weight valuecan make buses have higher average speed and the otherclasses of vehicles maybe achieve lower average speed Forthis issue the vehicle weight selection is important As shownin Figure 9 the different bus weight values are simulatedand they result in different average speed According to themultiple simulation results and the real traffic demand thevehicle weight can be determined

To sum up since the hierarchical multigranularity trafficmodel can represent the actual traffic network situationsaccurately and the EMP method can achieve traffic controlflexibly this paper may be a useful attempt in solving urbantraffic control problem

5 Conclusions

In this paper a hierarchical multigranularity traffic networkis proposed and on the basis of the proposed traffic networkan extended max-pressure traffic control method is imple-mented The simulation results illustrate that the proposedtraffic network model and EMP method achieve better

Average speed [Car]Average speed [Bus]Average speed [All vehicles]

24245

25255

26265

27

Aver

age s

peed

15 2 25 3 35 4 45 5 551Vehicle weight of Bus

Figure 9 Simulation results with different bus weight values

performance In the proposed model the traffic elementsin urban traffic network are represented using S-rough setin which the dynamic characteristics of traffic elementsare depicted This type of traffic element description canrepresent the traffic network situation accurately so as todesign the more efficient traffic control strategy

However the hierarchical multigranularity traffic net-work model is only a preliminary exploration and it needsto be further improved in the future research work such asthe traffic subregion formulation and the boundary controlof traffic subregion For the traffic subregion formulationthe correlation of downstream and upstream road segmentsshould be computed considering traffic states and the geo-graphical characteristics of each pair of upstream and down-stream segments The control strategy of subregion boundarycan compute the traffic pressure considering the remainingcapacity of downstream segment to speed up the vehicleflow release towards to the segment with larger remainingcapacity These issues will be studied in our further researchwork to improve the hierarchical multigranularity traffic net-workmodel Furthermore in the future work the completelymultigranularity traffic control strategies should be furtherresearched using granular computing theory to achieve thetraffic system multigranularity control completely

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial NaturalScience Foundation China [grant number ZR2018FM027]Project of Shandong Province Higher Educational Scienceand Technology Program [grant number J17KB181] KeyResearch and Development Project in Shandong ProvinceChina [grant numbers 2015GGX101048 2016GSF120009]

Mathematical Problems in Engineering 11

and National Natural Science Foundation of Youth FundProject [grant number 61501284]

References

[1] Q He K L Head and J Ding ldquoMulti-modal traffic signalcontrol with priority signal actuation and coordinationrdquoTrans-portation Research Part C Emerging Technologies vol 46 pp65ndash82 2014

[2] Y Yao ldquoInterpreting concept learning in cognitive informaticsand granular computingrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 39 no 4 pp 855ndash8662009

[3] J T Yao A V Vasilakos andW Pedrycz ldquoGranular computingperspectives and challengesrdquo IEEE Transactions on Cyberneticsvol 43 no 6 pp 1977ndash1989 2013

[4] H Liu andM Cocea ldquoGranular computing-based approach forclassification towards reduction of bias in ensemble learningrdquoGranular Computing vol 2 no 3 pp 131ndash139 2017

[5] X Ji L Liu and Q Wu ldquoTraffic management and decision-making support based on granular computingrdquo Journal ofHighway and Transportation Research and Development vol2008 no 10 pp 129ndash133 2008

[6] S Dewen C Xinhong C Jing and F Xujian ldquoGranularcomputing in intelligent transportation An exploratory studyrdquoCybernetics and Information Technologies vol 15 no 6 pp 122ndash134 2015

[7] C Yao J-C Gu and Q Yang ldquoTraffic congestion identificationand analysis of urban road network based on granular comput-ingrdquo Applied Mechanics and Materials vol 641-642 pp 916ndash922 2014

[8] T Li D Ruan Y Shen E Hermans and G Wets ldquoA newweighting approach based on rough set theory and granularcomputing for road safety indicator analysisrdquo InternationalJournal of Computational Intelligence vol 32 no 4 pp 517ndash5342016

[9] F He R Liu Y Zhang et al ldquoBGrR large-scale network routingspeedup techniques based on granular computingrdquo ComputerScience vol 41 no 11 pp 265ndash281 2014

[10] R Deb and A W-C Liew ldquoMissing value imputation forthe analysis of incomplete traffic accident datardquo InformationSciences vol 339 pp 274ndash289 2016

[11] S Hao L Yang and Y Shi ldquoData-driven car-following modelbased on rough set theoryrdquo IET Intelligent Transport Systemsvol 12 no 1 pp 49ndash57 2018

[12] L Livi and A Sadeghian ldquoGranular computing computationalintelligence and the analysis of non-geometric input spacesrdquoGranular Computing vol 1 no 1 pp 13ndash20 2016

[13] X Yang H Da-Wei S Bing and Z Duo-Jia ldquoCity traffic flowbreakdown prediction based on fuzzy rough setrdquo Open Physicsvol 15 no 1 pp 292ndash299 2017

[14] X Xiong L Chen and J Liang ldquoAnalysis of roadway trafficaccidents based on rough sets and Bayesian networksrdquo Promet- Traffic - Traffico vol 30 no 1 pp 71ndash81 2018

[15] M Deshpande and P Bajaj ldquoPerformance improvement of traf-fic flow prediction model using combination of support vectormachine and rough setrdquo International Journal of ComputerApplications vol 163 no 2 pp 31ndash35 2017

[16] Y Y Yao ldquoGranular computing basic issues and possible solu-tionsrdquo in Proceedings of the Fi13h International Conference onComputing and Information vol 1 pp 186ndash189 2000

[17] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Science vol 11 no 5 pp 341ndash356 1982

[18] K Shi ldquoS-rough sets and its applications in diagnosis-rec-ognition for diseaserdquo in Proceedings of the 1st InternationalConference on Machine Learning and Cybernetics pp 50ndash54IEEE Beijing China 2002

[19] K Q Shi and Y Q Cui ldquoS-Rough sets and their general struc-turesrdquo Journal of Shandong University Natural Science Shan-dong Daxue Xuebao Lixue Ban vol 37 no 6 pp 471ndash474 2002

[20] H Haiqing X Peijun and F Haiyan ldquoS-rough recognitionof knowledge and general threshold encryption authenticationscheme of recognition conclusionrdquo in Proceedings of the SixthInternational Conference on Intelligent Systems Design andApplications pp 816ndash824 Jian China October 2006

[21] S Hao L Yang and Y Shi ldquoReal-time data-driven trajectoryreconstruction based on rough set theoryrdquo Advances in Trans-portation Studies vol 44 pp 119ndash132 2018

[22] Y Yang L Jia Y Qin S Han and H Dong ldquoUnderstandingstructure of urban traffic network based on spatial-temporalcorrelation analysisrdquo Modern Physics Letters B CondensedMatter Physics Statistical Physics AtomicMolecular andOpticalPhysics vol 31 no 22 Article ID 1750230 19 pages 2017

[23] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[24] TWongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 USA September 2012

[25] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo IFAC Proceedings Volumes vol 47 no 3 pp11332ndash11337 2014

[26] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

MPFTEMP

05

101520253035404550

Que

ue L

engt

h (v

eh)

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000Time (s)

Figure 7 Average queue length of the three-traffic-light controlmethod

MPFTEMP

1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 36001000

Time (s)

0102030405060708090

100

Trav

el Ti

me (

s)

Figure 8 Average travel time of the three-traffic-light controlmethod

control for some particular vehicles can be achieved con-veniently since different classes of vehicles can be set withdifferent vehicle weight in the EMPmethod Different weightvalues can result in different traffic efficiency for differenttypes of vehicles For example when the type of bus is addedto the simulation traffic network the higher weight valuecan make buses have higher average speed and the otherclasses of vehicles maybe achieve lower average speed Forthis issue the vehicle weight selection is important As shownin Figure 9 the different bus weight values are simulatedand they result in different average speed According to themultiple simulation results and the real traffic demand thevehicle weight can be determined

To sum up since the hierarchical multigranularity trafficmodel can represent the actual traffic network situationsaccurately and the EMP method can achieve traffic controlflexibly this paper may be a useful attempt in solving urbantraffic control problem

5 Conclusions

In this paper a hierarchical multigranularity traffic networkis proposed and on the basis of the proposed traffic networkan extended max-pressure traffic control method is imple-mented The simulation results illustrate that the proposedtraffic network model and EMP method achieve better

Average speed [Car]Average speed [Bus]Average speed [All vehicles]

24245

25255

26265

27

Aver

age s

peed

15 2 25 3 35 4 45 5 551Vehicle weight of Bus

Figure 9 Simulation results with different bus weight values

performance In the proposed model the traffic elementsin urban traffic network are represented using S-rough setin which the dynamic characteristics of traffic elementsare depicted This type of traffic element description canrepresent the traffic network situation accurately so as todesign the more efficient traffic control strategy

However the hierarchical multigranularity traffic net-work model is only a preliminary exploration and it needsto be further improved in the future research work such asthe traffic subregion formulation and the boundary controlof traffic subregion For the traffic subregion formulationthe correlation of downstream and upstream road segmentsshould be computed considering traffic states and the geo-graphical characteristics of each pair of upstream and down-stream segments The control strategy of subregion boundarycan compute the traffic pressure considering the remainingcapacity of downstream segment to speed up the vehicleflow release towards to the segment with larger remainingcapacity These issues will be studied in our further researchwork to improve the hierarchical multigranularity traffic net-workmodel Furthermore in the future work the completelymultigranularity traffic control strategies should be furtherresearched using granular computing theory to achieve thetraffic system multigranularity control completely

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by Shandong Provincial NaturalScience Foundation China [grant number ZR2018FM027]Project of Shandong Province Higher Educational Scienceand Technology Program [grant number J17KB181] KeyResearch and Development Project in Shandong ProvinceChina [grant numbers 2015GGX101048 2016GSF120009]

Mathematical Problems in Engineering 11

and National Natural Science Foundation of Youth FundProject [grant number 61501284]

References

[1] Q He K L Head and J Ding ldquoMulti-modal traffic signalcontrol with priority signal actuation and coordinationrdquoTrans-portation Research Part C Emerging Technologies vol 46 pp65ndash82 2014

[2] Y Yao ldquoInterpreting concept learning in cognitive informaticsand granular computingrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 39 no 4 pp 855ndash8662009

[3] J T Yao A V Vasilakos andW Pedrycz ldquoGranular computingperspectives and challengesrdquo IEEE Transactions on Cyberneticsvol 43 no 6 pp 1977ndash1989 2013

[4] H Liu andM Cocea ldquoGranular computing-based approach forclassification towards reduction of bias in ensemble learningrdquoGranular Computing vol 2 no 3 pp 131ndash139 2017

[5] X Ji L Liu and Q Wu ldquoTraffic management and decision-making support based on granular computingrdquo Journal ofHighway and Transportation Research and Development vol2008 no 10 pp 129ndash133 2008

[6] S Dewen C Xinhong C Jing and F Xujian ldquoGranularcomputing in intelligent transportation An exploratory studyrdquoCybernetics and Information Technologies vol 15 no 6 pp 122ndash134 2015

[7] C Yao J-C Gu and Q Yang ldquoTraffic congestion identificationand analysis of urban road network based on granular comput-ingrdquo Applied Mechanics and Materials vol 641-642 pp 916ndash922 2014

[8] T Li D Ruan Y Shen E Hermans and G Wets ldquoA newweighting approach based on rough set theory and granularcomputing for road safety indicator analysisrdquo InternationalJournal of Computational Intelligence vol 32 no 4 pp 517ndash5342016

[9] F He R Liu Y Zhang et al ldquoBGrR large-scale network routingspeedup techniques based on granular computingrdquo ComputerScience vol 41 no 11 pp 265ndash281 2014

[10] R Deb and A W-C Liew ldquoMissing value imputation forthe analysis of incomplete traffic accident datardquo InformationSciences vol 339 pp 274ndash289 2016

[11] S Hao L Yang and Y Shi ldquoData-driven car-following modelbased on rough set theoryrdquo IET Intelligent Transport Systemsvol 12 no 1 pp 49ndash57 2018

[12] L Livi and A Sadeghian ldquoGranular computing computationalintelligence and the analysis of non-geometric input spacesrdquoGranular Computing vol 1 no 1 pp 13ndash20 2016

[13] X Yang H Da-Wei S Bing and Z Duo-Jia ldquoCity traffic flowbreakdown prediction based on fuzzy rough setrdquo Open Physicsvol 15 no 1 pp 292ndash299 2017

[14] X Xiong L Chen and J Liang ldquoAnalysis of roadway trafficaccidents based on rough sets and Bayesian networksrdquo Promet- Traffic - Traffico vol 30 no 1 pp 71ndash81 2018

[15] M Deshpande and P Bajaj ldquoPerformance improvement of traf-fic flow prediction model using combination of support vectormachine and rough setrdquo International Journal of ComputerApplications vol 163 no 2 pp 31ndash35 2017

[16] Y Y Yao ldquoGranular computing basic issues and possible solu-tionsrdquo in Proceedings of the Fi13h International Conference onComputing and Information vol 1 pp 186ndash189 2000

[17] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Science vol 11 no 5 pp 341ndash356 1982

[18] K Shi ldquoS-rough sets and its applications in diagnosis-rec-ognition for diseaserdquo in Proceedings of the 1st InternationalConference on Machine Learning and Cybernetics pp 50ndash54IEEE Beijing China 2002

[19] K Q Shi and Y Q Cui ldquoS-Rough sets and their general struc-turesrdquo Journal of Shandong University Natural Science Shan-dong Daxue Xuebao Lixue Ban vol 37 no 6 pp 471ndash474 2002

[20] H Haiqing X Peijun and F Haiyan ldquoS-rough recognitionof knowledge and general threshold encryption authenticationscheme of recognition conclusionrdquo in Proceedings of the SixthInternational Conference on Intelligent Systems Design andApplications pp 816ndash824 Jian China October 2006

[21] S Hao L Yang and Y Shi ldquoReal-time data-driven trajectoryreconstruction based on rough set theoryrdquo Advances in Trans-portation Studies vol 44 pp 119ndash132 2018

[22] Y Yang L Jia Y Qin S Han and H Dong ldquoUnderstandingstructure of urban traffic network based on spatial-temporalcorrelation analysisrdquo Modern Physics Letters B CondensedMatter Physics Statistical Physics AtomicMolecular andOpticalPhysics vol 31 no 22 Article ID 1750230 19 pages 2017

[23] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[24] TWongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 USA September 2012

[25] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo IFAC Proceedings Volumes vol 47 no 3 pp11332ndash11337 2014

[26] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

and National Natural Science Foundation of Youth FundProject [grant number 61501284]

References

[1] Q He K L Head and J Ding ldquoMulti-modal traffic signalcontrol with priority signal actuation and coordinationrdquoTrans-portation Research Part C Emerging Technologies vol 46 pp65ndash82 2014

[2] Y Yao ldquoInterpreting concept learning in cognitive informaticsand granular computingrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 39 no 4 pp 855ndash8662009

[3] J T Yao A V Vasilakos andW Pedrycz ldquoGranular computingperspectives and challengesrdquo IEEE Transactions on Cyberneticsvol 43 no 6 pp 1977ndash1989 2013

[4] H Liu andM Cocea ldquoGranular computing-based approach forclassification towards reduction of bias in ensemble learningrdquoGranular Computing vol 2 no 3 pp 131ndash139 2017

[5] X Ji L Liu and Q Wu ldquoTraffic management and decision-making support based on granular computingrdquo Journal ofHighway and Transportation Research and Development vol2008 no 10 pp 129ndash133 2008

[6] S Dewen C Xinhong C Jing and F Xujian ldquoGranularcomputing in intelligent transportation An exploratory studyrdquoCybernetics and Information Technologies vol 15 no 6 pp 122ndash134 2015

[7] C Yao J-C Gu and Q Yang ldquoTraffic congestion identificationand analysis of urban road network based on granular comput-ingrdquo Applied Mechanics and Materials vol 641-642 pp 916ndash922 2014

[8] T Li D Ruan Y Shen E Hermans and G Wets ldquoA newweighting approach based on rough set theory and granularcomputing for road safety indicator analysisrdquo InternationalJournal of Computational Intelligence vol 32 no 4 pp 517ndash5342016

[9] F He R Liu Y Zhang et al ldquoBGrR large-scale network routingspeedup techniques based on granular computingrdquo ComputerScience vol 41 no 11 pp 265ndash281 2014

[10] R Deb and A W-C Liew ldquoMissing value imputation forthe analysis of incomplete traffic accident datardquo InformationSciences vol 339 pp 274ndash289 2016

[11] S Hao L Yang and Y Shi ldquoData-driven car-following modelbased on rough set theoryrdquo IET Intelligent Transport Systemsvol 12 no 1 pp 49ndash57 2018

[12] L Livi and A Sadeghian ldquoGranular computing computationalintelligence and the analysis of non-geometric input spacesrdquoGranular Computing vol 1 no 1 pp 13ndash20 2016

[13] X Yang H Da-Wei S Bing and Z Duo-Jia ldquoCity traffic flowbreakdown prediction based on fuzzy rough setrdquo Open Physicsvol 15 no 1 pp 292ndash299 2017

[14] X Xiong L Chen and J Liang ldquoAnalysis of roadway trafficaccidents based on rough sets and Bayesian networksrdquo Promet- Traffic - Traffico vol 30 no 1 pp 71ndash81 2018

[15] M Deshpande and P Bajaj ldquoPerformance improvement of traf-fic flow prediction model using combination of support vectormachine and rough setrdquo International Journal of ComputerApplications vol 163 no 2 pp 31ndash35 2017

[16] Y Y Yao ldquoGranular computing basic issues and possible solu-tionsrdquo in Proceedings of the Fi13h International Conference onComputing and Information vol 1 pp 186ndash189 2000

[17] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Science vol 11 no 5 pp 341ndash356 1982

[18] K Shi ldquoS-rough sets and its applications in diagnosis-rec-ognition for diseaserdquo in Proceedings of the 1st InternationalConference on Machine Learning and Cybernetics pp 50ndash54IEEE Beijing China 2002

[19] K Q Shi and Y Q Cui ldquoS-Rough sets and their general struc-turesrdquo Journal of Shandong University Natural Science Shan-dong Daxue Xuebao Lixue Ban vol 37 no 6 pp 471ndash474 2002

[20] H Haiqing X Peijun and F Haiyan ldquoS-rough recognitionof knowledge and general threshold encryption authenticationscheme of recognition conclusionrdquo in Proceedings of the SixthInternational Conference on Intelligent Systems Design andApplications pp 816ndash824 Jian China October 2006

[21] S Hao L Yang and Y Shi ldquoReal-time data-driven trajectoryreconstruction based on rough set theoryrdquo Advances in Trans-portation Studies vol 44 pp 119ndash132 2018

[22] Y Yang L Jia Y Qin S Han and H Dong ldquoUnderstandingstructure of urban traffic network based on spatial-temporalcorrelation analysisrdquo Modern Physics Letters B CondensedMatter Physics Statistical Physics AtomicMolecular andOpticalPhysics vol 31 no 22 Article ID 1750230 19 pages 2017

[23] P Varaiya ldquoMax pressure control of a network of signalizedintersectionsrdquo Transportation Research Part C Emerging Tech-nologies vol 36 pp 177ndash195 2013

[24] TWongpiromsarn T Uthaicharoenpong Y Wang E Frazzoliand D Wang ldquoDistributed traffic signal control for maximumnetwork throughputrdquo in Proceedings of the 2012 15th Interna-tional IEEE Conference on Intelligent Transportation SystemsITSC 2012 pp 588ndash595 USA September 2012

[25] J Gregoire E Frazzoli A De La Fortelle and T Wong-piromsarn ldquoBack-pressure traffic signal control with unknownrouting ratesrdquo IFAC Proceedings Volumes vol 47 no 3 pp11332ndash11337 2014

[26] B Cesme and P G Furth ldquoSelf-organizing traffic signals usingsecondary extension and dynamic coordinationrdquo Transporta-tion Research Part C Emerging Technologies vol 48 pp 1ndash152014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom