Keywords: School bus problem, heuristic, harmony search

Transcription

1 Journal of Emergng Trends n Compung and Informaon Scences CIS Journal. All rghs reserved. hp:// Model and Algorhm for Solvng School Bus Problem 1 Taehyeong Km, 2 Bum-Jn Par 1 Senor Researcher, Korea Insue of Consrucon Technology, Korea 2 Senor Researcher, Korea Insue of Consrucon Technology, Korea ABSTRACT School bus roung problem has been a sgnfcan concern of mos people relaed o school and school bus sysem as one of vehcle roung problems. Mang an approprae problem formulaon depends on how o reflec he reales of he problem. And, as he problem scope becomes wder, he problem can be solved only wh he eac mehods. So, here s need o develop an effcen heursc mehod o solve more complcaed problem. In hs sudy, he model for school bus roung problem s proposed, and a heursc algorhm for solvng he proposed model s suggesed. The model s formulaed as a med-neger programmng problem. To valdae he model, several random small newor problems are solved by usng he commercal opmzaon pacage CPLEX. Also, a heurs algorhm based on harmony search s proposed o solve hs problem. The resuls of he heursc are compared wh he resuls obaned from eac soluon by CPLEX o valdae and evaluae he heursc algorhm. Compuaon resuls show ha he soluon by he heursc was eacly he same as ha of eac mehod usng CPLEX. Bu, he heursc produces he same resuls n a very shor me. Keywords: School bus problem, heursc, harmony search 1. INTRODUCTION SBRP (School bus roung problem) has been a sgnfcan concern of epers and researchers relaed o school and school bus sysem as one of VRP (vehcle roung problems). SBRP s o effcenly ranspor sudens from her orgns o a school usng a gven number of buses. As a school bus servce area ncreases, sudens and he number of buses ha are necessary for servce ncrease, and s hard o solve SBRP and ge he opmal soluon whn he desred me by eac mehod. Therefore, he heursc mehod s needed o solve comple problems effcenly. HS (Harmony Search) as one of he opmzaon algorhms developed by Geem e al. (2001) [1] has been wdely used n many areas such as musc composon, projec schedulng, unversy meablng, nerne roung, russ srucure desgn, waer newor desgn, medcal magng and asronomcal daa analyss. Also, he ecellence of HS has been proven hrough many pracces. However, here are few pracces of HS appled n he feld of ransporaon. Therefore, n hs sudy, SBPR model s proposed and a heursc algorhm s presened for solvng he proposed model. For hs wor, frs, a model of SBRP s formulaed as a med-neger programmng problem. To valdae he model, several random newor problems are solved by eac mehod usng he commercal opmzaon pacage CPLEX. Also, he same problems are solved by he proposed heursc algorhm usng harmony search and he resuls of he proposed heursc are compared wh he resuls obaned from eac mehod. 2. LITERATURE REVIEW The surveys on models and algorhm developed for DBRP are descrbed by Desrosers e al. (1981) [2] and Par and Km (2010) [3]. Accordng o Desrosers e al. (1981), SBRP can be dvded no 5 seps: daa preparaon, bus sop selecon, bus roue generaon, school bell me adjusmen and roue schedulng. Ths sudy belongs o bus roue generaon and roue schedulng seps among 5 seps. Benne and Gazs (1972) used savng algorhm o mnmze oal suden-dsance [4]. Bodn and Berman (1979) bul roues usng roue - frs cluser-second and hen mproved he roues usng 3- op algorhm [5]. Tsay and Frcer (1988) formulaed he model as mul-objecve funcon and solved he model hrough hree seps of processng [6]. Bowerman e al. (1995) appled cluser-frs roue-second o SBRP nsead of roue-frs cluser-second used by Bodn and Berman (1979) [7]. Braca e al. (1997) solved SBRP of New Yor cy for general and specal sudens o mnmze he number of buses needed for servce. Geem e al. (2005) appled HS o a es newor randomly generaed and compared he resul by HS wh ha by GA [9]. Consequenally, showed ha HS can ge beer soluon han GA whn shorer me. Recenly, mea-heurscs such as Smulaed Anealng (SA), Tabu Search (TS), Genec Algorhm (GA), An Colony Opmzaon (ACO) have been developed and appled n varous combnaoral opmzaon problems. However, here are no many pracces of mea-heurscs appled n SBRP, and s epeced ha many research on SBRP wll be done usng hese mea-heurscs n he near fuure. 2.1 School Bus Roung Problem Many opmzaon problems n varous felds have been solved usng opmzaon echnques, such as lnear programmng (LP), non-lnear programmng (NLP), and dynamc programmng (DP). However, her drawbacs generae demand for oher ypes of algorhms, such as heursc opmzaon approaches (smulaed annealng, abu search, and genec algorhm). However, here are sll some possbles of devsng new heursc algorhms based on analoges wh naural or arfcal phenomena. Geem e al. (2001) developed a new heursc algorhm mmcng he mprovsaon of musc players, named HS. 596

2 Journal of Emergng Trends n Compung and Informaon Scences CIS Journal. All rghs reserved. The basc concep, seps and srucures, and parameers of HS can be referred o Geem e al. (2001) [1] The Basc Concep HS was orgnaed from an arfcal phenomenon see he beer harmony on Jazz performance. Muscal performances see a bes sae (fanasc h armony) deermned by aeshec esmaon, as he opmzaon algorhms see a bes sae (global opmum) deermned by objecve funcon evaluaon. Aeshec esmaon s deermned by he se of he sounds played by joned nsrumens, jus as objecve funcon evaluaon s deermned by he se of he values produced by componen varables; he sounds for beer aeshec esmaon can be mproved hrough pracce afer pracce, jus as he values for beer objecve funcon evaluaon can be mproved eraon by eraon. A bref presenaon of hese observaons s shown Table 1. Table 1: Comparson beween opmzaon and muscal performance (Geem e al., 2001) Comparson Opmzaon Facor Process Performance Process Bes sae Global Opmum Fanasc Harmony Esmaed by Objecve Funcon Aeshec Sandard Esmaed wh Values of Varables Pches of Insrumens Process un Each Ieraon Each Pracce Seps and Srucures The seps n he procedure of Harmony Search are as follows: Sep 1: Inalze a Harmony Memory (HM) Sep 2: Improve a new harmony Sep 3: If he new harmony s beer han he wors harmony n HM, nclude he new harmony n HM Sep 4: If soppng crera are no sasfed, go o sep Parameers Of course, he above assumes ha all he pars of he global soluon es nally n HM. When hs s no he case, n order o fnd global opmum, Harmony Search naes a parameer, Harmony Memory Consderng Rae (HMCR), whch ranges from 0 o 1. If a unformly generaed value beween 0 and 1 occurs above he curren value of he HMCR, hen HS fnds noes randomly whn he possble playable range whou consderng HM. A HMCR of 0.95 means ha a he ne sep, he algorhm chooses a varable value from HM wh a 95% probably. For mprovng soluons and escapng local opma, ye anoher opon may be nroduced. Ths opon mmcs he pch adjusmen of each nsrumen for unng he ensemble. For compuaon, he pch hp:// adjusmen mechansm s devsed as shfng o neghborng values whn a range of possble values. A Pch Adjusmen Rae (PAR) of 0.10 means ha he algorhm chooses a neghborng value wh 10% probably (an upper value wh 5% or a lower value wh 5%). 3. MODEL FORMULATION FOR SCHOOL BUS PROBLEM 3.1 Assumpons In hs sudy, sngle depo and sngle school are consdered. A school has an arrval me wndow whch means he allowable arrval me. Therefore, a school bus mus arrve a school whn he me wndow, ha s, he bus mus no arrve a school earler han he earles arrval me and laer han he laes arrval me. I s assumed ha he wdh of me wndow s 15 mnues. For eample, f he arrval me wndow a school s (8, 8:15), he bus mus arrve beween 8 and 8:15. In he ypcal school bus roung problem, he locaons of bus sops are deermned by he way mnmzng he average walng dsance beween sudens homes and he neares bus sop. Therefore, we assume ha he locaons of bus sops are he same as hose of demand pons and n our problem, bus sop locaons are consdered o be gven daa. In addon, he avalable vehcles are consdered o be homogenous ype, whch means ha all vehcles have he same capacy. 3.2 Problem formulaon Ths secon provdes a mahemacal formulaon for school bus problem as a med-neger programmng problem. The decson varables, consans, daa ses used n hs model formulaon are defned follows Decson varables = 1 If vehcle ravels ln 0 Oherwse = acual arrval me of vehcle a node, TD, V Consans F = fed cos per school bus ($/bus) R = roung cos per un ravel me ($/mn) = ravel me beween node and node j (mn), DS, j SE p = demand (sudens) of node, S Q = capacy of vehcle, V b = average boardng me per 1 suden T = earles arrval me a school e T l = laes arrval me a school T = mamum n-vehcle me V, DS, j SE 597

3 Journal of Emergng Trends n Compung and Informaon Scences CIS Journal. All rghs reserved. hp:// Daa Ses S = demand nodes (bus sops) D = a vehcle s sarng node,.e. depo E = a vehcle s endng node,.e. school DS = all nodes whch perm on ou-flow,.e. depo and bus sops, D S SE = all nodes whch perm on n-flow,.e. school and bus sops, E S TD = he se of all nodes, D S E V = he vehcle se The proposed mahemacal formulaon s as follows: Mnmze subjec o Z = F + R (1) V V DS V D j S DS j SE V DS jse 1 1 p pj 0 jse 0 D js V S je V DS p X j SE j S (2) S (3) V, p S (4) (5) j p b p b j T Q M (1 X M( X e p b V js V js 1 1 E ) 1) l V (6) V, DS, j SE (7) V, DS, j SE (8) T (9) E T S, V (10) D (11) j E (12) Objecve Funcon The objecve of hs problem s o mnmze he oal cos ncurred n servng all sudens. Tha s, he oal cos consss of he fed coss of he number of vehcles and he roung coss Consrans In addon, here are many consrans for hs problem. Each demand node should be served by one vehcle (2-3). If a vehcle eners a node, mus e ha node (4). The number of vehcle leavng depo mus equal he number of vehcle enerng he school (5). There s a lm for he number of sudens on he bus as capacy consran (6). Arrval me wndow a school, whch means ha school bus arrve a school beween T e and T l (7-9). Earles deparure me a demand node (bus sops), by whch school bus canno leave each bus sop before he earles deparure me (10). Each vehcle can leave one depo and arrve a one school only once, whch means ha he number of roues should be less han or equal o he number of buses (11-12). 4. EXPERIMENTAL RESULTS AND COMPARISON 4.1 Tes Newors To valdae he proposed model n hs sudy, several random newor problems were made as shown n Table 2. There are 5 es newors and s assumed ha he fed cos of a bus s $100,000/bus and he ravelng cos s $105/mnue. Table 2: Tes newors for epermens NO. The number of The number of nodes(sops) avalable buses 1 6(4) 2 2 6(4) 3 3 9(7) (10) (18) Solvng problem by an eac mehod (Branch & Bound) The process of opmzng roues and schedulng for SBRP by eac mehod s dvded no wo seps. Frs, s necessary o prepare he program whch can generae and code correcly he objecve funcon and s consrans for each es newor as he forma of CPLEX Inpu. In hs sudy, C++ programmng language was used o generae a CPLEX npu fle. Ne, he oupu of hs program s pu no he CPLEX and a soluon s obaned hrough opmal process n CPLEX. 4.3 Solvng problem by a heursc (Harmony Search) The HS s appled o a SBRP as follows, Fg 2 The sze of Harmony memory s 10 and HMCR s

4 Journal of Emergng Trends n Compung and Informaon Scences CIS Journal. All rghs reserved. hp:// Inpu Newor Daa HMS = 10 (Mang 10 of Harmony Vecor) If consrans are sasfed Inal Harmony Memory Shores Dsance Mar Demand Mar Generang Random Numbers Compuaon Tme(sec) Number of Nodes EXACT HS Mang a new harmony (HMCR=0.9) Fg 2: The comparson of eac mehod and HS If consrans are sasfed Replacemen: Include a new harmony and Eclude he wors harmony STOP If a new harmony s beer han he wors harmony n HM If sop crera are sasfed Fg 1: Soluon seps of he proposed model usng HS 4.4 Comparson of resuls In hs secon, he resuls of a heursc usng HS are compared wh he resuls of eac mehod (Branch - and-bound) usng CPLEX. The HS was coded n Quc Basc and esed on 2.0 GHz Inel core 2 Duo CPU havng 3.0GB RAM. The compuaonal resuls are le Table 3. In case of HS, he values for he frs newor o he fourh newor n able are resuls of 200 eraons and hose for he ffh newor are resuls of 1200 eraons. Table 3: Compuaonal resuls by eac mehod and HS # # Eac mehod Harmony Search nodes avalable Cal. O.F. # Cal. O.F. # (sops) bus Tme(s) roues Tme(s) roues 6(4) , , (4) , , (7) , , (10) , (18) , CONCLUSION In hs sudy, he model for school bus roung problem s proposed, and a heursc algorhm for solvng he proposed model s suggesed. The model s formulaed as a med-neger programmng problem. To valdae he model, several random small newor problems are solved by eac mehod usng he commercal opmzaon pacage CPLEX. Also, a heursc algorhm based on harmony search s proposed o solve hs problem. The resuls of he heursc are compared wh he resuls obaned from eac soluon by CPLEX o valdae and evaluae he heursc algorhm. As a resul, he soluon (objecve funcon) by HS was eacly he same as ha of eac mehod. Bu, HS produces he same resuls n a very shor me. As he number of nodes eceeds 9, he compuaonal me by eac mehod eponenally ncreases and s mpossble o ge a soluon whn an approprae me. Bu, HS found he dencal feasble soluon only afer 1200 eraons and generaed alernave soluons. I shows ha a heursc usng HS can fnd a good soluon of SBRP wh a shor me. However, s no guaraneed ha he soluon of HS s he global opmal as he sze of newor ncreases. Therefore, s necessary o develop a mehod ha can fnd good lower bound of objecve funcon n a mahemacal model. In hs sudy, sngle depo and sngle school are consdered. However, mul depos and mul schools have o be consdered n real world. So, he modfcaon of he proposed model s nevable n order o reflec he realy of SBRP. ACKNOWLEDGEMENTS Ths research was suppored by a gran from a Sraegc Research Projec (Develop men of Real-me Traffc Tracng Technology Based on Vew Synhess) funded by he Korea Insue of Consrucon Technology. 599

5 Journal of Emergng Trends n Compung and Informaon Scences CIS Journal. All rghs reserved. REFERENCES [1] Z.W. Geem, J.H. Km, and G.V. Loganahan, A new heursc opmzaon algorhm: Harmony Search, Smulaon 76(2001), hp:// [8] J. Braca, J. Bramel, B. Posner and D. Smch-lev, A compuerzed approach o he new Yor cy school bus roung problem, IIE Transacons 29(1997), [2] M. Desrochers, J.A. Ferland, J.-M. Rousseau, G. Lapalme, L. Chapleau, An overvew of a school busng sysem, In: Jaswal, N.K. (Ed.), Scenfc Managemen of Transpor Sysem, Norh-Holland, Amserdam (1981), [3] J. Par and B. Km, The school bus roung problem: A revew, European Journal of Operaonal Research, Vol. 202(2010), [4] B. Benne and D. Gazs, School bus roung by compuer, Transporaon Research. Vol. 6(1972), [5] L.D. Bodn and L. Berman, Roung and schedulng of school buses by compuer, Transporaon Scence Vol.12, No. 2(1979), [6] H.-S. Tsay and J.D. Frcer, Praccal Approach for Solvng School Bus Problems, Transporaon Research Record, 1202(1988) [9] Z.W. Geem, K.S. Lee and Y. Par, Applcaon of harmony search o vehcle roung, Amercan Journal of Appled Scence 2, 12(2005), [10] Abou Harmony Search, hp://harmonysearch.nfo AUTHOR PROFILES Taehyeong Km receved he degree n ransporaon engneerng a he Unversy of Maryland n he U.S. Currenly, he s a senor researcher a Korea Insue of Consrucon Technology. Hs research neres covers opmzaon, pararans, logscs, and smulaon. Bum-Jn Par receved he degree n ransporaon engneerng a he Yonse Unversy n Korea. Currenly, he s a senor researcher a Korea Insue of Consrucon Technology. Hs research neres covers nellgen ransporaon sysems, raffc flow, and nformaon echnology. [7] R. Bowerman, B. Hall and P. Calama, A mulobjecve opmzaon approach o urban school bus roung: formulaon and soluon mehod, Transporaon Research Par A. Vol. 29A, No. 2(1995),