ResearchArticle A Traffic Restriction Scheme for Enhancing ...downloads.hindawi.com/journals/ddns/2017/9626938.pdf · DiscreteDynamicsinNatureandSociety 5 Table1:ODdemands. ODpair

Research ArticleA Traffic Restriction Scheme for Enhancing Carpooling

Dong Ding and Bin Shuai

School of Transportation and Logistic Southwest Jiaotong University Chengdu 610031 China

Correspondence should be addressed to Bin Shuai shuaibinhomeswjtueducn

Received 14 October 2016 Revised 23 January 2017 Accepted 20 February 2017 Published 12 March 2017

Academic Editor Gabriella Bretti

Copyright copy 2017 Dong Ding and Bin Shuai This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

For the purpose of alleviating traffic congestion this paper proposes a scheme to encourage travelers to carpool by traffic restrictionBy a variational inequity we describe travelersrsquo mode (solo driving and carpooling) and route choice under user equilibriumprinciple in the context of fixed demand and detect the performance of a simple network with various restriction links restrictionproportions and carpooling costs Then the optimal traffic restriction scheme aiming at minimal total travel cost is designedthrough a bilevel program and applied to a Sioux Fall network example with genetic algorithm According to various requirementsoptimal restriction regions and proportions for restricted automobiles are captured From the results it is found that trafficrestriction scheme is possible to enhance carpooling and alleviate congestion However higher carpooling demand is not alwayshelpful to the whole network The topology of network OD demand and carpooling cost are included in the factors influencingthe performance of the traffic system

1 Introduction

Traffic congestion is becoming aworld-wide problem Severalmeasures have emerged for alleviating traffic congestion suchas congestion pricing signal control and traffic restrictionCongestion pricing has been implemented in handful placessuch as Singapore London and Stockholmand received goodperformance as well as signal control and traffic restrictionHowever they still come with respective undesirable affectsFor example congestion pricing often results in rejectionsfrom the public signal control is difficult to implement in alarge network and normally traffic restriction inhibits traveldemand to large extent so that it brings inconvenience tothe population Carpooling is a better choice owing to therelationship between vehicles and travelers Namely it is ableto decrease the number of vehicles without reduced traveldemands Thus carpooling is supposed to be advocated foralleviating congestion

However a number of solo driving travelers remainpreference to keep driving alone [1 2] Possible factorsaffecting travelers solo driving behavior have been revealedAs an example Abrahamse andKeall [2] attributed the factorsto some sociodemographic characteristics such as income

age and gender Teal [3] indicated that the travelers withlonger travel distance and less access to a vehicle prefercarpooling Vanoutrive et al [4] analyzed the popularity ofcarpooling in Belgium and found that workplace locationorganization and promotion can explain carpooling behav-ior thereManners such as providing carpooling informationHOVHOT lanes have been taken to prompt carpoolingAbrahamse and Keall [2] investigated the effect of web-based carpooling in New Zealand and manifested that thenumber of carpoolers had increased to 27 with carpoolinginformation provided Nevertheless even in the presence ofcarpooling and measures prompting carpooling the perfor-mance of traffic system seems not desirable in some casesKwon and Varaiya [5] argued that HOV lanes failed to attractmore travelers to carpool so that overall congestion still existsWhat is more Dahlgren [6] showed that sometimes HOVlanes for carpooling perform worse than general purposelanes From the literatures above we attribute themain reasonto insufficient share of carpooling travelers As an alternativeHOT lane policy shows a better performance thanHOV lanesto some extent especially in North America Taking the I-394 HOT lane as an example Cho et al [7] revealed that

HindawiDiscrete Dynamics in Nature and SocietyVolume 2017 Article ID 9626938 9 pageshttpsdoiorg10115520179626938

2 Discrete Dynamics in Nature and Society

there are additional factors to influence time saving and thewillingness of travelers to pay to use HOT lanes Dahlgren[8] established a model to demonstrate the best situation ofsetting HOT lanes which is related to the initial delay andconstruct and operate cost Thus HOT lane is not always abetter choice to encourage carpooling to mitigate congestionMore effective measures need to be developed for enhancingcarpooling and the analysis of impacts of traffic restrictionon traffic system is necessary

Traffic restriction as a scheme forbidding automobilesto enter the given regions has been focused on in practiceand theory Daganzo [9] referred it as road spacing rationingand examined the effect of hybrid scheme of road spacingrationing and road pricing on the background of a bottleneckConsidering traffic equilibrium the efficiency traffic restric-tion was analyzed [9 10] Shi et al [11] established formula-tions to capture the optimal district and restriction ratio in areal network Traffic restriction hashad been implemented inmany areas around the world includingMexico City ManilaPhilippine and Columbia During 2008 Olympic Gamesthe traffic restriction scheme in Beijing received excellentperformance in alleviating urban congestion and reducingtraffic emission Regardless of the visible effect the drawbacksof traffic restriction are advocated For example Eskelandand Feyzioglu [12] pointed out that traffic restriction schemefailed in Mexico City since people always tried quite afew methods to avoid the restriction such as purchasingadditional vehicles driving with other license plates or onweekends Nie [13] indicated that traffic restriction schemeis a bad policy for social welfare unless some complementarymeasures are taken such as tradable credits scheme or controlof vehicle quota as the travelers probably buy another vehiclewith new road spacing qualification to avoid the regulationHowever purchasing another vehicle normally happens inthe places with low price of owing a vehicle It is a good policyunder the condition that people cannot afford additionalvehicles easily Thus the impacts of traffic restriction schemeon efficiency need to be analyzed

In conclusion traffic restriction scheme needs comple-mentarymeasures to optimize traffic systemwhile carpoolingneeds schemes to prompt Triggered by this we proposea scheme imposing traffic restriction scheme to enhancecarpooling Namely all restricted travelers have to eithercarpool to enter the restriction region or bypass it Apparentlythe analysis of the performance of this scheme is necessaryFor the purpose of minimizing total travel cost the optimalscheme to capture the restriction region and proportion isdesigned

The rest of this paper is organized as follows Section 2introduces the notations and assumptions of this paperand establishes the formulations of lower level Section 3investigates the impacts of various schemes to a simple net-work The optimal scheme is designed in Section 4 with thedemonstration of algorithm in Section 5 and applied to aSioux Falls network example in Section 6 The conclusionsand discussions are included in Section 7

2 Model Formulation

21 Notation and Assumptions Consider a strong connectednetwork (119866 119860) 119866 is the set of nodes and 119860 is the set oflinks119882 and119872 denote the set of OD pairs and travel modesrespectively 119871 is the set of links

We utilize the notation which represents the variables inthis paper in the notation section

For generality and simplicity we adopt the three assump-tions in this paper

(1) There exits two travelmodes in the network solo driv-ing and carpooling We use 119904 and 119888 to represent thesolo drivingmode and carpoolingmode respectivelyThus119898 = 119904 119888

(2) According to the investigation [14] the average num-ber of travelers in one vehicle is 16Thus in this paperwe assume two travelers in one vehicle for carpoolingmode [15]

(3) Travelers follow the traffic restriction regulationNamely all the restricted travelers have to bypass therestriction region or carpool Penalty cost does notexist in this paper because no one violates the restric-tion regulation

22 Cost Function Here we adopt the BPR function to com-pute the travel time

119905119886 = 1199050119886 [1 + 120573( V119888119886119862119886)120572] 119886 isin 119860 (1)

120572 and 120573 are constant parameters The path cost of solodriving travelers can bemeasured by the path travel time Forcarpooling travelers carpooling cost on each path is unavoid-able apart from the path travel time We use 998779 to representthe carpooling cost which may include the waiting timeinconvenience and the out-of-pocket savings and is set tobe a constant [15] Actually carpooling travelers will take thesame time with solo driving travelers on a link Thus the linkcost function on link 119886 can be expressed as follows

119888119904119886 = 119888119888119886 = 120582119905 sdot 119905119886 119886 isin 119860 (2)

Carpooling cost will happen before a trip if travelerschoose carpooling Then the cost function on path 119897 can bedenoted as follows

119888119904119908119897 = 120582119905 sdot sum119886

119905119886 sdot 120575119897119886 119886 isin 119860 119908 isin 119882 119897 isin 119871119888119888119908119897 = 998779 + 120582119905 sdot sum

119886

119905119886 sdot 120575119897119886 119886 isin 119860 119908 isin 119882 119897 isin 119871 (3)

where 120582119905 is the value of time For simplicity in this paper weassume 120582119905 is a constant which means all travelers have thesame value of time

23 The Model of Mode Split and User Equilibrium Basedon the aforementioned assumptions travelers choose either

Discrete Dynamics in Nature and Society 3

solo driving or carpoolingThe share of modes can be split byLogit formulation

119889119898119908 = 119889119908 sdot exp (minus120579120583119898119908 )sum119898 exp (minus120579120583119898119908 ) 119908 isin 119882 119898 isin 119872 (4)

where the 120583119898119908 is the minimal cost of travel mode 119898 betweenOD pair 119908 and 120579 is the sensitive coefficient

In a network with traffic restriction solo driving trav-elers have to bypass the district or carpool since they areforbidden to enter the restriction region When there is nopath connecting the OD pair out of the region they onlyhave carpooling choice Similar to [11 13] the minimal costfunction of solo driving travelers between the OD pair 119908 inwhich the restricted travelers are not fully blocked can becalculated as follows

120583119904119908 = (1 minus 120574) 120583119904119906119908 + 120574120583119904119903119908 119908 isin 119882 (5)

where 120583119904119906119908 represents the minimal cost of unrestricted solodriving travelers and 120583119904119903119908 denotes the minimal cost ofrestricted solo driving travelers 120574 is the restriction propor-tion of automobiles

Some OD pairs which are denoted as 119882119903 are notconnected by any path which is out of the restriction regionNamely the travelers between these OD pairs have to carpoolsince solo driving travelers cannot pass the region Theminimal cost function of the travelers between 119882119903 is asfollows

120583119904119908 = (1 minus 120574) 120583119904119906119908 + 120574120583119888119908 119908 isin 119882119903 (6)

The minimal cost function of carpooling travelers is

120583119888119908 = min (119888119888119908119897) 119908 isin 119882 119897 isin 119871 (7)

Then the following variational inequality (VI) program isorganized to express the mode split and user equilibrium

sum119908isin119882

sum119897isin119871

119888119888119908119897 (119891lowast) (119891119888119908119897 minus 119891119888lowast119908119897)+ sum119908isin119882

sum119897isin119871

119888119904119906119908119897 (119891lowast) (119891119904119906119908119897 minus 119891119904119906lowast119908119897 )+ sum119908isin119882119903

sum119897isin119871

119888119904119903119908119897 (119891lowast) (119891119904119903119908119897 minus 119891119904119903lowast119908119897 )

+ sum119908isin119882

1120579 ln

119889119888lowast119908119889119908 (119889119888

119908 minus 119889119888lowast119908 )

+ sum119908isin119882

1120579 ln

119889119904lowast119908119889119908 (119889119904

119908 minus 119889119904lowast119908 ) ge 0

(8)

The solutions (119891119888lowast119908119897 119891119904119906lowast119908119897 119891119904119903lowast119908119897 119889119888lowast119908 119889119904lowast119908 ) belong to the fol-lowing feasible region

119889119888119908 + 119889119904119908 = 119889119908 119908 isin 119882 (9)

sum119897isin119871

119891119904119906119908119897 = (1 minus 120574) 119889119904119908 119908 isin 119882 (10)

sum119897isin119871

119891119904119903119908119897 = 120574119889119904119908 119908 isin 119882119903 (11)

sum119897isin119871

119891119888119908119897 = 119889119888119908 119908 isin 119882 (12)

119891119904119903119908119897 ge 0 119908 isin 119882119903 119897 isin 119871 (13)

119891119904119906119908119897 ge 0 119908 isin 119882 119897 isin 119871 (14)

119891119888119908119897 ge 0 119908 isin 119882 119897 isin 119871 (15)

The complementarity conditions are

119891119894119908119897 sdot 120582119894119908 = 0 119894 = (119888 119904119906) 119908 isin 119882 119897 isin 119871 (16)

120582119894119908 ge 0 119894 = (119888 119904119906) 119908 isin 119882 (17)

119891119904119903119908119897 sdot 120582119904119903119908 = 0 119908 isin 119882119903 119897 isin 119871 (18)

120582119904119903119908 ge 0 119908 isin 119882119903 (19)

Proposition 1 The VI program above is equivalent to Logitmode split and user equilibrium

Proof According to KKT conditions we have the followingequations

119891119888119908119897 119888119888119908119897 minus 120583119888119908 minus 120582119888119908 = 0 119908 isin 119882 119897 isin 119871 (20)



119889119888119908 1120579 ln119889119888119908119889119908 minus 120596119908 + 120583

119888

119908 119908 isin 119882 (23)

119889119904119908 1120579 ln119889119904119908119889119908 minus 120596119908 + (1 minus 120574) 120583

119904119906

119908 + 120574120583119904119903119908 119908 isin 119882 (24)

From equations (16) and (22) we obtain the followingequations

119888119888119908119897 = 120583119888119908 if 119891119888119908119897 gt 0 119908 isin 119882 119897 isin 119871119888119888119908119897 ge 120583119888119908 if 119891119888119908119897 = 0 119908 isin 119882 119897 isin 119871119888119904119906119908119897 = 120583119904119906119908 if 119891119904119906119908119897 gt 0 119908 isin 119882 119897 isin 119871119888119904119906119908119897 ge 120583119904119906119908 if 119891119904119906119908119897 = 0 119908 isin 119882 119897 isin 119871119888119904119903119908119897 = 120583119904119903119908 if 119891119904119903119908119897 gt 0 119908 isin 119882119903 119897 isin 119871119888119904119903119908119897 ge 120583119904119903119908 if 119891119904119903119908119897 = 0 119908 isin 119882119903 119897 isin 119871

(25)


The equations above imply that the route choice ofusers follows user equilibrium which means travelers alwayschoose the route with the minimal travel cost and noindividual can curtain hisher travel cost by changing theroute

From equations (23) and (24) we can obtain the Logitformulation Hence the VI program meets the Logit modesplit and user equilibrium

When the restriction region is given the feasible path setsof solo driving travelers and carpooling travelers are fixedThe feasible region of the VI program is nonnegative andlinear while the VI formulation is continuous It means thatthere exists at least one solution for the VI program (8)Similar specific proof can be seen in [11 24]

Several approaches have emerged for solvingVI programsuch as Frank-Wolfe algorithm projection algorithm andsensitive-based analysis algorithm Here the block Gauss-Seidel decomposition approach together with the method ofsuccessive averages is applied to aforementionedVI programIn terms of this approach the demands of travel modes andflow patterns can be obtained

3 Impacts of Traffic Restriction Scheme

In this paper we adopt the total travel cost which is expressedlater to measure the performance of a network

119885 = 998779 sdot sum119908isin119882

119889119888119908 + sum119886isin119860

(119888119904119886 sdot V119904119886 + 119888119888119886 sdot V119888119886) (26)

For a traffic restriction scheme two components haveto be decided the traffic restriction region and proportionWith various regions and proportions the performances ofnetworks are different In addition other factors such ascarpooling cost can influence the efficiency of the schemeWe explore the impacts of a simple network under thescheme The simple network with six nodes and seven linksis illustrated in Figure 1 The OD demands are listed later aswell as the attributes of links in Tables 1 and 2 120579 = 005 120582119905 =2 120572 = 4 and 120573 = 015

Here are four types of scenarios no traffic restriction thewhole network region with proportion 1 given part of thenetwork region with the proportion 1 and given part of thenetwork region with given proportion less than 1 Table 3shows the results under the four scenarios With the samerestriction region the total travel cost under scheme 3 is lowerthan that under scheme 4 owing tomore carpooling travelersScheme 1 without traffic restriction and scheme 2 with allrestriction links and restricted solo driving travelers have thehighest and lowest total travel cost respectively

Figure 2 displays the respective trend of total travel costunder the four scenarios with various carpooling costs It isapparent that when carpooling cost increases all the totaltravel costs increaseNote that at the beginning the total travelcost under Scenario 2 is the lowest and the highest one isunder Scenario 1 However the curve under Scenario 2 hasthe fastest growing rate and the slowest growing rate followsScenario 1 When the carpooling cost equals 15 the totaltravel cost under Scenario 2 begins to be less than that under

1 2 3

4 5 6

Figure 1 A simple network

Scenario 3 The explanation is that when all the travelers areforbidden to enter the restriction region they have to carpoolleading to higher total carpooling cost So the total travel coststill increases although the travel time on links decreasesWhen the carpooling cost equals 25 the total travel costunder Scenario 4 is themaximal whichmeans the carpoolingcost plays the main role in the total travel cost When thecarpooling cost equals 5 the total travel cost without trafficrestriction scheme is minimal Thus if the carpooling cost istoo high the best choice is to abandon traffic restriction

Figure 3 shows the total travel costs with various carpool-ing costs and proportion when links 1 and 2 are restrictedThe total travel cost increases generally with the increasingcarpooling cost At some points the total travel costs withsmaller proportions are less than those with larger propor-tion For example when the carpooling cost varies from 3 to5 the total travel cost with proportion 03 is lower than thatwith proportion 04 From Figures 2 and 3 we know that thetotal travel cost is associated with factors such as topologyof the network travel demands carpooling cost and trafficrestriction scheme

4 Optimal Traffic Restriction Design

Under user equilibrium no travelers can curtail their travelcosts by changing routes which can be described as a Nashequilibrium By implementing a traffic restriction schemeour target is to minimize the total travel cost under theequilibrium state A bilevel program is applied and the upperlevel is organized as follows

min 119885 = 998779 sdot sum119908isin119882

119889119888119908 + sum119886isin119860

(119888119904119886 sdot V119904119886 + 119888119888119886 sdot V119888119886) st 0 le 120574 le 1

(27)

The lower level is the variational inequality (10) and theflow patterns can be obtained from it

In terms of the objective the selection of traffic restrictionlinks is a conventional problem However the restrictionlinks do not always assemble a connected region In realcircumstances road managers normally select an area as atraffic restriction region for convenience of implementationand the effect of alleviating congestion And if a special eventwill happen in a place the region including this place isexpected to be uncongested Thus this place is supposed tobe included in the restriction region In addition in manycities the restriction proportion is plate-number-based Thecommon methods are odd-and-even license plate rule and


Table 1 OD demands

OD pair 1-2 1ndash6 2-3 2ndash5 3ndash6 4-5 5-6Demand (vehh) 1000 2000 1000 800 600 1200 500

Table 2 Attributes of links

Link 1-2 2-3 1ndash3 2ndash5 3ndash6 4-5 5-61199050119886(min) 6 4 3 3 3 4 2

119862119886 (vehh) 2000 2000 1000 1000 1000 2000 2000

Table 3 Performance of scenarios

Scenario 1 Scenario 2 Scenario 3 Scenario 4Restriction links none all 1 2 1 2Proportion 0 1 1 03Total travel cost 90276 83186 84192 89299Carpooling demand 3506 7100 5252 4033

tail number restrictionThus the proportion can be 05 (odd-and-even license plate rule) 02 (every two tail numbers perday amongworkdays) and 01 (every tail number in turn) Forthese reasons we detect the scenarios that the links alwaysconstruct a connected region and with fixed proportionsThenwewill use an example to demonstrate the performance

5 Algorithm

Nonconvex property makes it difficult to solve the problemalthough some relevant algorithms are proposed Our targetis to capture the optimal district and the proportion For theoptimization of district almost all the previous studies denote119899119886 the binary variable to decide whether a link is selected Iflink a is selected 119899119886 equals 1 otherwise 0which is very similarto the discrete network design problem (DNDP) Besidescontinuous network design (CNDP) enlightens us with themethod to obtain the proportion Thus we are able to takemixed network design problem (MNDP) as the referenceTable 4 lists some algorithms of network design problemwith the expression of their superiorities and drawbacks Itdisplays that some algorithms are convenient to implementwithout guarantee of global optimal solutions while some ofthem are helpful for the global optimal solutions but com-putationally demanding For a large-scale network it mightbe extremely difficult to utilize the algorithms which captureglobal optimal solutions due to the very expensive com-putation such as Branch and Bound algorithm [20] andsimulation [25] In addition the difference between ourproblem and MNDP is that the optimal area is required tobe connected which may induce impossibility of some ofthe algorithms Friesz et al [25] pointed out that only whenthe global optimal solution or near global optimal solutionis essential can we adopt their algorithm In this paperthe objective is minimization of total travel cost and evena local optimal solution can provide reference to the man-agers Zhang and Yang [26] proposed a genetic-based algo-rithm for optimal congestion pricing cordon and charge

80000850009000095000

100000105000110000115000120000

05 1 15 2 25 3 35 4 45 5To

tal t

rave

l cos

tCarpooling cost

Scenario 1Scenario 2


Figure 2 The total travel costs with various carpooling costs

Tota

l tra

vel c

ost

11

105

10

95

9

85

81

times104

Proportion05

0 Carpooling cost0

12

34

5

Figure 3 The total travel costs with various carpooling costs andproportions

simultaneously in a large-size realistic network Thus for thefeasibility and convenience we adopt the genetic algorithm tooptimize the traffic restriction area and the proportion simul-taneously (see also [11])

6 Numerical Example

Here we take the Sioux Falls network presented in [27] as anexample to design the optimal traffic restriction scheme inthe presence of carpooling It is assumed that the influence ofthe pair of links between any two nodes is not mutual The


Table 4 Algorithms of network design problem

Approach Proposeruser Case Superiorities Drawbacks

Ant system Poorzahedy andAbulghasemi [16] Discrete network design Less computational demand and

simple implementationNo guarantee of global optimal

solution

Genetic algorithm Drezner andWesolowsky [17] Discrete network design Less computational demand and


solutionHybridmeta-heuristicalgorithm

Poorzahed andRouhani [18] Discrete network design Less computational demand and


solution

Support fuction Gao et al [19] Discrete network design Global optimal solution Expensive computationBranch and Boundalgorithm

Farvaresh andSepehri [20] Discrete network design Global optimal solution Expensive computation

Simulation Friesz et al (1992) Continuous networkdesign

Global optimal solution or beingvery close to global optimal

solutionExpensive computation

SO-relaxationbased method andUE-reductionbased method

Wang et al [21] Discrete network design Global optimal solution Expensive computation

Global optimalapproach Wang and Lo [22] Mixed network design Global optimal solution More restraints and expensive

computationMILP formulationapproach Luathep et al [23] Mixed network design Global optimal solution Expensive computation

attributes of links and the demands are the same with thosein [27]

Several scenarios exist when implementing traffic restric-tion In this example we propose three scenarios

Scenario 1 Normally the traffic restriction links constitute aconnected region In this scenario the links in an optimalconnected region and the proportion are captured

Scenario 2 Sometimes the uncongested places are necessarydue to some reasons for example when important eventhappens In this scenario we assume that node 10 is a requireduncongested place Namely all the traffic restriction schemesare supposed to include node 10Thenwe explore the optimalregion and proportion as well as the performance of thenetwork

Scenario 3 For the sake of feasibility not all methods canbe implemented in realistic world Thus in this scenario weassume the restriction proportion is 05 and 02 respectivelyThen we capture the optimal region and detect the perfor-mance

Scenario 4 is the situation without traffic restriction Fig-ure 4 illustrates the traffic restriction regions of scenarios Allregions are connected and a part of the whole network whichmeans the scenarios are convenient to be implemented inrealistic world Table 5 lists the performances of the four sce-narios Under various scenarios total travel costs under var-ious scenarios are less than that without traffic restriction aswell as the numbers of vehicles and the carpooling demandsare higher than that It displays that the traffic restriction isable to prompt carpooling and thereby alleviate congestionScenario 1 has the least total travel cost which shows that our

model and algorithm are effective Scenario 2 includes node10 which matches the requirement that node 10 is supposedto be in the restriction region In terms of the predefinedrestriction proportion the optimal region is captured in thecase of Scenario 3 It demonstrates that the scenarios with therequirements of specific region or proportion to encouragetravelers to carpool by traffic restriction are feasible It isworth noticing that in Table 5 the carpooling demand underScenario 1 is even less than that under Scenario 2 andthe vehicles under Scenario 1 are more than that underScenario 2 although the total travel cost under Scenario 1is less than that under Scenario 2 This is because the areasgive rise to different flow patterns leading to different car-pooling demand number of vehicles and total travel costsIt demonstrates that higher carpooling demand is not alwaysbeneficial to the network Although the restriction regionunder Scenario 1 is the same as that under Scenario 3(1)the different proportions lead to different total travel costsBesides the proportion under Scenario 1 is less than thatunder Scenario 3(1) It proves again that the performancedepends on various factors such as carpooling cost topologyof network and demand

Table 6 lists the weight effects on the components oftotal travel costs under various scenarios It is shown thatthe weights are similar Since the carpooling cost is low thecosts of total travel time play the main roles while the costs ofcarpooling are just small parts of the total travel costs How-ever carpooling cost still impact the mode choice due to thescale of the network and the demand Namely a large scale ofnetwork with a certain quantity of carpoolers may generatethe corresponding total travel time and thereby influences theperformance of the whole network although the total car-


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Figure 4 The traffic restriction regions under various scenarios

pooling cost is lower Therefore attributes of networkdemand and carpooling cost cannot be ignored

7 Conclusion

For the purpose of prompting carpooling this paper proposesa scheme that using traffic restriction for solo driving travel-ers to encourage them to carpool By a variational inequality

we examine the impacts of this scheme on carpooling in asimple network with fixed restriction links and proportionsthen taking Sioux falls network as an example we design theoptimal scheme to capture the optimal region and proportionunder various scenarios The algorithm based on geneticalgorithm is adopted for the solution The results show thatthis scheme can encourage more travelers to carpool andmitigate congestion with appropriate restriction linksregion


Table 5 Performances of the scenarios

Scenario 1 Scenario 2 Scenario 3(1) Scenario 3(2) No scenarioProportion 028 034 05 02 mdashTotal travel cost 365876 367071 366515 369409 370956Carpooling demand 10162 10539 9715 9387 8681Number of vehicles 12957 12769 13180 13344 13698

Table 6 Weight effects on components of total travel costs under the scenarios

Scenario 1 Scenario 2 Scenario 3(1) Scenario 3(2) No scenarioWeight of time cost () 9583 9569 9602 9619 9649Weight of carpooling cost () 417 431 398 381 351

and proportion which shows the feasibility of this schemeand the positive effects in realistic cases If the carpoolingcost is relatively too high normally this scheme cannot bringbenefits It is worth noticing that higher carpooling demandmay result in more total travel time In addition not onlycarpooling cost but also topology of network and traveldemand are the factors influencing the performance of thenetwork Hence we need to take into account the essentialfactors to analyze the specific cases

In this paper we just consider the travel time and carpool-ing cost as the components of cost function and the minimaltotal travel cost objective The future direction includes morefactors for cost function and more objectives for exampleenvironment and economics

Notation

1199050119886 The free flow travel time on link 119886 119886 isin 119860119905119886 The travel time on link 119886 119886 isin 119860119862119886 The capacity on link 119886 119886 isin 119860119889119908 The travel demand between OD pair 119908 119908 isin 119882119889119898119908 The travel demand of mode119898 between OD pair119908 119908 isin 119882 119898 isin 119872998779 Carpooling cost119891119898119908119897 The traffic flow of mode119898 on path 119897 between ODpair 119908 119908 isin 119882 119898 isin 119872 119897 isin 119871

V119898119886 The traveler flow of mode119898 on link119886 119898 isin 119872 119886 isin 119860V119888119886 The vehicle flow on link 119886 119886 isin 119860Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] M V Vugt ldquoCommuting by car or public transportation Asocial dilemma analysis of travel mode judgementsrdquo EuropeJournal of Operation Research vol 26 pp 373ndash395 1996

[2] W Abrahamse and M Keall ldquoEffectiveness of a web-basedintervention to encourage carpooling to work A Case Study ofWellington New Zealandrdquo Transport Policy vol 21 pp 45ndash512012

[3] R F Teal ldquoCarpooling who how and whyrdquo TransportationResearch Part A General vol 21 no 3 pp 203ndash214 1987

[4] T Vanoutrive E Van De Vijver L Van Malderen et al ldquoWhatdetermines carpooling to workplaces in Belgium locationorganisation or promotionrdquo Journal of Transport Geographyvol 22 pp 77ndash86 2012

[5] J Kwon and P Varaiya ldquoEffectiveness of Californiarsquos HighOccupancy Vehicle (HOV) systemrdquo Transportation ResearchPart C vol 16 no 1 pp 98ndash115 2008

[6] J Dahlgren ldquoHigh occupancy vehicle lanes not always moreeffective than general purpose lanesrdquo Transportation ResearchPart A Policy and Practice vol 32 no 2 pp 99ndash114 1998

[7] Y Cho R Goel P Gupta G Bogonko andMW Burris ldquoWhatare I-394 HOT lane drivers paying forrdquo in Proceedings of the90th Annual Meeting Transportation Research Board (TRB rsquo11)2011

[8] J Dahlgren ldquoHigh-occupancytoll lanes where should theybe implementedrdquo Transportation Research Part A Policy andPractice vol 36 no 3 pp 239ndash255 2002

[9] C F Daganzo ldquoA pareto optimum congestion reductionschemerdquo Transportation Research Part B vol 29 no 2 pp 139ndash154 1995

[10] D Han H Yang and XWang ldquoEfficiency of the plate-number-based traffic rationing in general networksrdquo TransportationResearch Part E Logistics and Transportation Review vol 46no 6 pp 1095ndash1110 2010

[11] F Shi G-M Xu B Liu and H Huang ldquoOptimization methodof alternate traffic restriction scheme based on elastic demandand mode choice behaviorrdquo Transportation Research Part CEmerging Technologies vol 39 pp 36ndash52 2014

[12] G S Eskeland and T Feyzioglu ldquoRationing can backfire theldquoday without a carrdquo in Mexico Cityrdquo World Bank EconomicReview vol 11 no 3 pp 383ndash408 1997

[13] M Y Nie ldquoWhy license plate rationing is a bad transportpolicyrdquo Working Paper

[14] Bundesamt fur Statistik Mobilitt in der Schweiz Ergebnisse desMikrozensus Mobilitt und Verkehr 2010 2012

[15] H Yang and H-J Huang ldquoCarpooling and congestion pricingin a multilane highway with high-occupancy-vehicle lanesrdquoTransportation Research Part A Policy and Practice vol 33 no2 pp 139ndash155 1999

[16] H Poorzahedy and F Abulghasemi ldquoApplication of Ant Systemto network design problemrdquo Transportation vol 32 no 3 pp251ndash273 2005

[17] Z Drezner and G O Wesolowsky ldquoNetwork design selec-tion and design of links and facility locationrdquo Transportation


Research Part A Policy and Practice vol 37 no 3 pp 241ndash2562003

[18] H Poorzahedy and O M Rouhani ldquoHybrid meta-heuristicalgorithms for solving network design problemrdquo EuropeanJournal of Operational Research vol 182 no 2 pp 578ndash5962007

[19] Z Gao J Wu and H Sun ldquoSolution algorithm for the bi-leveldiscrete network design problemrdquo Transportation Research PartB Methodological vol 39 no 6 pp 479ndash495 2005

[20] H Farvaresh and M M Sepehri ldquoA branch and bound algo-rithm for bi-level discrete network design problemrdquo Networksand Spatial Economics vol 13 no 1 pp 67ndash106 2013

[21] S Wang Q Meng and H Yang ldquoGlobal optimization meth-ods for the discrete network design problemrdquo TransportationResearch Part B Methodological vol 50 pp 42ndash60 2013

[22] D Z WWang and H K Lo ldquoGlobal optimum of the linearizednetwork design problem with equilibrium flowsrdquo Transporta-tion Research Part BMethodological vol 44 no 4 pp 482ndash4922010

[23] P Luathep A Sumalee W H K Lam Z-C Li and H KLo ldquoGlobal optimization method for mixed transportationnetwork design problem a mixed-integer linear programmingapproachrdquo Transportation Research Part B Methodological vol45 no 5 pp 808ndash827 2011

[24] J Zhu F Xiao and X Liu ldquoTaxis in road pricing zoneshould they pay the congestion chargerdquo Journal of AdvancedTransportation vol 49 no 1 pp 96ndash113 2015

[25] T L Friesz H-J Cho N J Mehta R L Tobin and GAnandalingam ldquoSimulated annealing approach to the networkdesign problem with variational inequality constraintsrdquo Trans-portation Science vol 26 no 1 pp 18ndash26 1992

[26] X Zhang and H Yang ldquoThe optimal cordon-based networkcongestion pricing problemrdquo Transportation Research Part BMethodological vol 38 no 6 pp 517ndash537 2004

[27] C-Y Yan R Jiang Z-Y Gao andH Shao ldquoEffect of speed lim-its in degradable transport networksrdquo Transportation ResearchPart C Emerging Technologies vol 56 pp 94ndash119 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


there are additional factors to influence time saving and thewillingness of travelers to pay to use HOT lanes Dahlgren[8] established a model to demonstrate the best situation ofsetting HOT lanes which is related to the initial delay andconstruct and operate cost Thus HOT lane is not always abetter choice to encourage carpooling to mitigate congestionMore effective measures need to be developed for enhancingcarpooling and the analysis of impacts of traffic restrictionon traffic system is necessary

Traffic restriction as a scheme forbidding automobilesto enter the given regions has been focused on in practiceand theory Daganzo [9] referred it as road spacing rationingand examined the effect of hybrid scheme of road spacingrationing and road pricing on the background of a bottleneckConsidering traffic equilibrium the efficiency traffic restric-tion was analyzed [9 10] Shi et al [11] established formula-tions to capture the optimal district and restriction ratio in areal network Traffic restriction hashad been implemented inmany areas around the world includingMexico City ManilaPhilippine and Columbia During 2008 Olympic Gamesthe traffic restriction scheme in Beijing received excellentperformance in alleviating urban congestion and reducingtraffic emission Regardless of the visible effect the drawbacksof traffic restriction are advocated For example Eskelandand Feyzioglu [12] pointed out that traffic restriction schemefailed in Mexico City since people always tried quite afew methods to avoid the restriction such as purchasingadditional vehicles driving with other license plates or onweekends Nie [13] indicated that traffic restriction schemeis a bad policy for social welfare unless some complementarymeasures are taken such as tradable credits scheme or controlof vehicle quota as the travelers probably buy another vehiclewith new road spacing qualification to avoid the regulationHowever purchasing another vehicle normally happens inthe places with low price of owing a vehicle It is a good policyunder the condition that people cannot afford additionalvehicles easily Thus the impacts of traffic restriction schemeon efficiency need to be analyzed

In conclusion traffic restriction scheme needs comple-mentarymeasures to optimize traffic systemwhile carpoolingneeds schemes to prompt Triggered by this we proposea scheme imposing traffic restriction scheme to enhancecarpooling Namely all restricted travelers have to eithercarpool to enter the restriction region or bypass it Apparentlythe analysis of the performance of this scheme is necessaryFor the purpose of minimizing total travel cost the optimalscheme to capture the restriction region and proportion isdesigned

The rest of this paper is organized as follows Section 2introduces the notations and assumptions of this paperand establishes the formulations of lower level Section 3investigates the impacts of various schemes to a simple net-work The optimal scheme is designed in Section 4 with thedemonstration of algorithm in Section 5 and applied to aSioux Falls network example in Section 6 The conclusionsand discussions are included in Section 7

2 Model Formulation

21 Notation and Assumptions Consider a strong connectednetwork (119866 119860) 119866 is the set of nodes and 119860 is the set oflinks119882 and119872 denote the set of OD pairs and travel modesrespectively 119871 is the set of links

We utilize the notation which represents the variables inthis paper in the notation section

For generality and simplicity we adopt the three assump-tions in this paper

(1) There exits two travelmodes in the network solo driv-ing and carpooling We use 119904 and 119888 to represent thesolo drivingmode and carpoolingmode respectivelyThus119898 = 119904 119888

(2) According to the investigation [14] the average num-ber of travelers in one vehicle is 16Thus in this paperwe assume two travelers in one vehicle for carpoolingmode [15]

(3) Travelers follow the traffic restriction regulationNamely all the restricted travelers have to bypass therestriction region or carpool Penalty cost does notexist in this paper because no one violates the restric-tion regulation

22 Cost Function Here we adopt the BPR function to com-pute the travel time

119905119886 = 1199050119886 [1 + 120573( V119888119886119862119886)120572] 119886 isin 119860 (1)

120572 and 120573 are constant parameters The path cost of solodriving travelers can bemeasured by the path travel time Forcarpooling travelers carpooling cost on each path is unavoid-able apart from the path travel time We use 998779 to representthe carpooling cost which may include the waiting timeinconvenience and the out-of-pocket savings and is set tobe a constant [15] Actually carpooling travelers will take thesame time with solo driving travelers on a link Thus the linkcost function on link 119886 can be expressed as follows

119888119904119886 = 119888119888119886 = 120582119905 sdot 119905119886 119886 isin 119860 (2)

Carpooling cost will happen before a trip if travelerschoose carpooling Then the cost function on path 119897 can bedenoted as follows

119888119904119908119897 = 120582119905 sdot sum119886

119905119886 sdot 120575119897119886 119886 isin 119860 119908 isin 119882 119897 isin 119871119888119888119908119897 = 998779 + 120582119905 sdot sum

119886

119905119886 sdot 120575119897119886 119886 isin 119860 119908 isin 119882 119897 isin 119871 (3)

where 120582119905 is the value of time For simplicity in this paper weassume 120582119905 is a constant which means all travelers have thesame value of time

23 The Model of Mode Split and User Equilibrium Basedon the aforementioned assumptions travelers choose either






120583119904119908 = (1 minus 120574) 120583119904119906119908 + 120574120583119904119903119908 119908 isin 119882 (5)



120583119904119908 = (1 minus 120574) 120583119904119906119908 + 120574120583119888119908 119908 isin 119882119903 (6)


120583119888119908 = min (119888119888119908119897) 119908 isin 119882 119897 isin 119871 (7)


sum119908isin119882

sum119897isin119871


sum119897isin119871


sum119897isin119871


+ sum119908isin119882

1120579 ln

119889119888lowast119908119889119908 (119889119888

119908 minus 119889119888lowast119908 )

+ sum119908isin119882

1120579 ln

119889119904lowast119908119889119908 (119889119904

119908 minus 119889119904lowast119908 ) ge 0

(8)


119889119888119908 + 119889119904119908 = 119889119908 119908 isin 119882 (9)

sum119897isin119871

119891119904119906119908119897 = (1 minus 120574) 119889119904119908 119908 isin 119882 (10)

sum119897isin119871

119891119904119903119908119897 = 120574119889119904119908 119908 isin 119882119903 (11)

sum119897isin119871

119891119888119908119897 = 119889119888119908 119908 isin 119882 (12)

119891119904119903119908119897 ge 0 119908 isin 119882119903 119897 isin 119871 (13)

119891119904119906119908119897 ge 0 119908 isin 119882 119897 isin 119871 (14)

119891119888119908119897 ge 0 119908 isin 119882 119897 isin 119871 (15)


119891119894119908119897 sdot 120582119894119908 = 0 119894 = (119888 119904119906) 119908 isin 119882 119897 isin 119871 (16)

120582119894119908 ge 0 119894 = (119888 119904119906) 119908 isin 119882 (17)

119891119904119903119908119897 sdot 120582119904119903119908 = 0 119908 isin 119882119903 119897 isin 119871 (18)

120582119904119903119908 ge 0 119908 isin 119882119903 (19)






119889119888119908 1120579 ln119889119888119908119889119908 minus 120596119908 + 120583

119888

119908 119908 isin 119882 (23)

119889119904119908 1120579 ln119889119904119908119889119908 minus 120596119908 + (1 minus 120574) 120583

119904119906

119908 + 120574120583119904119903119908 119908 isin 119882 (24)



(25)








119885 = 998779 sdot sum119908isin119882

119889119888119908 + sum119886isin119860

(119888119904119886 sdot V119904119886 + 119888119888119886 sdot V119888119886) (26)




1 2 3

4 5 6






min 119885 = 998779 sdot sum119908isin119882

119889119888119908 + sum119886isin119860


(27)




Table 1 OD demands




119862119886 (vehh) 2000 2000 1000 1000 1000 2000 2000




5 Algorithm


80000850009000095000

100000105000110000115000120000

05 1 15 2 25 3 35 4 45 5To

tal t

rave

l cos

tCarpooling cost




Tota

l tra

vel c

ost

11

105

10

95

9

85

81

times104

Proportion05

0 Carpooling cost0

12

34

5



6 Numerical Example







solution






solution



















1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

24 2248

26 47

17

20

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471

7246 67

45

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

2422

4826 47

17

20

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471 72 46 67

45

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

31

13

2527

2321

16

24 2248

26 47

1720

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

3837

4244

7172

46 6745

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

2422

4826 47

1720

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471

7246 67

45

59 6163

6865

62647539

74 66

69

70

7673

18 54

9 12

3

4 14




7 Conclusion










Notation




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














120583119904119908 = (1 minus 120574) 120583119904119906119908 + 120574120583119904119903119908 119908 isin 119882 (5)



120583119904119908 = (1 minus 120574) 120583119904119906119908 + 120574120583119888119908 119908 isin 119882119903 (6)


120583119888119908 = min (119888119888119908119897) 119908 isin 119882 119897 isin 119871 (7)


sum119908isin119882

sum119897isin119871


sum119897isin119871


sum119897isin119871


+ sum119908isin119882

1120579 ln

119889119888lowast119908119889119908 (119889119888

119908 minus 119889119888lowast119908 )

+ sum119908isin119882

1120579 ln

119889119904lowast119908119889119908 (119889119904

119908 minus 119889119904lowast119908 ) ge 0

(8)


119889119888119908 + 119889119904119908 = 119889119908 119908 isin 119882 (9)

sum119897isin119871

119891119904119906119908119897 = (1 minus 120574) 119889119904119908 119908 isin 119882 (10)

sum119897isin119871

119891119904119903119908119897 = 120574119889119904119908 119908 isin 119882119903 (11)

sum119897isin119871

119891119888119908119897 = 119889119888119908 119908 isin 119882 (12)

119891119904119903119908119897 ge 0 119908 isin 119882119903 119897 isin 119871 (13)

119891119904119906119908119897 ge 0 119908 isin 119882 119897 isin 119871 (14)

119891119888119908119897 ge 0 119908 isin 119882 119897 isin 119871 (15)


119891119894119908119897 sdot 120582119894119908 = 0 119894 = (119888 119904119906) 119908 isin 119882 119897 isin 119871 (16)

120582119894119908 ge 0 119894 = (119888 119904119906) 119908 isin 119882 (17)

119891119904119903119908119897 sdot 120582119904119903119908 = 0 119908 isin 119882119903 119897 isin 119871 (18)

120582119904119903119908 ge 0 119908 isin 119882119903 (19)






119889119888119908 1120579 ln119889119888119908119889119908 minus 120596119908 + 120583

119888

119908 119908 isin 119882 (23)

119889119904119908 1120579 ln119889119904119908119889119908 minus 120596119908 + (1 minus 120574) 120583

119904119906

119908 + 120574120583119904119903119908 119908 isin 119882 (24)



(25)








119885 = 998779 sdot sum119908isin119882

119889119888119908 + sum119886isin119860

(119888119904119886 sdot V119904119886 + 119888119888119886 sdot V119888119886) (26)




1 2 3

4 5 6






min 119885 = 998779 sdot sum119908isin119882

119889119888119908 + sum119886isin119860


(27)




Table 1 OD demands




119862119886 (vehh) 2000 2000 1000 1000 1000 2000 2000




5 Algorithm


80000850009000095000

100000105000110000115000120000

05 1 15 2 25 3 35 4 45 5To

tal t

rave

l cos

tCarpooling cost




Tota

l tra

vel c

ost

11

105

10

95

9

85

81

times104

Proportion05

0 Carpooling cost0

12

34

5



6 Numerical Example







solution






solution



















1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

24 2248

26 47

17

20

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471

7246 67

45

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

2422

4826 47

17

20

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471 72 46 67

45

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

31

13

2527

2321

16

24 2248

26 47

1720

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

3837

4244

7172

46 6745

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

2422

4826 47

1720

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471

7246 67

45

59 6163

6865

62647539

74 66

69

70

7673

18 54

9 12

3

4 14




7 Conclusion










Notation




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119885 = 998779 sdot sum119908isin119882

119889119888119908 + sum119886isin119860

(119888119904119886 sdot V119904119886 + 119888119888119886 sdot V119888119886) (26)




1 2 3

4 5 6






min 119885 = 998779 sdot sum119908isin119882

119889119888119908 + sum119886isin119860


(27)




Table 1 OD demands




119862119886 (vehh) 2000 2000 1000 1000 1000 2000 2000




5 Algorithm


80000850009000095000

100000105000110000115000120000

05 1 15 2 25 3 35 4 45 5To

tal t

rave

l cos

tCarpooling cost




Tota

l tra

vel c

ost

11

105

10

95

9

85

81

times104

Proportion05

0 Carpooling cost0

12

34

5



6 Numerical Example







solution






solution



















1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

24 2248

26 47

17

20

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471

7246 67

45

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

2422

4826 47

17

20

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471 72 46 67

45

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

31

13

2527

2321

16

24 2248

26 47

1720

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

3837

4244

7172

46 6745

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

2422

4826 47

1720

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471

7246 67

45

59 6163

6865

62647539

74 66

69

70

7673

18 54

9 12

3

4 14




7 Conclusion










Notation




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 1 OD demands




119862119886 (vehh) 2000 2000 1000 1000 1000 2000 2000




5 Algorithm


80000850009000095000

100000105000110000115000120000

05 1 15 2 25 3 35 4 45 5To

tal t

rave

l cos

tCarpooling cost




Tota

l tra

vel c

ost

11

105

10

95

9

85

81

times104

Proportion05

0 Carpooling cost0

12

34

5



6 Numerical Example







solution






solution



















1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

24 2248

26 47

17

20

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471

7246 67

45

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

2422

4826 47

17

20

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471 72 46 67

45

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

31

13

2527

2321

16

24 2248

26 47

1720

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

3837

4244

7172

46 6745

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

2422

4826 47

1720

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471

7246 67

45

59 6163

6865

62647539

74 66

69

70

7673

18 54

9 12

3

4 14




7 Conclusion










Notation




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














solution






solution



















1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

24 2248

26 47

17

20

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471

7246 67

45

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

2422

4826 47

17

20

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471 72 46 67

45

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

31

13

2527

2321

16

24 2248

26 47

1720

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

3837

4244

7172

46 6745

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

2422

4826 47

1720

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471

7246 67

45

59 6163

6865

62647539

74 66

69

70

7673

18 54

9 12

3

4 14




7 Conclusion










Notation




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

24 2248

26 47

17

20

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471

7246 67

45

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

2422

4826 47

17

20

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471 72 46 67

45

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

31

13

2527

2321

16

24 2248

26 47

1720

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

3837

4244

7172

46 6745

59 6163

6865

62647539

74 66

6970

7673

18 54

9 12

3

4 14


1 2

3 4 5 6

9 8 7

101112 16

17

1514

2223

2124

19

2013

18

152

8 11 156

7 35 10

33

3113

2527

2321

16

2422

4826 47

1720

19

5550

605658

52492951

3043

5753

28

32

40

41

34

36

383742

4471

7246 67

45

59 6163

6865

62647539

74 66

69

70

7673

18 54

9 12

3

4 14




7 Conclusion










Notation




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Notation




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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