, pp.-20 http://dx.do.org/0.4257/t.204.2..0 Performance of a Genetc Algorthm for Solng the Mult-Obecte, Multmodal Transportaton Netork Desgn Problem Tes Brands and Erc C. an Berkum Centre for Transport Studes, Unersty of Tente Postbus 27, 7500AE Enschede, the Netherlands t.brands@utente.nl, e.c.anberkum@utente.nl Abstract The optmzaton of nfrastructure plannng n a multmodal netork s defned as a multobecte netork desgn problem, th accessblty, use of urban space by parkng, operatng defct and clmate mpact as obectes. Decson arables are the locaton of park and rde facltes, tran statons and the frequency of publc transport lnes. For a case study the Pareto set s estmated by the Non-domnated Sortng Genetc Algorthm (NSGA-II). Such a Pareto set s one specfc outcome of the optmzaton process, for a specfc alue of the parameters generaton sze, number of generatons and mutaton rate and for a specfc outcome of the Monte Carlo smulaton thn NSGA-II. Smlar ssues exst for many other metaheurstcs. Hoeer, hen appled n practce, a polcy maker desres a result that s robust for these unknon aspects of the method. In ths paper Pareto sets from arous runs of the NSGA-II algorthm are analyzed and compared. To compare the alues of the decson arables n the Pareto sets, ne methods are necessary, so these are defned and appled. The results sho that the dfferences concernng decson arables are consderably larger than the dfferences concernng obectes. Ths ndcates that the randomness of the algorthm may be a problem hen determnng the decsons to be made. Furthermore, t s concluded that aratons caused by dfferent parameters are comparable th the aratons caused by randomness thn the algorthm. Keyords: Mult-obecte optmzaton, genetc algorthm, NSGA-II, multmodal passenger transportaton netorks, Pareto set comparson. Introducton Hghly urbanzed regons n the orld noadays face ell-knon sustanablty problems n traffc systems, lke congeston, use of scarce space n ctes by ehcles and the emsson of greenhouse gases. A shft from car to publc transport (PT) modes s lkely to alleate these problems. Extenson of the PT netork may contrbute to ths shft by enhancng the qualty of PT trps. Hoeer, nestments n PT nfrastructure requre large fnancal resources. Better utlzaton of the exstng PT nfrastructure thout the need for bg nestments n large nfrastructural deelopments may be acheed by facltatng an easy transfer from prate modes (bcycle and car) to PT modes (bus, tram, metro, tran). Ths transfer can be made easer by netork deelopments that enable multmodal trps, lke openng ne Park and Rde (P&R) facltes, openng ne tran statons or changng exstng or openng ne PT serce lnes. When such transportaton netork deelopments are planned, the decson s often based on an ealuaton of a fe pre-defned scenaros. The composton of the scenaros s usually based on expert udgment, and the assessment s usually based on a mult crtera analyss. ISSN: 2287-7940 IJT Copyrght c 204 SERSC
Hoeer, there s a far chance that the scenaro that s selected as the best can stll be mproed. For ths reason applyng an optmzaton technque s preferred. In ths paper, a multobecte netork optmzaton approach s adopted, hch enables us to dentfy trade-offs beteen obectes (see also []), because the feld of transportaton plannng s, lke many other felds, a typcal example of a stuaton that noles more than one obecte [2]. Netork desgn problems (NDPs) hae receed a lot of attenton n the lterature, n many dfferent ersons. One subclass of problems s the transt netork desgn problem, hch has been studed n arous ays, as reeed by [3]. Ths ncludes greedy algorthms, eolutonary algorthms and desgn meetngs nolng expert udgments. Another subclass of problems s the unmodal road netork desgn problem, hch has also been studed n arous ays [4]. Applcatons n a multmodal context are less common, but they do exsts, for example road lnk capacty and bus routes [5] or prcng of prate and publc lnks [6]. The NDP s consdered a mult-obecte problem n only a lmted number of cases. Examples of obectes beng consdered smultaneously are trael tme and constructon costs [7], trael tme and trael tme araton [8], trael tme and CO emssons [9, 0] and socal elfare mproement, reenue generaton and equty []. Fnally, [5] smultaneously maxmze user benefts, passenger share of the bus mode, serce coerage, hle mnmzng aerage generalzed trael cost for a bus mode trp. To our knoledge [5] s the only paper combnng mult-obecte optmzaton th a multmodal NDP, consderng both ne street constructon and lane addtons / allocatons as ell as redesgn of bus routes. The focus s on the deelopment of the metaheurstcs to sole the problem: the performance of a hybrd genetc algorthm and a hybrd clonal selecton algorthm are compared, usng multple test netorks. In the loer leel, car and bus are dstngushed as separate modes. A number of papers appled metaheurstcs for solng the transt netork desgn problem, for example [2-4] use a genetc algorthm, [5] uses smulated annealng, [6] tests both scatter search as a genetc algorthm and [7] uses a metaheurstc called GRASP (Greedy Randomzed Adapte Search Procedure). The randomness n such algorthms s usually addressed by dong multple runs and presentng aerage alues, for example hen comparng arous (arants of) soluton algorthms (as s done for example n [2]). An aspect that s less addressed s the araton of the results thn one arant of a soluton algorthm, th respect to the araton of the results beteen to arants of a soluton algorthm. Another aspect s the comparson of decson arables of optmzaton outcome. To be able to compare arous results n the case of mult-obecte optmzaton, one must be able to compare Pareto sets th each other, both concernng obecte alues as ell as alues of decson arables. Frstly, the contrbuton of ths paper s to formalze methods to compare Pareto sets based on the alues of decson arables. Secondly, the mpact of randomness and parameter settngs n the applcaton of a metaheurstc lke NSGA-II on the outcome of the optmzaton process s assessed, for the case of a netork desgn problem th multmodal decson arables. Therefore, exstng ndcators to compare the obecte alues of dfferent Pareto sets are used, n addton to the nely deeloped methods to compare the alues of decson arables. Secton 2 defnes the problem n more detal and descrbes the soluton method appled. Secton 3 defnes methods to compare Pareto sets th each other, to be used n Secton 4, here the outcomes of a real orld case study are analyzed. The case study s descrbed and the alues of arous outcomes of the optmzaton process are compared th each other. Ths leads to conclusons n the fnal secton. 2 Copyrght c 204 SERSC
2. Problem Defnton 2.. B-leel Problem The transportaton netork desgn problem s often soled as a b-leel optmzaton problem, to correctly ncorporate the reacton of the transportaton system users to netork changes, as s argued by [8-20]. In our research, the netork desgn problem s regarded as a b-leel system as ell (see Fgure ). The upper leel represents the behaor of the netork authorty, optmzng system obectes. In the loer leel the traelers mnmze ther on generalzed costs (e.g., trael tme, cost), by makng nddually optmal choces n the multmodal netork, consderng arety n trael preferences among traelers. The netork desgn n the upper leel nteracts th the behaor of the traelers n the netork: the loer leel. Ths s put nto operaton by a transport model, hch assumes a stochastc user equlbrum (no drer can unlaterally change routes to mproe hs/her perceed generalzed trael costs). For any netork desgn the planner chooses, the transport model yelds a netork state (e.g., trael tmes and loads), from hch the alues of all obecte functons can be dered. The equlbrum n the loer leel s a constrant for the upper leel problem. Upper leel: Mnmzaton system obectes concernng sustanablty Netork state Loads, speeds Loer leel: User equlbrum problem: multmodal assgnment Netork desgn: Multmodal facltes 2.2. Netork and Demand Defnton Fgure. The B-leel Optmzaton Problem The multmodal transportaton netork s defned as a drected graph G, consstng of node set N, lnk set A, a lne set L and a stop set U (see Table ). For each lnk a one or more modes are defned that can traerse that lnk th a certan speed and capacty: the lnk characterstcs C. Orgns r R and destnatons s S a are a subset of N and act as orgns Copyrght c 204 SERSC 3
Table. Notaton NSGA-II MOOP NDP PT P&R IC Table 2. Abbreatons Non-domnated Sortng Genetc Algorthm (second erson) Mult-Obecte Optmzaton Problem Netork Desgn Problem Publc Transport Park and Rde Intercty (tran) 4 Copyrght c 204 SERSC
and destnatons. Total fxed transportaton demand q s stored n a matrx th sze R S. rs Furthermore, transt serce lnes l L are defned as ordered subsets A thn A and can be l stop serces or express serces. PT flos can only traerse transt serce lnes. Transt statons or stops u U are defned as a subset thn N. Consequently, a lne l traerses seeral stops. The trael tme beteen to stops and the frequency of a transt serce lne l are stored as lne characterstcs C. Access / egress modes and PT are only connected l through these stops. A combnaton of usng a specfc access mode, PT and a specfc egress mode s defned as a mode chan M. Whether a lne calls at a stop u or not, s ndcated by stop characterstcs C. All together, the transportaton netork s defned by G ( N, A, L, U ), u here A, L and U are further specfed by C, C and C. a l u 2.3. Optmzaton Problem We defne a decson ector y (or a soluton), that conssts of V decson arables: y y,, y,, y V. Y s the set of feasble alues for the decson ector y (also called decson space). The obecte ector Z (consstng of W obecte functons, Z Z,, Z,, Z W ) depends on the alue of the decson ector y. Eery Z s part of the so called obecte space, and n prncple Z may be any alue n W, but dependng on ts meanng, an obecte functon may be subect to natural bounds. The resultng mult-obecte optmzaton problem (MOOP) s defned n program -5. The loer leel program n 3-5 s based on the formulaton n [2]. The characterstcs C, C and l C of A, L and U depend on the decson ector y. These characterstcs defne the u multmodal transportaton netork G. A fxed demand rs q s assgned to ths transportaton netork, such that satsfes the constrants (2-3). Feasblty constrant (2) should be satsfed to defne a transportaton netork that s physcally thn reach. Constrant (3) represents the loer leel optmzaton problem, that optmzes modal splt and flos n the netork from the traeler s perspecte and has equatons (4-5) as constrants. Constrant (4) ensures that the sum of car route flos and transt demand equal total demand. Constrant (5) s a non-negatty constrant for car route flos. Note that PT route flos are nonnegate by the defntons n constrants (6-7). (6) ensures poste PT demand, but bounded by the total demand. (7) dstrbutes PT demand oer arous mode chans and PT routes, usng the fractons n constrants (2-3): (3) defnes the route fractons for PT trps per mode chan: the PT model contans multple routng, here trps are dstrbuted among sensble routes usng a logt model. Constrant (2) defnes the mode chan fractons for PT trps. Constrant (0) calculates the OD based generalzed costs per mode chan as a eghted aerage of route costs, to correctly take nto account combned atng costs of multple transt lnes. Constrant (9) calculates the logsum for the mode chans, to represent OD based PT costs. Constrants (8) and () defne the relaton beteen lnk costs and route costs, for car and for PT. Fnally, constrants (4-5) defne the ehcle and passenger lnk flos based on car and PT route flos. Note that PT routes may contan car lnks, n the case of the use of a P&R faclty n the route (see constrant 4). On the other hand, car routes neer nclude PT passenger lnks, as can be obsered n constrant (8). a Copyrght c 204 SERSC 5
Based on the resultng ehcle and passenger flos n the netork G, an obecte ector s calculated for a decson ector y n a general formulaton (). The case specfc formulaton of obecte functons s gen separately n Secton 4.. 2.4. Mult-obecte Optmzaton: Pareto Optmalty Mathematcally, the concept of Pareto optmalty s as follos. If e assume to decson ectors y, y Y, then y s sad to strongly domnate y ff Z ( y ) Z ( y ) ' ' (also rtten as y y ) Addtonally, ' y s sad to coer or eakly domnate y ff Z ( y ) Z ( y ) (also rtten as y y ). All solutons that are not ' ' ' eakly domnated by another knon soluton are possbly optmal for the decson maker: these solutons form the Pareto-optmal set P. ' 6 Copyrght c 204 SERSC
2.5. Soluton Method The MOOP defned n (-5) s an NP-hard problem, snce the b-leel lnear programmng problem s already an NP-hard problem [22]. Therefore, the problem s computatonally too expense to be soled exactly for larger netorks, so e rely on a heurstc. For solng mult-obecte problems, the class of genetc algorthms s often used, because these algorthms hae a lo rsk of endng up n a local mnmum, do not requre the calculaton of a gradent and are able to produce a derse Pareto set [23]. More specfcally, e use the NSGA-II algorthm, deeloped n [24]. NSGA-II has been successfully appled by researchers to sole mult-obecte optmzaton problems n traffc engneerng and proed to be effcent for ths type of problems [2, 8]. It optmzes multple obectes smultaneously, searchng for a set of non-domnated solutons,.e. the Pareto optmal set. It s a genetc algorthm, based on the prncples of natural selecton thn eoluton, combnng solutons to ne solutons (crossoer), here the solutons th a hgher ftness alue hae a larger chance to sure oer orse solutons. In the next generaton, these enhanced solutons are recombned agan, untl no progress s made any more or untl the maxmum number of generatons s reached. Wthn NSGA-II, the matng selecton s done by bnary tournament selecton th replacement. In addton to ths matng process, a random mutaton operator s appled to a (small) fracton of solutons from each generaton, to promote the exploraton of unexplored regons n the decson space. These aspects of the algorthm make that the result s an approxmaton of the true Pareto set. The random processes n the algorthm cause that from sub sequental runs of the same algorthm, dfferent Pareto sets result. Furthermore, NSGA-II contans eltsm, to presere good solutons n an arche. If the number of non-domnated solutons gros bgger than the arche, the arche only contans the best non-domnated solutons based on the defned ftness alue. The ftness alue s calculated n to steps. In the frst step (non-domnated sortng), the solutons are ranked based on Pareto domnance. All solutons n the Pareto set recee rank. In the next step, these solutons are extracted from the set and all Pareto solutons n the remanng set recee rank 2, etc. In the second step, the solutons are sorted thn these ranks based on ther crodng dstance. Crodng dstance calculaton requres sortng of the populaton accordng to each obecte alue. The extreme alues for each obecte are assgned an nfnte alue, assurng that these alues sure. All ntermedate solutons are assgned a alue equal to the dfference n the normalzed functon alues of to adacent solutons. Concludng, the crodng dstance alue (and thus the ftness alue) s hgher f a solutons s more solated, promotng a more derse Pareto optmal set. The algorthm n steps (for more detals, the reader s referred to [24]): Step : Intalzaton: Set populaton sze p, hch s equal to the arche sze a, the maxmum number of generatons H, and generate an ntal populaton. Set. 0 h 0 and. 0 Step 2: Ftness assgnment: Combne arche and offsprng, formng and h h h h h calculate ftness alues of solutons by domnance rankng and croded dstance sortng. Step 3: Enronmental selecton: Determne ne arche h by selectng the best a solutons out of based on ther ftness. h Copyrght c 204 SERSC 7
Step 4: Termnaton: If h H or another stoppng crtera s satsfed, determne Pareto set P from the set of all calculated solutons 0 H (non-domnated solutons). Step 5: Matng selecton: Perform bnary tournament selecton th replacement on h to determne matng pool of parents. h Step 6: Varaton: Apply recombnaton and mutaton operators to the matng pool g to create offsprng. h Set h h and go to step 2. 2.6. Output Defntons The set s defned as all decson ectors (or solutons) that are calculated durng one optmzaton process, so,,, 2 N P y y y H a. The set of N solutons s defned as the Pareto set resultng from process, hch ncludes all non-domnated solutons th respect to all solutons n s no y P : y y, y. ' ', or mathematcally there P s the -th outcome of our MOOP (): one approxmaton of the Pareto-optmal set. The -th executon of the algorthm s also denoted as run. Pareto set P of all calculated solutons { J } s denoted as the superset, and s the best knon Pareto set based on J runs. Please recall that eery element or soluton y represents one transportaton netork desgn that s defned by the alues of V decson arables n,,,,, 2 V. For eery element y y y y y y : y a ector of obecte functons Z ( y ) s calculated, that conssts of W obecte functons, denoted by ndex : Z ( y ) Z ( y ), Z ( y ),, Z ( y ) W. 3. Comparng Multple Optmzaton Outcomes When comparng Pareto sets, t s common to assess the qualty of the Pareto sets by ndcators. These ndcators are usually based on the obecte alues of the solutons n the Pareto set, to measure ether the extent to hch the set s close to the real Pareto-optmal front or the extent to hch the set s derse [23]. Hoeer, another aspect that s releant hen comparng dfferent outcomes of a randomzed algorthm lke NSGA-II (and many other MO optmzaton algorthms), s the extent to hch to Pareto sets are smlar n the decson space. If a Pareto set s used as a decson support tool, the leel of uncertanty n the results s mportant for the decson maker. Especally the uncertanty n the decson space s of mportance, because that s the range he or she can choose from,.e., ths determnes f he or she has proper nformaton to make the rght decson. The parameter alues n the algorthm (n the case of NSGA-II, H a and ) may be another source of uncertanty, hch s ndcated by the senstty of the results for these parameters. To be able to assess ths smlarty formally for our case study n Secton 4, n the next secton comparson ndcators are defned. 8 Copyrght c 204 SERSC
3.. Methods for Comparson A dstncton may be made based on to man characterstcs of the method. The frst s a dstncton beteen the number of Pareto sets to be compared: a unary ndcator can be calculated for one Pareto set on ts on, a bnary ndcator can be calculated for to Pareto sets [25]. The so called attanment functon can be calculated for J Pareto sets. The second dstncton s hether the method compares the Pareto sets n the obecte space or n the decson space: the frst type of method s calculated based on the obecte functon alues Z of the solutons n the set(s), hle the second type of method s calculated based on the decson alues y of the solutons n the set(s). For comparson n the obecte space, a selecton of ndcators from the lterature s presented. For comparson n the decson space, ne ndcators are defned. An oere of the ndcators used n ths paper s gen n Table 3. These ndcators use a dstance functon, hch can be any dstance functon defned for ' to solutons as ( d y, y ). In ths paper the dstance functon s defned as n formula 6. ' V ' ' ' '. (6) d ( y, y ) y y 4. Case Study We apply the mult-obecte optmzaton frameork to a real lfe multmodal transportaton netork, stuated n the area around the Dutch ctes of Amsterdam and Haarlem (see Fgure 2). Frst, the obectes and decson arables for ths specfc case are descrbed. After that, the dfferent runs of the NSGA-II algorthm are lsted, ncludng ther scores on unary performance ndcators. Fnally, the runs are compared th each other by usng the bnary ndcators and the attanment functon. Fgure 2. Map of the Study Area, Contanng Transportaton Zones, Ralays and Roads Copyrght c 204 SERSC 9
Table 3. Oere of Indcators to Compare Pareto Sets Formula Type Descrpton N Unary Cardnalty of Pareto set. The bgger a Pareto set, the more complex t s to analyze for a decson maker. Furthermore, th more solutons n the set t s easer to fulfll other crtera, lke hyperolume. M IN ( P ) m n Z ( y ), y P M A X ( P ) m a x Z ( y ), R M I k S S C ( P ) z ( y) 2 ( P ) S S C y P M IN ( P ) M IN ( P ) P k M A X ( P ) M IN ( P ) z ( y) Fg. 3: hyperolume 2-D sualzaton ' C T S ( P, P ) ' ' y P ; y P : y y ' ' ' A D ( P, P ) d ( y, y ) ' N N N N ' ' ' ' A N D ( P, P ) m n (, ) d y y ' ' N '.. N N N V ' ' A F D ( P, P ) V N N ( y ), th 0 f y 0 ( y ) o th er se ', th ' Unary, obecte space Unary, obecte space Bnary, obecte space Bnary, decson space Bnary, decson space Bnary, decson space The mnmum obecte alue attaned by Pareto set for eery obecte functon, th respect to the range Z coered by the superset. For that, the mnmum and maxmum alues are determned for the superset, as ell as the mnmum alue for set. Ths ndcates the extent to hch set coers the entre Pareto front for eery obecte dmenson. The space coerage. Implemented as n [26], also knon as S-metrc or hyperolume. In the 2-dmensonal case t determnes the area that s coered by the Pareto set th respect to a reference pont (the star n fg. 3). The reference pont represents the upper bound of all obectes: the reference pont s defned such that t s domnated by all solutons n the Pareto set. Because the true maxmum alues of the obecte functons are not knon, e choose a conserate pont, based on the ealuated solutons. In the 3 dmensonal case area s replaced by olume, and n the more dmensonal case by hyperolume. The set coerage or C-metrc, see [27]. The leel n ' hch the solutons n P are eakly domnated by at least one soluton n set P, so a hgher alue ndcates ' a better set coerage of P oer P. The aerage dstance (n the decson space) beteen the ' solutons n to Pareto sets P and P (here may also be equal to ). Ths ndcator s used to check hether the araton thn a Pareto set s dfferent from the araton beteen Pareto sets. The dstance of an element n P to the closest element (.e. ' most smlar n decson space) n Pareto set P, aeraged oer all elements P. A lo alue ndcates hgher smlarty (the extreme case A N D ( P, P ) 0 holds). Comparson of fractons of nonzero decson arables. The fracton of solutons n Pareto set that hae a poste alue for decson arable y characterzes the decson space of the set: decson arable y represents the exstence of a measure n the transportaton netork, so a hgher fracton for arable y mples a larger ablty to attan Pareto solutons hen that arable has a poste alue. The functon defnes the nonzero relaton. To ndcate the dfference beteen Pareto set and, the 0 Copyrght c 204 SERSC
J ( Z ) I ( y P : y Z ) J dfferences beteen these fractons are calculated, aeraged oer all V decson arables. Compare Attanment functon, as defned n [25]. For each obecte J Pareto ector Z ths functon returns the probablty that Z s sets, attaned by a fracton of J Pareto sets. It s approxmated obecte on the bass of the approxmaton set sample. Functon space I returns f ts argument s true and 0 f t s false. 4.. Obectes In ths paper e consder 4 polcy obectes related to sustanablty, concernng accessblty, use of urban space by parkng, clmate mpact and costs. These obectes and ther operatonalzaton are shon n table 4. Varables that are not defned earler, are defned belo the table. All 4 obectes are to be mnmzed. In our case of a fxed total demand, total trael tme ges the same results as aerage trael tme ould hae gen. The urban space used by parkng s represented by the number of car trps to or from zones that are classfed as hghly urban, because such a trp requres a parkng space that cannot be used for other urban land uses. These alternate land uses ge addtonal alue to property [28]. CO 2 emssons are calculated usng the ARTEMIS traffc stuaton based emsson model, here emsson factors for cars on a lnk depend on free flo speed and olume / capacty rato [29]. Emsson factors for PT depend on ehcle type. Operatng defct of the PT system s formulated as an obecte, rather than as a budget constrant, to prode explct nsght n the relaton beteen costs and other obectes. Cost parameters follo from Dutch PT operatng practce. 4.2. Decson Varables Decson arables n ths case study are related to transfer facltes or to PT facltes. For eery potental netork deelopment, a decson arable y s defned n adance (see Table 5). Netork deelopments are only ncluded as a canddate locaton f spatal and capacty constrants are met. For example, a P&R locaton s only opened f the correspondng staton s sered by PT. The characterstcs of lnks, lnes and stops that are not canddate locatons are fxed at one alue. Furthermore, the car and bcycle netorks are assumed to be fxed. In ths case, the feasble regon Y contans approxmately 0 9 possble decson ectors. Frequency of PT Serce Lnes A part of the decson ector y represents frequences of PT serce lne, ncludng tran lnes (local trans stop at all statons they pass through, Intercty (IC) trans or express trans only stop at IC statons) and man bus lnes (th multple stops). In our case, the frequences can be chosen from 0, 2, 4, 6, so the alue may also be 0: the lne does not exst n that case. To achee comparable alues of y oer the entre the decson space, the correspondng alues that y are normalzed as 0,,,. Tram lne extensons are defned as a bnary 2 3 3 arable, ndcatng hether the lne s extended or not (th the orgnal frequency). A feeder bus netork s aalable n the netork, but s assumed to be fxed n ths case. Exstence of Tran Statons For the exstence of tran statons y s defned as a bnary arable, that ndcates hether transt ehcles call at a stop u or not (.e. exchange beteen the alk and bcycle netorks Copyrght c 204 SERSC
and the PT netork). Note that for bus stops and for exstng tran statons the alue of fxed to : these PT stops can n all solutons be reached by bcycle and alkng. Express status of tran staton u For the express tran status of tran statons, y s defned as a bnary arable, that ndcates hether transt ehcles of express lnes call at a stop u or not. Note that at exstng express stops ths arable s fxed to. Table 4. Obectes and ther Operatonalzaton n Words and n Mathematcal Formulaton y s Park and Rde faclty at stop u For Park and Rde, y s defned as a bnary arable that ndcates hether t s possble to reach a stop by car, by changng the characterstcs of the lnk connectng the stop to the access netork. Note that there are also exstng P&R facltes that are fxed to,.e., PT stops that can n all solutons be reached by car. 4.3. Pareto Set Approxmatons The NSGA-II algorthm as executed 3 tmes n total (see Table 6). The frst 5 runs ere executed th dentcal parameter settngs, to nestgate the mpact of randomness n the algorthm. After that, n to more groups of 3 runs, 2 dfferent combnatons of parameter settngs ere chosen. Fnally, to unque runs ere executed, th a dfferent mutaton rate. A set of runs th the same parameter settngs s further denoted as a group of runs. The groups are hghlghted n Table 6. All parameter settngs ere chosen n such a ay that 2 Copyrght c 204 SERSC
computaton tmes ere stll reasonable (one eek per run), hle conergence as reached or almost reached n terms of attaned hyperolume. Furthermore, for a far comparson, the total computaton tme (ndcated by the total number of calculated solutons H ) as a approxmately the same for all runs. Decson arable ndex Table 5. The Defnton of 26 Decson Varables n the Case Study Possble alues of y Represents real alue Descrpton of decson arable 2, 2, 4 Frequency of ntercty tran Amsterdam - Haarlem - 3 3 Leden 2 0, 0, 2 Frequency of local tran Amsterdam - Haarlem Leden 3 2 0,,, 0, 2, 4, 6 3 3 Frequency of ne tran lne Utgeest - Haarlem - Amsterdam Sloterdk - Zud Blmer 4 0, 0, 2 Frequency of tran lne Utgeest - Haarlem - Amsterdam Centraal Fracton of nonzero alues n superset 00% 5 0, 0, 2 Frequency of tran lne Haarlem Leden 4% 6 2 0,,, 0, 2, 4, 6 3 3 7 2 0,,, 0, 2, 4, 6 3 3 8 2 0,,, 0, 2, 4, 6 3 3 9 2 0,,, 0, 2, 4, 6 3 3 0 2 0,,, 0, 2, 4, 6 3 3 2 0,,, 0, 2, 4, 6 3 3 Frequency of bus lne 76: Amsterdam Zud Haarlem Frequency of bus lne 83: IJmuden A dam Sloterdk Frequency of ne bus lne Hoofddorp - Amsterdam West - Sloterdk Frequency of bus lne 75: Haarlem - Amsteleen - Amsterdam Zudoost Frequency of bus lne 80: Zandoort - Haarlem - Amsterdam Marnxstraat Frequency of ne bus lne 'Westtangent': Amsterdam Sloterdk Amsterdam West Schphol 2 0, Exstence Extenton of tram 3 n Geuzeneld 42% 3 0, Exstence Extenton of express tram Amsteleen to Uthoorn 62% 4 0, Exstence Ne tran staton Halfeg-Zanenburg % 5 0, Exstence Ne tran staton Haarlem Zud 36% 6 0, Exstence Ne tran staton Amsterdam Geuzeneld 0% 7 0, no ICs, ICs Intercty status of staton Hoofddorp 88% 8 0, no ICs, ICs Intercty status of staton Heemstede-Aardenhout 7% 9 0, no ICs, ICs Intercty status of staton Amsterdam Lelylaan 22% 20 0, no ICs, ICs Intercty status of staton Duendrecht 45% 2 0, Exstence P&R at tran staton Halfeg-Zanenburg 0% 22 0, Exstence P&R Velsen Zud (at bus lne 83) 30% 23 0, Exstence P&R at tram staton Amsteleen Oranebaan 2% 24 0, Exstence P&R at tran staton Geuzeneld and tram 3 5% 25 0, Exstence P&R Badhoeedorp at ne bus lne 2% 26 0, Exstence P&R Schphol Noord at bus lnes 'Westtangent' and 'Zudtangent' 6% 75% 5% 76% 99% 53% 6% 74% 62% 30% Copyrght c 204 SERSC 3
Table 6. Parameter Settngs and Unary Indcators of the 3 Runs of the Optmzaton Process Set number arche sze a Number of generatons H Mutaton rate Number of solutons n set N Hyperolume S S C ( P ) (*0 2 ) 80 36 0.03 339 9.57 2 80 36 0.03 282 9.78 3 80 36 0.03 30 9.84 4 80 36 0.03 364 9.76 5 80 36 0.03 29 9.82 6 50 60 0.03 287 9.80 7 50 60 0.03 329 9.88 8 50 60 0.03 340 9.76 9 00 30 0.03 363 9.86 0 00 30 0.03 255 9.79 00 30 0.03 32 9.85 2 80 36 0. 59 9.28 3 80 36 0.0 337 9.80 - - - 09 0.2 4.4. Unary Performance Indcators All 3 runs ere able to attan consderably larger hyperolumes (Table 6) than the random startng populatons (the aerage alue of hyperolume coered by the startng J populatons to as 7.3*0 2 ). Furthermore, the nddual runs are not too far 0 0 from the best knon Paretoset, the superset P, takng nto account that the superset contans a larger number of solutons than the nddual runs. When e compare the number of attaned Pareto solutons, only run 2 shos a large dfference th all other runs. The araton thn the number of Pareto solutons s smlar thn groups of runs and beteen groups of runs. The same holds for the hyperolume that s coered by the Pareto sets. In addton, run has a relately lo alue for hyperolume, hch seems to be an outler resultng from the random optmzaton process. Next, the mnmum alues attaned by the dfferent runs R M I ( P ) for all 4 obecte functons are compared. Some obseratons from Fgure 4 are: - Trael tme has the most (relate) araton from all obectes - Run 2 does not prode (near) optmal solutons for any obecte - Run 7 performs excellent, acheng 3 mnmal alues and aerage alue Although no hard conclusons can be dran based on one optmzaton outcome, the unary 2 ndcators sho that P has a orse performance than the other Pareto sets. Ths s also 2 shon by the set coerage th respect to P : the aerage alue oer, 2 of 2 C T S ( P, P ) s 0.94, hle the aerage alue oer, 2 of 2 C T S ( P, P ) s 0.43. Ths ndcated that a mutaton rate of 0. (as as appled n run 2) s too hgh for ths type of problem. 4 Copyrght c 204 SERSC
RMI(P) Internatonal Journal of Transportaton 4% 2% 0% 08% 06% 04% Trael tme Urban space used Operaton defct CO2 emssons 02% 00% 2 3 4 5 6 7 8 9 0 2 3 Fgure 4. The Relate Values R M I ( P ), for 3 Runs and for 4 Obectes 4.5. Comparng Pars of Pareto Sets In ths secton a parse comparson beteen Pareto sets s made. Frstly, Pareto sets 2 and 3 are plotted as an example n fg. 5. Ths shos that the sets hae comparable shapes, but the extremes are dfferent, and one Pareto set contans more solutons n certan areas than the other set. Next, the ndcators defned n Secton 3. are used to compare the Pareto sets. ' Table 7 shos the alues of ( A D P, P ) for each par of Pareto sets. The dagonal of the table contans the alues for the aerage dstance thn the same Pareto set. No clear dfferences can be obsered n the table. Hoeer, there s a small dfference beteen cases here ' and cases here ' : oerall, these aerage alues are 6.5 resp. 6.8. Furthermore, there s a small dfference beteen the aerage alue thn groups of runs (6.7) and beteen groups of runs (6.9). These dfferences seem too small to be substantal, although one ould expect that a par of Pareto sets th the same parameter settngs ould hae more smlartes than a par of Pareto sets th dfferent parameter settngs. Fgure 5. 2 Dmensonal Plots of To Dfferent Outcomes of the Optmzaton Process, for Obecte and 3 (Left) and for Obecte 2 and 4 (Rght) Copyrght c 204 SERSC 5
Table 7. The Aerage Dstance beteen Pareto Solutons ' A D ( P, P ) 2 3 4 5 6 7 8 9 0 2 3 6.4 6.9 6.9 6.9 7. 6.7 7.2 6.9 7. 7.0 7.0 7.6 7.0 2 6.5 6.5 6.6 6.7 6.6 7.0 6.8 6.7 6.8 6.7 7.5 6.8 3 5.8 6.4 6.3 6.4 6.6 6.6 6.4 6.2 6.4 7.3 6.4 4 6.4 6.7 6.7 6.9 6.9 6.8 6.7 6.7 7.6 6.8 5 6.4 6.6 6.8 6.7 6.5 6.7 6.6 7.6 6.8 6 6. 6.7 6.5 6.6 6.6 6.7 7.4 6.6 7 6.6 6.9 6.8 6.9 6.8 7.7 7.0 8 6.4 6.6 6.8 6.7 7.5 6.8 9 6.2 6.7 6.5 7.4 6.8 0 6.5 6.7 7.4 6.7 6.5 7.6 6.8 2 7.8 7.5 3 6.6 Table 8 shos the alues of ' ( A N D P, P ) for each par of Pareto sets. Note that the ' dagonal only contans zeros: n the same Pareto set an dentcal soluton th ( d y, y ) = 0 can alays be found. Another obseraton s that the ndcator s not symmetrcal,.e., ' ' (, ) (, A N D P P A N D P P ) : a soluton y n set that acts as nearest soluton for soluton ' y may not hae soluton ' ' y as nearest soluton n set. Agan, only mnor dfferences can ' ' be obsered n the table. Hoeer, the absolute alues of ( A N D P, P ) hae an mplcaton: on aerage, the closest soluton n another Pareto optmal set can be found at a dstance of 2.3. Ths mples that on aerage more than 2 of 26 the decson arables are dfferent f a soluton from another random outcome of the optmzaton process s used, hch s a substantal dfference. Furthermore, also n ths ndcator there s a small dfference beteen the aerage alue thn groups of runs (2.2) and beteen groups of runs (2.3). These dfferences seem too small to substantal. ' Table 8. The Aerage Dstance to the Nearest Pareto Soluton ' A N D ( P, P ) 2 3 4 5 6 7 8 9 0 2 3 0.0 2.3 2.7 2.2 2.6 2.4 2.6 2.5 2.5 2.6 2.5 3. 2.4 2 2.3 0.0 2.3 2. 2.2 2.2 2.4 2.2 2.2 2.3 2. 2.9 2. 3 2.5 2.0 0.0.9.8 2.0 2.0 2.0 2.0.9.9 2.7.8 4 2.3 2.2 2.2 0.0 2.2 2.4 2.4 2.4 2.3 2.2 2. 2.9 2.2 5 2.4 2.0 2. 2.0 0.0 2.2 2.0 2.0.9 2.3.8 3.0 2. 6 2.3 2. 2. 2.2 2.2 0.0 2. 2.0 2. 2.2 2. 2.7 2.0 7 2.4 2.3 2.3 2. 2. 2.2 0.0 2.3 2.2 2.4 2. 2.9 2.3 8 2.4 2. 2.2 2.3 2. 2.0 2.2 0.0.8 2.4 2.0 2.8.9 9 2.5 2. 2. 2..9 2. 2..8 0.0 2.2.8 2.7 2.0 0 2.5 2.3 2. 2. 2.2 2.3 2.4 2.2 2. 0.0 2.2 2.7 2.0 2.5 2. 2. 2.0 2.0 2.2 2.3 2.2.9 2.2 0.0 2.8 2. 2 3.2 3.2 3.2 3.0 3.2 3. 3. 3.0 2.9 3.0 3.0 0.0 3.0 3 2.3 2. 2. 2.2 2.2 2. 2.3 2.0 2. 2.0 2. 2.7 0.0 6 Copyrght c 204 SERSC
Table 9. The Dfference beteen Fractons of Nonzero Decson Varables n the ' Pareto Sets A F D ( P, P ) 2 3 4 5 6 7 8 9 0 2 3 9.6% 2.5% 8.7% 2.0% 9.3% 3.7%.7% 4.6% 9.4%.7% 9.8% 9.8% 2 8.3% 5.6% 7.7% 7.5% 8.3% 9.9% 9.7% 7.8% 5.9% 9.5% 6.7% 3 7.0% 6.4% 8.9% 8.7% 9.5% 9.4% 5.4% 6.%.7% 6.9% 4 6.8% 8.3% 9.6% 0.5% 0.0% 5.9% 6.5% 0.8% 6.2% 5 7.5% 7.6% 8.% 7.4% 7.6% 4.5%.7% 8.5% 6 8.0% 7.6% 9.7% 7.% 7.8% 9.% 7.4% 7 9.6% 9.0% 9.3% 6.7% 0.8% 9.2% 8 7.% 0.0% 8.4% 9.7% 8.3% 9 0.2% 7.%.3% 9.9% 0 6.8% 7.7% 5.8% 0.5% 7.% 2 9.4% ' Table 9 contans the alues for ( A F D P, P ). Relately large dfferences beteen pars of Pareto sets can be obsered. The absolute alue of ths ndcator has an mplcaton as ell: on aerage, the fracton of Pareto solutons n hch a certan decson arable s Pareto optmal, dffers 8.7%. Ths can be consdered as substantal, takng nto account that some decson arables are (almost) alays equal to 0, or (almost) alays nonzero (see table 5). Furthermore, also n ths ndcator there s a small dfference beteen the aerage alue thn groups of runs (8.4%) and beteen groups of runs (8.8%). These dfferences seem too small to be substantal. 4.6. Comparng Multple Pareto Sets Fnally, Fgure 6 shos the attanment plots of the 5 dentcal parameter settngs, for the obecte combnatons of total trael tme and CO 2 emssons resp. urban space used and operatng beneft. For the frst combnaton of obectes, some clear araton n results can be obsered, manly n the extreme ends of the Pareto set. Ths could already be seen n Fgure 5. The second combnaton of obectes only shos a small boundary of uncertanty, also because the tradeoff regon of the Pareto set s much bgger th these obectes. 5. Conclusons In ths paper the NSGA-II algorthm s appled to the multmodal transportaton netork desgn problem, as an example applcaton of a mult-obecte eolutonary algorthm. The performance of seeral runs of the optmzaton process s llustrated by a case study. The dfferences beteen multple random outcomes of the algorthm are analyzed, both th a sngle parameter settng and th dfferent parameter settngs. For ths analyss, arous ndcators and methods are used to assess both the dfferences beteen the attaned obecte alues as ell as the dfferences beteen decson arables of the Pareto sets. Especally the nely defned ndcators to compare Pareto sets n the decson space appeared to be useful to prode nsght n the structure of the Pareto sets. Copyrght c 204 SERSC 7
( Z ) =0; ( Z ) =/5 ( Z ) =2/5 ( Z ) =3/5 ( Z ) =4/5 ( Z ) = Fgure 6. The Attanment Plots of the Outcome of Optmzaton Processes -5. The Blue Surface s attaned by none of the Pareto Sets, hle the Dark Red Surface s attaned by all Pareto Sets The frst concluson from ths analyss s that the dfferences n the attaned obecte alues are not too bg among the runs: nether thn runs th the same parameters nor beteen runs th dfferent parameter settngs. Hoeer, the mutaton rate seems to be of nfluence on the result, so t seems se to try at least a fe dfferent alues for ths parameter hen settng up a ne case study. Secondly, based on the aerage dstance to the nearest soluton n the decson space and dfference beteen the fractons of nonzero decson arables n to dfferent Pareto sets, t can be concluded that the decsons to be taken are substantally dfferent n all analyzed pars of Pareto sets, so the adsed decsons are nfluenced by the randomness of the algorthm. When takng decsons based on Pareto sets that are constructed by such an algorthm, more knoledge s needed on the senstty of decson arables th respect to the obecte alues, to dentfy mportant and less mportant decson arables. Thrdly, aratons caused by dfferent parameters n the NSGA-II algorthm are slghtly hgher than the aratons caused by randomness thn the algorthm, based on seeral methods for comparson of Pareto sets. Hoeer, because ths dfference s too small to be substantal, e conclude that the nfluence of the parameter alues to be chosen n the algorthm s not bg, mplyng that the rsk for runnng the algorthm th bad parameter alues s small. Acknoledgements We thank The Netherlands Organsaton for Scentfc Research (NWO) for fundng ths research n the program Sustanable Accessblty of the Randstad (SAR): ths research s part of the proect Strategy toards Sustanable and Relable Multmodal Transport n the Randstad (SRMT). Furthermore, e thank the Cty Regon of Amsterdam for the use of the Venom data. 8 Copyrght c 204 SERSC
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[27] E. Ztzler and L. Thele, "Multobecte eolutonary algorthms: A comparate case study and the strength pareto approach", IEEE transactons on eolutonary computaton, no. 3, no. 4, (999). [28] J. Luttk, "The alue of trees, ater and open space as reflected by house prces n the Netherlands", Landscape and Urban Plannng, ol. 48, no. 3-4, (2000). [29] INFRAS, "Artems: Assessment and relablty of transport emsson models and nentory systems", Road Emsson Model Model Descrpton. Workpackage 00. Delerable, no. 3, INFRAS, (2007). Authors Tes Brands, s a PhD canddate at the Centre for Transport studes at the Unersty of Tente (The Netherlands). He receed hs BSc degree (Cl Engneerng) n 2004 and hs to MSc degrees (Cl Engneerng and Appled Mathematcs) n 2008. Hs MSc thess as on optmzaton of dynamc road prcng and as conducted at the Dutch consultancy company Goudappel Coffeng. After graduaton, Tes started orkng at Goudappel Coffeng as a publc transport consultant, th specalzaton n publc transport modelng and data analyss. Parallel to hs ork as a consultant, he started a PhD proect at the Centre for Transport Studes on mult-obecte optmzaton of multmodal passenger transportaton netorks. Erc an Berkum n the early 80 s Erc an Berkum receed hs BSc and MSc n Appled Mathematcs at the Unersty of Tente (The Netherlands). After a relately short career as a softare engneer at Cap Gemn he moed n 988 to Goudappel Coffeng, a traffc and transport consultancy company n Deenter. He orked there for 2 years n hch he has had seeral postons n ths company. Hs man nterest has been on transport modelng, netorks, trael behaour and traffc management. In 998 he became a parttme professor n traffc management at the Centre for Transport Studes. In 2009 he left Goudappel Coffeng and became a full professor and also head of the Centre for Transport Studes at the Unersty of Tente. 20 Copyrght c 204 SERSC