Research Article Managing Rush Hour Congestion with Lane

Research ArticleManaging Rush Hour Congestion withLane Reversal and Tradable Credits

Qing Li12 and Ziyou Gao1

1 School of Traffic and Transportation Beijing Jiaotong University Beijing 100044 China2Department of Statistics College of Mathematics and Systems Science Shandong University of Science and TechnologyQingdao 266590 China

Correspondence should be addressed to Qing Li i liqing163com

Received 11 April 2014 Revised 21 June 2014 Accepted 1 July 2014 Published 4 August 2014

Academic Editor X Zhang

Copyright copy 2014 Q Li and Z Gao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Within the morning and evening rush hour the two-way road flows are always unbalanced in opposite directions In order to makefull advantage of the existing lanes the two-way road lane has to be reallocated to play the best role in managing congestion Onthe other hand an effective tradable credit scheme can help to reduce the traffic demand and improve fairness for all travelersSo as to alleviate the commute congestion in urban transportation network a discrete bilevel programming model is establishedin this paper In the bilevel model the government at the upper level reallocates lanes on the two-way road to minimize the totalsystem cost The traveler at the lower level chooses the optimal route on the basis of both travel time and credit charging for thelanes involved A numerical experiment is conducted to examine the efficiency of the proposed method

1 Introduction

In many big cities a large number of people commute fromresidential areas to their workplaces in the central businessdistrict (CBD) in the morning It leads to the phenomenonthat the lanes are congested in the direction from residentialareas to CBD while the other lanes in the opposite directionare free But in the evening all of that will be reversed Inorder to adjust the asymmetric amounts of traffic flow inthe two directions of one road section the lane reversal isemployed as a traffic control technique It is an effectiveway tomake full use of existing road resource and can increase trafficsupply

Lane reversal has been used for many years in managingroad congestion In recent years lane reversal has beenintroduced in Chinese big cities such as Beijing to relievemorning and evening commute congestion In the last yearChaoyang Road a big thoroughfare on the cityrsquos east side hasbeen tested as the first lane reversal in Beijing to allow trafficto travel in either direction depending on certain conditionsAccording to the practice the lane reversal can help toalleviate traffic congestion to some extent

The implement of lane reversal is a network designproblem (NDP) A plenty of research works are focused onthe effectiveness feasibility and safety of implementing lanereversal [1ndash5] The development of techniques applicationsand the engineering practices of lane reversal also attractsmuch attention [6] In practice improper use of lane reversalmay lead to a worse condition of the whole road networkCongestion on the road with lane reversal may be relievedbut other parts of the network will be more congestedThere-fore the lane reversal must be put into use on the basis ofadjustment of the whole urban road network [7 8] A lot ofresearch onNDP is concernedwith uncertain demand [9ndash12]For simplicity the commute traffic demand is assumed to befixed demand Apparently the number of reallocated lanesis integer As a result the implement of lane reversal is adiscrete network design problem (DNDP) In essence thelane reversal contributes to increasing road supply

In addition to increasing supply another efficient way tomanage congestion is reducing traffic demand It is widelyacknowledged that congestion pricing is helpful in trafficdemand management but the equity debates confined itsadoption all over the world Without taking into account

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 132936 5 pageshttpdxdoiorg1011552014132936

2 Mathematical Problems in Engineering

commutersrsquo income level trip purposes and valuation oftime congestion pricing scheme might increase individualsrsquotravel costs Though it can improve the system performanceit often has to face public resistance

According to the latest research on congestion controltradable credit scheme is proved to be a fairer more effec-tive and more practical congestion management schemeIt should be noticed that Yang and Wang first explored atradable credit scheme in a general transportation networkequilibrium context with homogeneous and heterogeneoustravelers respectively [13 14] Under their credit scheme thesocial planner is assumed to develop an initial distributionof the credits to all eligible travelers and link-specific chargesto travelers on that link Credits can be traded freely amongtravelers It is the competitive market rather than the plannerthat can determine the price of the credit An optimizationmodel subject to a total credit consumption constraint is for-mulated to find the equilibrated credit price With an appro-priate distribution of credits among travelers and correctselection of link-specific rates the results of the traditionalcongestion pricing scheme can be duplicated Furthermorethe combined tradable credit scheme is shown to be bothsystem-optimal and a Pareto-improvement in a revenue-neutral manner The same context of research has beendone on the tradable credits scheme employed in managingrush hour travel choice and bottleneck congestion [15ndash18]

In this paper a discrete bilevel programming model isconstructed for managing rush hour congestion caused byunderutilization of the existing road resource The proposedmodel employs tradable credit scheme and lane reversalfor increasing traffic supply and decreasing traffic demandrespectively At the upper level the government chooses opti-mal lanes to be reallocated tominimize the total system costsTaking into account the generalized travel cost includingtravel time and link-specific charges for using the links thetravelers at the lower level will choose the optimal route tominimize it

This paper is organized as follows In Section 2 a discretebilevel programming model is established with lane reversaland tradable credit scheme A chaotic algorithm is employedto solve the proposed model In Section 3 numerical experi-ments and analysis results are illustratedwith a basic networkIn Section 4 conclusion of this paper is presented

2 Discrete Bilevel Programming Model withLane Reversal and Tradable Credits Scheme

Consider a two-way network 119866 = (119873119860) with a set 119873 ofnodes and a set 119860 of directed links Link in one direction ofa road section is denoted by 119886 isin 119860 and link in the oppositedirection is 1198861015840 isin 119860 Let119882 and 119877

119908denote the set of O-D pairs

and the set of all routes between an O-D pair 119908 isin 119882 Thetravel demand for each O-D pair 119908 isin 119882 denoted by 119889

119908

119889119908gt 0 is given and fixedTo all directed links in the roadnetwork119899

119886and 1198991198861015840 denote

the number of lanes on link 119886 isin 119860 and 1198861015840 isin 119860 For simplicitylet 119899119886= 1198991198861015840 before lane reversal The capacity of each lane on

link 119886 isin 119860 is assumed to be equal to unity which is denoted

by 119888119886 That is to say the traffic capacity of link 119886 isin 119860 is 119899

119886sdot 119888119886

before lane reversal After lane reversal 119906119886is used to denote

the number of lanes in the opposite direction on link 1198861015840 isin 119860to be added to link 119886 isin 119860 If 119906

119886gt 0 it means that 119906

119886lanes on

the opposite direction will add to link 119886 isin 119860 and its capacityis equal to (119899

119886+119906119886) sdot 119888119886 On the other hand if 119906

119886lt 0 it means

that 119906119886lanes on link 119886 isin 119860 will be added to link 1198861015840 isin 119860 on

the opposite direction Relative to other means for networkcapacity enhancement the investment of lane reversal can beignored

For simplicity consider a separable link travel cost func-tion 119905

119886(V119886 119906119886) which is assumed to be nonnegative contin-

uously differentiable convex and monotonically increasingwith respect to the amount of aggregate traffic flow V

119886on link

119886 isin 119860 and 119906119886 the number of lanes to be reallocated In gen-

eral 120597119905119886(V119886 119910119886)120597119910119886is assumed to be continuous too Assume

also that travelers are homogeneous which means thatthey have the same value of time (VOT)

Tradable credit scheme is characterized by its initial dis-tribution and the charging scheme To minimize complexitythe initial distribution schemes considered here will be O-Dspecific for a given and fixed demand 119889

119908 119896 = 119896

119908 is used to

denote the credit distribution scheme Here 119896119908is the amount

of credit distributed to each traveler over theO-Dpair119908 isin 119882Let119870 denote the total amount of credits for all links which ispredetermined and obviously 119870 = sum

119908isin119882119896119908119889119908 The credits

are month specific so as no one can benefit from the trade ofcredit Analysis here is restricted to link-specific credit charg-ing Let 120581 = 120581

119886 119886 isin 119860 denote the charging scheme where

120581119886is the credit charge for using link 119886 isin 119860 Then (119870 120581) will

be used to represent a credit charging scheme 120581 with a totalnumber of credits 119870 for all links

Let119891119908119903denote the traffic flow on route 119903 isin 119877

119908betweenO-

D pair 119908 isin 119882 119891 is a path flow vector 119891 = (119891119908

119903 119903 isin 119877

119908 119908 isin

119882) and Ω119891represents the set of feasible path flow patterns

defined as follows

Ω119891= 119891 | 119891

119908

119903ge 0 sum

119903isin119877119908

119891119908

119903= 119889119908 119903 isin 119877

119908 119908 isin 119882 (1)

Let V = (V119886 119886 isin 119860) denote the link flow vector and

Ω119891represents the set of feasible link flow patterns defined as

follows

ΩV = V | V119886 = sum

119908isin119882

sum

119903isin119877119908

119891119908

119903120575119886119903 119891 isin Ω

119891 119886 isin 119860 (2)

and 120575119886119903= 1 if route 119903 uses link 119886 and 0 otherwise

It has been proved that not all (119870 120581) can guarantee theexistent feasible network flow patternsThe amount of creditsmight not be big enough for supporting all travelers goingthrough the network even if all of themchoose the least-creditpaths In order to ensure the existence of feasible network flowpatterns the feasible set of credit schemes denoted by Ψ asfollows

Ψ = (119870 120581) | exist119891 isin Ω119891such that sum

119886isin119860

V119886120581119886le 119870 V isin ΩV

(3)

Ψ is assumed to be nonempty

Mathematical Problems in Engineering 3

The following bilevel programmingmodel is to minimizethe sum of total system costs while the travelers choose theoptimal route for minimizing the generalized travel costsincluding both travel time and link-specific credit charges forusing the links The model can be described by

(BLP) min119880

SC = sum

119886isin119860

V119886119905119886(V119886 119906119886) (4)

subject to

119906119886isin minus119899

119886 minus (119899119886minus 1) minus1 0 1 119899

119886minus 1 119899119886 119886 isin 119860

119906119886+ 1199061198861015840 = 0 119886 isin 119860 119886

1015840isin 119860

(5)

where 119881 = (V119886(119906119886)) is the solution of the next problem

min119881

sum

119886isin119860

int

V119886(119906119886)

0

119905119886(120579 119906119886) 119889120579 (6)

subject to

sum

119886isin119860

V119886120581119886le 119870

sum

119903isin119877119908

119891119908

119903= 119889119908 119903 isin 119877

119908 119908 isin 119882

V119886= sum

119908isin119882

sum

119903isin119877119908

119891119908

119903120575119886119903 119886 isin 119860

119891119908

119903ge 0 forall119908 isin 119882 119903 isin 119877

119908

(7)

The model is a mixed integer nonlinear programmingprogramThe decision variable at upper level is integer whilevariable at lower level is real number Chaos algorithm hereis adopted to solve the proposed model [7] The steps of thealgorithm are as follows

Step 1 Assume chaos variable is denote by random number1199100

119886isin [0 1] 119886 isin 119860 Let the initial optimal solution 1199060

119886= 0

119886 isin 119860 SC0 = +infin Check set is denoted by Φ and it is nullset Counter119898 = 1

Step 2 Chaos variable 119910119898

119886is generated by the following

equation

119910119898

119886= 4119910119898minus1

119886(1 minus 119910

119898minus1

119886) 119886 isin 119860 (8)

Step 3 Carrier can be produced by the following equation

119898

119886= minus119899119886minus 1205761+ (3119899119886+ 1205762) 119910119898

119886 119886 isin 119860 (9)

where 1205761and 1205762are all small enough positive numbers with

1205761lt 1205762 Let 119906119898

119886be equal to rounding 119898

119886 119886 isin 119860 If 119906119898

119886notin Φ

119886 isin 119860 add 119906119898119886to Φ then turn to Step 4 otherwise turn to

Step 2

Step 4 For given 119906119898119886 119886 isin 119860 solve the users equilibrium (UE)

assignment model at the lower levels (6)-(7) The solution isthe link flow vector V119898 = (V119898

119886 119886 isin 119860) at UE

4

2

3

1

21

4

3

6 5

8

7

9

10

Figure 1 Original structure of the test network before lane reversal

Table 1 Input data for the test network

119905119886(V119886 119906119886) = 1199050

119886(1 + 015(V

119886119862119886)4

)

Link 119886 1199050

119886119862119886

119899119886

119888119886

1 and 2 40 40 2 203 and 4 60 40 2 205 and 6 20 60 3 207 and 8 50 40 2 209 and 10 30 40 2 20

Table 2 Transportation condition before lane reversal and tradablecredit scheme

Link 119886 Link flow V119886

Link utilization rate 120594119886= V119886119862119886

1 and 2 525322 and 315193 13133 and 078803 and 4 474678 and 284807 11867 and 071205 and 6 157200 and 134320 02620 and 022397 and 8 468122 and 280873 11703 and 070229 and 10 531878 and 319127 13297 and 07978System cost SC = 18240 times 10

3

Step 5 Solve the model at the upper level and get SC119896 IfSC119898 lt SC0 let SC0 = SC119898 1199060

119886= 119906119898

119886 119886 isin 119860

Step 6 If the termination condition is met output theoptimal solution 1199060

119886 119886 isin 119860 otherwise turn to Step 2

3 Numerical Experiments

In this paper a basic two-way road network as shown inFigure 1 is employed to validate the efficiency of the proposedmodel The network consists of 4 nodes 10 links and 2 O-Dpairs One O-D pair is from node 1 to node 4 and the other isfrom node 4 to node 1 In the morning rush hour the trafficdemand of the two O-D pairs is 100 and 60 in an hour

The link cost function 119905119886(V119886 119906119886) used here is classical BPR

function Initial capacity of link 119886 isin 119860 which has 119899119886lanes

before lane reversal and each lanersquos capacity is 119888119886 is denoted

by 119862119886= 119899119886119888119886 The number of 119862

119886 119888119886 119899119886 and the link free flow

travel time 1199050119886 119886 isin 119860 are all illustrated in Table 1

The traffic flows under UE before lane reversal can becalculated and are shown in Table 2 It shows that the two-way road traffic flows are unbalanced in opposite directionsA parameter 120594

119886= V119886119862119886is introduced here to estimate the


Table 3 Results after lane reversal

Lane reversal SC1199061and 119906

21199063and 119906

41199065and 119906

61199067and 119906

81199069and 119906

10

0 and 0 1 and minus1 0 and 0 1 and minus1 0 and 0 1680

Table 4 Transportation condition after lane reversal

Link 119886 Link flow 119909119886

Link utilization ratio 120594119886= V119886119870119886


3

4

2

3

1

21

43

6 5

8

7

910

Figure 2 The structure of the test network after lane reversal

utilization ratio of link 119886 isin 119860 As is shown Table 2 theutilization ratios of links 1 3 7 and 9 are bigger than 1 Thatmeans that these links have been overused Otherwise theutilization ratios of the links in opposite directions are farsmaller than them It can be derived from Table 2 that theutilization ratio of link 2 is slightly bigger than half of thatof link 1 And the same happens between link 10 and link 9

Solve the models (4)ndash(7) with the use of the algorithm inSection 2 The amounts of credit charging for using links areset as follows 120581

1= 1205815= 1 120581

8= 2 and 120581

119886= 0 for other

links 119886 isin 119860 The parameter 120579 = 005 transfers money intoequivalent time value The strategy of the lane reversal andthe system cost can be obtained as shown in Table 3

It is shown in Table 3 that one lane of link 4 and one laneof link 8 have been reversed in their directions and addedto links 3 and link 7 respectively Other lanes of other linksdo not have any changes It can be found with the systemcost illustrated in Table 4 that the scheme of lane reversaland tradable credits is efficient for managing rush hourcongestion

The transportation condition after lane reversal is shownin Figure 2 The lanes of links 3 and 4 and links 7 and 8have been adjustedThe traffic flows after lane reversal can becalculated and are shown inTable 4 From the utilization ratioit can be concluded that most of the links have been madefull use of And the system cost is lower than that before lanereversal

4 Conclusions

With the development of economics and city scales the roadbecomes more and more congested in morning and eveningrush hour In order to achieve better effect in solving the con-gestion problem traffic demand and increasing road supplyare all considered at the same time in this paper A bilevel pro-gramming model is proposed to deal with the two-way roadunbalance usage problem In order to make full advantage ofthe existing lanes the two-way road lanes have to be reallo-cated to play the best role in managing congestion An effec-tive tradable credit scheme is also employed to help to allevi-ate the commute congestionwith lane reversal in urban trans-portation networkThemodels and the algorithm are demon-strated with the basic two-way road network example

In the future research the heterogeneous users should beconsidered Users with different job and income may havedifferent value of time so it will be helpful for transportationplanning to simulate the real situation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was jointly supported by the National BasicResearch Program of China (2012CB725401) and theNational Natural Science Foundation of China (71271205 and71322102)

References

[1] J Hemphill and V H Surti ldquoA feasibility study of a facility fora Denver street corridorrdquo Transportation Research Record vol514 pp 29ndash32 1974

[2] R J Caudill and N M Kuo ldquoDevelopment of an interactiveplanning model for contraflow lane evaluationrdquo TransportationResearch Record vol 906 pp 47ndash54 1983

[3] Y Jiang and L Bao ldquoStudy on setting of variable lanes nearintersection between one-way and two-way trafficrdquo Journal ofShanghai Jiaotong University vol 45 no 10 pp 1562ndash1566 2011

[4] F Ni and Z Liu ldquoThe optimal location of variablemessage signrdquoInformation and Control vol 32 no 5 pp 395ndash398 2003

[5] X Gong and S Kang ldquoStudy and application of traffic directionchanging algorithm for urban tide traffic situationrdquo Journal ofTransportation Systerns Engineering and Information Technol-ogy vol 6 no 6 pp 33ndash40 2006

[6] P B Wolshon and L Lambert Convertible Roadways andLanes A Synthesis of Highway Practice NCHRP Synthesis 340National Cooperative Highway Research Program Transporta-tion Research Board Washington DC USA 2004


[7] H Zhang and Z Gao ldquoOptimaization approach for traffic roadnetwork design problemrdquo Chinese Journal of Management Sci-ence vol 15 no 2 pp 86ndash91 2007

[8] X Liang C Hu C Ma and B Mao ldquoEmpirical study onvariable lanes design of chaoyangnorth street inBeijingrdquo inPro-ceedings of the 30th Chinese Control Conference (CCC rsquo11) pp5527ndash5531 July 2011

[9] H Shao W H K Lam and M L Tam ldquoA reliability-basedstochastic traffic assignment model for network with multipleuser classes under uncertainty in demandrdquoNetworks and SpatialEconomics vol 6 no 3-4 pp 173ndash204 2006

[10] H Shao Q Tian X Yuan and G Wang ldquoRisk-taking pathchoice behaviors under ATIS in transportation networks withdemanduncertaintyrdquo Journal of SoutheastUniversity vol 24 pp43ndash48 2008

[11] H Shao W H K Lam Q Meng and M L Tam ldquoTravel timereliability-based traffic assignment problemrdquo Journal of Mange-ment Sciences in China vol 12 no 5 pp 27ndash35 2009

[12] H ShaoW H K Lam A Sumalee and A Chen ldquoJourney timeestimator for assessment of road network performance underdemand uncertaintyrdquo Transportation Research C EmergingTechnologies vol 35 pp 244ndash262 2013

[13] H Yang and X Wang ldquoManaging network mobility with trad-able creditsrdquo Transportation Research B Methodological vol45 no 3 pp 580ndash594 2011

[14] X Wang H Yang D Zhu and C Li ldquoTradable travel creditsfor congestion management with heterogeneous usersrdquo Trans-portation Research E Logistics and Transportation Review vol48 no 2 pp 426ndash437 2012

[15] Y M Nie and Y Yin ldquoManaging rush hour travel choices withtradable credit schemerdquoTransportationResearch BMethodolog-ical vol 50 pp 1ndash19 2013

[16] F Xiao Z Qian and H M Zhang ldquoManaging bottleneckcongestion with tradable creditsrdquo Transportation Research BMethodological vol 56 pp 1ndash14 2013

[17] L Tian H Yang and H Huang ldquoTradable credit schemesfor managing bottleneck congestion and modal split withheterogeneous usersrdquo Transportation Research E vol 54 pp 1ndash13 2013

[18] G Wang Z Gao M Xu and H Sun ldquoModels and a relaxationalgorithm for continuous network design problem with atradable credit scheme and equity constraintsrdquo Computers ampOperations Research vol 41 pp 252ndash261 2014

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Stochastic AnalysisInternational Journal of


commutersrsquo income level trip purposes and valuation oftime congestion pricing scheme might increase individualsrsquotravel costs Though it can improve the system performanceit often has to face public resistance

According to the latest research on congestion controltradable credit scheme is proved to be a fairer more effec-tive and more practical congestion management schemeIt should be noticed that Yang and Wang first explored atradable credit scheme in a general transportation networkequilibrium context with homogeneous and heterogeneoustravelers respectively [13 14] Under their credit scheme thesocial planner is assumed to develop an initial distributionof the credits to all eligible travelers and link-specific chargesto travelers on that link Credits can be traded freely amongtravelers It is the competitive market rather than the plannerthat can determine the price of the credit An optimizationmodel subject to a total credit consumption constraint is for-mulated to find the equilibrated credit price With an appro-priate distribution of credits among travelers and correctselection of link-specific rates the results of the traditionalcongestion pricing scheme can be duplicated Furthermorethe combined tradable credit scheme is shown to be bothsystem-optimal and a Pareto-improvement in a revenue-neutral manner The same context of research has beendone on the tradable credits scheme employed in managingrush hour travel choice and bottleneck congestion [15ndash18]

In this paper a discrete bilevel programming model isconstructed for managing rush hour congestion caused byunderutilization of the existing road resource The proposedmodel employs tradable credit scheme and lane reversalfor increasing traffic supply and decreasing traffic demandrespectively At the upper level the government chooses opti-mal lanes to be reallocated tominimize the total system costsTaking into account the generalized travel cost includingtravel time and link-specific charges for using the links thetravelers at the lower level will choose the optimal route tominimize it

This paper is organized as follows In Section 2 a discretebilevel programming model is established with lane reversaland tradable credit scheme A chaotic algorithm is employedto solve the proposed model In Section 3 numerical experi-ments and analysis results are illustratedwith a basic networkIn Section 4 conclusion of this paper is presented

2 Discrete Bilevel Programming Model withLane Reversal and Tradable Credits Scheme

Consider a two-way network 119866 = (119873119860) with a set 119873 ofnodes and a set 119860 of directed links Link in one direction ofa road section is denoted by 119886 isin 119860 and link in the oppositedirection is 1198861015840 isin 119860 Let119882 and 119877

119908denote the set of O-D pairs

and the set of all routes between an O-D pair 119908 isin 119882 Thetravel demand for each O-D pair 119908 isin 119882 denoted by 119889

119908

119889119908gt 0 is given and fixedTo all directed links in the roadnetwork119899

119886and 1198991198861015840 denote

the number of lanes on link 119886 isin 119860 and 1198861015840 isin 119860 For simplicitylet 119899119886= 1198991198861015840 before lane reversal The capacity of each lane on

link 119886 isin 119860 is assumed to be equal to unity which is denoted

by 119888119886 That is to say the traffic capacity of link 119886 isin 119860 is 119899

119886sdot 119888119886

before lane reversal After lane reversal 119906119886is used to denote

the number of lanes in the opposite direction on link 1198861015840 isin 119860to be added to link 119886 isin 119860 If 119906

119886gt 0 it means that 119906

119886lanes on

the opposite direction will add to link 119886 isin 119860 and its capacityis equal to (119899

119886+119906119886) sdot 119888119886 On the other hand if 119906

119886lt 0 it means

that 119906119886lanes on link 119886 isin 119860 will be added to link 1198861015840 isin 119860 on

the opposite direction Relative to other means for networkcapacity enhancement the investment of lane reversal can beignored

For simplicity consider a separable link travel cost func-tion 119905

119886(V119886 119906119886) which is assumed to be nonnegative contin-

uously differentiable convex and monotonically increasingwith respect to the amount of aggregate traffic flow V

119886on link

119886 isin 119860 and 119906119886 the number of lanes to be reallocated In gen-

eral 120597119905119886(V119886 119910119886)120597119910119886is assumed to be continuous too Assume

also that travelers are homogeneous which means thatthey have the same value of time (VOT)

Tradable credit scheme is characterized by its initial dis-tribution and the charging scheme To minimize complexitythe initial distribution schemes considered here will be O-Dspecific for a given and fixed demand 119889

119908 119896 = 119896

119908 is used to

denote the credit distribution scheme Here 119896119908is the amount

of credit distributed to each traveler over theO-Dpair119908 isin 119882Let119870 denote the total amount of credits for all links which ispredetermined and obviously 119870 = sum

119908isin119882119896119908119889119908 The credits

are month specific so as no one can benefit from the trade ofcredit Analysis here is restricted to link-specific credit charg-ing Let 120581 = 120581

119886 119886 isin 119860 denote the charging scheme where

120581119886is the credit charge for using link 119886 isin 119860 Then (119870 120581) will

be used to represent a credit charging scheme 120581 with a totalnumber of credits 119870 for all links

Let119891119908119903denote the traffic flow on route 119903 isin 119877

119908betweenO-

D pair 119908 isin 119882 119891 is a path flow vector 119891 = (119891119908

119903 119903 isin 119877

119908 119908 isin

119882) and Ω119891represents the set of feasible path flow patterns

defined as follows

Ω119891= 119891 | 119891

119908

119903ge 0 sum

119903isin119877119908

119891119908

119903= 119889119908 119903 isin 119877

119908 119908 isin 119882 (1)

Let V = (V119886 119886 isin 119860) denote the link flow vector and

Ω119891represents the set of feasible link flow patterns defined as

follows

ΩV = V | V119886 = sum

119908isin119882

sum

119903isin119877119908

119891119908

119903120575119886119903 119891 isin Ω

119891 119886 isin 119860 (2)

and 120575119886119903= 1 if route 119903 uses link 119886 and 0 otherwise

It has been proved that not all (119870 120581) can guarantee theexistent feasible network flow patternsThe amount of creditsmight not be big enough for supporting all travelers goingthrough the network even if all of themchoose the least-creditpaths In order to ensure the existence of feasible network flowpatterns the feasible set of credit schemes denoted by Ψ asfollows

Ψ = (119870 120581) | exist119891 isin Ω119891such that sum

119886isin119860

V119886120581119886le 119870 V isin ΩV

(3)

Ψ is assumed to be nonempty



(BLP) min119880

SC = sum

119886isin119860

V119886119905119886(V119886 119906119886) (4)

subject to

119906119886isin minus119899


119886minus 1 119899119886 119886 isin 119860

119906119886+ 1199061198861015840 = 0 119886 isin 119860 119886

1015840isin 119860

(5)


min119881

sum

119886isin119860

int

V119886(119906119886)

0

119905119886(120579 119906119886) 119889120579 (6)

subject to

sum

119886isin119860

V119886120581119886le 119870

sum

119903isin119877119908

119891119908

119903= 119889119908 119903 isin 119877

119908 119908 isin 119882

V119886= sum

119908isin119882

sum

119903isin119877119908

119891119908

119903120575119886119903 119886 isin 119860

119891119908


119908

(7)




119886= 0




equation

119910119898

119886= 4119910119898minus1

119886(1 minus 119910

119898minus1

119886) 119886 isin 119860 (8)


119898

119886= minus119899119886minus 1205761+ (3119899119886+ 1205762) 119910119898

119886 119886 isin 119860 (9)


1205761lt 1205762 Let 119906119898


119886 119886 isin 119860 If 119906119898

119886notin Φ


Step 2



119886 119886 isin 119860) at UE

4

2

3

1

21

4

3

6 5

8

7

9

10



119905119886(V119886 119906119886) = 1199050

119886(1 + 015(V

119886119862119886)4

)

Link 119886 1199050

119886119862119886

119899119886

119888119886






3


119886= 119906119898

119886 119886 isin 119860








by 119862119886= 119899119886119888119886 The number of 119862








21199063and 119906

41199065and 119906

61199067and 119906

81199069and 119906

10



Link 119886 Link flow 119909119886



3

4

2

3

1

21

43

6 5

8

7

910




1= 1205815= 1 120581

8= 2 and 120581

119886= 0 for other




4 Conclusions





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(BLP) min119880

SC = sum

119886isin119860

V119886119905119886(V119886 119906119886) (4)

subject to

119906119886isin minus119899


119886minus 1 119899119886 119886 isin 119860

119906119886+ 1199061198861015840 = 0 119886 isin 119860 119886

1015840isin 119860

(5)


min119881

sum

119886isin119860

int

V119886(119906119886)

0

119905119886(120579 119906119886) 119889120579 (6)

subject to

sum

119886isin119860

V119886120581119886le 119870

sum

119903isin119877119908

119891119908

119903= 119889119908 119903 isin 119877

119908 119908 isin 119882

V119886= sum

119908isin119882

sum

119903isin119877119908

119891119908

119903120575119886119903 119886 isin 119860

119891119908


119908

(7)




119886= 0




equation

119910119898

119886= 4119910119898minus1

119886(1 minus 119910

119898minus1

119886) 119886 isin 119860 (8)


119898

119886= minus119899119886minus 1205761+ (3119899119886+ 1205762) 119910119898

119886 119886 isin 119860 (9)


1205761lt 1205762 Let 119906119898


119886 119886 isin 119860 If 119906119898

119886notin Φ


Step 2



119886 119886 isin 119860) at UE

4

2

3

1

21

4

3

6 5

8

7

9

10



119905119886(V119886 119906119886) = 1199050

119886(1 + 015(V

119886119862119886)4

)

Link 119886 1199050

119886119862119886

119899119886

119888119886






3


119886= 119906119898

119886 119886 isin 119860








by 119862119886= 119899119886119888119886 The number of 119862








21199063and 119906

41199065and 119906

61199067and 119906

81199069and 119906

10



Link 119886 Link flow 119909119886



3

4

2

3

1

21

43

6 5

8

7

910




1= 1205815= 1 120581

8= 2 and 120581

119886= 0 for other




4 Conclusions





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












21199063and 119906

41199065and 119906

61199067and 119906

81199069and 119906

10



Link 119886 Link flow 119909119886



3

4

2

3

1

21

43

6 5

8

7

910




1= 1205815= 1 120581

8= 2 and 120581

119886= 0 for other




4 Conclusions





Acknowledgments


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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