Robust Evaluation for Transportation Network Capacity...

Research ArticleRobust Evaluation for Transportation NetworkCapacity under Demand Uncertainty

Muqing Du1 Xiaowei Jiang2 Lin Cheng2 and Changjiang Zheng1

1College of Civil and Transportation Engineering Hohai University 1 Xikang Rd Nanjing Jiangsu 210098 China2School of Transportation Southeast University 35 Jinxianghe Rd Nanjing Jiangsu 210096 China

Correspondence should be addressed to Muqing Du dumuqinggmailcom

Received 28 April 2017 Accepted 17 July 2017 Published 10 September 2017

Academic Editor Dongjoo Park

Copyright copy 2017 Muqing Du et al 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

As more and more cities in worldwide are facing the problems of traffic jam governments have been concerned about how todesign transportation networkswith adequate capacity to accommodate travel demands To evaluate the capacity of a transportationsystem the prescribed origin and destination (O-D) matrix for existing travel demand has been noticed to have a significant effecton the results of network capacity models However the exact data of the existing O-D demand are usually hard to be obtained inpractice Considering the fluctuation of the real travel demand in transportation networks the existing travel demand is representedas uncertain parameters which are defined within a bounded set Thus a robust reserve network capacity (RRNC) model usingminndashmax optimization is formulated based on the demand uncertainty An effective heuristic approach utilizing cutting planemethod and sensitivity analysis is proposed for the solution of the RRNC problem Computational experiments and simulationsare implemented to demonstrate the validity and performance of the proposed robustmodel According to simulation experimentsit is showed that the link flow pattern from the robust solutions to network capacity problems can reveal the probability of highcongestion for each link

1 Introduction

The capacity of transportation network reflects the supplyability of its infrastructure and service to the travel demandwhich is generated from the zones covered by the transporta-tion system in a specific period For many years transporta-tion planners andmanagers wanted to understand howmanytrips can be accommodated at the most by the current ordesigned network in a certain period of time This need ismore necessary in those developing regions which are con-fronted with rapid growth of private vehicles and increasedurban congestion Meanwhile the researchers made a long-term effort to model and estimate the maximum throughputof transportation networks The achievements include max-flow min-cut theorem [1] incremental assignment approach[2] and later bilevel programming models [3ndash5]

For the network capacitymodel themost popular formu-lation in passenger transportation system is the bilevelmodelwhich maximizes the traffic flows under the equilibriumconstraints Wong and Yang [3] first incorporated the reserve

capacity concept into a traffic signal control network Thereserve capacity is defined as the largest multiplier appliedto a given O-D demand matrix without violating capacityconstraints so the solution is significantly affected by thepredetermined O-D matrix Ziyou and Yifan [6] extendedthe reserve capacity model by considering O-D specificdemand multipliers and all demand multipliers should beensured not lower than a predetermined minimum valueIn order to avoid assuming that all O-D flows increasein a same rate another concept of ultimate capacity wasproposed [5] But it assumes that the O-D distribution istotally variable which may produce unrealistic results thatcause the trip productions at some origins below their currentlevels Furthermore Yang et al [4] suggested that the newincreased O-D demand pattern should be variable in bothlevel and distribution while the current travel demand isfixed Later Yangrsquos model was also referred to as the practicalcapacity byKasikitwiwat andChen [5] In summary althoughunrealistic the reserve capacity model is more easy-to-use

HindawiJournal of Advanced TransportationVolume 2017 Article ID 9814909 11 pageshttpsdoiorg10115520179814909

2 Journal of Advanced Transportation

and has been adopted widely in many researches [7ndash10]The ultimate capacity and practical capacity model are morepractical but have more parameters to be calibrated whenapplied and the formulated models are still difficult to solve[11]

While the deterministic network capacity problem hasbeen explored extensively few studies have investigated theissue of uncertainties in demand data associated with thisproblem The ultimate capacity and practical capacity modelare only concerned with the uncertainties related to the newincreased travel demandby using combinedmodels [5] whilethe uncertainties in the current (or existing) demand arenot considered In reality travel demands in transportationsystem are always fluctuant day by day even hour by hourBesides errors of survey data also affect the accuracy of theexisting O-D matrix As a consequence the existing traveldemands are usually difficult to be obtained in actual trans-portation projects and then are not easy to be representedusing fixed values As the existing O-D matrix is usuallyused as the reference matrix in reserve capacity or practicalcapacity model and its pattern significantly influences theresult of the models we first consider it as an uncertainvariable in this study And thus the network capacity model isextended to be an optimization with parameter uncertainty

Researches on other areas of transportation networkoptimization typically adopted two methods to address theuncertain O-D demand [12] (i) stochastic optimization aimsat maximizing the expected profit by assuming that thedemand follows a known probability distribution (ii) robustoptimization aims at maximizing the profit with the worst-case scenario of the demand pattern Considering the exactprobability distribution of the O-D demand is still hard to beobtained the robust optimization is more effective in dealingwith this problem If a limited number of discrete scenarios ofO-Ddemand patterns are detected the scenario-based robustoptimization [13] is conducted which is a practical approachusually implemented in transportation projects It is moregeneral to assume the possibility of the travel demand to be acontinuous variable within a bounded set and the set-basedrobust optimization can be used for decision-making [14]The uncertainty set is constructed to include most of possiblevalues of the travel demandThe decision-makersrsquo attitudes torisk should be considered as well when deciding the shapesand size of the uncertainty sets It is important to make atrade-off between the system performance and the level ofrobustness achieved [13]

In this study we propose a robust optimization modelfor the network capacity problem by using the existing O-D travel demands as uncertain parameters The existingdemand between each O-D pair is assumed to be variablebetween its upper and lower limits Besides three typicaluncertainty regions are introduced to provide a bounded setfor the uncertain demand A heuristic solution is developedfor the solution to the robust network capacity model In thenext section the concept of network spare capacity is revisitedbased on the reserve capacity model Then the robustmodel for network capacity estimation is presented andthe three typical uncertainty sets of existing travel demandare defined After that the solution algorithm is described

Computational experiments show the validation and justifi-cation of the robust model Conclusions and perspectives forfurther research are provided in the last section

2 Network Spare Capacity and Its Flexibility

The reserve capacity was proposed as the largest multiplier120583 applied to a given existing O-D demand matrix that canbe allocated to a transportation network without violatingany individual link capacity [3] The product of the largestmultiplier and the existing O-D demand (represented byvector q) gives the maximum travel demand which canbe loaded to the network For clarity sake we refer to themaximum travel demand as the value of network capacity inrest of this paper For passenger network it is well known thatmultiple O-D pairs exist and demands between different O-D pairs are not exchangeable or substitutableThus the traveldemand pattern ormatrix reflects both its quantity and spatialdistributionThemethod of reserve capacity assumes that theexisting O-D demand is scaled with a uniform O-D growthThe largest value of 120583 indicates whether the current networkhas spare capacity or not So the network spare capacity isgenerally explained as follows if 120583 gt 1 then the network canbe loaded more travel demand and the additional demandcan be accommodated by the network which is (120583 minus 1)qotherwise that is 120583 lt 1 the network is overloaded and theexisting O-D demand should decrease by (1 minus 120583)q to satisfythe capacity constraints [10] In some researches the demandmultiplier 120583 is regarded as the uncertainties in the future O-Ddemand [9 15]

The classical model of reserve network capacity (RNC) isdefined as follows

RNC max120583

120583 (1)

st V119886 (120583q) le 119862119886 forall119886 isin 119860 (2)

where V119886(120583q) is obtained by solving the following userequilibrium problem

minf

sum119886

intV119886

0119905119886 (119909) 119889119909 (3)

st sum119903isin119877119894119895

119891119894119895119903 = 120583119902119894119895 forall119894 isin 119868 119895 isin 119869 (4)

V119886 = sum119894

sum119895

sum119903

119891119894119895119903 sdot 120575119894119895119886119903 forall119886 isin 119860 (5)

119891119894119895119903 ge 0 forall119894 isin 119868 119902 isin 119869 119903 isin 119877119894119895 (6)

where 120583 is the O-D demand multiplier to all O-D demands119877 is the set of all routes in the network 119894 is the origin index119894 isin 119868 and 119868 is the set of all origin nodes 119895 is the destinationindex 119895 isin 119869 and 119869 is the set of all destination nodes 119862119886 isthe capacity of link 119886 V119886 is the flow on link 119886 119886 isin 119860 v is thevector of all link flows 119902119894119895 is the existing trip demand betweenO-D pair 119894119895 q is the vector of all O-D demand 119891119894119895119903 is the flowon route 119903 119903 isin 119877 between O-D pair 119894119895 associated with 119902119894119895f is the vector of flows of all route in 119877 120575119894119895119886119903 is the link-route

Journal of Advanced Transportation 3

Total travel demand

Parameters

Existing demands with different patterns

Network capacities

Robust network capacity

Possiblerange ofpatterns

Capacityflexibility

Max demand

Existing demand

Pattern i middot middot middot middot middot middot Pattern k middot middot middot middot middot middot Pattern j

Figure 1 Robust solution of network capacities with different demand pattern

incidence indicator 1 if link 119886 is on route 119903 between O-D pair119894119895 and 0 otherwise 119905119886(V119886) is the travel cost function for link 119886In the abovemodel the upper-level model maximizes the

O-D matrix multiplier without violating the capacity con-straints (2) for every individual link The parameter q givesthe prescribed O-D travel demand in the network whichcan be obtained according to the current trip demand or apredicted demandpattern accordingly Route choice behaviorand congestion effect are considered in the user equilibrium(UE) model as the lower-level model in (3)ndash(6) Generallyother traffic assignment methods such as stochastic userequilibrium (SUE) model can be used in place of the abovedeterministic UE model as required [10]

The result of the reserve capacity model which is con-sidered may underestimate the capacity of the passengernetwork because only the existing O-D demand pattern thatis more congruous with the network topology would achievea higher value of network capacity [16] Basically the reservecapacity depends on the initial O-D demand patterns androute choice behavior of the users Given the lower-leveltraffic assignment method the existing O-D demand shouldbe the only determinant to the result of the above model Itmeans that if the given O-D matrix is not consistent withthe network the reserve capacity model will produce a resulthaving a low level of maximum demand Otherwise if theO-D pattern is determined according to the network spatialstructure the travel demand can grow to a very high amount

Directly applying the result of the reserve capacity mayhave the following problems (i) It is hard to decide anexact existing (or predetermined) O-D matrix because thereal travel demand pattern is changing at different hoursevery day and different days every week Also it is stillvery difficult to obtain the full data of the O-D demandscoveringmany different hours (ii) In real-world applicationsdecision-makers tend to be risk averse and may be moreconcerned with the worst cases Using only a few situationsof the O-D demand pattern may not provide a robust answer

to the network capacity estimation Conversely as long as thesystem performance reaches an acceptable level it does notmatter how much it changes above that level Thus it may bemore desirable to have an optimization result that performsbetter in the worst case

When estimating the capacity of transportation systemsdecision-makers are not only concerned with the extremeresults that the total trips can be allocated to a transportationnetwork but also need to evaluate the unknown situationsresulted from the fluctuation of the travel demand Thus tomeasure the ability of transportation networks that can dealwith the variation of travel demand Chen and Kasikitwiwat[16] discussed the concept of the network capacity flexibilityusing three typical network capacity models The networkcapacity flexibility is defined as the ability of a transportsystem to accommodate changes in traffic demand whilemaintaining a satisfactory level of performance [16 17] Inthis study integrated with the uncertainties from the existingdemand in transportation networks the network capacityflexibility is further illustrated in Figure 1 On the basis of thisthe robust estimation of network capacity is defined as themaximum travel demand can be allocated to a transportationnetwork when satisfying all the possibilities of the uncertainchanges in the quantitative and spatial demand pattern Therobust value of the network capacity is also illustrated inFigure 1

In this study we extended the reserve capacity model byconsidering the existing O-D demands as uncertain param-eters within a certain bounded region Robust solutions tothe network capacity can be conducted using the robustoptimization We utilize the classical reserve capacity modelto conduct the robust network capacity for two reasons (i) thereserve capacity is easy to solve and the O-D travel demandis allowed either increasing or decreasing by applying anO-Dmatrix multiplier greater than one or less than one (ii) as theexistingO-Dmatrix is extended to be an uncertain parameterin the reserve capacity model the O-D distribution is no


longer fixed but a variable pattern within some range givenby the uncertainty set

3 Robust Network Capacity Estimationunder Demand Uncertainty

In this section we assume that the prescribed O-D tripdemand is unknown but bounded within an uncertainty set119876 Mathematically the uncertainty set should be closed andconvex In practice the set of uncertain demand should bederived based on the transportation plannersrsquo knowledgeon the uncertainty associated with both the current andfuture O-D travel demand Nevertheless it is very difficult toobtain the exact probability distribution of the trip demandbetween all O-D pairs If the random demand follows acontinuous distribution defined from zero to infinity forexample normal distribution [12] this would require aninfinity capacity to meet all possible demand realizationsThus the uncertainty set of travel demand is defined as abounded region typically utilizing the highest travel demand119902119880119894119895 and the lowest demand 119902119871119894119895 for each O-D pair Using thebounded uncertainty region the events with low probabilitiescan be excluded and then the robust optimization would notprovide overly conservative results

In this study three typical uncertainty sets were con-structed for the existing travel demands

(1) Interval Constraint [13] The travel demand between eachO-D pair which is assumed varies independently within agiven interval of 119876119894119895 = [119902119871119894119895 119902119880119894119895 ] The interval could be theconfidence interval of an estimated demand obtained froma survey or by using an O-D estimation model Withoutadditional restraints the whole uncertainty set will be a boxcentered at the average travel demand In this case it is simpleto set 119902119894119895 fl 119902119880119894119895 for all O-Dpairs and solve the resulting reservecapacity problem The capacity value would be estimated inthe worst case which is too conservative In reality it isnever possible that the demand for all O-D pairs reachestheir estimated upper bound at the same time The traveldemand pattern is always fluctuant among the O-D pairs indifferent directions Therefore it is more reasonable to set anupper limit to the summation of the existing travel demandin network that is sum119894119895 119902119894119895 le 119863 The upper limits 119863 could beestimated by using themaximum trip volume investigated forthe existing travel demand

(2) Ellipsoid [18] An ellipsoidal set is generally defined asfollows

119876 = q | sum119894119895 (119902119894119895 minus 119902119894119895(12) (119902119880119894119895 minus 119902119871119894119895))

2

le 1205792 (7)

where 119902119894119895 = (119902119880119894119895 + 119902119871119894119895)2 the average O-D demand 120579 is aparameter that reflects decision-makersrsquo attitudes to risk andlarger 120579 indicates that it is more adverse to risk The value of120579 is from zero to radic|119882| where |119882| denotes the number ofO-D pairs When 120579 = 1 the uncertainty region is the largest

ellipsoid contained in the box region 119876 = q | 119902119871119894119895 le 119902119894119895 le119902119880119894119895 forall119894 119895(3) Polyhedron The polyhedron is a set of a finite numberof linear equalities and inequalities that restrains the traveldemand It is a generalized form of the box uncertaintyset For example Sun et al [19] constructed the followingpolyhedron region for uncertain O-D demand

119876

=q |

1199020119894119895 minus 1205741199020119894119895 le 119902119894119895 le 1199020119894119895 + 1205741199020119894119895 120574 isin [0 1] forall119894 119895sum119894

119902119894119895 = sum119894

1199020119894119895 forall119895sum119895

119902119894119895 = sum119895

1199020119894119895 forall119894

(8)

where 1199020119894119895 is the nominal value of the travel demand for O-D pair 119894119895 It may choose the mean value of the interval con-straints or an observed result from a survey This uncertaintyset allows the demand pattern varying entirely around itsnominal value and involves the implicit possible interactionsamong O-D demands Therefore the overly conservativeresultsmay be avoidedThe last two sets of constraints requirethat the uncertain travel demands meet the conservationcondition with the nominal demand matrix at the zonalproduction and attraction

Note that the shape of uncertainty set affects the efficiencyand robustness of network capacity value Ben-Tal andNemirovski [14] suggested applying the minndashmax optimiza-tion model Once the uncertainty set of the travel demand119876 is determined the minndashmax model will find a robustsolution that tolerates changes in travel demand up to thegiven bound Using any type of the uncertainty sets 119876 therobust reserve network capacity (RRNC) problem can beformulated as followsRRNC max

120583minq

120583st V119886 (120583q) le 119862119886

forall119886 isin 119860 such that q isin 119876(9)

where V119886(120583q) solvesminf

sum119886

intV119886

0119905119886 (119909) 119889119909

st sum119903isin119877119894119895

119891119894119895119903 = 120583119902119894119895 forall119894 isin 119868 119895 isin 119869 119902119894119895 isin 119876V119886 = sum

119894

sum119895

sum119903

119891119894119895119903 sdot 120575119894119895119886119903 forall119886 isin 119860119891119894119895119903 ge 0 forall119894 isin 119868 119895 isin 119869 119903 isin 119877119894119895

(10)

The above model is referred to as the robust counterpartof the original reserve network capacity problem The solu-tion of the robust counterpart results in a maximum totaltravel demand scheme under the corresponding worst-casedemand pattern


4 Solution Algorithm

A heuristic algorithm is proposed to solve the above robustoptimization model It takes a similar framework as theprocedure presented in [18] which is referred to as the cut-ting plane algorithm to robust optimization The algorithminvolves an iterative procedure to solve two inner optimiza-tion problems alternately until the convergence criterion issatisfied The algorithm is presented as follows

Step 0 (initialization) Give the initial values of the O-Ddemand q(0) isin 119876 (usually 119902(0)119894119895 = 119902119871119894119895 forall119894 119895) and solve the fol-lowing reserve network capacity (RNC) problem to producean initial demand multiplier 120583(0)

RNC max120583

120583subject to V119886 (120583q(119899)) le 119862119886 forall119886 isin 119860 (3)ndash(6) (11)

Set the iteration counter 119899 fl 0Step 1 (direction finding)

Step 11 Solve the following inner (worst-case scenario(WCS)) problem with the determined 120583(119899) to obtain theworst-case demand scenario

WCS maxq

sum119886

(max (0 V119886 (120583(119899)q) minus 119862119886))subject to q isin 119876 (10)

(12)

Step 12 Formulate a RNC problem with the scheme ofexisting demand 119902(119899)119894119895 produced from the WCS problem inStep 11 Solve this inner problem to find a search directionof the maximum demand multiplier 120583(119899)Step 2 (move) Compute 120583(119899) fl 120583(119899minus1) + 120572(119899)(120583(119899) minus 120583(119899minus1))where 120572(119899) is the step length In this study the step lengthis chosen as 120572(119899) = 1119899 which is used in the method ofsuccessive averages

Step 3 (convergence check) If the objective value of theWCSproblem 119910 gt 120576 or 119899 reaches the maximum iterations thenstop where there is a predetermined convergence criterionOtherwise denote the solution of theWCS problem as 119902(119899+1)119894119895 and go to Step 1 set 119899 fl 119899 + 1Remark 1 In the above steps theWCS problem is formulatedto find a solution of q isin 119876 in which case the traffic flowson all links which exceed their capacities the most If q isan optimal solution to the RRNC the corresponding optimalobjective value of theWCS problemmust be zero Otherwisean improved solution may be obtained by solving the RNCproblems in Step 12 which is a relaxation of the RRNCproblems with a specific demand pattern In the process ofthe algorithm each of these relaxed RRNC problems canapproximate the original RRNC better than its predecessorsAlthough it is still difficult in practice to find a global

optimum of the relaxed RRNC and WSC Yin et al showedthat the cutting plane algorithm is effective in providing agood solution to the robust optimization problem [18] Therelaxed RRNC problems and the WSC problems are solvedby the sensitivity analysis based (SAB) algorithm [20]

Remark 2 The second inner problem is a standard RNCmodel when the existing O-D demand is determined TheRNC can be solved efficiently by applying the SAB algorithm[3] The SAB algorithm locally approximates the originalbilevel problem as a single-level optimization by using first-order Taylor expansion The derivatives of lower-level deci-sion variables with respect to upper-level ones are utilized forthe linear approximation The derivatives can be conductedfrom the sensitivity analysis of the lower-level model

In this study we used the restriction approach for the sen-sitivity analysis of the lower-level UE model The restrictionapproach was proposed by Tobin and Friesz [21] and thencorrected by Yang and Bell [22] for its flaws on selecting thenondegenerate extreme point One can also refer to Du etal [23] for the details of this approach In this section somenecessary results are present without proof

For the reserve capacity model the link flows in upper-level V119886(120583q) are represented as an implicit function of theO-D matrix multiplier 120583 as constraint (2) shows Using thefirst-order Taylor expansion it can be approximated as

V119886 (120583q) asymp V119886 (120583lowastq) + [ 120597V119886 (120583q)120597120583100381610038161003816100381610038161003816100381610038161003816120583=120583lowast] (120583 minus 120583lowast)

forall119886 isin 119860(13)

where 120583lowast is the given solution of the O-D demand multiplierat the current iteration of SAB algorithm

From the results in Tobin and Friesz [21] the derivativesof the route flows f with respect to the O-D demandmultiplier 120583 are derived as follows

[nabla120583f0nabla120583120587] = [Δ0119879nablakt (klowast 0) Δ0 minusΛ0119879minusΛ0 119874 ]minus1 [ 119874

nabla120583 (120583q)] (14)

where the superscript ldquo0rdquo denotes that the variables ormatrices are only associated with the restricting subproblemderived by the restriction approach (applying the correctionin Yang and Bell [22]) and the superscript ldquo119879rdquo represents thetransposed matrix Other notations are defined as follows

Δ = [120575119894119895119886119903] is the link-route incidence matrix

Λ = [120582119894119895119903 ] is the O-D-route incidence matrix where120582119894119895119903 equals 1 if O-D pair 119894119895 is connected by route 119903 and0 otherwise120587 is the Lagrangian multiplier associated with con-straint (4)t(klowast 0) is the vector of the travel cost function ofall links with the equilibrium link flow vlowast for theperturbation parameters at 0

Thus the derivatives of the link flows to themultiplier areobtained by nabla120583k = Δ0 sdot nabla120583f0 Based on the above derivationsthe SAB algorithm can be used for solving the RNC problem


Remark 3 The WCS problem is also formulated as a bilevelprogramming using equilibrium constraints so the SABmethod can also be modified for its solution The implicitrelationship V119886(120583q) is first-order approximated as

V119886 (120583q) asymp V119886 (120583qlowast)+sum119894isin119868

sum119895isin119869

[ 120597V119886 (120583q)1205971199021198941198951003816100381610038161003816100381610038161003816100381610038161003816q=qlowast] (119902119894119895 minus 119902119894119895

lowast) forall119886 isin 119860

(15)

where qlowast is the solution of the existing O-D demand at thecurrent iteration of SAB algorithm

The derivatives of the route flows f with respect tothe existing O-D demand q are derived from the followingequation

[nablaqf0nablaq120587] = [Δ0119879nablakt (klowast 0) Δ0 minusΛ0119879minusΛ0 119874 ]minus1 [ 119874

nablaq (120583q)] (16)

Because in this inner problem the value of 120583 is fixed thederivative nablaq(120583q) will be a diagonal matrix with the value of120583 on its diagonalThen the derivatives of v to 120583 are calculatedby nablaqk = Δ0 sdotnablaqf0Thus at each iteration of the SABmethodthe approximating WCS problem is

maxq

sum119886isin119860

(max (0 nablaqV119886q + Vlowast119886 minus nablaqV119886qlowast minus 119862119886))subject to q isin 119876 (17)

Note that the above localized approximation problem is anonlinear problem and a number of optimization tools incommercial software packages could be used for its solutionIn this study the approximate WCS problem is solved usingMATLAB built-in functions which were converted to a NETcomponent and used in our solution program inC languageSimilar to [18] we randomly generate 100 vectors of q fromthe uncertainty region 119876 For each q the user equilibriumproblem with 120583(119899) is solved Let q be the random q withthe maximum objective value of the WCS problem anddenote the objective value as 119910 If 119910 gt 120576 then set q(119899+1) flq Otherwise use the MATLAB functions to solve theapproximate WCS problem with q as an initial solution If itgives a solution with an objective larger than 120576 the solution isused as q(119899+1) for the next iteration otherwise the algorithmis terminated with an optimal solution q

5 Computational Experiments

51 Experiment 1 Nguyen-Dupuis Network Computationalexperiments are presented in this section to illustrate theresults of the robust network capacity model The example isbased on a road network which is adopted from Nguyen andDupuis [24] as Figure 2 shows It consists of 13 nodes 19 linksand 4 O-D pairs The nominal value of the existing traveldemand is given by the O-D matrix in Figure 2 (denoted byq0) The characteristics of the links are listed in Table 1 The

Table 1 Link characteristics of the example network

Link number 119886 Free-flow time 1199050119886 Capacity 1198621198861 70 8002 90 4003 90 2004 120 8005 30 3506 90 4007 50 8008 130 2509 50 25010 90 30011 90 55012 100 55013 90 60014 60 70015 90 50016 80 30017 70 20018 140 40019 110 600

1

4 5 6 7 8

12

9 10 11 2

313

3

1

2

18

5 7 9

17

64

12 14

8 10

15

11

1613

19

5 65

Link number

Node number

DO 2 3

1 400 800

4 600 200

Existing O-D demand

Figure 2 An example network

Bureau of Public Road link performance function was usedin the experiments

119905119886 (V119886) = 1199050119886 [1 + 015 sdot ( V119886119862119886)4] (18)

where 1199050119886 is the free-flow travel time for link 119886We applied the proposed approach for robust network

spare capacity estimation with the three typical uncertaintysets which are described in this paper Assume that the inter-vals for the O-D travel demands are [300 600] [600 1050][400 950] and [100 375] for O-D pairs 1-2 1-3 4-2 and 4-3respectively


Table 2 Robust reserve capacities with ellipsoidal region and polyhedral region

Ellipsoid PolyhedronParameter 120579 Max 120583 Robust capacity Parameter 120574 Max 120583 Robust capacity00 01795 39263 00 01944 3888902 01727 39103 025 01892 3783805 01633 38871 05 01843 3685410 01498 38546 075 01795 3590120 01286 37668 10 01753 35054

Table 3 Robust reserve capacity estimations and the performances

Network Nguyen-Dupuis Nominal Conservative Robust-e Robust-pReserve capacity estimation 38889 36342 38546 36854Travel demand multiplier 01944 01346 01498 01843Percentage of meeting capacity constraints () 172 1000 984 724Percentage of above-robust-estimation () 518 1000 546 988

Network capacityMaximum multiplier

35000

37000

39000

41000

43000

45000

Trav

el d

eman

d

1600 1800 2000 2200 2400 2600 2800 30001400Maximal total existing demand

01

015

02

025

03

O-D

mat

rix m

ultip

lier

Figure 3 Robust reserve capacities with interval constraints underdifferent maximal total existing demand

(1) Interval Region Theoretically under the interval con-strains for each O-D pair the total of the existing O-Ddemand can vary from 1400 to 2975 which covers a widerange However according to our practical experience thetravel demands between the O-D pairs may not reach theirmaximum simultaneously Thus an additional constraintsum119894119895 119902119894119895 le 119863 was introduced to give an upper bound ofthe total existing demand To be consistent with the intervalconstraints 119863 was set to change from 1400 to 3000 Figure 3illustrates how uncertainty of the existing travel demandaffects the reserve capacity results of the example networkThe reserve capacity value is calculated by 120583lowastsum119894119895 119902lowast119894119895 where120583lowast is the robust solution of the RRNC problem and 119902lowast119894119895 is thecorresponding travel demand pattern Note that the reservecapacity value and the 120583lowast decrease synchronously along withthe growth of 119863 Because when 119863 is increased the vectorof travel demand pattern q will have more space to changeits spatial distribution and this makes the existing demandeasier to archive a demand pattern whose correspondingmultiplier can reach a smaller value Besides when119863 exceeds

2700 the value of 120583lowast cannot get worse At this value120583lowast = 01346 the corresponding existing travel demand isqlowast = (600 1050 950 100) which produces a smallest reservecapacity value 36342 (referred to as the conservative solutionin the rest part of this section) Furthermore for the robustsolutions when119863 gt 2700 the change of 11990243 will have no effecton the value of120583lowast Here we use the lower bound of 11990243 to con-duct the worst-case results of the reserve capacity Accordingto the properties of the robust optimization the solutions ofthe existing O-D demand may not be unique so no specificsolution of the existing demand is presented in this paper

(2) Ellipsoidal Region For the ellipsoidal uncertainty set theparameter 120579 is set to be 0 02 05 10 and 20 The centerof the ellipsoid is decided by the boundary for each O-D demand (not the nominal value for this example) Thecomputational results in Table 2 show that the robust solutionto 120583 changes between 01795 and 01286 with a descendingtrend as well as the corresponding network capacity value

(3) Polyhedral Region For the robust network capacity esti-mation with the polyhedral set described in previous sectionthe robust results at 120574= 0 025 05 075 and 10 are computedand presented in Table 2 The nominal demand (given inFigure 2) is used in this type of uncertainty The value of120583 declines from 01944 to 01753 The same trend has beenobserved on the robust values of network capacity whichvaries from 38889 to 35054

Table 3 reports the solutions of the reserve capacity valuefrom two RNC problems with q0 = (400 800 600 200) andq = (600 1050 950 100) separately and two RRNC problemswith 119876 defined in the ellipsoid region (120579 = 10) and thepolyhedral region (120574 = 05)We refer to the first two solutionsas ldquonominalrdquo and ldquoconservativerdquo estimation respectively andthe latter two as ldquorobust-erdquo and ldquorobust-prdquo separately For theresults the robust solution for the O-D demand multiplierindicates that all the values of the existing demand withinthe uncertainty set can be applied by a multiplier larger thanthis robust value without violating the capacity constraints A


Table 4 Network capacity estimations on Sioux-Falls

Network Sioux-Falls Nominal Conservative Robust-e Robust-pReserve capacity estimation 2090291 1950658 2013667 1768503Travel demand multiplier 05797 04291 05588 04904

corresponding maximum travel demand is produced as therobust network capacity solution for the same uncertaintyset Since the samemaximum travel demandmay produce bydifferent combinations of the multiplier and feasible existingdemand the proposed algorithm only finds the smallestvalue of the maximum multiplier Besides if the shape ofthe uncertainty set changes different solutions for the robustmultiplier and maximum demand will be produced Table 3shows these features

To evaluate the four estimations of the network capacitywe randomly generate 500 samples of O-D demand as thepossible realizations of the existing travel demand patternThe samples are from the normal distribution119873(119902119894119895 0251199022119894119895)within its interval [119902119871119894119895 119902119880119894119895 ] for all O-D pairs Then for eachsample one has the following

Firstly the user equilibrium assignment associated witheach reserve network capacity estimation (ie the largest 120583)is computed The number of failures is counted wheneverthe link flow exceeds its capacity at each sample In the endof the simulation the successful rates are computed as [1 minustotal number of failures(number of samples times number oflinks)] The results are referred to as ldquopercentage of meetingcapacity constraintsrdquo in Table 3 Obviously the conservativeestimation gives the worst value of the network capacity thenominal estimation produces a medium level of result Bycomparison the robust results from the ellipsoid and polyhe-dron are more reasonable Note that all samples generated inthe above simulations belong to a box region comprised of theintervals for O-D demands The aforementioned uncertaintysets are used as replacements of the box regions so as toprevent overly conservative robust resolutions The results inTable 3 indicate that the ellipsoid approximate the box regionbetter than the polyhedron because more random sampleswill not exceed the network capacity after they are scaled bythe robust multiplier 120583 = 01498 On the other hand although120583 = 01843 is the worst case in the polyhedral set when theboundary is extended to the box region only 724 of therandom samples can be covered

Furthermore considering that the demand multiplierin essence is a relative value the reserve capacity resultsare derived Therefore the reserve capacity problem withthe every random existing O-D demand is solved in ourtest For each estimation value of the reserve capacity thenumber of successes is countedwhenever the reserve capacityvalue of the sample exceeds the robust capacity estimationConsequently the successful rates are computed as [totalnumber of successesnumber of samples] These results arealso presented in Table 3 as ldquopercentage of above-robust-estimationrdquo From this aspect the polyhedron provides amore robust estimation of the network capacity which can be

met bymost realizations of the existing travel demand pattern(988 versus 546 compared with the ellipsoid) We maysuppose that the worst cases of network capacity values existin the corner area of the box-shaped uncertain region whereit is not easy to be covered by the corresponding ellipsoidalset Consequently the choice of the robust results dependson the decision-makersrsquo attitude to risk and the usage ofthe robust results From the computational perspective therobust optimizations with polyhedral uncertainty sets havemore advantages More effective solution methods could bedeveloped in future studies

52 Experiment 2 Sioux-Falls Network Experiments are fur-ther presented on the Sioux-Falls network [25] The networkcontains 24 nodes 76 links and 528 O-D pairs The charac-teristics of the links and travel demands are also provided inBar-Gera [25] In this experiment we used the default valuesof the travel demand in as the nominal value of the existingdemand in the network If 119902119894119895 ge 1000 then the demand for O-D pair 119894119895 is uncertain and its upper limit and lower limit areset to 119902119871119894119895 = 05119902119894119895 and 119902119880119894119895 = 15119902119894119895 separately Hence there are104O-D pairs with uncertain existing demands According tothe previous section four estimation methods are employedto evaluate the network capacity of the Sioux-Falls

Table 4 reports the solutions of the reserve capacity fromthe four estimation methods Note that the robust-e solutiongives a moderate robust result compared to the others Therobust-p solution provides the lowest estimation of networkcapacity One may use this lowest value as the worst-caseperformance that the network can serve the travel demandBesides note that the lowest estimation of the networkcapacity was not derived from the conservative solutionsin which the uncertain O-D demands are set to its upperlimits Although the conservative solution is correspondingto the lowest demand multiplier it may not reflect the mostunfavorable situationwhich is possible to be resulted from thechanges of the network demands

We selected the conservative robust-e and robust-psolutions to further inspect the link flow patterns at themaximum travel demand situations (ie the reserve capac-ity) The link flow patterns are shown in Figures 4(a) 4(b)and 4(c) The width of the line indicates the traffic volumethrough the link The red lines show the links whose VC(volumecapacity) ratio is greater than 09 and the blacklines denote the links with zero flow As a reference wealso randomly generate 500 samples of O-D demand whichare from the normal distribution 119873(119902119894119895 0251199022119894119895) within itsinterval [119902119871119894119895 119902119880119894119895 ] forall119894 119895 For each sample the reserve capacityproblem is solved Then at each link the highly saturatednumber is counted whenever its VC ratio is greater than


(a) Conservative solution (b) Robust solution with ellipsoid (120579 = 10)

(c) Robust solution with polyhedron (120574 = 05) (d) Simulation results

Figure 4 Link flow pattern associated with different maximum travel demands (a b and c) and probabilities that links are congested (d)

09 In consequence the link saturation rates are computed as[total the saturated numbernumber of samples] The resultsare also shown in Figure 4(d) The width of the line indicatesthe saturation rate of the linkThe red lines indicate links withsaturation rates greater than 50 The orange lines denotelinks with saturation rates lower than 50 but greater thanzero The black lines mean the links have no chance to beblocked From Figure 4 we conclude that the solution of therobust-e will produce a link flow pattern most likely to berealized when the network reaches its capacity (comparedto the simulation results) The robust-p solution provides anextreme result that the network is congested only because ofa very few links These links could be considered as the mostcritical links which restrict the capacity of the entire roadnetwork By contrast the conservative solution seems not tofit the simulation results very well In conclusion the robustestimation to the network capacity problem could be muchmore practical than the results from any specific scenario

6 Conclusions and Perspectives

In this study a robust network capacity model with uncer-tain demand has been proposed The robust optimizationis formulated using the minndashmax model with a boundeduncertainty set of the existing O-D travel demands Withthe uncertainty set the low-probability realizations of thetravel demand pattern are excluded and thus the robustmodel can produce a proper estimation of network capacitywhich can be achieved with a large probability Then aheuristic algorithm has been proposed for the proposedrobust model It solves two inner problems iteratively one isthe worst-case scenario problem and the other is the relaxedrobust optimization namely the standard reserve capacitymodel At each iteration the cutting plane method has beenadopted to generate the worst-case demand scenario and thesensitivity analysis based approach has been developed forthe solution of the worst-case model and the reserve capac-ity model The validity and performance of the proposed


robust model have been demonstrated in the computationalexperiments Different results under three typical uncertaintysets say interval ellipsoidal and polyhedral region havebeen conducted and compared The interval set is simplebut easy to produce too conservative results the ellipsoidalset is a good approximation to the uncertain region andproduces results with moderate robustness but its solutionis more complicated due to the nonlinear constraints thepolyhedral set is considered if a high level of robustness isrequired and its linear formulation makes the robust modeleasier to solve Furthermore by conducting computationalexperiments on the Sioux-Falls network robust solutionsshown can provide more practical results of the link flowpatterns In applications these uncertainty sets and theirparameters should be selected according to the desired levelof robustness

The robustmodel based on the reserve capacitymodel hasbeen proposed and explored in this study Future researchesshould focus on more efficient solution approaches for therobust problem with the minndashmax model The experimentson large-scale networks are also needed Besides the demanduncertainties existing in other network capacity models arealso expected to be detected and discussed Alterative trafficassignment model such as the stochastic user equilibriumcould also be discussed for the network capacity problemsThe robust solution of network capacity gives a lower boundto the possible schemes of the maximum demand in atransportation networkThese possibilities constitute a rangewhere the robust solution can be most likely to be reachedin reality Therefore the robust solution to network capacityproblems needs to receive more attentions in transportationplanning applications

Disclosure

An initial version of this paper was presented at the Trans-portation Research Board (TRB) 96th Annual Meeting Ittherefore also appears in the proceedings of the TRB 96thAnnual Meeting Compendium of Papers

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by ldquothe National Natural ScienceFoundation of China (no 51508161)rdquo ldquothe Natural ScienceFoundation of Jiangsu Province (no BK20150817)rdquo and ldquotheFundamental Research Funds for the Central Universities(no 2017B12414)rdquo

References

[1] R K Ahuja T LMagnanti and J OrlinNetwork FlowsTheoryAlgorithms and Applications Prentice-Hall New Jersey NJUSA 1993

[2] Y Asakura ldquoMaximumcapacity of road network constrained byuser equilibrium conditionsrdquo in Proceedings of the 24th AnnualConference of the UTSG 1992

[3] S CWong andHYang ldquoReserve capacity of a signal-controlledroad networkrdquo Transportation Research Part B Methodologicalvol 31 no 5 pp 397ndash402 1997

[4] H Yang M G H Bell and Q Meng ldquoModeling the capacityand level of service of urban transportation networksrdquo Trans-portation Research Part B Methodological vol 34 no 4 pp255ndash275 2000

[5] P Kasikitwiwat and A Chen ldquoAnalysis of transportation net-work capacity related to different system capacity conceptsrdquoJournal of the Eastern Asia Society of Transportation Studies vol6 pp 1439ndash1454 2005

[6] G Ziyou and S Yifan ldquoA reserve capacity model of optimal sig-nal control with user-equilibrium route choicerdquo TransportationResearch B Methodological vol 36 no 4 pp 313ndash323 2002

[7] H Ceylan and M G H Bell ldquoReserve capacity for a roadnetwork under optimized fixed time traffic signal controlrdquoJournal of Intelligent Transportation Systems vol 8 no 2 pp87ndash99 2004

[8] S-W Chiou ldquoReserve capacity of signal-controlled road net-workrdquo Applied Mathematics and Computation vol 190 no 2pp 1602ndash1611 2007

[9] A Chen P Kasikitwiwat and C Yang ldquoAlternate capacityreliability measures for transportation networksrdquo Journal ofAdvanced Transportation vol 47 no 1 pp 79ndash104 2013

[10] X XuA Chen S JansuwanKHeaslip andC Yang ldquoModelingtransportation network redundancyrdquo Transportation ResearchProcedia vol 9 pp 283ndash302 2015

[11] M Du L Cheng X Jiang and Z Li ldquoSensitivity basedheuristics for network capacity estimation in transportationrdquo inProceedings of the 94th Transportation Research Board AnnualMeeting Washington DC USA 2015

[12] K An and H K Lo ldquoTwo-phase stochastic program for transitnetwork design under demand uncertaintyrdquo TransportationResearch Part B Methodological vol 84 pp 157ndash181 2016

[13] Y Yin S M Madanat and X-Y Lu ldquoRobust improvementschemes for road networks under demand uncertaintyrdquo Euro-pean Journal of Operational Research vol 198 no 2 pp 470ndash479 2009

[14] A Ben-Tal and A Nemirovski ldquoRobust optimizationmdashmethodology and applicationsrdquo Mathematical Programmingvol 92 no 3 pp 453ndash480 2002

[15] S-W Chiou ldquoOptimization of robust area traffic control withequilibrium flow under demand uncertaintyrdquo Computers andOperations Research vol 41 no 1 pp 399ndash411 2014

[16] A Chen and P Kasikitwiwat ldquoModeling capacity flexibility oftransportation networksrdquoTransportation Research A Policy andPractice vol 45 no 2 pp 105ndash117 2011

[17] E K Morlok and D J Chang ldquoMeasuring capacity flexibility ofa transportation systemrdquo Transportation Research Part A Policyand Practice vol 38 no 6 pp 405ndash420 2004

[18] Y Yin S Lawphongpanich and Y Lou ldquoEstimating investmentrequirement for maintaining and improving highway systemsrdquoTransportation Research Part C Emerging Technologies vol 16no 2 pp 199ndash211 2008

[19] H Sun Z Gao and J Long ldquoThe robust model of contin-uous transportation network design problem with demanduncertaintyrdquo Journal of Transportation Systems Engineering andInformation Technology vol 11 no 2 pp 70ndash76 2011

[20] T L Friesz R L Tobin H-J Cho and N J Mehta ldquoSensitivityanalysis based heuristic algorithms for mathematical programswith variational inequality constraintsrdquoMathematical Program-ming vol 48 no 2 pp 265ndash284 1990


[21] R L Tobin and T L Friesz ldquoSensitivity analysis for equilibriumnetwork flowrdquo Transportation Science vol 22 no 4 pp 242ndash250 1988

[22] H Yang and M G H Bell ldquoSensitivity analysis of networktraffic equilibria revisited the corrected approachrdquo in SelectedProceedings of the 4th IMA International Conference on Mathe-matics in Transport pp 373ndash395 2007

[23] M Du L Cheng and H Rakha ldquoSensitivity analysis ofcombined distribution-assignment model with applicationsrdquoTransportation Research Record no 2284 pp 10ndash20 2012

[24] S Nguyen and C Dupuis ldquoAn efficient method for computingtraffic equilibria in networks with asymmetric transportationcostsrdquo Transportation Science vol 18 no 2 pp 185ndash202 1984

[25] H Bar-Gera TransportationNetworkTest Problems 2001 2017httpwwwbguacilsimbargeratntp

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



and has been adopted widely in many researches [7ndash10]The ultimate capacity and practical capacity model are morepractical but have more parameters to be calibrated whenapplied and the formulated models are still difficult to solve[11]

While the deterministic network capacity problem hasbeen explored extensively few studies have investigated theissue of uncertainties in demand data associated with thisproblem The ultimate capacity and practical capacity modelare only concerned with the uncertainties related to the newincreased travel demandby using combinedmodels [5] whilethe uncertainties in the current (or existing) demand arenot considered In reality travel demands in transportationsystem are always fluctuant day by day even hour by hourBesides errors of survey data also affect the accuracy of theexisting O-D matrix As a consequence the existing traveldemands are usually difficult to be obtained in actual trans-portation projects and then are not easy to be representedusing fixed values As the existing O-D matrix is usuallyused as the reference matrix in reserve capacity or practicalcapacity model and its pattern significantly influences theresult of the models we first consider it as an uncertainvariable in this study And thus the network capacity model isextended to be an optimization with parameter uncertainty

Researches on other areas of transportation networkoptimization typically adopted two methods to address theuncertain O-D demand [12] (i) stochastic optimization aimsat maximizing the expected profit by assuming that thedemand follows a known probability distribution (ii) robustoptimization aims at maximizing the profit with the worst-case scenario of the demand pattern Considering the exactprobability distribution of the O-D demand is still hard to beobtained the robust optimization is more effective in dealingwith this problem If a limited number of discrete scenarios ofO-Ddemand patterns are detected the scenario-based robustoptimization [13] is conducted which is a practical approachusually implemented in transportation projects It is moregeneral to assume the possibility of the travel demand to be acontinuous variable within a bounded set and the set-basedrobust optimization can be used for decision-making [14]The uncertainty set is constructed to include most of possiblevalues of the travel demandThe decision-makersrsquo attitudes torisk should be considered as well when deciding the shapesand size of the uncertainty sets It is important to make atrade-off between the system performance and the level ofrobustness achieved [13]

In this study we propose a robust optimization modelfor the network capacity problem by using the existing O-D travel demands as uncertain parameters The existingdemand between each O-D pair is assumed to be variablebetween its upper and lower limits Besides three typicaluncertainty regions are introduced to provide a bounded setfor the uncertain demand A heuristic solution is developedfor the solution to the robust network capacity model In thenext section the concept of network spare capacity is revisitedbased on the reserve capacity model Then the robustmodel for network capacity estimation is presented andthe three typical uncertainty sets of existing travel demandare defined After that the solution algorithm is described

Computational experiments show the validation and justifi-cation of the robust model Conclusions and perspectives forfurther research are provided in the last section

2 Network Spare Capacity and Its Flexibility

The reserve capacity was proposed as the largest multiplier120583 applied to a given existing O-D demand matrix that canbe allocated to a transportation network without violatingany individual link capacity [3] The product of the largestmultiplier and the existing O-D demand (represented byvector q) gives the maximum travel demand which canbe loaded to the network For clarity sake we refer to themaximum travel demand as the value of network capacity inrest of this paper For passenger network it is well known thatmultiple O-D pairs exist and demands between different O-D pairs are not exchangeable or substitutableThus the traveldemand pattern ormatrix reflects both its quantity and spatialdistributionThemethod of reserve capacity assumes that theexisting O-D demand is scaled with a uniform O-D growthThe largest value of 120583 indicates whether the current networkhas spare capacity or not So the network spare capacity isgenerally explained as follows if 120583 gt 1 then the network canbe loaded more travel demand and the additional demandcan be accommodated by the network which is (120583 minus 1)qotherwise that is 120583 lt 1 the network is overloaded and theexisting O-D demand should decrease by (1 minus 120583)q to satisfythe capacity constraints [10] In some researches the demandmultiplier 120583 is regarded as the uncertainties in the future O-Ddemand [9 15]

The classical model of reserve network capacity (RNC) isdefined as follows

RNC max120583

120583 (1)

st V119886 (120583q) le 119862119886 forall119886 isin 119860 (2)

where V119886(120583q) is obtained by solving the following userequilibrium problem

minf

sum119886

intV119886

0119905119886 (119909) 119889119909 (3)

st sum119903isin119877119894119895

119891119894119895119903 = 120583119902119894119895 forall119894 isin 119868 119895 isin 119869 (4)

V119886 = sum119894

sum119895

sum119903

119891119894119895119903 sdot 120575119894119895119886119903 forall119886 isin 119860 (5)

119891119894119895119903 ge 0 forall119894 isin 119868 119902 isin 119869 119903 isin 119877119894119895 (6)

where 120583 is the O-D demand multiplier to all O-D demands119877 is the set of all routes in the network 119894 is the origin index119894 isin 119868 and 119868 is the set of all origin nodes 119895 is the destinationindex 119895 isin 119869 and 119869 is the set of all destination nodes 119862119886 isthe capacity of link 119886 V119886 is the flow on link 119886 119886 isin 119860 v is thevector of all link flows 119902119894119895 is the existing trip demand betweenO-D pair 119894119895 q is the vector of all O-D demand 119891119894119895119903 is the flowon route 119903 119903 isin 119877 between O-D pair 119894119895 associated with 119902119894119895f is the vector of flows of all route in 119877 120575119894119895119886119903 is the link-route


Total travel demand

Parameters


Network capacities



Capacityflexibility

Max demand

Existing demand

















119876 = q | sum119894119895 (119902119894119895 minus 119902119894119895(12) (119902119880119894119895 minus 119902119871119894119895))

2

le 1205792 (7)



119876

=q |


119902119894119895 = sum119894

1199020119894119895 forall119895sum119895

119902119894119895 = sum119895

1199020119894119895 forall119894

(8)



120583minq

120583st V119886 (120583q) le 119862119886



sum119886

intV119886

0119905119886 (119909) 119889119909

st sum119903isin119877119894119895


119894

sum119895

sum119903


(10)






RNC max120583




WCS maxq

sum119886


(12)








forall119886 isin 119860(13)




nabla120583 (120583q)] (14)








sum119895isin119869



(15)




nablaq (120583q)] (16)


maxq

sum119886isin119860







1

4 5 6 7 8

12

9 10 11 2

313

3

1

2

18

5 7 9

17

64

12 14

8 10

15

11

1613

19

5 65

Link number

Node number

DO 2 3

1 400 800

4 600 200

Existing O-D demand



119905119886 (V119886) = 1199050119886 [1 + 015 sdot ( V119886119862119886)4] (18)









35000

37000

39000

41000

43000

45000

Trav

el d

eman

d


01

015

02

025

03

O-D

mat

rix m

ultip

lier




























Disclosure




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Total travel demand

Parameters


Network capacities



Capacityflexibility

Max demand

Existing demand

















119876 = q | sum119894119895 (119902119894119895 minus 119902119894119895(12) (119902119880119894119895 minus 119902119871119894119895))

2

le 1205792 (7)



119876

=q |


119902119894119895 = sum119894

1199020119894119895 forall119895sum119895

119902119894119895 = sum119895

1199020119894119895 forall119894

(8)



120583minq

120583st V119886 (120583q) le 119862119886



sum119886

intV119886

0119905119886 (119909) 119889119909

st sum119903isin119877119894119895


119894

sum119895

sum119903


(10)






RNC max120583




WCS maxq

sum119886


(12)








forall119886 isin 119860(13)




nabla120583 (120583q)] (14)








sum119895isin119869



(15)




nablaq (120583q)] (16)


maxq

sum119886isin119860







1

4 5 6 7 8

12

9 10 11 2

313

3

1

2

18

5 7 9

17

64

12 14

8 10

15

11

1613

19

5 65

Link number

Node number

DO 2 3

1 400 800

4 600 200

Existing O-D demand



119905119886 (V119886) = 1199050119886 [1 + 015 sdot ( V119886119862119886)4] (18)









35000

37000

39000

41000

43000

45000

Trav

el d

eman

d


01

015

02

025

03

O-D

mat

rix m

ultip

lier




























Disclosure




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















119876 = q | sum119894119895 (119902119894119895 minus 119902119894119895(12) (119902119880119894119895 minus 119902119871119894119895))

2

le 1205792 (7)



119876

=q |


119902119894119895 = sum119894

1199020119894119895 forall119895sum119895

119902119894119895 = sum119895

1199020119894119895 forall119894

(8)



120583minq

120583st V119886 (120583q) le 119862119886



sum119886

intV119886

0119905119886 (119909) 119889119909

st sum119903isin119877119894119895


119894

sum119895

sum119903


(10)






RNC max120583




WCS maxq

sum119886


(12)








forall119886 isin 119860(13)




nabla120583 (120583q)] (14)








sum119895isin119869



(15)




nablaq (120583q)] (16)


maxq

sum119886isin119860







1

4 5 6 7 8

12

9 10 11 2

313

3

1

2

18

5 7 9

17

64

12 14

8 10

15

11

1613

19

5 65

Link number

Node number

DO 2 3

1 400 800

4 600 200

Existing O-D demand



119905119886 (V119886) = 1199050119886 [1 + 015 sdot ( V119886119862119886)4] (18)









35000

37000

39000

41000

43000

45000

Trav

el d

eman

d


01

015

02

025

03

O-D

mat

rix m

ultip

lier




























Disclosure




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













RNC max120583




WCS maxq

sum119886


(12)








forall119886 isin 119860(13)




nabla120583 (120583q)] (14)








sum119895isin119869



(15)




nablaq (120583q)] (16)


maxq

sum119886isin119860







1

4 5 6 7 8

12

9 10 11 2

313

3

1

2

18

5 7 9

17

64

12 14

8 10

15

11

1613

19

5 65

Link number

Node number

DO 2 3

1 400 800

4 600 200

Existing O-D demand



119905119886 (V119886) = 1199050119886 [1 + 015 sdot ( V119886119862119886)4] (18)









35000

37000

39000

41000

43000

45000

Trav

el d

eman

d


01

015

02

025

03

O-D

mat

rix m

ultip

lier




























Disclosure




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












sum119895isin119869



(15)




nablaq (120583q)] (16)


maxq

sum119886isin119860







1

4 5 6 7 8

12

9 10 11 2

313

3

1

2

18

5 7 9

17

64

12 14

8 10

15

11

1613

19

5 65

Link number

Node number

DO 2 3

1 400 800

4 600 200

Existing O-D demand



119905119886 (V119886) = 1199050119886 [1 + 015 sdot ( V119886119862119886)4] (18)









35000

37000

39000

41000

43000

45000

Trav

el d

eman

d


01

015

02

025

03

O-D

mat

rix m

ultip

lier




























Disclosure




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















35000

37000

39000

41000

43000

45000

Trav

el d

eman

d


01

015

02

025

03

O-D

mat

rix m

ultip

lier




























Disclosure




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






























Disclosure




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Robust Evaluation for Transportation Network Capacity...

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