A new regret insertion heuristic for solving large-scale dial-a-ride problems with time windows. Marco Diana

Transcription

1 A new regret nserton heurstc for solvng large-scale dal-a-rde problems wth tme wndows Marco Dana Dpartmento d Idraulca, Trasport e Infrastrutture Cvl Poltecnco d Torno Corso Duca degl Abruzz, 24 Torno, Italy I Tel Fax E-mal dana@polto.t Maged M. Dessouy** Department of Industral and Systems Engneerng Unversty of Southern Calforna 3715 McClntoc Avenue Los Angeles, CA , USA Tel Fax E-mal maged@usc.edu ** Correspondng Author

2 Abstract In ths paper we present a parallel regret nserton heurstc to solve a dal-a-rde problem wth tme wndows. A new route ntalzaton procedure s mplemented, that eeps nto account both the spatal and the temporal aspects of the problem, and a regret nserton s then performed to serve the remanng requests. The consdered operatng scenaro s representatve of a large-scale dal-a-rde program n Los Angeles County. The proposed algorthm was tested on data sets of 500 and 1000 requests bult from data of paratranst servce n ths area. The computatonal results show the effectveness of ths approach n terms of tradng-off soluton qualty and computatonal tmes. The latter measure beng especally mportant n large-scale systems where numerous daly requests need to be processed. Keywords Publc transportaton; Paratranst servces; Dal-a-rde problem; Heurstcs 2

3 1. Introducton Hstorcally, dal-a-rde servces were large-scale systems desgned n the seventes to serve the general populaton of large urban metropoltan areas. These system soon met wth fnancal problems and were ether dsmssed or radcally transformed (Lave et al., 1996). Recently, they are almost exclusvely used n partcular stuatons, for example for servces n rural areas or for users wth partcular needs. In the Unted States, due to the passage of the Amercan wth Dsabltes Act, most of the exstng servces are for dsabled and elder ctzens; however there s also a flourshng maret for feeder servces to arports (Cervero, 1997). In Europe there s growng nterest n mplementng technologcally advanced systems, and plot studes such as SAMPO and SAMPLUS nvestgated ther feasblty under varous operatng scenaros. There s a sgnfcant body of wor n the lterature on schedulng and routng dal-a-rde systems. The Dal-a-Rde Problem s smlar to the Pcup and Delvery Problem wth the added constrant of restrctng the maxmum passenger rde tme. These constrants are added to lmt the nconvenence to the passengers. Desaulners et al. (2000) and Savelsbergh and Sol (1995) provde a detaled revew of the Pcup and Delvery Problem and ts related problems. We brefly summarze the wor n ths area. Poneer research on the Dal-a-Rde Problem dates bac to the seventes. Theoretcal studes for the sngle-vehcle case nclude the wor by Psarafts (1980, 1983a), Sexton and Bodn (1985a, 1985b), Sexton and Cho (1986), Desrosers et al. (1986) for exact algorthms and the wor of Psarafts (1983b, 1983c) for heurstc approaches. Sten (1978a, 1978b) developed a probablstc analyss of the problem, and Daganzo (1978) presented a model to evaluate the performance of a dal-a-rde system. Heurstcs to solve mult-vehcle problems have been then proposed by Psarafts (1986), Jaw et al. (1986), Bodn and Sexton (1986) and Desrosers et al. (1988). Mn (1989) consders a vehcle routng problem wth smultaneous pcup and delveres, that nvolves the defnton of a capacty constrant. Dumas et al. (1991) present a column generaton scheme for optmally solvng the Pcup and Delvery Problem wth tme wndows. Madsen et al. (1995), Ioachm et al. (1995), Toth and Vgo 3

4 (1997) and Borndörfer et al. (1999) propose heurstcs to solve a transportaton problem of handcapped persons. Savelsbergh and Sol (1995, 1998) study a general verson of the Pcup and Delvery Problem n whch each request can have more than one delvery pont. Local search procedures are reported n Van Der Bruggen et al. (1993) and Healy and Moll (1995). A tabu search technque has been appled by Nanry and Barnes (2000), whereas Teodorovch and Radvojevc (2000) use a fuzzy logc approach. Exact procedures to solve small problems can be found n Ruland and Rodn (1997) and Lu and Dessouy (2003). Recent papers focus on the desgn of dal-a-rde servces on a technologcally advanced bass. Dal (1995) proposes the mplementaton of a decentralzed control strategy for a fleet of vehcles. Horn (2002b) develops an algorthm for the schedulng and routng of a fleet of vehcles that s embedded n a modellng framewor for the assessment of the performance of a general publc transport system, wth the latter beng presented n Horn (2002a). A smulaton model for paratranst servces can also be found n Fu (2002). From ths short revew, the prevously developed algorthms can be classfed prmarly n three areas: exact, nserton heurstcs, and local neghbourhood search technques. The exact approaches provde theoretcal nsght to the problem. The nserton heurstcs, whch ncludes the wor of Jaw et al. (1986) and Madsen et al. (1995), are computatonally fast. However, they may not provde as good of a soluton as local search technques such as the tabu method. On the other hand, local search technques may not be computatonally feasble when a large number of requests need to be scheduled n a dynamc envronment, and they generally requre extensve computatonal tests to set up a number of parameters that are hghly case-senstve. Developng fast robust schedulng algorthms s becomng of ncreasng mportance to ths ndustry due to the dffuson of low cost nformaton technologes. For example, Access Servces Inc, ASI, the agency responsble for coordnatng paratranst servces n Los Angeles County s equppng most of ther fleet wth global postonng systems (GPS) and moble data termnals. Wth the ntroducton of these technologes t s possble to trac the vehcles n real-tme wth capabltes to schedule the requests n a real-tme dynamc mode. In ASI, 50% of the customers mae ther reservatons on the same day of the requested pcup tme and n some cases the 4

5 requests are made only hours before the desred pcup tme. Furthermore, the average daly volume ranges from around 250 to 2000 requests per day dependng on the regon wthn Los Angeles County. Wth ths hgh volume and the requrement to fnd solutons qucly, there s a need to develop algorthms wth the computatonal effcency of the nserton heurstcs but wth the soluton qualty of the local search technques. In ths paper we present a parallel regret nserton heurstc to solve a dal-a-rde problem wth tme wndows. A new route ntalzaton procedure s mplemented, that eeps nto account both the spatal and the temporal aspects of the problem, and a regret nserton s then performed to serve the remanng requests. As opposed to the nserton heurstcs whose computatonal complexty s of O(n 2 ) where n s the number of requests, the computatonal complexty of the regret nserton heurstc s of O(n 3 ). Thus, t s slower than the classcal nserton heurstcs. However, on sample data sets representatve of paratranst operatons n Los Angeles County consstng of 500 and 1000 daly requests, we show that the regret nserton heurstc can provde sgnfcantly superor solutons n terms of total vehcle mles and fleet sze. Although computatonally slower than the nserton heurstcs, the regret nserton heurstc s computatonally faster than the local search procedures. Furthermore, ts computatonal CPU soluton tme s much more predctable than the local search procedures where the soluton tmes are extremely dependent on the structure of the data sets. Ths can be an mportant characterstc to transportaton planners who may need to now how long t taes to obtan a soluton, especally when operatng n a dynamc mode. The remander of the paper s organzed as follows. A detaled descrpton of the studed problem s gven n secton 2. The proposed soluton methodology s descrbed n secton 3. In secton 4 we present the computatonal results obtaned on varous large szed data sets representatve of dal-a-rde operatons n Los Angeles County. Fnally, some concludng remars and drectons for future research are contaned n secton Modelng the operaton of a paratranst system 2.1. Servce features and related constrants 5

6 As prevously stated, our effort s drected at studyng a problem that could realstcally model the operaton of a paratranst system. In the followng we wll partally adopt the operatng scenaro descrbed by Jaw et al. (1986). When mang a reservaton, the customer has to specfy the orgn and the destnaton of the trp, as well as the number of passengers. He can also specfy ether the pcup or the delvery tme; on the other hand, the operator fxes (or negotates) the maxmum rde tme and the maxmum wat tme WT at the pcup pont (for customers that specfy the pcup tme) or the maxmum advance tme AT at the delvery pont (for customers that specfy the delvery tme). The maxmum rde tme for each customer (MRT ) s usually set as an ncreasng functon of ts drect rde tme DRT. We use the followng defnton for MRT, where a and b are two parameters that are specfed by the scheduler: MRT = max (a DRT max (a DRT + b, DRT + b, DRT + WT) + AT) for requests wth specfed pcup tme for requests wth specfed delvery tme It s convenent to merge these constrants, related to the qualty of the servce to be provded, nto the defnton of the tme wndows for all the pcup and delvery nodes. Let EPT be the earlest pcup tme for customer f specfed or LDT be the latest delvery tme for customer f specfed. Then, let (EPT, LPT ) and (EDT, LDT ) be the tme wndows assocated wth the pcup and delvery tmes for customer, respectvely. When EPT s specfed by the user, the tme wndows are computed as follows. LPT = EPT + WT EDT = EPT + DRT LDT = EPT + MRT When the customer specfes LDT, the tme wndows are computed n the followng manner. EPT LPT = LDT MRT = LDT DRT EDT = LDT AT 6

7 In fgure 1, we llustrate how ths computaton s performed for both nds of requests. Usng ths defnton for tme wndows, the number of tentatve nsertons that must be performed s dramatcally reduced by a pror dscardng those that would be nfeasble as regards to some of these constrants. Unle the defnton proposed by Jaw et al. (1986), t can be seen that our tme wndows also mply the respect of the maxmum rde tme. The most serous drawbac s that the tme wndow related to the delvery pont (for pcup-tme specfed requests) or to the pcup pont (for delvery-tme specfed requests) s to some extent unnecessarly narrowed, and ths could n turn mae t more dffcult to dentfy a feasble soluton. On the other hand, narrowng these tme wndows leads to solutons of hgh qualty for the traveler. Ths aspect wll be nvestgated when we present the results of the computatonal tests. Fg. 1. In addton, we assocate wth each request a servce tme s both at the pcup and at the delvery node. Ths servce feature s usually consdered only n the case of the desgn of a paratranst system for dsabled persons. In fact, when modelng a hgh qualty servce, n whch the temporal constrants are very tght, the length of operatons such as boardng and payng the fare cannot be overlooed Idle tmes wthn the schedule As prevously mentoned, we are dealng wth a hghly constraned problem. In order to enlarge the soluton space wthout affectng the qualty of the servce, the vehcles are allowed to stop and dle at any pcup locaton, watng to serve the followng request, f only no passengers are onboard. Ths modfcaton of the standard dal-a-rde problem ncreases the possblty of nsertng new requests, especally when the tme wndows are narrow and the number of requests per unt of tme s low. Furthermore, on an operatonal pont of vew, the presence of these pauses could greatly smplfy the crew roster desgn, as a multtude of ponts n whch a drver turnover s possble, s created. On the other hand, the mplementaton of ths possblty drves to a more 7

8 complcated algorthm desgn, as the nserton of new requests across an dle tme mght cause a passenger to be onboard whle the vehcle s dlng. The easest method to avod ths drawbac s to prevent the algorthm from performng nsertons of requests across one or more pauses. Ths lmtaton s somewhat arbtrary, as there could be the possblty of operatng a shft n the schedule n such a manner that the ncluded dle tmes are elmnated, thus mang the nserton possble. Our algorthm performs ths chec, and n our computatonal tests we measured to what extent ths added capablty mproves the qualty of the soluton (see secton 4) Objectve functon When operatng a publc transport servce there are always two conflctng ponts of vew to be consdered. The servce provder s nterested n the economc effcency of the system, whereas the customer loos at the servce qualty. The objectve functon to mnmze s a weghted sum of three elements that represent these dfferent ponts of vew: (1) the total dstance traveled by all the vehcles, (2) the excess rde tme over the drect tme for all the customers and (3) the total length of the dle tmes wthn the schedule. The latter can also not be consdered f t s stated that a vehcle can dle wthout passengers onboard at no cost. In our smulaton we used 0.45, 0.55 and 0.05 as weghts for the three components, not consderng the scalng factors. The number of vehcles to be used s not mnmzed durng the optmzaton process; but t s an nput of the algorthm. To test the performance of the proposed heurstc, n our computatonal tests we teratvely run each scenaro to determne the mnmum number of vehcles requred to servce all the requests. In order to do ths, we performed the frst run of the algorthm wth a very hgh number of vehcles, and we progressvely lowered ths number n the successve runs untl some requests could not be scheduled. 3. The proposed regret nserton heurstc 8

9 From the dscusson presented n the prevous secton we can conclude that the studed problem s an extremely constraned problem, and therefore the feasblty regon can be very lmted, even compared to the classcal pcup and delvery problem wth tme wndows. On the other hand, from the pont of vew of the professonal scheduler of a paratranst system, the qualty of the soluton n terms of the mnmzaton of the objectve functon mght not be the only desred feature. In an operatve context, even f real-tme requests are not allowed, there are always some elements that mae the problem dynamc (no shows, vehcle breadowns ). Thus, t s also useful to have a soluton that allows some degree of flexblty. In other words, operatng changes n the current schedule, such as addng or removng requests or changng the travel tmes of some arcs, should not always cause the schedule to become nfeasble. These consderatons led us to develop a heurstc prmarly focusng on the maxmzaton of the feasblty of the soluton found. On the other hand, a consderable amount of past research (Solomon, 1987) has shown that the nserton methods perform best when we face a routng problem wth tme wndows. Furthermore, Lu and Shen (1999) for example show that parallel nserton procedures outperform sequental approaches. Our proposal s to adopt a parallel nserton heurstc wth an approprate metrc, amed at mprovng the myopc behavor that s often the drawbac of such methods. The metrc we use n our algorthm s the generalzed regret measure, a technque that has been already employed wth nterestng results for the study of the standard vehcle routng problem wth tme wndows (Potvn and Rousseau, 1993; Lu and Shen, 1999). The regret metrc s partcularly useful n fndng feasble solutons for hghly constraned problems The seed request choce Assumng there are m vehcles, the ntal request to be servced for each of the m tours needs to be determned. Prevous research has shown the senstvty and the mportance of the ntalzaton of the m tours of a parallel constructon heurstc n order to obtan good solutons. One of the smplest and most ntutve ways to ntalze the routes s to choose the m requests wth the earlest pcup tme. Ths ntalzaton rule however does not eep nto account the routng aspect of the problem, and could perform poorly when solvng nstances n whch the 9

10 requests are spread over the terrtory. In fact, t has been ponted out that n ths stuaton the farthest requests tend to be nserted last, and ths behavor clearly worsens the fnal soluton. One way to overcome ths flaw s to eep nto account the spatal poston of the requests when mang the choce of the seeds, tryng to consder the ones that are more decentralzed, as well as the ones wth the earlest pcup tme. The underlyng dea of ths ntalzaton strategy s consstent wth that of the regret nserton method that we wll ntroduce later, snce n both cases we try to antcpate the nserton of a request that would be hard or not convenent to consder later. Dfferent proposals can be found n the lterature on the Vehcle Routng Problem for effcently choosng the seed requests. From the routng aspect, the most popular approaches are varants of the one orgnally proposed by Fsher and Jaumar (1981) that parttons the plane n m cones, whose vertex are the central depot, and chooses the seed customers on the bass of ther dstance from t. Ths dea s also used by many authors to determne n whch order the request must be nserted (Russell, 1995; Caseau and Laburthe, 1999), but n our case ths s rrelevant, snce the nserton order s determned by the regret heurstc. For mxed routng and schedulng problems, usually a ranng ndex s defned on the bass of two aspects. Russell (1995) taes nto account the temporal locaton and the length of the tme wndows along wth the dstance from the depot, whereas Toth and Vgo (1997) propose a more elaborate methodology that consders the loadng and unloadng tmes of the customers and the travel tmes from the pcup and delvery ponts of the request under consderaton to all the other nodes. As our prmary concern s to provde a hgh qualty servce, n our case the tme wndows are qute narrow. Furthermore, we study the feasblty of paratranst systems n an urban envronment, where the densty of requests could be relatvely hgh. The combnaton of these two factors maes the schedulng aspect of the problem preponderant on the routng, far beyond the cases addressed n the cted studes. Hence, we bascally eep the dea of ranng the requests n ascendng pcup tme order, but we operate two major modfcatons. We try to avod to ntalze a route wth a request that can be easly nserted after a prevously chosen seed, and we 10

11 allow some swaps n the ranng order to gve preference to the requests that could be dffcult to nsert later, due to ther spatal poston. We llustrate our procedure through an example. In fgure 2, we represent on a tme lne the pcup (p) and delvery (d) ponts assocated to the earlest fve requests of a set; we omt to represent the assocated tme wndows for smplcty. We want to choose three seed requests among these snce we assume there are three vehcles, and agan for smplcty we suppose that the dstance between any par of nodes assocated to these fve requests s the same. Accordng to the earlest pcup tme ranng, requests 1, 2, and 3 should be the ntal requests. It can however be seen that these three are much more spaced n terms of pcup and delvery tmes (and so a vehcle could easly servce all of them), whereas requests 3, 4 and 5 are so close n ther pcup and delvery tmes that they cannot be served by the same vehcle. As a consequence, t would be more effcent to consder for example requests 1, 4 and 5 as seed canddates, as shown n the fgure. To tae these cases nto account, we consder each par of consecutve requests and +1. If the th request has already been chosen as seed and f the followng nequalty s verfed LDT + TTD(),P(+ 1) EPT+ 1 then the (+1) th request s not taen as seed, and we consder the th and the (+2) th requests. The quantty TT D(),P(+1) s the travel tme between the delvery pont of request and the pcup pont of the followng, whereas LDT s the latest delvery tme of request and EPT +1 s the earlest pcup tme of request +1, as defned by the assocated tme wndows. Fg. 2 We also try to consder the spatal aspect of the problem. To do ths, we compute for each request a decentralzaton ndex (D ) gven by the followng expresson, n whch TD represents the travel dstance between the specfed par of nodes and n s the total number of requests: 11

12 D = n n ( TDP( ), P( j) + TDP( ), D( j) + TDD( ), P( j) + TDD( ), D( j) ) n ( TDP( ), P( j) + TDP( ), D( j) + TDD( ), P( j) + TDD( ), D( j) ) j j The range of D s between 0 and 1. A hgher value ndcates that the th request s relatvely more decentralzed than the others. After ths, we consder agan each par of consecutve requests -1 and, accordng to the prevously defned ranng order. If the followng nequalty s verfed D 1 α D 1 then the two requests are swapped, and we consder the th and the (-2) th requests. The parameter α taes on values between 0 and 1. By ncreasng t we put a greater emphass on the spatal aspect of the problem. The best choce depends on the practcal problem that must be addressed: generally speang, α should ncrease as the tme wndows are wdened and the densty of requests becomes lower. Fnally, the frst m requests of the lst, ordered by earlest pcup tme and processed accordng to the two aforementoned procedures, are taen as seeds and nserted n the empty routes The parallel regret nserton procedure The remanng requests are nserted followng the regret nserton scheme. Ths methodology dates bac to the seventes, but more recently t has been appled to solvng the basc vehcle routng problem wth tme wndows by Potvn and Rousseau (1993), Kontoravds and Bard (1995) and Lu and Shen (1999). These authors propose a modfcaton n the computaton of the regret measure that we also adopted. We refer the reader to the Potvn and Rousseau (1993) paper for more detals, as n the followng we wll only recall the most mportant concepts. The basc dea s to fnd for each unrouted request ts best nserton (.e., the one that mnmzes the related cost, consdered as an ncrement of the value of the objectve functon) n each 12

13 tnerary. In ths manner, we buld an ncremental cost matrx n whch the rows represent the requests and the columns the routes. If a request has no feasble nserton n a route, the correspondng ncremental cost s set to an arbtrarly large value. After that, we compute for each request ts regret, gven by the sum of the dfferences between all the elements of the correspondng row and the mnmum one. The request wth the largest regret wll be nserted n the prevously computed poston. These steps are terated untl all the requests are nserted or untl all the regret costs are zero. In the latter case, the correspondng requests cannot be nserted n any of the exstng routes. The regret cost s a measure of the potental prce that could be pad f a gven request were not mmedately nserted. As t can be seen, whenever a request cannot be nserted n a route, the related regret cost s greatly ncremented. Ths feature s partcularly useful for hghly constraned problems, as t drves the algorthm towards the search of feasble solutons. The man focus s to lmt as much as possble the myopc behavor of the classcal nserton procedures, whch for mxed schedulng and routng problems wth addtonal constrants s partcularly harmful. In summary, both the ntalzaton and the nserton procedures have been desgned to consstently pursue the same goals Qucly checng the feasblty of an nserton The regret nserton algorthm requres at each step to tentatvely chec the nserton of each unrouted request (that s, both the pcup and the delvery pont) n every feasble poston of all the vehcles. Its computatonal complexty s of O(n 3 ). For ths, t s even more mportant than for classcal nserton heurstcs, whose computatonal complexty s usually O(n 2 ), to develop methods for rapdly checng the feasblty of the nserton of a request n a predetermned poston. The most dffcult part s to control the devaton from the orgnal route that s needed to serve a customer wthout causng a volaton of the tme wndows of any prevously nserted requests. Of course ths s not a suffcent condton to have a feasble nserton, but t s the most dffcult one to chec, and perhaps also the most mportant, as when modelng the problem we ncorporated n the tme wndow defnton the constrants related to both the schedulng of the 13

14 request and to the qualty of the servce. In the followng we wll not dscuss the more trval checs related to the respect of the capacty and of the couplng constrants. Jaw et al. (1986) perform the tme wndows control by decomposng each route n dfferent schedule blocs, defned by the correspondng tme ntervals n whch the vehcle s servng requests wthout dlng. They then develop a procedure to chec f the nserton of a request n a schedule would cause a volaton of the tme wndows of any prevously nserted request wthn the schedule bloc. To accomplsh ths tas, four quanttes are computed for each node of the schedule sequence: BUP, BDOWN, AUP and ADOWN, representng the maxmum tme nterval by whch the nodes n the schedule bloc precedng and followng ( s ncluded) can be pushed bacward and forward. If the addtonal tme requred to reach a pont s larger than BUP + ADOWN +1, then the nserton of ths pont between and +1 s not feasble, as t would cause the volaton of some tme wndows of the nodes n the schedule. We remar that assocated wth each request are two nodes, one representng the pcup pont and the other representng the delvery pont. It can be seen that ths procedure focuses only on the sngle schedule bloc. Durng the nserton process, as more requests are nserted, the pauses between schedule blocs decrease. To prevent two or more schedule blocs from overlappng, the above four measures are defned so that the feasble shfts are bounded by the length of the dle tmes wthn them. Ths s an addtonal constrant not requred by the orgnal problem, that could be removed f the shfts needed to serve a request could propagate across dfferent blocs. For example, n fgure 3 we can see that the pcup pont of the request number 3 can be nserted between p(2) and d(2) only f we elmnate the pause between the two blocs and allow the whole schedule to be pushed bacward. Fg. 3 Generally speang, t would be nce to chec the feasblty of the nserton of a node only on the bass of the tme wndows of the entre route, automatcally creatng and mergng the dfferent schedule blocs. Ths mprovement s of great mportance when we loo for solutons of good qualty for the customers, as the average number of pauses per route grows when the tme 14

15 wndows are narrowed. Usng the above mentoned technque would lead to a great dffculty n nsertng new requests, gven by our feasblty chec method and not mpled n the problem tself. In order to avod ths problem, n our algorthm we use dfferent statstcs, so that t s possble to chec n one step f the nserton of a pcup or delvery pont causes some tme wndow volaton over the entre schedule of the vehcle. For each node we defne four quanttes: BTOP and BBOTTOM (the maxmum tme nterval by whch and all the precedng nodes n the vehcle schedule can be pushed bacward and forward, respectvely), ATOP and ABOTTOM (the maxmum tme nterval by whch and all the followng nodes n the vehcle schedule can be pushed bacward and forward, respectvely). To understand how can we mathematcally defne these statstcs, let us focus our attenton on a generc node of a schedule bloc h delmted by the dle tmes ds h and ds h+1. If we consder the quantty BBOTTOM, t s evdent that t s not nfluenced by the tme wndows of the nodes precedng ds h, as a forward shft of the nodes between ds h and can smply be compensated by ncreasng ds h of the correspondng quantty. The same happens wth ATOP. The tme wndows only wthn the same schedule blocs must be consdered. In summary, the defnton of BBOTTOM and ATOP s the same as the one for BDOWN and AUP as proposed by Jaw et al. (1986). We report t n the followng recursve form. ATOP BBOTTOM = = AT s ET for the frst node of each schedule bloc mn (ATOP + 1, AT s ET ) otherwse LT AT mn (BBOTTOM 1,LT AT ) for the last node of each schedule bloc otherwse In the precedng equatons we ndcated wth AT the leavng tme currently scheduled for the node after that the servce has been exploted, whereas ET and LT are the earlest and latest scheduled tme gven by the assocated tme wndow and s s the needed servce tme. Let us now consder BTOP. It s evdent that ths quantty s nfluenced by the tme wndows of all the precedng nodes of the schedule, as well by all the precedng pauses. In the same manner, 15

16 ABOTTOM depends on the tme wndows of all the followng nodes and the followng pauses. They can be computed as follows. BTOP ABOTTOM = = mn (BTOP mn (BTOP 1 1 mn (ABOTTOM mn (ABOTTOM + ds h, AT s ET ), AT s ET ) ds h+ 1, LT AT ), LT AT ) for the frst node of each schedule bloc h otherwse for the last node of each schedule bloc h otherwse Fnally, we perform the nserton feasblty chec of a node between and +1, loong f the requred devaton s less than BTOP + ABOTTOM +1. If ths occurs, the node can be nserted n ths poston wthout volatng any of the tme wndows n the schedule, even f t s necessary to merge two or more schedule blocs. As prevously stated, ths s the most mportant and computatonally heavy feasblty chec that s performed by the algorthm. In our operatng scenaro, t guarantees the satsfacton of both the requred pcup or delvery tmes and the qualty of the servce for the customers already n the schedule. There are often several ways to nsert a request n a specfed poston. We can shft bacward the precedng nodes, or move forward the followng ones, or fnally do a mx of the two untl the requred detour s covered. A specal procedure has been mplemented to eep nto account these two correlated aspects, that for brefness s not exposed here; t s however reported n Dana (2003). Concernng ths pont, t s however mportant to note that the dfferent ways n whch an nserton between two gven nodes can be performed only affects the term of the objectve functon related to the mnmzaton of the dle tme. As ths s less mportant of a performance crtera than the other two measures (total vehcle dstance and excess rde tme), n our algorthm we chose to push bacward the schedule as much as possble, regardless of ts mpacts on the value of the objectve functon. We beleve that ths s the best strategy to follow n a true operatonal settng, as t has the hghest potental to mprove the behavor of the system n a dynamc context, for example n case of delays. 4. Computatonal tests 16

17 4.1. Generatng random samples The algorthm presented n secton 3 has been tested smulatng a realstc dal-a-rde system wth data provded by Access Servces, Inc. (ASI). ASI s the transt agency desgnated to coordnate paratranst servces for elderly and dsabled persons wthn Los Angeles County. About 6000 requests, spread on a terrtory of more than 10,500 square lometers, are served each weeday through varous provders, each one operatng n a dfferent regon of the County. The average daly number of requests servced by each provder depends on the regon and vares from 250 to 2000 requests per day. By performng a statstcal analyss of the data provded by ASI over a three-day perod of servce operaton, Dessouy and Adam (1998) suggest technques for generatng random samples that are representatve of the possble servce requests both n spatal and n temporal terms; the nterested reader s referred to that wor for more detals, as n the followng we wll only recall the man steps. We determned the coordnates (x,y) of the pcup locaton of each request by dvdng the servce area n 15 bns of 10x10 mles gvng a total servce area of 150x150 mles. In order to represent clusters that are exhbted n the actual data, the probablty a request s wthn each bn s not unform. The dstrbuton of the requests n the dfferent bns was statstcally nduced from the real data, and then samples were drawn to determne n whch bn each pcup pont s located. After determnng the bn, two samples were drawn from a unform (-5,5) dstrbuton to determne the coordnates (x,y). The delvery pont was selected by addng to the pcup pont the travel length, whch was sampled from the dstrbuton of the travel lengths. The drecton of addton to determne the delvery pont was sampled from a unform (0, 2π) dstrbuton. The dstrbuton from whch the pcup tmes of the samples were drawn were based on the emprcal dstrbutons derved from Los Angeles County (see Dessouy and Adam, 1998). After that, the delvery tmes were determned on the bass of the travel dstance assumng a constant speed of travel (20 mph). The servce tmes at the nodes, expressed n mnutes, were drawn from a unform (1,3) dstrbuton for passengers wth wheelchar and were fxed to 30 seconds for the other requests. The probablty to serve a passenger wth a wheelchar was set to 0.20, and the number of 17

18 accompanyng passengers was obtaned as a sample of the followng cumulatve dstrbuton: ; ; For requests wthout passengers on a wheelchar, the probablty of havng two people to transport was The remanng requests travel alone. All these assumptons are based on the analyss of the ASI data reported by Dessouy and Adam (1998). We generated two dstnct sets of problem nstances, each one havng fve random samples. In the frst set all the samples contan 500 requests, n the second 1000 requests. For each scenaro, we smulate 24 hours of provded servce. These parameters represent a farly large-scale nstance of ths nd of system Implementaton of the algorthm and computatonal results Accordng to the problem defnton that we ntroduced n secton 2, we need to specfy some parameters that assure an acceptable qualty of servce for all the customers. As our fnal am s to chec f dal-a-rde servces can be feasble also for the general publc, we were partcularly restrctve n the selecton of the parameters related to the qualty of the servce. For ths, we fxed the tme wndow span to 10 mnutes, n order to mae the average watng tme smlar to a typcal transt lne, and we mposed that the maxmum rde tme for each request must not exceed 1.3 tmes the drect rde tme plus 10 mnutes. Note that wthn these tme wndows the above defned servce tmes must be comprsed, so that, for example, a servce tme of 2 mnutes n a tme wndow of 10 reduces the feasble shft of the schedule to 8 mnutes. In secton 4.3 we wll present the results of a senstvty analyss on the tghtness of the schedulng constrants. In order to realze f our methodology performs better than other possble approaches, we mplemented fve dfferent varants and we tested them on the same problem nstances. In algorthm 1 the seed requests are chosen exclusvely on the bass of the pcup tme and a parallel verson of the classcal Solomon sequental nserton s performed (Solomon, 1987). Thus, the request to be nserted s the one that causes the least ncrement of the value of the objectve functon. Algorthm 2 s very smlar to algorthm 1, but the nserton of a request across dfferent schedule blocs s also allowed, accordng to the descrpton n secton 3.3. As mentoned n Jaw et al. (1986), whose heurstc does not provde for ths capablty, from a conceptual pont of vew 18

19 ths should allow for a better exploraton of the feasblty regon of the problem, thus, mprovng the qualty of the soluton found. Algorthm 3 uses the seed request choce procedure descrbed n secton 3.1, but performs the Solomon nserton. Algorthm 4 ntalzes the routes le algorthms 1 and 2, but the regret nserton s performed. Fnally, algorthm 5 represents our proposal, as t uses both the new seed request choce and the regret nserton. All these algorthms were coded n C++ and executed on a Personal Computer wth a Pentum III processor. The mplementaton of all these varants allows us to see the effectve behavor of each alternatve, compared to the classcal base case represented by algorthm 1. The computatonal results for the sets of 500 requests and of 1000 requests are shown n tables 1 and 2, respectvely. The reported numbers refer to the mean values obtaned from ndependent runs on fve dfferent random samples, as earler descrbed. Each table reports the followng nformaton: the frst column ndcates the used algorthm and the second the mnmum number of vehcles needed to serve all the requests. After that, the overall number of mles that are covered and the mles that are traveled wth no passengers onboard (excludng the trps to and from the depot) are shown. The ffth and the sxth columns contan the total length of the schedule,.e. the sum of the tme ntervals from leavng to returnng to the depot of all the vehcles, and the total length of all the dle tmes. Fnally, we report a rdeshare measure,.e. the rato between the number of served requests and the number of trps started wth no passengers onboard, and the average ncrease of the rde tme as regards to the drect rde tme. Tables 1 and 2 Several comments are possble on the bass of these data. Algorthms 1 and 2 perform almost the same when solvng problems wth 500 requests, whereas there s an mprovement of the latter for the bgger nstances, moreover f we consder the number of vehcles needed. In those cases algorthm 2 has a slghtly better performance, although on an ntutve pont of vew the possblty of nsertng requests across dfferent schedule blocs should have had a much deeper mpact. A possble explanaton s that t s lely that elmnatng the pauses leads to a less flexble schedule, that n turn worsens the qualty of the nserton for the later requests. 19

20 Algorthm 3 allows us to evaluate the behavor of the new methodology used to choose the seed request that was presented n secton 3.1. It can be seen that t mproves the schedules over the Solomon nserton manly by reducng the number of mles that are traveled wthout customers onboard. On the other hand, consderng the computatonal results of algorthm 4 we can conclude that the regret nserton method alone gves solutons that use around 6-7% less vehcles and 6-7% less vehcle mles traveled and total rde hours. Algorthm 5, whch combnes the new seed request choce wth the regret nserton method, clearly outperforms all the others, and becomes partcularly effectve n reducng the number of requred vehcles especally when the dmenson of the problem s ncreasng. For the consdered data sets, the fleet sze s decreased on average by 8.17% over the base case. The comparson of algorthms 3, 4 and 5 gves us an expermental confrmaton that the proposed method for ntalzng the routes and the regret nserton wor well together. The mprovement of the schedule over algorthm 1 manly seems to be caused by the dramatc decrease on the empty vehcle mles (the mles traveled wth no passenger onboard), whch n turn causes an ncrement of the average vehcle occupancy and of rdeshare. Also, algorthm 5 slghtly ncreases the dle tmes over the base case. Ths result cannot be generalzed, as t may be due to the small penalty (0.05) that was gven to ths measure n the objectve functon (secton 2.3). The excess of the rde tme over the drect trp tme obvously ncreases wth rdeshare. A more detaled analyss of the results provded by algorthm 5 shows that on average 64% of the requests are served drectly when the number of requests s 500; ths percentage s decreased to 53% wth 1000 requests. Snce these users have no devatons, the rde tme concdes wth the drect rde tme, and for them the system wors as a taxcab. The remanng requests have one or more devatons (.e. the vehcle vsts one or more nodes between the pcup and the delvery pont of the request), and for these requests, ther rde tme s ncreased over the drect tme of about 19%. Only for few requests the ncrease of the rde tme s near the upper bound set by the qualty constrant, and typcally these are among the ones wth 2 to 4 devatons. Qute surprsngly, the requests wth the hghest number of devatons (more than 4) often do not have the hghest excess of rde tme. Ths seems to confrm that f the demand densty ncreases t s 20

21 possble to acheve hgher rdeshares and servce effcences wthout worsenng the qualty of the servce for the passengers. Another nterestng result can be found by performng an analyss on the qualty of the soluton among the dfferent nstances n terms of the requred fleet sze. In table 3 we report the number of requred vehcles for each algorthm n each problem nstance and the resultng sample standard devatons. It can be seen that these latter are consstently reduced when usng the regret nserton method n smaller data sets, whereas they only slghtly ncrease when consderng the larger ones. In all the consdered cases ther value s always less than 3 vehcles. On an operatonal pont of vew, ths means that when the servce has to be mplemented the fleet can be more effcently dmensoned a pror, as the fluctuatons on the number of requred vehcles for a gven level of demand are less evdent. Table 3 Fnally, the computatonal tmes of the nserton procedure for each run on the two data sets are reported n table 4, together wth the respectve mean values and standard devatons. For brefness we report the results only for algorthm 5, but they are almost the same for algorthm 4, as they both use the regret nserton method. Consderng the hghly constraned nature of the problem and the dmenson of the problem nstances, these computatonal tmes are wthn a practcally useful range. Of course, the computatonal tmes can be further mproved by usng a faster processng computer. However, the most mportant thng that has to be ponted out s that these tmes are rather robust gven the dmenson of the nstances and the smulated road networ. The standard sample devatons are about 14 seconds on a length of 26 mnutes and less than 2 mnutes for the larger problems, that were solved n 195 mnutes on average. Ths result s partcularly strng when compared to the performance of local search or metaheurstc procedures, that often have runnng tmes that can vary one degree of magntude for dfferent samples wth the same dmensons. The elmnaton of ths uncertanty can be helpful to the servce provder for plannng purposes, as the mpossblty to foresee how long the schedulng phase wll be n any gven day of servce could be a serous drawbac n a real operatonal envronment. 21

22 Table Senstvty analyss on the tghtness of the schedulng constrants As we mentoned n secton 2.3, n a DRT system the nterests of the customers and of the servce provder are contrastng. Hence, t s essental for ts success to fnd the rght balance between these opposng ponts of vew. Ths can be done by nvestgatng the relatonshp between the ncrease of the servce qualty and the correspondng ncrease of operatng costs. Thus, t s nterestng to evaluate the behavor of the proposed heurstc also under dfferent operatng scenaros. In order to do ths, we solved agan the fve larger problems, nvolvng 1000 requests, under dfferent condtons. We defned three scenaros beyond the one studed n the precedng secton, each of these beng characterzed by a certan mnmum tme wndow span and maxmum rde tme. We report n table 5 the settngs that have been used. Scenaro L has characterstcs that are smlar to a typcal paratranst servce for dsabled people, whereas the qualty of scenaro H can be comparable to that of a taxcab, except for the possblty of rdesharng. Table 5 We focus our attenton on varants 1 and 5 of our algorthm, representng respectvely the standard soluton method and our new proposal. For these, the mean computatonal results over the fve samples for the defned scenaros are reported n table 6. It can be seen that algorthm 5 mproves over algorthm 1 n all the consdered scenaros. However, the regret nserton algorthm performs best aganst the classcal nserton heurstc under medum to small tme wndow constrants. When the tme wndow s large, the beneft of the regret procedure s reduced. When the tme wndow s very small (scenaro H ), the gap between the regret nserton and algorthm 1 also narrows snce n ths case the number of feasble alternatves s rather small. Thus, there are few feasble nserton schedulng alternatves to consder. 22

23 Table 6 5. Conclusons In ths paper we presented a new heurstc desgned to solve large problem nstances of a realstc formulaton of the dal-a-rde problem wth tme wndows. The heurstc was tested on a seres of nstances that were bult from data concernng three days of paratranst operaton n Los Angeles County. The results of these experments showed that the qualty of the soluton s consstently mproved wth respect to a classcal nserton heurstc. Our efforts were drected at fndng a good balance between the need of developng a tool that could be used n practce, the qualty of the soluton and the assocated computatonal burden of the heurstc. The regret nserton scheme furthermore does not have parameters to be set, whch s n contrast to local search and metaheurstc procedures. Our heurstc can manage nstances of bg dmensons for the studed problem. If we compare the outputs for the two data sets, t s evdent that bgger nstances are harder to solve, but n turn the proposed heurstc consderably outperforms the classcal nserton heurstc for the large problem nstances. On the other hand, prelmnary experments wth only 100 requests showed no sgnfcant dfferences between the two approaches n terms of the qualty of the soluton. The feasblty of the mplementaton of a demand-responsve transt system for the general populaton s currently under consderaton n the cty of Turn (Italy), as an advanced Intellgent Transport System s already n use both for traffc management and for the operaton of the publc transport lnes. Research wor s n progress on the topc, but the need of a relable algorthm for the operaton of the servce s one of the prmary concerns. The next phase of our research wll be targeted at adaptng the schedulng capabltes of the presented algorthm to ths transt envronment, ncludng the possblty of nsertng real-tme requests. Acnowledgements 23

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