2.1 Conventional Analysis and Quick Response Models

Transcription

1 2 LITERATURE REVIEW This chpter presents survey of theories, lgorithms, techniques nd methods used for estimting O-D mtrices. The literture review begins with discussion of generl historicl developments of the subject. A vriety of models for estblishing O-D trip tbles bsed on trffic volume re then ddressed. The Equilibrium-bsed model nd Liner Progrmming (LP) model, on which this reserch is bsed, re then studied in some detil. A short summry is provided in the lst section tht evlutes the different models. Typiclly, trnsporttion models cn be divided into dynmic nd sttic stedy-stte representtions. The results on sttic O-D trip tbles re only considered in this survey, since this is the emphsis of the present reserch effort. 2.1 Conventionl Anlysis nd Quic Response Models Before the 1970s, O-D trip tbles were obtined vi sttisticl surveys, such s home interviews, license plte surveys nd rodside surveys. The methods tht use such survey dt to determine rel trip distributions re now clled conventionl nlysis. The first lrge-scle cordon count (O-D tble) ws conducted in Chicgo in 1916 (Es 1993). Prior to World Wr II, informtion on the distribution of urbn trffic ws obtined using rodside interviews. Since surveys were conducted through smpling, it is impossible to determine the rel trip informtion. Furthermore, with the evolution of society nd with rpid chnges in trnsporttion demnds, these surveys becme hrder to perform, nd expensive with respective to time, mnpower, money nd effort. Another drwbc of conventionl nlysis is tht, s the lnd-use chnges, these dt soon become out-dted. In ddition, in most cses, n ssignment of this mtrix to the networ cnnot reproduce observed flows. Three min types of models re considered in the conventionl nlysis: Frtr models, 2. Literture Review 6

2 opportunity models nd grvity models. These re evluted by Es (1993), nd the reder is referred to this pper for detiled illustrtion of chrcteristics, dvntges nd limittions of these models. The emphsis on trnsporttion system mngement in the erly 1970s incresed the need for studying smll urbn res in detil. Some cheper nd quicerresponse theory nd methods for synthesizing trip tbles from more conveniently vilble informtion hve been developed since then. According to the purpose of these models, they cn be clssified into the following subgroups: Ntionl Coopertive Highwy Reserch Progrm (NCHRP) Simplified Techniques, Trffic Count-Bsed Models, Self-Clibrting Grvity Models, Prtil Mtrix Techniques, models using GIS dt, Heuristic Methods nd Fcility Forecsting Techniques. There re lso some specil ppliction models, for exmple, freewy trip distribution, pedestrin trip distribution nd specil purpose trip-distribution models. Exmples of specil purpose trip-distribution models include choice models (employing individul trvelers insted of the zones s the unit of observtion), continuous models (tht ignore the zones ltogether when the chnges in lnd-use ptterns re smll), simultneous models (tht simultneously nlyze trip distributions nd other plnning steps). We do not discuss the differences mong these subgroups in detil. The interested reder is recommended to refer to the review ppers by Es (1993b), Sivnndn (1991), nd Sivnndn et l. (1996) nd the references listed therein. For review nd evlution of some microcomputer progrms in quic-response nlysis, the reder is referred to Es s pper. 2.2 Trffic Count-Bsed Models In relity, the problem on given networ is dynmic (time-dependent) system. The nlyticl trffic flow process on such networs cn be simplified to study some density function tht stisfies certin differentil or difference equtions of the continution type (Lyrintzis et l. 1994, Zhng et l. 1995), or cn be treted by 2. Literture Review 7

3 dynmic progrmming (DP) pproches (Jnson 1995). In prctice, however, this dynmic system is very hrd to study to obtin desired solution. In order to derive O-D trip tbles from count-bsed informtion, the trffic flow is usully considered in its sttic (time-independent) or sttionry stte. There is some evidence to support such n ide. First, the study of O-D trip tbles is motivted by the purpose of reducing the congestion problem. In the cse of congestion, however, the distribution of trffic flow is dominted by the user equilibrium principle discussed in the next section. Moreover, it cn be shown tht the flow pttern is independent of time if the system stisfies n equilibrium condition. Second, if the system is considered during very short time period, it cn be pproximted s sttic system. In ddition, the sttionry tretment of dynmic systems cn be thought s first step towrd problem simplifiction, nd frequently, this revels more detiled structure of the problem. Among ll types of esily derived dt, trffic counts (lin volumes in networ) perhps contins the most importnt informtion bout O-D distributions. A vriety of nlytic models hve been developed to estblish O-D trip tbles bsed on trffic counts long with other informtion. Bsed on the theory (or principle, or hypothesis) used, these models my be divided into the following types (Es 1993b, CTR 1995). I. Grvity-Bsed Models These models re sometimes clled Prmeter Clibrtion models, nd represent the originl ide of estblishing trip distributions. In these models, the entries of the O-D mtrix re ssumed to be function of the trffic count nd other prmeters. Regression techniques nd the flow conservtion lw re pplied to clibrte the prmeters such tht the differences between observed volumes nd estblished volumes re minimized. The models re divided into liner (Low 1972, Holm et l. 1976, Gudry nd Lmrre 1978, Smith nd McFrlne 1978) nd nonliner (Rolillhrd 1975, Hogbg 1976) regression models. 2. Literture Review 8

4 II. Equilibrium Models These models re bsed on the principle of user optimiztion of trffic flow, clled the Equilibrium Principle or Wrdrop s Principle (Wrdrop 1952). These include LINKOD (Nguyen 1977-b, Gur 1980), SMALD (Kurth et l. 1979), nd LP (Sivnndn 1991, Sherli et l b). The equilibrium-bsed models will be discussed in detil in the following sections. III. Entropy Models Minimum Informtion nd Mximum Entropy models re included in this group nd cn be converted to type of grvity models. In these models, the probbility of prticulr trip distribution occurring is ssumed to be proportionl to the number of the sttes (entropy or disorder) of the system. The derived O-D tble is purported to be the most liely one tht is consistent with informtion such s length nd free speed of the lins contined in the lin flows. The pioneers of these models, ME2, re Wilumsen (1978) nd Vn Zuylen (1978, 1979). Mny improvements nd combintions with other theory hve been conducted since then, nd gret del of testing on these models hs been performed. These results hve been summrized in the review ppers of Es (1993b) nd CTR (1995). IV. Sttisticl Models These models te into ccount inccurcies on the observed O-D flows, row nd column sums nd trffic counts. This group includes the constrined generlized lest squres (CGLS) model (McNeil 1983), nd constrined mximum-lielihood (CML) models (Gev 1983, Spiess 1987, Wlting nd Mher 1988, Wlting nd Grey 1991). Another model, clled MEUSE, stnding for Mtrix Estimtion Using Structure Explicitly (Bierlire nd Toint 1995), which uses both historic dt nd pring dt s input, cn be prtilly included in this subgroup. Sttisticl models re not s populr in prctice s compred with equilibrium nd entropy models. 2. Literture Review 9

5 V. Neurl Networ Models Muller nd Reinbrdt (1990) introduced the neurl networ pproch to determine O- D trip tble from trffic counts. In loose sense, this pproch is bsed on the concepts derived from reserch into the nture of the brin. The procedure in this pproch includes lerning nd optimiztion components. The model my be mthemticlly described s directed grph with three chrcteristics. 1) A stte of vrible ssocited with ech node; 2) weight ssigned to ech lin; nd 3) trnsfer function defined for determining the stte of ech node s function of its bis nd weights of its incoming lins. Yng, Aiymm nd Ssi (1992) dopted feed-forwrd neurl networ for synthesizing O-D flow for four-wy intersection nd short freewy segment. Chin, Hwng nd Pei (1994) described neurl networ model for generting O-D informtion from flow volumes. There still exists need for further testing these models. VI. Fuzzy Weight Models Insted of the ll or nothing ssumption mde in most models, Fuzzy Weight pproches pply some ind of fuzziness to the lin dt (Xu nd Chn 1992-b). Fuzziness indictes probbility, but is quite different in nture. Different types of fuzziness hve been tested on the networ of the Estern Highwy Corridor. The model is reltively new, nd dditionl cse studies nd experimenttion re recommended for evlution. 2. Literture Review 10

6 2.3 Equilibrium-Bsed Models The User Equilibrium Principle, lso nown s Wrdrop s (first) principle or useroptiml ssignment principle, ws originlly used to guide the trffic flow ssignment process. It requires tht ll routes hving positive flows between ny O-D pir should hve equl trffic cost, nd this cost must not exceed the trffic cost on ny other unused route between this O-D pir. Mthemticlly, estimting the O-D trip tbles from lin volumes cn be thought of s n inverse problem of trnsporttion ssignment. Therefore, the equilibrium-bsed O-D mtrix estimtion problem my be described s determining n O-D mtrix such tht, when this mtrix is user-optimlly ssigned to the networ, it reproduces the observed O-D trvel times. Since the correspondence between equilibrium lin flow ptterns nd equilibrium O - D trvel times for the stndrd equilibrium problem is one-to-one (Yng et l. 1994), it consequently reproduces the observed lin flows. Nguyen (1977) exploits Wrdrop s user-equilibrium principle for route choice nd formultes deterministic networ equilibrium pproch. He shows tht, if the set of observed lin flows is t equilibrium, the solution mtrix cn be found by solving the following optimiztion problem with respect to vribles v nd t Nguyen s Model 0 (NM0) Min F( v, t) = v, t v 0 A f ( x) dx i j u t [2.1] subject to: r h = t, i, j [2.1b] r r d h = v,, r r [2.1c] 2. Literture Review 11

7 i j u t = f ( v ) v [2.1d] h 0, t 0. r Here f (x) is the volume-dely function for lin, t is the trvel demnd between origin i nd destintion j, v is the ssigned flow on lin, r connecting origin i nd destintion j, h r is the flow on route d, r 1 = 0 if lin is on route r between i nd otherwise, j v is the observed lin flow, nd u, the observed inter-zonl ccessibility (trvel time on ny used route between zone i nd zone j due to the equilibrium ssumption). As shown by Nguyen (1984), for ech lin A, the problem NM0 hs unique optiml solution v, becuse the objective function is strict convex in v. The optiml solution v is consistent with the observed lin flow v, A, if the observed lin flow pttern is t equilibrium. However, the problem NM0 my not hve unique O-D mtrix solution becuse tht it is not strictly convex in the vrible t (Sheffi 1985). Nmely, there re mny possible O-D mtrices tht produce the sme set of observed O-D trvel times of observed lin flows when ssigned to the networ. Nguyen s theoreticl model ws opertionlized during the course of Federl Highwy Administrtion Project in which the LINKOD system of models were developed (Turnquist nd Gur 1979; Gur et l. 1980). The LINKOD system is comprised of two mjor components: SMALD nd ODLINK. SMALD (Kurth et t. 1979), is smll re trip distribution model tht determines trip tble for subre. This tble is used to overcome the under-specifiction problem of Nguyen s formultion nd hs been extensively tested nd verified by Hn, Dowling, Sullivn nd My (1981), Hn nd Sullivn (1983), nd Dowling nd My (1984). To obtin unique O-D mtrix, trget O-D mtrix t is ssumed to be vilble. Different 2. Literture Review 12

8 criteri hve been suggested for choosing mong ll the O-D mtrices tht produce the observed lin flows. In the LINKOD model (Gur 1980), the estimtion problem is formulted s NM1 min 2 ( t t ) [2.2] subject to t being optiml to NM0. Another model is suggested by Jornsten nd Nguyen (1980) tht chooses the most liely mtrix mong the optiml solution set by solving the following optimiztion problem, NM2 mx t t log 1 t [2.3] subject to t being optiml to NM0. In the existing literture, two requirements, nmely, to reproduce the observed O-D trvel times nd to reproduce the observed lin flows, hve been used. These two requirements re equivlent when the observed networ is in equilibrium. The equivlence reltionship, however, will not hold when the set of observed lin flows is not n equilibrium solution (Yng et l. 1994). Both NM1 nd NM2 hve bi-level progrmming structure tht poses computtionl difficulties for lrge-scle networs (Fis 1989, Fis & Boyce 1983). LeBlnc nd Frhngin (1982) suggested prtil duliztion method for solving NM1 (Sheffi 1985). The prtil duliztion method involves updting Lgrnge multiplier by itertively solving Lgrngin minimiztion problem. A similr method for solving 2. Literture Review 13

9 NM2 ws lso suggested by Jornsten nd Nguyen (1980) nd Nguyen (1984). Both of these methods require itertively solving progrm NM0, nd re computtionl demnding. Yng et l. (1992b) hve shown how to integrte existing methods such s the generlized lest squres technique with n equilibrium trffic ssignment pproch using Stcelberg leder-follower optimiztion model. An ttempt is mde for uncertinties in both the trget O-D mtrix nd in the trffic lin counts, nd heuristic solution method is proposed becuse of the inherent difficulty in solving moderte to lrge sized problems of this type. It hs been shown by Yng et l. (1994) tht Nguyen s bi-level optimiztion models cn be trnsformed into single convex progrm. When the observed lin flow pttern is in equilibrium, the originl model is demonstrted to be equivlent to reduced system of liner equtions. By exploiting the properties of the system s fesible region, simpler methods, such s lest squres technique, cn be used to obtin n O-D mtrix tht, when user-optimlly ssigned to the networ, will reproduce the observed lin flows. The models mentioned bove re bsed on improving nd modifying Nguyen s systems. Other models bsed directly on the user-optiml principle re lso developed. One of them is to pply Liner Progrmming (LP) theory to estimte O - D trip tbles from the observed lin flows (Sherli et l. 1994, Sivnndn 1991). The detiled studies on LP models re presented in the next section. 2.4 Liner Progrmming Models Liner progrmming theory nd technique hve been successfully pplied to vrious trnsporttion problems lmost since its erly beginning. A fmous exmple is given by Dntzig [1951] to dpt his simplex method to solve (Hitchcoc s) trnsporttion problem. The terminology, such s trnsporttion/ssignment problems, nd lloction problems, hve become stndrd in these contexts since then (Bzr et l. 1990). 2. Literture Review 14

10 Liner progrmming methods were first used to study O-D distributions in the 1970s (Colston 1970). The developed pproches, unfortuntely, were not successful in prctice. An lterntive liner progrmming pproch hs been proposed recently (Sivnndn 1991, Sherli et l b) bsed on non-proportionl ssignment nd user-equilibrium principle. The bsic ide nd technique used in this modeling pproch my be described s follows. Consider rod networ. Let N be the set of nodes including centroids, A be the set of corresponding directed lins or rcs, nd let OD be the set of O-D pirs tht comprise the trip tble to be estimted. Let x be the estimtion of flow originting t i nd destined for j, nd f be the observed lin flow for ech lin A = 1, L, n, represent ll the n pths between ech O-D pir OD, nd the contribution of x to the pth p for ech. Let p, x be = 1, L, n. Assume tht the lin trvel time/cost c ( v ) is (strictly) incresing function of flow v, let c = c p denote the cost on route between O-D pir ( i, for = 1, L, n, nd let c = min{ c, = 1, L, n }. Then bsed on the user-optiml principle, the following liner progrmming model my be constructed. Minimize n cˆ x OD = 1 [2.4] subject to n OD = 1 ( p ) x = f [2.4b] x 0 [2.4c] where cˆ c = M c if K 1 if K [2.4d] 2. Literture Review 15

11 nd where M > 1 1 is constnt, nd K is the observed inter-zonl ccessibility defined by K { {1, L, n } : c = c }, for ech OD. = It hs been shown by Sherli et l. tht the observed lin flow pttern is t n equilibrium if nd only if the objective vlue of [2.4] is equl to C totl = c f, the totl observed system cost. In order to te the internl inconsistencies of input dt nd the prior trip tble into ccount, some penlized terms re dded to the objective function nd the following LP nd LP(TT) models re considered, respectively. LP: Minimize n j ) OD = 1 cˆ x Me ( y y ) [2.5] subject to n ( p ) x ( y y ) = OD = 1 f [2.5b] x 0, y 0, y 0. [2.5c] LP(TT): Minimize n OD = 1 cˆ x Me ( y y ) M ( Y σ OD Y ) [2.6] subject to n ( p ) x ( y y ) = OD = 1 f [2.6b] n = 1 x ( Y Y ) = Q ( i, OD [2.6c] Here, x 0, y 0, y 0, Y 0, Y 0. [2.6d] OD OD is subset of O-D pirs for which prtil prior (trget) trip tble informtion is vilble, nd M nd M σ re some positive penlty prmeters. The 2. Literture Review 16

12 uthors investigted on how lrge these penlty coefficients should be for the purpose of chieving desired solutions. Insted of using the (stndrd) Simplex Method, Column Genertion Algorithm (CGA) is developed for solving LP(TT), in order to reduce the computtionl effort for prcticl problems. The models hve been tested on rel networ of Northern Virgini. In prctice, it is commonly the cse tht not ll of the lin volumes of the networ re vilble. The LP models hve been improved to te cre of the cse of missing volume (Sherli et l. 1994b). The ide used in this sitution is to updte the trvel time/cost by solving some liner nd nonliner progrmming subproblems itertively. Both the originl nd improved versions hve been tested nd evluted on some rel networs (Sivnndn et l. 1996). Some of these test results will be presented in Chpter Summry of Literture Review Different pproches hve been investigted for estimting O-D trip tbles in the lst severl decdes. Conventionl nlysis is very expensive in prctice. It is hrd to reproduce the observed flows using these techniques, nd the trip tbles often become outdted. As consequence, quic response models become of gret relevnce in trnsporttion plnning. Among quic response models, the one bsed on trffic counts is very populr nd pervsive. The problem is beset with gret del of complexities, nd vrious pproches hve been employed to overcome them. Grvity-bsed models require considerble dt, nd re reltively more liely to hve their results become outdted. This mes these models unttrctive. LINKOD type models incorporte the desired equilibrium ssignment concept, but their nonliner nture leds to the issue of excessive computtionl effort for deriving cceptble solutions to prcticl problems. The entropy-bsed models pose restrictions on dt, give little weight to prior informtion, nd need refinements for 2. Literture Review 17

13 incorporting the equilibrium principle. Sttisticl models te into ccount the stochstic nture of the dt nd the problem. However, they hve not been dequtely tested. In ddition, the stochstic theory used itself sometimes mes the problem more complicted for prcticl purposes. Both neurl networ nd fuzzy set pproches still need to be verified regrding their prcticl vibility. Moreover, indepth theoreticl studies re themselves needed before these pproches cn be justified for use in prctice. Liner progrmming models nd lgorithms hve been widely used in vrious pplictions, including trnsporttion nd ssignment problems. Using this pproch to estimte the O-D trip mtrix from lin volumes, however, is reltively new. The pproches hve some dvntges such s simpler formultion, nd for the cse of ll lin volumes being vilble, n estblished theory gurnteeing finite convergence. On the other hnd, the pproch pproximtes the rndom nture of the dt nd the problem, nd the resulting O-D tble often hs mny zeros becuse of the extreme point optimlity principle. (This is somewht llevited when using prior trip tble informtion.) Moreover, in the cse of missing volumes, the pproch itertively updtes the trvel cost on missing dt lins nd then minimizes the totl cost, which leds to problem of excessive computtionl effort. Therefore, there is need to improve the LP model so tht it cn te cre of inccurcies in input dt in prctice, interpret both user behvior nd user-optiml principles in resonble mnner, nd reduce the computtionl effort in generting prcticl cceptble solutions. With this motivtion, liner progrmming model nd pproch is proposed in the next chpter. 2. Literture Review 18