Hindawi Jounal of Advanced Tanspotation Volume 2017, Aticle ID 9814909, 11 pages https://doi.og/10.1155/2017/9814909 Reseach Aticle Robust Evaluation fo Tanspotation Netwok Capacity unde Demand Uncetainty Muqing Du, 1 Xiaowei Jiang, 2 Lin Cheng, 2 and Changjiang Zheng 1 1 College of Civil and Tanspotation Engineeing, Hohai Univesity, 1 Xikang Rd, Nanjing, Jiangsu 210098, China 2 School of Tanspotation, Southeast Univesity, 35 Jinxianghe Rd, Nanjing, Jiangsu 210096, China Coespondence should be addessed to Muqing Du; dumuqing@gmail.com Received 28 Apil 2017; Accepted 17 July 2017; Published 10 Septembe 2017 Academic Edito: Dongjoo Pak Copyight 2017 Muqing Du et al. This is an open access aticle distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. As moe and moe cities in woldwide ae facing the poblems of taffic jam, govenments have been concened about how to design tanspotation netwoks with adequate capacity to accommodate tavel demands. To evaluate the capacity of a tanspotation system, the pescibed oigin and destination (O-D) matix fo existing tavel demand has been noticed to have a significant effect on the esults of netwok capacity models. Howeve, the exact data of the existing O-D demand ae usually had to be obtained in pactice. Consideing the fluctuation of the eal tavel demand in tanspotation netwoks, the existing tavel demand is epesented as uncetain paametes which ae defined within a bounded set. Thus, a obust eseve netwok capacity (RRNC) model using min max optimization is fomulated based on the demand uncetainty. An effective heuistic appoach utilizing cutting plane method and sensitivity analysis is poposed fo the solution of the RRNC poblem. Computational expeiments and simulations ae implemented to demonstate the validity and pefomance of the poposed obust model. Accoding to simulation expeiments, it is showed that the link flow patten fom the obust solutions to netwok capacity poblems can eveal the pobability of high congestion fo each link. 1. Intoduction The capacity of tanspotation netwok eflects the supply ability of its infastuctue and sevice to the tavel demand which is geneated fom the zones coveed by the tanspotation system in a specific peiod. Fo many yeas, tanspotation plannes and manages wanted to undestand how many tips can be accommodated at the most by the cuent o designed netwok in a cetain peiod of time. This need is moe necessay in those developing egions which ae confonted with apid gowth of pivate vehicles and inceased uban congestion. Meanwhile, the eseaches made a longtem effot to model and estimate the maximum thoughput of tanspotation netwoks. The achievements include maxflow min-cut theoem [1], incemental assignment appoach [2], and late bilevel pogamming models [3 5]. Fo the netwok capacity model, the most popula fomulation in passenge tanspotation system is the bilevel model, which maximizes the taffic flows unde the equilibium constaints. Wong and Yang [3] fist incopoated the eseve capacity concept into a taffic signal contol netwok. The eseve capacity is defined as the lagest multiplie applied to a given O-D demand matix without violating capacity constaints, so the solution is significantly affected by the pedetemined O-D matix. Ziyou and Yifan [6] extended the eseve capacity model by consideing O-D specific demand multiplies, and all demand multiplies should be ensued not lowe than a pedetemined minimum value. In ode to avoid assuming that all O-D flows incease in a same ate, anothe concept of ultimate capacity was poposed [5]. But it assumes that the O-D distibution is totally vaiable, which may poduce unealistic esults that cause the tip poductions at some oigins below thei cuent levels. Futhemoe, Yang et al. [4] suggested that the new inceased O-D demand patten should be vaiable in both level and distibution, while the cuent tavel demand is fixed. Late, Yang s model was also efeed to as the pactical capacity by Kasikitwiwat and Chen [5]. In summay, although unealistic, the eseve capacity model is moe easy-to-use
2 Jounal of Advanced Tanspotation and has been adopted widely in many eseaches [7 10]. The ultimate capacity and pactical capacity model ae moe pactical but have moe paametes to be calibated when applied, and the fomulated models ae still difficult to solve [11]. While the deteministic netwok capacity poblem has been exploed extensively, few studies have investigated the issue of uncetainties in demand data associated with this poblem. The ultimate capacity and pactical capacity model ae only concened with the uncetainties elated to the new inceased tavel demand by using combined models [5], while the uncetainties in the cuent (o existing) demand ae not consideed. In eality, tavel demands in tanspotation system ae always fluctuant day by day, even hou by hou. Besides, eos of suvey data also affect the accuacy of the existing O-D matix. As a consequence, the existing tavel demands ae usually difficult to be obtained in actual tanspotation pojects and then ae not easy to be epesented using fixed values. As the existing O-D matix is usually used as the efeence matix in eseve capacity o pactical capacity model and its patten significantly influences the esult of the models, we fist conside it as an uncetain vaiable in this study. And thus the netwok capacity model is extended to be an optimization with paamete uncetainty. Reseaches on othe aeas of tanspotation netwok optimization typically adopted two methods to addess the uncetain O-D demand [12]: (i) stochastic optimization aims at maximizing the expected pofit by assuming that the demand follows a known pobability distibution; (ii) obust optimization aims at maximizing the pofit with the wostcase scenaio of the demand patten. Consideing the exact pobability distibution of the O-D demand is still had to be obtained, the obust optimization is moe effective in dealing with this poblem. If a limited numbe of discete scenaios of O-D demand pattens ae detected, the scenaio-based obust optimization [13] is conducted, which is a pactical appoach usually implemented in tanspotation pojects. It is moe genealtoassumethepossibilityofthetaveldemandtobea continuous vaiable within a bounded set, and the set-based obust optimization can be used fo decision-making [14]. The uncetainty set is constucted to include most of possible values of the tavel demand. The decision-makes attitudes to isk should be consideed as well when deciding the shapes and size of the uncetainty sets. It is impotant to make a tade-off between the system pefomance and the level of obustness achieved [13]. In this study, we popose a obust optimization model fo the netwok capacity poblem by using the existing O- D tavel demands as uncetain paametes. The existing demand between each O-D pai is assumed to be vaiable between its uppe and lowe limits. Besides, thee typical uncetainty egions ae intoduced to povide a bounded set fo the uncetain demand. A heuistic solution is developed fo the solution to the obust netwok capacity model. In the next section, the concept of netwok spae capacity is evisited based on the eseve capacity model. Then, the obust model fo netwok capacity estimation is pesented, and the thee typical uncetainty sets of existing tavel demand ae defined. Afte that, the solution algoithm is descibed. Computational expeiments show the validation and justification of the obust model. Conclusions and pespectives fo futhe eseach ae povided in the last section. 2. Netwok Spae Capacity and Its Flexibility The eseve capacity was poposed as the lagest multiplie μ applied to a given existing O-D demand matix that can be allocated to a tanspotation netwok without violating any individual link capacity [3]. The poduct of the lagest multiplie and the existing O-D demand (epesented by vecto q) gives the maximum tavel demand which can be loaded to the netwok. Fo claity sake we efe to the maximum tavel demand as the value of netwok capacity in est of this pape. Fo passenge netwok, it is well known that multiple O-D pais exist and demands between diffeent O- D pais ae not exchangeable o substitutable. Thus, the tavel demand patten o matix eflects both its quantity and spatial distibution. The method of eseve capacity assumes that the existing O-D demand is scaled with a unifom O-D gowth. The lagest value of μ indicates whethe the cuent netwok has spae capacity o not. So the netwok spae capacity is geneally explained as follows: if μ>1, then the netwok can be loaded moe tavel demand and the additional demand can be accommodated by the netwok which is (μ 1)q; othewise, that is, μ<1, the netwok is oveloaded and the existing O-D demand should decease by (1 μ)q to satisfy the capacity constaints [10]. In some eseaches, the demand multiplie μ is egaded as the uncetaintiesin the futueo-d demand [9, 15]. The classical model of eseve netwok capacity (RNC) is defined as follows: RNC: max μ μ, (1) s.t. V a (μq) C a, a A, (2) whee V a (μq) is obtained by solving the following use equilibium poblem: min f s.t. a V a t a (x) dx, (3) 0 f ij =μq ij, i I, j J, R ij V a = i j f ij (4) δij a,, a A, (5) f ij 0, i I, q J, R ij, (6) whee μ is the O-D demand multiplie to all O-D demands; R is the set of all outes in the netwok; i is the oigin index, i I,andI is the set of all oigin nodes; j is the destination index, j J,andJ is the set of all destination nodes; C a is the capacity of link a; V a is the flow on link a, a A; v is the vecto of all link flows; q ij is the existing tip demand between O-D pai ij; q is the vecto of all O-D demand; f ij is the flow on oute, R, between O-D pai ij associated with q ij ; f isthevectoofflowsofallouteinr; δ ij a, is the link-oute
Jounal of Advanced Tanspotation 3 Total tavel demand Max demand Capacity flexibility Robust netwok capacity Netwok capacities Existing demand Existing demands with diffeent pattens Possible ange of pattens Patten i Patten k Patten j Paametes Figue 1: Robust solution of netwok capacities with diffeent demand patten. incidence indicato: 1 if link a is on oute between O-D pai ij and 0 othewise; t a (V a ) is the tavel cost function fo link a. In the above model, the uppe-level model maximizes the O-D matix multiplie without violating the capacity constaints (2) fo evey individual link. The paamete q gives thepescibedo-dtaveldemandinthenetwok,which can be obtained accoding to the cuent tip demand o a pedicted demand patten accodingly. Route choice behavio and congestion effect ae consideed in the use equilibium (UE) model as the lowe-level model in (3) (6). Geneally, othe taffic assignment methods, such as stochastic use equilibium(sue)model,canbeusedinplaceoftheabove deteministic UE model as equied [10]. The esult of the eseve capacity model which is consideed may undeestimate the capacity of the passenge netwok, because only the existing O-D demand patten that is moe conguous with the netwok topology would achieve a highe value of netwok capacity [16]. Basically, the eseve capacity depends on the initial O-D demand pattens and oute choice behavio of the uses. Given the lowe-level taffic assignment method, the existing O-D demand should be the only deteminant to the esult of the above model. It meansthatifthegiveno-dmatixisnotconsistentwith the netwok, the eseve capacity model will poduce a esult having a low level of maximum demand. Othewise, if the O-D patten is detemined accoding to the netwok spatial stuctue, the tavel demand can gow to a vey high amount. Diectly applying the esult of the eseve capacity may have the following poblems. (i) It is had to decide an exact existing (o pedetemined) O-D matix, because the eal tavel demand patten is changing at diffeent hous evey day and diffeent days evey week. Also, it is still vey difficult to obtain the full data of the O-D demands coveing many diffeent hous. (ii) In eal-wold applications, decision-makes tend to be isk avese and may be moe concened with the wost cases. Using only a few situations oftheo-ddemandpattenmaynotpovideaobust answe to the netwok capacity estimation. Convesely, as long as the system pefomance eaches an acceptable level, it does not matte how much it changes above that level. Thus, it may be moe desiable to have an optimization esult that pefoms bette in the wost case. When estimating the capacity of tanspotation systems, decision-makes ae not only concened with the exteme esults that the total tips can be allocated to a tanspotation netwok but also need to evaluate the unknown situations esulted fom the fluctuation of the tavel demand. Thus, to measue the ability of tanspotation netwoks that can deal with the vaiation of tavel demand, Chen and Kasikitwiwat [16] discussed the concept of the netwok capacity flexibility using thee typical netwok capacity models. The netwok capacity flexibility is defined as the ability of a tanspot system to accommodate changes in taffic demand while maintaining a satisfactoy level of pefomance [16, 17]. In this study, integated with the uncetainties fom the existing demand in tanspotation netwoks, the netwok capacity flexibility is futhe illustated in Figue 1. On the basis of this, the obust estimation of netwok capacity is defined as the maximum tavel demand can be allocated to a tanspotation netwok when satisfying all the possibilities of the uncetain changes in the quantitative and spatial demand patten. The obust value of the netwok capacity is also illustated in Figue 1. In this study, we extended the eseve capacity model by consideing the existing O-D demands as uncetain paametes within a cetain bounded egion. Robust solutions to the netwok capacity can be conducted using the obust optimization. We utilize the classical eseve capacity model to conduct the obust netwok capacity fo two easons: (i) the esevecapacityiseasytosolve,andtheo-dtaveldemand is allowed eithe inceasing o deceasing by applying an O-D matix multiplie geate than one o less than one; (ii) as the existing O-D matix is extended to be an uncetain paamete in the eseve capacity model, the O-D distibution is no
4 Jounal of Advanced Tanspotation longe fixed but a vaiable patten within some ange given by the uncetainty set. 3. Robust Netwok Capacity Estimation unde Demand Uncetainty In this section, we assume that the pescibed O-D tip demand is unknown but bounded within an uncetainty set Q. Mathematically, the uncetainty set should be closed and convex. In pactice, the set of uncetain demand should be deived based on the tanspotation plannes knowledge on the uncetainty associated with both the cuent and futue O-D tavel demand. Nevetheless, it is vey difficult to obtain the exact pobability distibution of the tip demand between all O-D pais. If the andom demand follows a continuous distibution defined fom zeo to infinity, fo example, nomal distibution [12], this would equie an infinity capacity to meet all possible demand ealizations. Thus, the uncetainty set of tavel demand is defined as a bounded egion, typically utilizing the highest tavel demand, q U ij, and the lowest demand, ql ij,foeacho-dpai.usingthe bounded uncetainty egion, the events with low pobabilities can be excluded, and then the obust optimization would not povide ovely consevative esults. In this study, thee typical uncetainty sets wee constucted fo the existing tavel demands. (1) Inteval Constaint [13]. The tavel demand between each O-D pai which is assumed vaies independently within a given inteval of Q ij = [q L ij,qu ij ].Theintevalcouldbethe confidence inteval of an estimated demand obtained fom asuveyobyusingano-destimationmodel.without additional estaints, the whole uncetainty set will be a box centeed at the aveage tavel demand. In this case, it is simple to set q ij fl q U ij fo all O-D pais and solve the esulting eseve capacity poblem. The capacity value would be estimated in the wost case, which is too consevative. In eality, it is neve possible that the demand fo all O-D pais eaches thei estimated uppe bound at the same time. The tavel demand patten is always fluctuant among the O-D pais in diffeent diections. Theefoe, it is moe easonable to set an uppe limit to the summation of the existing tavel demand in netwok, that is, i,j q ij D.TheuppelimitsD could be estimated by using the maximum tip volume investigated fo the existing tavel demand. (2) Ellipsoid [18]. An ellipsoidal set is geneally defined as follows: Q= { q ij q ij { q ( i,j (1/2) (q { ij U ql ij )) 2 θ 2} }, (7) } whee q ij = (q U ij +ql ij )/2, the aveage O-D demand; θ is a paamete that eflects decision-makes attitudes to isk; and lage θ indicatesthatitismoeadvesetoisk.thevalueof θ is fom zeo to W, whee W denotes the numbe of O-D pais. When θ = 1, the uncetainty egion is the lagest ellipsoid contained in the box egion Q={q q L ij q ij q U ij, i, j}. (3) Polyhedon. Thepolyhedonisasetofafinitenumbe of linea equalities and inequalities that estains the tavel demand. It is a genealized fom of the box uncetainty set. Fo example, Sun et al. [19] constucted the following polyhedon egion fo uncetain O-D demand: Q { = q { { q 0 ij γq0 ij q ij q 0 ij +γq0 ij, γ [0, 1], i,j i j q ij = i q ij = j q 0 ij, j q 0 ij, i }, whee q 0 ij isthenominalvalueofthetaveldemandfoo- Dpaiij. It may choose the mean value of the inteval constaints o an obseved esult fom a suvey. This uncetainty set allows the demand patten vaying entiely aound its nominal value and involves the implicit possible inteactions among O-D demands. Theefoe, the ovely consevative esults may be avoided. The last two sets of constaints equie that the uncetain tavel demands meet the consevation condition with the nominal demand matix at the zonal poduction and attaction. Note that the shape of uncetainty set affects the efficiency and obustness of netwok capacity value. Ben-Tal and Nemiovski [14] suggested applying the min max optimization model. Once the uncetainty set of the tavel demand, Q, is detemined, the min max model will find a obust solution that toleates changes in tavel demand up to the given bound. Using any type of the uncetainty sets, Q, the obust eseve netwok capacity (RRNC) poblem can be fomulated as follows: RRNC: max min μ q μ, whee V a (μq) solves min f s.t. a s.t. V a (μq) C a, V a t a (x) dx, 0 } } a A, such that q Q, f ij =μq ij, i I, j J, q ij Q, R ij V a = i j f ij δij a,, a A, f ij 0, i I, j J, R ij. (8) (9) (10) The above model is efeed to as the obust countepat of the oiginal eseve netwok capacity poblem. The solution of the obust countepat esults in a maximum total tavel demand scheme unde the coesponding wost-case demand patten.
Jounal of Advanced Tanspotation 5 4. Solution Algoithm A heuistic algoithm is poposed to solve the above obust optimization model. It takes a simila famewok as the pocedue pesented in [18], which is efeed to as the cutting plane algoithm to obust optimization. The algoithm involves an iteative pocedue to solve two inne optimization poblems altenately until the convegence citeion is satisfied. The algoithm is pesented as follows. Step 0 (initialization). Give the initial values of the O-D demand q (0) Q(usually q (0) ij =q L ij, i, j) andsolvethefollowing eseve netwok capacity (RNC) poblem to poduce an initial demand multiplie μ (0) : RNC: max μ μ, subject to V a (μq (n) ) C a, a A, (3) (6). Set the iteation counte n fl 0. Step 1 (diection finding). (11) Step 1.1. Solve the following inne (wost-case scenaio (WCS)) poblem with the detemined μ (n) to obtain the wost-case demand scenaio: WCS: max q (max (0, V a (μ (n) q) C a )) a subject to q Q, (10). (12) Step 1.2. Fomulate a RNC poblem with the scheme of existing demand q (n) ij poduced fom the WCS poblem in Step 1.1. Solve this inne poblem to find a seach diection ofthemaximumdemandmultiplie μ (n). Step 2 (move). Compute μ (n) fl μ (n 1) +α (n) ( μ (n) μ (n 1) ), whee α (n) is the step length. In this study, the step length is chosen as α (n) = 1/n,whichisusedinthemethodof successive aveages. Step 3 (convegence check). If the objective value of the WCS poblem y > εo n eaches the maximum iteations, then stop, whee thee is a pedetemined convegence citeion. Othewise, denote the solution of the WCS poblem as q (n+1) ij, and go to Step 1; set n fl n+1. Remak 1. In the above steps, the WCS poblem is fomulated to find a solution of q Q,inwhichcasethetafficflows on all links which exceed thei capacities the most. If q is an optimal solution to the RRNC, the coesponding optimal objectivevalueofthewcspoblemmustbezeo.othewise, an impoved solution may be obtained by solving the RNC poblems in Step 1.2 which is a elaxation of the RRNC poblems with a specific demand patten. In the pocess of the algoithm, each of these elaxed RRNC poblems can appoximate the oiginal RRNC bette than its pedecessos. Although it is still difficult in pactice to find a global optimum of the elaxed RRNC and WSC, Yin et al. showed that the cutting plane algoithm is effective in poviding a good solution to the obust optimization poblem [18]. The elaxed RRNC poblems and the WSC poblems ae solved by the sensitivity analysis based (SAB) algoithm [20]. Remak 2. The second inne poblem is a standad RNC model when the existing O-D demand is detemined. The RNC can be solved efficiently by applying the SAB algoithm [3]. The SAB algoithm locally appoximates the oiginal bilevel poblem as a single-level optimization by using fistode Taylo expansion. The deivatives of lowe-level decision vaiables with espect to uppe-level ones ae utilized fo the linea appoximation. The deivatives can be conducted fom the sensitivity analysis of the lowe-level model. In this study, we used the estiction appoach fo the sensitivity analysis of the lowe-level UE model. The estiction appoach was poposed by Tobin and Fiesz [21] and then coected by Yang and Bell [22] fo its flaws on selecting the nondegeneate exteme point. One can also efe to Du et al. [23] fo the details of this appoach. In this section, some necessay esults ae pesent without poof. Fo the eseve capacity model, the link flows in uppelevel, V a (μq), ae epesented as an implicit function of the O-D matix multiplie μ as constaint (2) shows. Using the fist-ode Taylo expansion, it can be appoximated as V a (μq) V a (μ q)+[ V a (μq) μ μ=μ ](μ μ ), a A, (13) whee μ is the given solution of the O-D demand multiplie at the cuent iteation of SAB algoithm. Fom the esults in Tobin and Fiesz [21], the deivatives of the oute flows, f, with espect to the O-D demand multiplie μ, ae deived as follows: [ μf 0 0T μ π ] = [Δ k t (k,0)δ 0 Λ 0T Λ 0 O ] 1 [ O μ (μq) ], (14) whee the supescipt 0 denotes that the vaiables o maticesaeonlyassociatedwiththeestictingsubpoblem deived by the estiction appoach (applying the coection in Yang and Bell [22]) and the supescipt T epesentsthe tansposed matix. Othe notations ae defined as follows: Δ=[δ ij a, ] is the link-oute incidence matix; Λ=[λ ij ] is the O-D-oute incidence matix, whee λ ij equals 1 if O-D pai ij is connected by oute,and 0othewise; π is the Lagangian multiplie associated with constaint (4); t(k,0) is the vecto of the tavel cost function of all links with the equilibium link flow v fo the petubation paametes at 0. Thus, the deivatives of the link flows to the multiplie ae obtained by μ k =Δ 0 μ f 0. Based on the above deivations, the SAB algoithm can be used fo solving the RNC poblem.
6 Jounal of Advanced Tanspotation Remak 3. The WCS poblem is also fomulated as a bilevel pogamming using equilibium constaints, so the SAB method can also be modified fo its solution. The implicit elationship V a (μq) is fist-ode appoximated as V a (μq) V a (μq ) + i I j J [ V a (μq) ](q q ij q ij ), ij q=q a A, (15) whee q is the solution of the existing O-D demand at the cuent iteation of SAB algoithm. The deivatives of the oute flows, f, withespectto the existing O-D demand q, ae deived fom the following equation: [ qf 0 0T q π ] = [Δ k t (k,0)δ 0 Λ 0T Λ 0 O ] 1 [ O q (μq) ]. (16) Because in this inne poblem the value of μ is fixed, the deivative q (μq) will be a diagonal matix with the value of μ on its diagonal. Then, the deivatives of v to μ ae calculated by q k =Δ 0 q f 0. Thus, at each iteation of the SAB method, the appoximating WCS poblem is max q subject to (max (0, q V a q + V a qv a q C a )) a A q Q. (17) Note that the above localized appoximation poblem is a nonlinea poblem and a numbe of optimization tools in commecial softwae packages could be used fo its solution. In this study the appoximate WCS poblem is solved using MATLAB built-in functions which wee conveted to a.net component and used in ou solution pogam in C# language. Simila to [18], we andomly geneate 100 vectos of q fom the uncetainty egion Q. Foeachq, theuseequilibium poblem with μ (n) is solved. Let q be the andom q with themaximumobjectivevalueofthewcspoblem,and denote the objective value as y. If y >ε,thensetq (n+1) fl q. Othewise, use the MATLAB functions to solve the appoximate WCS poblem with q as an initial solution. If it gives a solution with an objective lage than ε,thesolutionis used as q (n+1) fo the next iteation; othewise, the algoithm is teminated with an optimal solution q. 5. Computational Expeiments 5.1. Expeiment 1: Nguyen-Dupuis Netwok. Computational expeiments ae pesented in this section to illustate the esults of the obust netwok capacity model. The example is based on a oad netwok which is adopted fom Nguyen and Dupuis [24] as Figue 2 shows. It consists of 13 nodes, 19 links, and 4 O-D pais. The nominal value of the existing tavel demand is given by the O-D matix in Figue 2 (denoted by q 0 ). The chaacteistics of the links ae listed in Table 1. The Table 1: Link chaacteistics of the example netwok. Link numbe a Fee-flow time t 0 a Capacity C a 1 7.0 800 2 9.0 400 3 9.0 200 4 12.0 800 5 3.0 350 6 9.0 400 7 5.0 800 8 13.0 250 9 5.0 250 10 9.0 300 11 9.0 550 12 10.0 550 13 9.0 600 14 6.0 700 15 9.0 500 16 8.0 300 17 7.0 200 18 14.0 400 19 11.0 600 1 1 2 12 3 5 7 9 4 5 6 7 8 4 6 Existing O-D demand: D O 2 3 1 400 800 4 600 200 12 14 15 9 10 11 2 13 13 17 18 8 10 19 Figue 2: An example netwok. 3 16 5 5 6 Node numbe 11 Link numbe Bueau of Public Road link pefomance function was used in the expeiments: t a (V a ) =t 0 a [1 + 0.15 ( V 4 a ) ], (18) C a whee t 0 a is the fee-flow tavel time fo link a. We applied the poposed appoach fo obust netwok spae capacity estimation with the thee typical uncetainty sets which ae descibed in this pape. Assume that the intevals fo the O-D tavel demands ae [300, 600], [600, 1050], [400, 950],and[100, 375] fo O-D pais 1-2, 1-3, 4-2, and 4-3, espectively.
Jounal of Advanced Tanspotation 7 Table 2: Robust eseve capacities with ellipsoidal egion and polyhedal egion. Ellipsoid Polyhedon Paamete θ Max μ Robust capacity Paamete γ Max μ Robust capacity 0.0 0.1795 392.63 0.0 0.1944 388.89 0.2 0.1727 391.03 0.25 0.1892 378.38 0.5 0.1633 388.71 0.5 0.1843 368.54 1.0 0.1498 385.46 0.75 0.1795 359.01 2.0 0.1286 376.68 1.0 0.1753 350.54 Table 3: Robust eseve capacity estimations and the pefomances. Netwok: Nguyen-Dupuis Nominal Consevative Robust-e Robust-p Reseve capacity estimation 388.89 363.42 385.46 368.54 Tavel demand multiplie 0.1944 0.1346 0.1498 0.1843 Pecentage of meeting capacity constaints (%) 17.2 100.0 98.4 72.4 Pecentage of above-obust-estimation (%) 51.8 100.0 54.6 98.8 Tavel demand 450.00 430.00 410.00 390.00 370.00 350.00 0.1 1400 1600 1800 2000 2200 2400 2600 2800 3000 Maximal total existing demand Netwok capacity Maximum multiplie 0.3 0.25 0.2 0.15 Figue 3: Robust eseve capacities with inteval constaints unde diffeent maximal total existing demand. (1) Inteval Region. Theoetically, unde the inteval constains fo each O-D pai, the total of the existing O-D demand can vay fom 1400 to 2975, which coves a wide ange. Howeve, accoding to ou pactical expeience, the tavel demands between the O-D pais may not each thei maximum simultaneously. Thus, an additional constaint i,j q ij D was intoduced to give an uppe bound of the total existing demand. To be consistent with the inteval constaints, D was set to change fom 1400 to 3000. Figue 3 illustates how uncetainty of the existing tavel demand affects the eseve capacity esults of the example netwok. The eseve capacity value is calculated by μ i,j q ij,whee μ is the obust solution of the RRNC poblem and q ij is the coesponding tavel demand patten. Note that the eseve capacity value and the μ decease synchonously along with the gowth of D. BecausewhenDis inceased, the vecto of tavel demand patten, q, willhavemoespacetochange its spatial distibution, and this makes the existing demand easie to achive a demand patten whose coesponding multiplie can each a smalle value. Besides, when D exceeds O-D matix multiplie 2700, the value of μ cannot get wose. At this value μ = 0.1346, the coesponding existing tavel demand is q = (600, 1050, 950, 100), which poduces a smallest eseve capacity value 363.42 (efeed to as the consevative solution intheestpatofthissection).futhemoe,fotheobust solutions when D>2700, the change of q 43 will have no effect on the value of μ. Hee, we use the lowe bound of q 43 to conduct the wost-case esults of the eseve capacity. Accoding to the popeties of the obust optimization, the solutions of the existing O-D demand may not be unique, so no specific solution of the existing demand is pesented in this pape. (2) Ellipsoidal Region. Fotheellipsoidaluncetaintyset,the paamete θ issettobe0,0.2,0.5,1.0,and2.0.thecente of the ellipsoid is decided by the bounday fo each O- D demand (not the nominal value fo this example). The computational esults in Table 2 show that the obust solution to μ changes between 0.1795 and 0.1286 with a descending tend, as well as the coesponding netwok capacity value. (3) Polyhedal Region. Fo the obust netwok capacity estimation with the polyhedal set descibed in pevious section, the obust esults at γ = 0, 0.25, 0.5, 0.75, and 1.0 ae computed and pesented in Table 2. The nominal demand (given in Figue 2) is used in this type of uncetainty. The value of μ declines fom 0.1944 to 0.1753. The same tend has been obseved on the obust values of netwok capacity, which vaies fom 388.89 to 350.54. Table 3 epots the solutions of the eseve capacity value fom two RNC poblems with q 0 = (400, 800, 600, 200) and q = (600, 1050, 950, 100) sepaately and two RRNC poblems with Q defined in the ellipsoid egion (θ = 1.0) and the polyhedal egion (γ = 0.5). We efe to the fist two solutions as nominal and consevative estimation, espectively, and thelattetwoas obust-e and obust-p sepaately.fothe esults, the obust solution fo the O-D demand multiplie indicates that all the values of the existing demand within the uncetainty set can be applied by a multiplie lage than this obust value without violating the capacity constaints. A
8 Jounal of Advanced Tanspotation Table 4: Netwok capacity estimations on Sioux-Falls. Netwok: Sioux-Falls Nominal Consevative Robust-e Robust-p Reseve capacity estimation 209,029.1 195,065.8 201,366.7 176,850.3 Tavel demand multiplie 0.5797 0.4291 0.5588 0.4904 coesponding maximum tavel demand is poduced as the obust netwok capacity solution fo the same uncetainty set.sincethesamemaximumtaveldemandmaypoduceby diffeent combinations of the multiplie and feasible existing demand, the poposed algoithm only finds the smallest value of the maximum multiplie. Besides, if the shape of the uncetainty set changes, diffeent solutions fo the obust multiplie and maximum demand will be poduced. Table 3 shows these featues. To evaluate the fou estimations of the netwok capacity, we andomly geneate 500 samples of O-D demand as the possible ealizations of the existing tavel demand patten. The samples ae fom the nomal distibution, N(q ij, 0.25q 2 ij ), within its inteval [q L ij,qu ij ] foallo-dpais.then,foeach sample, one has the following. Fistly, the use equilibium assignment associated with each eseve netwok capacity estimation (i.e., the lagest μ) is computed. The numbe of failues is counted wheneve the link flow exceeds its capacity at each sample. In the end ofthesimulation,thesuccessfulatesaecomputedas[1 totalnumbeoffailues/(numbeofsamples numbe of links)]. The esults ae efeed to as pecentage of meeting capacity constaints in Table 3. Obviously, the consevative estimation gives the wost value of the netwok capacity; the nominal estimation poduces a medium level of esult. By compaison, the obust esults fom the ellipsoid and polyhedon ae moe easonable. Note that all samples geneated in the above simulations belong to a box egion compised of the intevals fo O-D demands. The afoementioned uncetainty setsaeusedaseplacementsoftheboxegions,soasto pevent ovely consevative obust esolutions. The esults in Table 3 indicate that the ellipsoid appoximate the box egion bette than the polyhedon, because moe andom samples will not exceed the netwok capacity afte they ae scaled by the obust multiplie μ = 0.1498. On the othe hand, although μ = 0.1843 is the wost case in the polyhedal set, when the bounday is extended to the box egion, only 72.4% of the andom samples can be coveed. Futhemoe, consideing that the demand multiplie in essence is a elative value, the eseve capacity esults ae deived. Theefoe, the eseve capacity poblem with the evey andom existing O-D demand is solved in ou test. Fo each estimation value of the eseve capacity: the numbe of successes is counted wheneve the eseve capacity value of the sample exceeds the obust capacity estimation. Consequently, the successful ates ae computed as [total numbe of successes/numbe of samples]. These esults ae also pesented in Table 3 as pecentage of above-obustestimation. Fom this aspect, the polyhedon povides a moe obust estimation of the netwok capacity, which can be met by most ealizations of the existing tavel demand patten (98.8% vesus 54.6% compaed with the ellipsoid). We may suppose that the wost cases of netwok capacity values exist in the cone aea of the box-shaped uncetain egion, whee it is not easy to be coveed by the coesponding ellipsoidal set. Consequently, the choice of the obust esults depends onthedecision-makes attitudetoiskandtheusageof the obust esults. Fom the computational pespective, the obust optimizations with polyhedal uncetainty sets have moeadvantages.moeeffectivesolutionmethodscouldbe developed in futue studies. 5.2. Expeiment 2: Sioux-Falls Netwok. Expeiments ae futhe pesented on the Sioux-Falls netwok [25]. The netwok contains 24 nodes, 76 links, and 528 O-D pais. The chaacteistics of the links and tavel demands ae also povided in Ba-Gea [25]. In this expeiment, we used the default values ofthetaveldemandinasthenominalvalueoftheexisting demand in the netwok. If q ij 1000, then the demand fo O- Dpaiij is uncetain and its uppe limit and lowe limit ae set to q L ij = 0.5q ij and q U ij = 1.5q ij, sepaately. Hence, thee ae 104 O-D pais with uncetain existing demands. Accoding to the pevious section, fou estimation methods ae employed to evaluate the netwok capacity of the Sioux-Falls. Table 4 epots the solutions of the eseve capacity fom the fou estimation methods. Note that the obust-e solution gives a modeate obust esult compaed to the othes. The obust-p solution povides the lowest estimation of netwok capacity. One may use this lowest value as the wost-case pefomance that the netwok can seve the tavel demand. Besides, note that the lowest estimation of the netwok capacity was not deived fom the consevative solutions, in which the uncetain O-D demands ae set to its uppe limits. Although the consevative solution is coesponding to the lowest demand multiplie, it may not eflect the most unfavoablesituationwhichispossibletobeesultedfomthe changes of the netwok demands. We selected the consevative, obust-e, and obust-p solutions to futhe inspect the link flow pattens at the maximum tavel demand situations (i.e., the eseve capacity). The link flow pattens ae shown in Figues 4(a), 4(b), and 4(c). The width of the line indicates the taffic volume though the link. The ed lines show the links whose V/C (volume/capacity) atio is geate than 0.9, and the black lines denote the links with zeo flow. As a efeence, we also andomly geneate 500 samples of O-D demand which ae fom the nomal distibution, N(q ij, 0.25q 2 ij ),withinits inteval [q L ij,qu ij ], i, j. Fo each sample, the eseve capacity poblem is solved. Then, at each link, the highly satuated numbe is counted wheneve its V/C atio is geate than
Jounal of Advanced Tanspotation 9 (a) Consevative solution (b) Robust solution with ellipsoid (θ =1.0) (c) Robust solution with polyhedon (γ =0.5) (d) Simulation esults Figue 4: Link flow patten associated with diffeent maximum tavel demands (a, b, and c) and pobabilities that links ae congested (d). 0.9. In consequence, the link satuation ates ae computed as [total the satuated numbe/numbe of samples]. The esults aealsoshowninfigue4(d).thewidthofthelineindicates the satuation ate of the link. The ed lines indicate links with satuation ates geate than 50%. The oange lines denote links with satuation ates lowe than 50% but geate than zeo.theblacklinesmeanthelinkshavenochancetobe blocked. Fom Figue 4, we conclude that the solution of the obust-e will poduce a link flow patten most likely to be ealized when the netwok eaches its capacity (compaed to the simulation esults). The obust-p solution povides an exteme esult that the netwok is congested only because of a vey few links. These links could be consideed as the most citical links which estict the capacity of the entie oad netwok. By contast, the consevative solution seems not to fit the simulation esults vey well. In conclusion, the obust estimation to the netwok capacity poblem could be much moe pactical than the esults fom any specific scenaio. 6. Conclusions and Pespectives In this study, a obust netwok capacity model with uncetain demand has been poposed. The obust optimization is fomulated using the min max model with a bounded uncetainty set of the existing O-D tavel demands. With the uncetainty set, the low-pobability ealizations of the tavel demand patten ae excluded, and thus the obust model can poduce a pope estimation of netwok capacity which can be achieved with a lage pobability. Then, a heuistic algoithm has been poposed fo the poposed obust model. It solves two inne poblems iteatively: one is the wost-case scenaio poblem; and the othe is the elaxed obust optimization, namely, the standad eseve capacity model. At each iteation, the cutting plane method has been adopted to geneate the wost-case demand scenaio, and the sensitivity analysis based appoach has been developed fo the solution of the wost-case model and the eseve capacity model. The validity and pefomance of the poposed
10 Jounal of Advanced Tanspotation obust model have been demonstated in the computational expeiments. Diffeent esults unde thee typical uncetainty sets, say inteval, ellipsoidal, and polyhedal egion, have been conducted and compaed. The inteval set is simple buteasytopoducetooconsevativeesults;theellipsoidal set is a good appoximation to the uncetain egion and poduces esults with modeate obustness, but its solution is moe complicated due to the nonlinea constaints; the polyhedal set is consideed if a high level of obustness is equied, and its linea fomulation makes the obust model easie to solve. Futhemoe, by conducting computational expeiments on the Sioux-Falls netwok, obust solutions shown can povide moe pactical esults of the link flow pattens. In applications, these uncetainty sets and thei paametes should be selected accoding to the desied level of obustness. The obust model based on the eseve capacity model has been poposed and exploed in this study. Futue eseaches should focus on moe efficient solution appoaches fo the obust poblem with the min max model. The expeiments on lage-scale netwoks ae also needed. Besides, the demand uncetainties existing in othe netwok capacity models ae also expected to be detected and discussed. Alteative taffic assignment model, such as the stochastic use equilibium, could also be discussed fo the netwok capacity poblems. The obust solution of netwok capacity gives a lowe bound to the possible schemes of the maximum demand in a tanspotation netwok. These possibilities constitute a ange wheetheobustsolutioncanbemostlikelytobeeached in eality. Theefoe, the obust solution to netwok capacity poblems needs to eceive moe attentions in tanspotation planning applications. Disclosue An initial vesion of this pape was pesented at the Tanspotation Reseach Boad (TRB) 96th Annual Meeting. It theefoe also appeas in the poceedings of the TRB 96th Annual Meeting Compendium of Papes. 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