Analysis of Transportation Network Design Strategies for Forced Transfer Busing

Transcription

1 Analyss of Transportaton Networ Desgn Strateges for Forced Transfer Busng Scott J. Mason, PhD (PI) Edward A. Pohl, PhD (Co-PI) MBTC 3011 December 2009 DISCLAIMER The contents of ths report reflect the vews of the authors, who are responsble for the facts and the accuracy of the nformaton presented heren. Ths document s dssemnated under the sponsorshp of the Department of Transportaton, Unversty Transportaton Centers Program, n the nterest of nformaton exchange. The U.S. Government assumes no lablty for the contents or use thereof.

2 MBTC 3011 Fnal Report: Analyss of Transportaton Networ Desgn Strateges for Forced Transfer Busng Faculty Investgators Scott J. Mason, PhD (PI) Edward A. Pohl, PhD (Co-PI) Research Assstants Jun Ja, PhD (GRA) Behrooz Kamal, BSIE (GRA) Elzabeth A. McCallon (UGRA) Natassa D. Taylor (UGRA) Abstract Forced transfer busng occurs prmarly at the elementary school level when students are bused to an alternate school when ther geographcally-assgned school s full at ther specfc grade level. Ineffectve forced transfer busng can result n extra student travel tme and neffcent use of often scarce transportaton resources. In fact, some force transferred students regularly arrve to ther alternate school after mornng classes have started due to neffcent transportaton practces. We examne varous forced transfer busng networ desgn strateges usng actual publc school system data from two school dstrcts to assess varous proposed soluton methodologes effectveness at developng practcally mplementable busng solutons n a realstc amount of tme. In addton, prelmnary models and analyss are presented for specal needs busng problems n one local school dstrct such that student travel tme s mnmzed for these often medcally-fragle chldren.

3 Table of Contents 1 Introducton Prevous Research Hub-and-Spoe Lterature Pure Hub-and-Spoe Networs Hybrd Hub-and-Spoe Networs Crcut Routng Lterature The Vehcle Routng Problem wth Pcups and Delveres and Tme Wndows The Dal-A-Rde Problem Mathematcal Models for Forced Transfer Busng Hybrd Hub-and-Spoe Model Notaton Mxed-Integer Program Crcut Routng Model Notaton Mxed-Integer Program Forced Transfer Busng Case Studes Fort Smth Publc Schools FSPS Hybrd Hub-and-Spoe Model FSPS Crcut Model Sprngdale Publc Schools SPS Hybrd Hub-and-Spoe Model SPS Crcut Model Specal Needs Student Transportaton Problem Descrpton Mathematcal Model... 43

4 5.2.1 Notaton Mxed-Integer Program Sprngdale Publc Schools Specal Needs Case Study Conclusons and Future Wor References... 49

5 1 Introducton When a cty s growth rate sgnfcantly ncreases n a short perod of tme, the school dstrct of that cty s often faced wth dffcultes allocatng students to schools. Usually, a school wll reach maxmum capacty before all students n that school s zone have been enrolled. The remanng students are then forced to attend a school n a dfferent school zone. Transferrng these students not only requres extra buses and thus extra costs, but also frequently results n thrty to sxty mnutes of lost nstructonal tme for the students. Forced transfer students are usually requred to leave for school up to two hours before ther school start tme, wat on ther bus and at other schools for long perods of tme, and then stll arrve late to ther own school. Unfortunately, forced transfer students are generally elementary school students. Therefore, these students, who travel and wat for nearly two hours wth lttle supervson, range n age from fve to twelve. In Sprngdale, Aransas there are approxmately 170 of these students. The Sprngdale School Dstrct s currently the thrd largest school dstrct n the state. The dstrct has grown 62% n the last ten years, wth most of the growth concentrated on the east sde of town. Sprngdale currently has twenty-two schools, sxteen of whch are elementary schools. To accommodate the growth, the cty s plannng to buld one new school per year for the next ten years. In the meantme, the dstrct s strugglng to manage school capacty, bus routng and schedulng. Other large school dstrcts n the Northwest Aransas area use advanced GPS technology to effectvely route ther buses; however, Sprngdale lacs the avalable resources for ths technology and resorts to less effectve modes of routng. Ths research s an attempt to fnd a less expensve soluton to the problem of routng forced transfer students n school dstrcts wth fewer avalable means. 1

6 In Fort Smth, Aransas there are currently 185 students beng force transferred at the elementary level. The Fort Smth School Dstrct s rght behnd Sprngdale, as the fourth largest dstrct n Aransas. The problems Fort Smth must address are smlar to those n Sprngdale. Populaton growth, along wth parental requests for transfer, have caused demand to exceed capacty for many of the nneteen elementary schools. Ths defct s overcome by transferrng students across town, ncurrng addtonal costs and resultng n lost educaton tme. Wth the currently sufferng economy, t s as mportant as ever to use resources wsely and effcently, and student classroom tme s always a prorty. Ths research attempts to determne the best way to mnmze transportaton costs wthout sacrfcng classroom tme. 2 Prevous Research In conductng ths research, prevously researched networ desgn strateges were frst nvestgated. Two of these strateges were then used wth developed models to analyze real world data. The followng descrbes the networ desgn strateges studed and outlnes the two models chosen to analyze the data. Prevous forced transfer research has conssted of four man networ desgn strateges. These strateges are drect parwse, hub-and-spoe, hybrd and crcut. Examples of each desgn are shown below n Fgure 1. In the drect parwse strategy, a bus s sent from each school to every alternate school where demand occurs. In the hub-and-spoe strategy, buses are sent from each school to a common locaton, one of the schools, whch s called the hub. Buses then travel from the hub to all other schools as demand occurs. The hybrd strategy s a combnaton of the drect parwse and hub-and-spoe strateges. Demand for transfer from a student s home school to an alternate school s met ether by drect transfer or through the hub. In crcut routng, buses 2

7 travel n a loop pattern from school to school. Each school can be vsted only once n a crcut. The most basc crcut route that satsfes demand to and from each school has two buses wth bus one travelng from school 1 to school 2, 3, 4, etc. and the second bus smultaneously travelng n the reverse order Drect Parwse Hub and Spoe Hybrd Crcut Fgure 1. Networ Desgn Strateges 2.1 Hub-and-Spoe Lterature Pure Hub-and-Spoe Networs Research concernng hub-and-spoe networs s manly focused on the hub locaton problem. The hub locaton problem nvolves two decsons: locatng hubs and assgnng non-hub nodes to hubs. In addton, there typcally are three assumptons for the hub locaton problem: The networ s complete wth a ln between every node par. There are economes of scale expressed as a dscount factor (α) for usng nter-hub connectons. No drect servce (e.g., between two non-hub nodes) s permtted. Bryan and O Kelly (1999) classfy the hub-and-spoe problem further nto two classes: sngle assgnment and multple assgnments. Sngle assgnment means that each node can only be 3

8 connected to one hub and vce versa, whereas multple assgnments allow each node to be connected to more than one hub. For both types of hub-and-spoe problems, nter-hub connecton s also permtted when multple hubs exst. Bryan and O Kelly (1999) summarze the hub-and-spoe lterature to date n Table 1. Table 1. Analytcal Research on Hub-and-Spoe Networs (Bryan and O Kelly, 1999) 4

9 O Kelly (1987) frst models the sngle assgnment hub locaton problem as a quadratc nteger program to mnmze total networ cost under the condton that all nter-hub lns are fully nterconnected. The networ costs are descrbed n three parts: 1) the travel cost from orgn to hub, 2) the cost of travelng across the nter-hub ln (f necessary), and 3) the travel cost from hub to destnaton. In O Kelly (1987), travel costs on nter-hub lns are ndependent of the amount of flow travelng across the ln. In partcular, although there can be more than one hub n the networ, every node s assgned to a sngle hub. Many heurstcs have snce been developed to solve ths quadratc program: Exchange heurstc of Klncewcz (1991) exchange hubs wth non-hub nodes Clusterng heurstc of Klncewcz (1992) frst cluster nodes nto groups, then assgn a hub to each group Greedy exchange based on maxmum flow or mnmum transportaton cost by Campbell (1996) Smulated annealng by Abdnnour-Helm and Venataramanam (1992) Genetc algorthm by Abdnnour-Helm and Venataramanam (1993) Tabu search by Sorn-Kapov and Sorn-Kapov (1994) To date, the Tabu search of Sorn-Kapov and Sorn-Kapov (1994) has acheved the best solutons. Rather than analyze the quadratc program drectly, another popular approach s to lnearze the quadratc model (Campbell (1994), Sorn-Kapov et al. (1996), O Kelly (1996), Ernst and Krshnamoorthy (1996)). Tght lnearzatons of the quadratc program often can provde nteger solutons wthout enforcng ntegralty requrements. However, the lnearzed model can only obtan exact solutons on relatvely small problem nstances (e.g., 25 nodes). Irnch (2000) expands the sngle assgnment problem to nclude mult-depot pcup and delvery, narrow tme wndows, and heterogeneous vehcles. The tas s to fnd a mnmal cost set of trps. Irnch (2000) proposes a two-phase algorthm to solve the problem wheren phase 5

10 one enumerates all possble route combnatons, then phase two assgns transportaton requests to the combnatons. The method of Irnch (2000) s lmted by the number of hub stops on the route ths must be farly small so that all possble routes can be enumerated easly Hybrd Hub-and-Spoe Networs The hybrd hub-and-spoe networ desgn problem can be classfed as ether a locatonroutng problem or a pure routng problem. For the locaton-routng problem, Ayn (1995) nvestgates an expanson locaton-routng problem n whch non-stop servces are permtted. He bulds a nonlnear model to mnmze total dstance traveled, and then decomposes the model nto two subproblems: nown hub locaton and nown servce type. On the bass of these two subproblems, Ayn (1995) proposes a heurstc that fnds hub locatons frst, and then assgns nodes to hubs whle smultaneously determnng delvery routngs. The pure routng problem deals wth fndng optmal routes for nown hubs. Lu et al. (2003) develop a heurstc procedure based on the Clare-Wrght heurstc (1964) to solve a hybrd hub-and-spoe networ wth ml runs (.e., more than one stop can be made durng a collecton or delvery trp). In ther research, Lu et al. (2003) assume there s no fxed cost for operatng a hub, nor any varable cost ncurred when enterng or leavng the hub. Further, Lu et al. (2003) assume that hub locaton s nown and that there s an nfnte supply of vehcles. Lu et al. (2003) prove that a hybrd system s better than both pure systems and that the demand dstrbuton s the most mportant factor n hybrd system desgn. Fnally, Chong et al. (2006) develop a heurstc procedure to solve the problem of schedulng and routng shpments n a hybrd system when a set of feasble, dscrete nter-shpment tmes s requred. In ther heurstc procedure, the hybrd networ desgn s determned va an enumeratve strategy. 6

11 2.2 Crcut Routng Lterature Another transportaton strategy used n practce today to deal wth forced transfer busng ssues s to use addtonal, specalzed bus routes to transport only forced transfer students. In contrast to regular bus routes whch drop off students at ther base school, these bus routes travel from school to school wthn the dstrct, pcng up and droppng off only transferred students. In ths crcut routng strategy, the forced transfer buses travel n a cycle or loop pattern, and each school can be vsted at most one tme on any crcut. Therefore, students cannot be transferred to an alternate school prevously vsted on a gven bus s route. Consder a smple crcut bus route wth the route A B C. Ths means that the bus frst vsts School A, followed by School B, and then School C. On ths route, students can be transported from A to B, from A to C, and from B to C. However, f a student needs to be transferred from School C to School A, another bus route would be requred, as ths s not possble on ths current smple route. Therefore, all combnatons of orgns and destnatons could be acheved by havng two crcut routes runnng n opposte drectons (.e., A B C and C B A ). Whle ths soluton would clearly acheve the obectve of mnmzng the number of bus routes requred, as more and more schools are added to the crcut bus routes, addtonal buses may be requred n order to mnmze total dstance traveled and/or student wat tme The Vehcle Routng Problem wth Pcups and Delveres and Tme Wndows The forced transfer busng problem s smlar to the Vehcle Routng Problem wth Pcups and Delveres and Tme Wndows (VRPPDTW). In the VRPPDTW, all routes are requred to begn and end at a common depot, and each vehcle has a capacty that cannot be 7

12 exceeded. Also, tme constrants are added that requre servce at any partcular stop to occur durng a specfc tme nterval. Nagy and Salh (2006) developed an ntegrated heurstc to solve the Vehcle Routng Problem wth Pcups and Delveres (VRPPD) n order to fnd a set of routes whch mnmze the total dstance traveled by the vehcles, subect to maxmum dstance and capacty constrants. The basc ntegrated heurstc conssts of four phases durng whch an ntal feasble route s constructed and then mproved upon usng several mprovement technques. In Phase 1, a wealy feasble soluton s found usng a route constructon heurstc. In ths wealy feasble soluton, both the total pcup and the total delvery of every route are below the maxmum capacty constrant. In Phase 2, ths ntal soluton s mproved upon whle mantanng wea feasblty. In Phase 3, the soluton becomes strongly feasble, whch refers to a soluton n whch the load for each ndvdual arc does not exceed the vehcle capacty. Fnally, n Phase 4, ths soluton s mproved upon whle mantanng strong feasblty. Ths basc heurstc can be adapted to solve problems wth multple depots by applyng the dea of borderlne customers as used n Salh and Sar (1997). In ths sense, a customer s consdered borderlne when they are located approxmately half-way between two depots. An ntal soluton to the VRPPD s found by frst separatng the borderlne and non-borderlne customers. All non-borderlne customers are assgned to the nearest depot and wealy feasble solutons are found for each resultng VRPPD. Borderlne customers are then nserted nto the routes to provde an ntal feasble soluton, whch can be used n Phase 2 of the ntegrated heurstc. Several other heurstc soluton approaches exst to solve the VRPPDTW. A local search heurstc was presented by Van der Bruggen et al. (1993) for the sngle vehcle problem wth 8

13 mnmzng route duraton as the obectve. Ths method nvolves two phases: constructng a feasble route and then teratvely mprovng upon the soluton. The wor of Ioachm et al. (1995) focused on a clusterng algorthm n whch customer proxmty s used to group customers n order to smplfy the routng problem. Lastly, Bent and Hentenryc (2006) ntroduced a two-stage hybrd algorthm for the VRPPDTW. The frst stage ams to mnmze the number of routes requred usng smulated annealng, whle the second stages attempts to mnmze travel costs usng a large neghborhood search The Dal-A-Rde Problem The Dal-A-Rde (DARP) problem s smlar to the VRPPDTW, except that the requested transport nvolves persons, rather than goods. Ths s a typcal problem whch apples to the transportaton of the elderly or dsabled n urban areas. In the DARP, users mae requests for transportaton from a specfc orgn to a specfc destnaton, and transportaton s carred out by vehcles based at a common depot. Also, tme wndows are pre-specfed whch bound the arrval and/or departure tme of the users. The DARP s NP-hard because t generalzes both the VRPPD and the Travelng Salesman Problem wth Tme Wndows (TSPTW) (Cordeau 2006). DARP problems are unque n that operatng costs and user nconvenence often are weghted aganst each other when desgnng a soluton. When referrng to operatng costs, fleet sze and travel dstance are mnmzed. When referrng to user nconvenence, devatons from desred pcup and drop-off tmes and excess rde tme are mnmzed. Typcally, these competng obectves are balanced by mnmzng operatng costs subect to servce qualty constrants. Therefore, the overall goal of the DARP s to desgn a set of least cost vehcle routes whch satsfy capacty, duraton, tme wndow, and rde-tme constrants. 9

14 Several route constructon and mprovement heurstcs have been developed for the DARP. Jaw et al. (1986) develop an nserton heurstc whch taes tme wndows drectly nto account by balancng the preferences of the users wth the costs of operaton. In ths heurstc algorthm, transfer requests are selected and nserted nto the vehcle schedule n order of ncreasng earlest pcup tmes. Cordeau and Laporte (2005) proposed a tabu search metaheurstc for the DARP. Ther algorthm begns wth an ntal feasble soluton, and then moves to the best soluton wthn the neghborhood of the current one. Neghborhood evaluaton s based on mnmum route duraton and mnmum rde tmes. Cyclng bac to prevously vsted solutons s avoded by preventng the algorthm from proceedng to a soluton that s consdered tabu. Cordeau (2006) ntroduces a mxed-nteger programmng formulaton to fnd a set of routes that mnmze total routng cost whle satsfyng capacty and servce constrants. Several vald nequaltes whch strengthen the LP-relaxaton of the model are also descrbed for use n a branch-and-cut algorthm. Before the branch-and-cut algorthm s appled, several preprocessng technques are performed to reduce problem nstance sze. Next, the algorthm frst solves the LP-relaxaton. If the relaxaton soluton s nteger, then the optmal soluton has been dentfed; otherwse, an enumeraton tree s developed, and separaton heurstcs are used at each node n the tree to dentfy volated vald nequaltes. If one or more volated nequaltes are dentfed, the cuts are appled and the relaxaton model s solved agan. If no volated nequaltes can be found at a partcular node, then the algorthm stops processng at that node. Ths branch-and-cut algorthm s not sutable for large-scale problem nstances; however, on small to medum-sze nstances, the algorthm reduces both computaton tme as well as the sze of the branch-andbound tree. 10

15 3 Mathematcal Models for Forced Transfer Busng 3.1 Hybrd Hub-and-Spoe Model Based on our lterature revew, t s clear that hybrd hub-and-spoe networs are at least as good as (and often better than) pure hub-and-spoe and drect busng networ desgns, as the hybrd strategy accommodates both of these approaches n a sngle methodology. Therefore, we now develop a mathematcal program for hybrd hub-and-spoe forced transfer busng that allows for both hub-and-spoe and drect bus transportaton between schools (nodes). In ths ntal model, we assume that buses do not mae any ntermedate stops between leavng ther orgn and arrvng at ther ntended destnaton (.e., the hub or another school when performng a drect transport). For both obectve functons, we see to mnmze the performance measure. Therefore, two ndependent optmzaton models, each wth ts own obectve functon, have been developed. In both model cases, the followng assumptons are made: Each bus has a sngle, prmary pc-up and drop-off locaton. There exsts a fnte number of buses avalable for routng, each of whch has an nfnte capacty for students. Demand between each par of schools cannot be splt across multple transportaton resources (e.g., all demand from locaton to locaton must be accommodated on the same bus) Notaton Sets and Parameters N Set of all nodes, N 1,..., n d, Dstance (n mles) from node to node ( N, N ) t, Bus travel tme from node to node ( N, N ) Q, =1 f there exsts student forced transfer demand from node to node ; otherwse, =0 ( N, N,, 0 ) Q 11

16 b Tme a bus leaves school ( N ) s Start tme for all schools r Number of buses avalable n Number of demand pars d (Q, ) Decson Varables x, =1 f students are transferred from node to node drectly; otherwse, =0 ( N, N,, 0 ) x y, =1 f students are transferred from node to node through the hub; otherwse, =0 ( N, N,, 0 ) y z =1 f node s chosen as the hub node; otherwse, =0 ( N ) o, =1 f a bus s used to transfer students from node to hub ; otherwse, =0 ( N, N, ) e, =1 f a bus s used to transport students from hub to node ; otherwse, =0 ( N, N, ) v Total transportaton mles, 0 Hub departure tme of bus bound for school ( N w ) under the leave-when-ready hub polcy, 0 δ Tme buses leave the hub under the leave-smultaneously hub polcy, 0 f Number of buses used at the hub, 0 h Number of buses used for drect transport, 0 L Maxmum lateness of all the buses, 0 u Arrval tme of bus travellng from node to the hub ( N ), 0, Arrval tme of bus transportng demand par (, ) drectly ( N, N ), 0 g, Arrval tme of bus transportng demand par (, ) through the hub ( N, N ), 0 a, Number of mnutes bus travellng from node to node s late ( N, N ), Mxed-Integer Program The obectve s to mnmze total transportaton mles as shown n obectve (1) or mnmze maxmum bus lateness (#2). The secondary obectve terms n each obectve functon are scaled by as a means of breang tes when alternatve optmal solutons exst. mn v L (1) 12

17 mn L v (2) Constrant set (3) guarantees exactly one and only one hub exst. z 1 (3) In addton, a lower bound on the number of hubs s gven by constrant set (4). z ( y ) n (4), d Constrant set (5) assures student forced transfer demand must be satsfed ether va drect or ndrect (.e., hub-and-spoe) bus transportaton. N, N (5) y, x, Q, Upper and lower bounds for decson varable o, s computed by constrant sets (6) and (7). o,, l N N l y e,, N N l yl, (6), (7) Upper and lower bounds for decson varable e, s computed by constrant sets (8) and (9)., y l o, l z 1 N N n, y l e, l z 1 N N n, (8), (9) Next, constrant set (10) allows that at most one bus can leave from each node, whle constrant set (11) permts at most one bus to arrve at each node. o, 1 N (10) 13

18 e, 1 N (11) On the bass of the values of the decson varables o, and, the total number of buses used e, n hub-and-spoe transportaton s calculated by constrant sets (12) and (13). f, (12) o f e, (13) Smlarly, constrant set (14) calculates the number of buses used for drect transportaton. h, (14) x Constrant set (15) guarantees that the total number of buses used does not exceed the number of buses avalable. f h r (15) Constrant sets (16) and (17) calculate the arrval tme of the bus travellng from orgn locaton to the hub. Specfcally, constrant set (16) focuses on the demand pars orgnatng at schools rather than the hub, whle constrant set (17) pertans to hub-orgnatng demand pars. u o, l b t, l l N (16) u b z N (17) The hub departure tme of the bus headed to school s calculated by constrant set (18) when the leave-when-ready hub polcy s n effect. In (18), the value of M ( bg M ) s set as the largest possble value of w, whch s the sum of the maxmum school ready tme and the longest amount of tme spent on any bus route from a school to the hub ( M maxb 1 max N t, N, N, ). w 1 y, M 1 u N, N (18) 14

19 On the bass of constrant set (18), the leave smultaneously tme s obtaned n (19). w N (19) Constrant sets (20) and ether (21) or (22) calculate upper and lower bounds on each bus s arrval tme to ts destnaton for demand pars transported through the hub, dependng on the hub polcy beng used. Under the leave smultaneously polcy, the lower bound s computed by (21), whle the lower bound for the leave-when-ready polcy s computed usng (22). In constrant sets (20), (21), and (19-lwr) the value of M s set to the bggest possble bus arrval tme, whch s the sum of the largest school ready tme and the longest tme spent on a bus route b t 2 max 2 max ). M, ( N N, N, g My, N N,, (20) g t e M 1 y ) N, N (21), l, l, l 2 (, l g w t e M ( 1 y, ), l, l, l 2 l N, N (22) Fnally, consderng the two ndvdual obectve functons of nterest, addtonal computatons are requred to properly compute our tme-based metrc of maxmum bus lateness tme n (23)- (25). b t, x,, a N N g,, s, N N, (23), (24) L a, N N, (25) 15

20 3.2 Crcut Routng Model Based on our lterature revew, we are motvated to assess the effcacy of crcut routng strateges for the forced transfer busng problem. Therefore, we now develop a mathematcal program for the crcut routng case Notaton Sets and Parameters ) ) n total number of bus stops, a s the number of buses avalable. h, Dstance between node and node, I, I I: Set of nodes ( I 1,...,n R: set of routes ( R 1,...,a Q, 1, f there s at least one student transferred from to, otherwse 0, I, I C Canddates of orgn pont, I Decson Varables x r,, 1, If node s followed by on route r, otherwse 0, I, I,( ), r R y r 1, f route r s used, otherwse 0, r R z, 1, f node s the orgn pont on route r, otherwse 0, I, r R r o, 1, f node s the destnaton pont on route r, otherwse 0, I, r R r d, 1, f node s on route r, otherwse 0, I, r R r t r total dstance/tme for each route r R U r, p, r, 1 f there s at least one student need to be transferred from to on the route r, otherwse 0, I, I,( ), r R postons of node on the route r, the bg number means the former poston, I, r R Mxed-Integer Program The obectve s to mnmze total bus dstances: Mnmze t r (26) r 16

21 Frst, ctng constrant set (27), orgn ponts must be selected from canddate pont. Ths constrant s used to reduce the range of orgn ponts. o, r C I r (27) Constrant sets (28) and (29) guarantee that the total number of orgns/destnatons s equal to the number of routes. o, r yr R r (28) d, r yr R r (29) In addton, constrant sets (30) and (31) confrm that the orgn and destnaton of each route s on the route. o r z, r, I, r R (30) d r z, r, I, r R (31) The total number of orgns for all the used routes s equal to the total number of destnatons for all the used routes determned n constrant set (32). r r o, y (32) r r Constrant set (33) certfes that the total number of orgns for all the routes s equal to the total number of routes used. r r o d, r r, (33) The total number of orgnal ponts s at least equal to 1 as demonstrated n constrant set (34). r o, 1 (34) r Constrant set (35) guarantees that all the nodes must be assgned on at least one route. 17

22 z, r 1 I, r R (35) Constrant set (36) s used to decde the logc relatons between x and the other decson varables. The bass s that there s sequence between and only when both and are on the route r. x z z 2 I, I,( ), r R (36) r,,, r, r At the same tme, there s no sequence between and on route r f route r s not used as expressed by constrant set (37). x y r,, r I I,( ), r R, (37) Constrant sets (38) and (39) assure the orgnal ponts must be followed by other nodes and the destnaton pont must have nodes n front of t. x r,, o, r I r R x, d, r, r I r R, (38), (39) Constrant sets (40) and (41) mae sure that only one node can be connected wth the other node on one route. r,, I r R x 1 x 1 r,, I r R, (40), (41) Further, one and only one node can be connected to the other (former or latter) on the condton that the node s on the route: x r, o, r z, r I r R, (42) x r, d, r z, r I r R, (43) 18

23 Constrant set (44) requres that each connecton s drected and has only one drecton: x x 1 I, I,( ), r R (44) r,, r,, Further, an arc can t be used by more than one route. r x 1 I, I,( ) (45) r,, Constrant set (46) s used to calculate the total tme/dstance spent on each route: t r xr, h,, * r R (46) In addton to the other constrant descrbed above for the crcut routng method, we need addtonal constrant sets to model the contnuty of student travel. The poston of nodes on each route s resolved by constrant set (47): p p 1 x M I, I,( ), r R (47), r, r 1 r,, * In addton, the poston number of destnaton node s set to 1 n constrant set (48). p r d, r, I, r R (48) The U r,, values are decded by poston number wth the followng constrant set: U r,, U U z z, r, r, r, r I, I,( ), r R (49) M 2 Furthermore, f at least one student needs to be transferred from to on route r, then node and node must both assgned on the route r: U z z 2 I, I,( ), r R (50) r,,, r, r The fnal constrant sets maes sure to meet all the student transfer requrements. U r,, Q, I, I,( ) r (51) 19

24 4 Forced Transfer Busng Case Studes Forced transfer problems occur prmarly at the elementary level, so ths research uses data from the elementary schools n both the Fort Smth and Sprngdale school dstrcts. The demand data provded by both school dstrcts wth the correspondng dstance and tme data was frst mplemented n the hybrd hub-and-spoe model developed by Ja (2008) to determne the best hub locaton, the total bus mles, the maxmum tme a student wll arrve late to school, the average lateness, the number of buses used and the number of drect par transfers. The data was analyzed wth three dfferent methodologes. In the frst method, each bus was allowed to leave when ready, or when all students had boarded that partcular bus and the obectve was to mnmze total bus mles ( Dst obectve). The data was then analyzed wth the buses stll allowed to leave when ready, but wth the obectve to mnmze maxmum lateness ( Late- LWR obectve). Fnally, the data was run wth the buses requred to leave at the same tme and wth the obectve set to mnmze the maxmum late tme a student arrves ( Late-DS obectve). The bus start tme was smulated throughout the expermentaton. The demand and dstance data was then used n the crcut routng model above to determne the best crcut route combnatons, the number of buses needed, and total mles. The results from testng these models wth data from both Fort Smth and Sprngdale are detaled n the followng sectons. 4.1 Fort Smth Publc Schools The Fort Smth School Dstrct currently has nneteen elementary schools, located as shown n Fgure 2. The current bus routes are shown n Fgure 3. Four buses run dfferent crcut routes to meet all elementary-level demand. The demand data to be analyzed s shown below n Table 2. 20

25 Fgure 2 Fort Smth Elementary Schools (50.3 m 2 ) Fgure 3: Current Ft. Smth Routes 21

26 Table 2: Fort Smth Elementary Demand Data Table 2 shows demand from the schools n the left column to the schools along the top row. For example, the demand from Farvew to Beard s three (.e., 3 students must be transferred from Farvew to Beard) FSPS Hybrd Hub-and-Spoe Model We have used the hybrd hub-and-spoe model to analyze an alternatve busng strategy for Fort Smth. We hope to offer dstrct decson maers choces n ther approach to forced transfer busng, and hopefully suggest alternatves that wll save transportaton costs and student rde and wat tme. No data was provded from Fort Smth descrbng exactly when a transfer bus s ready to leave a partcular school (after all regular bus routes have arrved). We created fve dfferent data sets to be analyzed n the model based on our estmates for these tmes. All data sets are based on a school start tme of 8:00 a.m. The frst data set assumes all buses can leave as early as 7:30 and the second assumes all buses are ready to leave at 8:00. These gve us a frame of reference: one assumes an unusually early leave tme and the other, a very late leave tme. In the remanng three data sets, tmes wthn 5 mnutes of 7:45 are randomly assgned to each school. Also note that we dd not receve any data tellng us whch 22

27 schools could feasbly be the hub. Thus we assumed all schools were elgble. Tables 3 through 7 show the model results usng each of the fve descrbed data sets. We obtan the suggested hub, total bus mles and the maxmum lateness of any bus used to transport students. We changed the number of avalable buses we allowed the model untl we obtaned feasble results. As we changed the number of buses from ths feasble number (17), results ether became nfeasble or dd not change. Therefore, only the model results usng 17 buses are shown below. Table 3. Buses leave at 7:30 Obectve # Buses Hub Total mles # drect # hub Max Lateness Dst 17 Ballman Late-LWR 17 Ballman Late-DS 17 Ballman Table 4. Buses leave at 8:00 Obectve # Buses Hub Total mles # drect # hub Max Lateness Dst 17 Ballman Late-LWR 17 Euper Lane Late-DS 17 Euper Lane Table 5. Buses leave around 7:45a Obectve # Buses Hub Total mles # drect # hub Max Lateness Dst 17 Ballman Late-LWR 17 Orr Late-DS 17 Euper Lane Table 6. Buses leave around 7:45b Obectve # Buses Hub Total mles # drect # hub Max Lateness Dst 17 Pe Late-LWR 17 Sutton Late-DS 17 Sutton

28 Table 7. Buses leave around 7:45c Obectve # Buses Hub Total mles # drect # hub Max Lateness Dst 17 Ballman Late-LWR 17 Euper Lane Late-DS 17 Euper Lane Tang a closer loo at Table 3, we see that for all three obectves, the suggested hub s Ballman, there are 2 buses used for drect transportaton and 15 for demand through the hub, the maxmum student lateness s 0 (all arrve on tme), and the total mleage s mles. Smlar observatons can be made by examnng each of the remanng tables. Note that n other scenaros the data vares wth the dfferent obectves. For example, for the 8:00 ready-tme case, f the obectve s to mnmze dstance, Ballman s the suggested hub. However, f the obectve s to mnmze student lateness, Euper Lane s the suggested hub. Notce that for the 7:30 readytme case, no bus arrves late. Ths s because ths case assumes all regular bus routes arrve to the home schools at least 30 mnutes before the school day begns. When the ready-tme s ncreased to 8:00, maxmum lateness ncreases as much as 30 mnutes because by the tme these buses are ready to leave, the students are already late. The suggested Fort Smth hubs are shown below n Fgure 4. 24

29 Fgure 4: Fort Smth Suggested Hubs The next step n ths research s to meet wth Fort Smth offcals to compare model results wth actual results. When the model performs le the current system, t s valdated. If t does not, the model wll be adusted so that better results can be obtaned. We also plan to learn whch schools are feasble to be the hub. There are a varety of reasons a school may not be allowed to be the hub. For example, there may be lmted parng space, not enough room for a bus to turn around, etc. After talng to FSPS offcals, we wll lmt the schools allowed to be the hub n our model and reanalyze the data. We can then compare our new results wth the ntal results to see how much the hub lmtaton affects our obectves. Part of ths analyss wll nclude changng the number of buses we provde to the model to see f any new solutons result. We wll also compare the number of buses needed to mae the model feasble to the current number used by Fort Smth. We wll also compare the model results wth current practces to loo for ways to mprove Fort Smth s busng strategy n terms of cost 25

30 and student rde tme. We hope to offer them alternate solutons so that decson maers can be well-nformed and allocate ther resources wsely FSPS Crcut Model Fort Smth currently uses a crcut strategy for forced transfer busng. The four current elementary-level routes are shown above n Fgure 3. In a crcut strategy, a bus vsts a seres of schools and students can be pced up or dropped off at any school vsted along the route. We developed a crcut routng model to analyze Fort Smth s data. Our model s capable of analyzng problems nstances over a varety of number of bus cases. We analyze ths problem for the smallest feasble number of buses case, two, and are able to produce the optmal soluton as gven n Tables 8 and 9. Table 8. FSPS Two Bus Optmal Crcut Soluton Frst Bus Poston School Mles Travel # Students # Students Current # of Traveled Tme (mn) On Off Students 1 Barlng Woods Euper Bonnevlle Pe Sunnymede Sutton Spradlng Morrson Trusty Howard Tlles Ballman Farvew Orr Carnall Cavanaugh Beard Coo

31 Table 9. FSPS Two Bus Optmal Crcut Soluton Second Bus Poston School Mles Travel # Students # Students Current # of Traveled Tme (mn) On Off Students 1 Beard Cavanaugh Carnall Orr Farvew Ballman Tlles Howard Trusty Morrson Spradlng Sutton Sunnymede Pe Bonnevlle Euper Woods Examnng the results n Table 8 reveals that an exceedngly large number of students are requred to be on the crcut bus at the same tme (a maxmum of 66). In addton, the 70 mnutes of requred travel tme are excessve n that students may be forced to arrve at ther destnaton school late or early stop bus rders wll be ased to arrve at ther embaraton pont very early n the mornng. Conversatons wth FSPS personnel confrmed these shortcomngs of the two bus optmal crcut routng solutons. Wth ths n mnd, we analyzed the FSPS problem usng three crcut buses. The optmal results from ths analyss are gven n Tables 10, 11, and 12 for each of the resultng crcut bus routes. 27

32 Table 10. FSPS Three Bus Optmal Crcut Soluton Frst Bus Poston School Mles Travel # Students # Students Current # of Traveled Tme (mn) On Off Students 1 Tlles Howard Trusty Morrson Spradlng Sutton Sunnymede Pe Table 11. FSPS Three Bus Optmal Crcut Soluton Second Bus Poston School Mles Travel # Students # Students Current # of Traveled Tme (mn) On Off Students 1 Barlng Woods Euper Bonnevlle Pe Sunnymede Sutton Spradlng Morrson Trusty Howard Tlles Ballman Farvew Orr Carnall Cavanaugh Beard Coo

33 Table 12. FSPS Three Bus Optmal Crcut Soluton Thrd Bus Poston School Mles Travel # Students # Students Current # of Traveled Tme (mn) On Off Students 1 Beard Cavanaugh Carnall Orr Farvew Ballman Bonnevlle Euper Woods Observng the results n Table 11, we stll see a large number of students present on the bus n the mddle of ts route. Ths optmal result combnes wth the Table 10 and Table 12 bus recommendatons, whch are far less crowded and much shorter routes. In fact, when we run the optmzaton model to add a fourth possble bus, we obtan the same soluton as ths three bus case. Pe, Sunnymede, and Sutton schools have sgnfcant numbers of student that board at each school, leadng to ths large number of students on the bus. In the future, we need to add addtonal model constrants to attempt to mnmze and/or balance the number of students on each crcut transfer bus route, as well as to mantan some maxmum acceptable length of bus rdng tme for the students. 4.2 Sprngdale Publc Schools The Sprngdale Publc School Dstrct currently has sxteen elementary schools, whch are dvded nto eght east and eght west schools by Hghway 71 Busness. The locaton of these schools s depcted below n Fgure 5. 29

34 Fgure 5. Sprngdale Elementary Schools Sprngdale s current polcy does not allow any forced transfers to occur across Hghway 71 Busness; therefore, they presently use two hub-and-spoe networs: one for the eght east schools and one for the eght west schools. Consequently, we performed our analyss wth both models by separatng the data nto east and west regons. The east hub s currently Harp Elementary and the west hub s Smth Elementary. These schools were chosen by Sprngdale because they are centrally located, allow for ease of traffc and have adequate space for bus parng. In each nstance bus mleage ncreased as a result of lmtng the hubs, but there was not a hugely sgnfcant ncrease n lateness. As llustrated n Fgure 6, the current Sprngdale east sde polcy assgns Harp Elementary as the hub wth seven buses travelng to the other seven schools wth no drect pars. The current west route s dsplayed n Fgure 7 (where the blac arrows represent the frst set of buses travelng to the hub and the blue arrows represent the buses leavng the hub). 30

35 Fgure 6: Sprngdale East Sde Current Route Fgure 7: Sprngdale West Sde Current Route 31

36 For each set of eght schools (east and west), data on the dstance and tme between each school was collected as well as the transfer demand between each school and the number of students transferred. The demand data s dsplayed below n Table 13 and Table 14. Table 13. Sprngdale East Schools Transfer Demand From/To Bayyar George Harp Jones Lee Montor Parson Hlls Turnbow Bayyar George Harp Jones Lee Montor Parson Hlls Turnbow Table 14. Sprngdale West Schools Transfer Demand From/To Elmdale Hunt Shaw Smth Tyson Waler Westwood Young Elmdale Hunt Shaw Smth Tyson Waler Westwood Young SPS Hybrd Hub-and-Spoe Model Ths research frst used the Sprngdale data n the hybrd hub-and-spoe model to test Sprngdale s current polcy and attempt to fnd an mproved soluton. Sprngdale provded a lst of schools that were deemed nfeasble hub locatons due to locaton or bus parng capacty. Wth each of the three methods of testng the data, the data was run wth all hubs possble and wth the hub lmted for Sprngdale. The east results for mnmzng total bus mles ( Dst 32

37 obectve) are gven n Table 15. For the mnmzng maxmum bus lateness obectve, Table 16 (Table 17) dsplays the results for the east schools under the Late-LWR (Late-DS) obectve. Table 15. East Schools: Leave When Ready, Obectve: Mnmze Mles All Hubs Possble # Buses Max Late Avg Late Mles Hub # Drect Jones 0 Hubs lmted to Bayyar, Harp, and Turnbow # Buses Max Late Avg Late Mles Hub # Drect Harp 0 Table 16. East Schools: Leave When Ready, Obectve: Mnmze Maxmum Late All Hubs Possble # Buses Max Late Avg Late Mles Hub # Drect Jones Jones 0 Hubs lmted to Bayyar, Harp, and Turnbow # Buses Max Late Avg Late Mles Hub # Drect Harp Harp Harp 0 33

38 Table 17. East Schools: Leave Smultaneously, Obectve: Mnmze Maxmum Late All Hubs Possble # Buses Max Late Avg Late Mles Hub # Drect Jones Parson Hlls 0 Hubs lmted to Bayyar, Harp, and Turnbow # Buses Max Late Avg Late Mles Hub # Drect Harp Harp Harp 0 When the hubs were lmted n the model, Sprngdale s current method was found to be the best route (fewest buses) n each nstance. Ths valdates the model s effectveness n real stuatons. When all hubs were allowed, however, the model found a route that used the same number of buses but wth 7.1 fewer mles and a decreased maxmum lateness of two mnutes when Jones s set as the hub (Fgure 8). Nonetheless, Sprngdale consders Jones an nfeasble hub locaton because t ether does not have the necessary parng capacty or s located n an area that numerous buses would have dffcultly accessng at the same tme. 34

39 Fgure 8: Sprngdale East Sde wth Jones as the Hub The same three methods of testng the data were used for the west data, wth the correspondng results beng gven n Tables 18 through 20. Table 18. West Schools: Leave When Ready, Obectve: Mnmze Mles All Hubs Possble # Buses Max Late Avg Late Mles Hub # Drect Smth 1 Hubs lmted to Hunt, Smth, Waler, and Young # Buses Max Late Avg Late Mles Hub # Drect Smth 1 35

40 Table 19. West Schools: Leave When Ready, Obectve: Mnmze Maxmum Late All Hubs Possble # Buses Max Late Avg Late Mles Hub # Drect Shaw Smth Smth 1 Hubs lmted to Hunt, Smth, Waler, and Young # Buses Max Late Avg Late Mles Hub # Drect Hunt Smth Smth 1 Table 20. West Schools: Leave Smultaneously, Obectve: Mnmze Maxmum Late All Hubs Possble # Buses Max Late Avg Late Mles Hub # Drect Hunt Smth Smth 1 Hubs lmted to Hunt, Smth, Waler, and Young # Buses Max Late Avg Late Mles Hub # Drect Hunt Smth Smth 1 The results when the hubs were lmted and not lmted n the west were almost dentcal. The only dfference occurred n Table 19 where Shaw was elmnated as a possble hub. However, maxmum lateness was the only factor that ncreased (fve mnutes). Smth Elementary s currently the Sprngdale west hub and was found most frequently to be the best hub by the model, thereby valdatng the model once agan. However, the model requres fve buses and one drect transfer (Fgure 9) whle Sprngdale currently only uses four buses on the west sde and no 36

41 drect pars. Ths s possble because the transfer demand on the west sde s consderably lower than on the east (132 students n the east versus only 35 n the west) and s acheved by usng a mult-stop approach n whch one bus wll stop at two schools before gong to the hub. A prelmnary mult-stop model was developed to examne ths phenomenon and ndeed, we confrmed the optmalty of ths approach as well. Fgure 9: Sprngdale West Sde wth Smth as the Hub and one Drect Par SPS Crcut Model When evaluatng the Sprngdale data wth the crcut model, the sxteen elementary schools were once agan dvded nto eght east and eght west schools and analyzed separately. For each sde, the model was frst run wth the number of buses lmted to eght then to seven and so on to ultmately two buses. In both nstances only two routes (usng only two buses) were 37

42 found, no matter what the route capacty was set to. On the east sde, route two was smply the reverse of route one. Table 21 shows the two routes for the east sde, mles traveled, travel tme, and the bus capacty feasblty. Route 1 Table 21. East Crcut Routes and Bus Capacty Feasblty Poston School Mles Traveled Travel Tme (mn) # Students On # Students Off 8 Turnbow Harp Current # of Students Parson Hlls George Jones Lee Bayyar Montor Route 2 Poston # Students On # Students Off School Mles Traveled Travel Tme (mn) 6 Montor Bayyar Lee Jones George Current # of Students Parson Hlls Harp Turnbow The maxmum number of students at any pont n tme s 52 on bus 1 and 39 on bus two. The two buses travel a total of 27.6 mles wth a maxmum tme of 35 mnutes. Fgure 10 below 38

43 shows the Sprngdale east sde crcut routes where the blac arrows represent route one and the blue arrows represent route two. Fgure 10: Sprngdale East Crcut Routes On the west sde only two routes were found agan; however, route two only ncluded three schools whle route one vsted every school. Table 22 shows the routes, mles, tme traveled, and bus capacty for the west sde. 39

44 Route 1 Poston Table 22. West Crcut Routes and Bus Capacty Feasblty # Students On # Students Off Current # of Students School Mles Traveled Travel Tme (mn) 3 Shaw Smth Hunt Elmdale Westwood Tyson Waler Young Route 2 Poston # Students On # Students Off School Mles Traveled Travel Tme (mn) 5 Tyson Elmdale Smth Current # of Students The maxmum number of students at any pont n tme s 19 on bus 1 and 6 on bus two. The two buses travel a total of 19.6 mles wth a maxmum tme of 44 mnutes. Fgure 11 depcts the two west routes. Agan, the blac arrows are route one and the blue arrows are route two. 40

45 Fgure 11: Sprngdale West Crcut Routes The crcut routes for both the east and west sdes produced lower total bus mle results than the hybrd hub-and-spoe model. These results were then presented to the Sprngdale Transportaton Department; however, Sprngdale reected the proposton. In a hub-and-spoe system, students rde a bus from ther home to the school n ther zone, catch a hub-and-spoe bus to the hub and then tae another bus to ther destnaton school. In a crcut model, parents must provde transportaton to the school n ther zone because the crcut buses must leave before the normal buses arrve due to the tme requred to complete the crcut. Snce the maorty of transfers n Sprngdale are mandated by the school dstrct, the dstrct must provde transportaton for equalty reasons; therefore, requrng that parents provde transportaton to the base school would be met wth consderable opposton. Sprngdale prefers to use ther current 41

46 Hub-and-Spoe routng method; however, further research could further mprove upon ther current soluton. 5 Specal Needs Student Transportaton 5.1 Problem Descrpton "Specal needs" students are those chldren who, for varous reasons, cannot attend/partcpate n "normal" classes for ther grade level at ther neghborhood school. Each specal needs chld has some type of dsablty, such as autsm, speech mparment, learnng dsablty, and/or bran nury, whch prevents them from beng able to learn effectvely n a normal classroom envronment. Accordng to hs or her specfc needs, each specal needs chld s assgned to a desgnated specal needs class wthn the school dstrct. Specal needs classes typcally are desgnated accordng to the student to teacher rato (e.g., 6:1, 10:1, etc.). The needed specal needs class may or may not be located at the specal needs student's neghborhood school. School dstrcts typcally provde door-to-door transportaton for all specal needs chldren. Each specal needs student s pced up at home and taen to the assgned school, regardless of how far the chld s home s from the school. In our research, we assess the mpact of reassgnng specal needs classrooms to schools based on each school's proxmty to the students requrng the class's specfc servces and functons. Our goal s to optmze the travel experence of specal needs students by mnmzng the amount of bus travel requred, and therefore, student bus rdng tmes and hopefully, gasolne consumpton and transportaton costs. Specal needs busng s approached dfferently than forced transfer busng. Each student s pced up at hs or her home and drven to the school provdng that student s requred class or 42