A MULTIPERIOD EXPECTED COVERING LOCATION MODEL FOR DYNAMIC REDEPLOYMENT OF AMBULANCES

Transcription

1 Advanced OR and AI Mehods in Transporaion A MULTIPERIOD EXPECTED COVERING LOCATION MODEL FOR DYNAMIC REDEPLOYMENT OF AMBULANCES Hari K. RAJAGOPALAN, Cem SAYDAM, Jing XIAO 2 Absrac. Emergency response adminisraors ofen face he difficul ask of locaing a limied number of ambulances in a manner ha will yield he bes service o a consiuen populaion. In his sudy we ry o deermine he minimum number of ambulances ha mee or exceed a predeermined coverage requiremen for dynamic redeploymen of ambulances in response o flucuaing demands hroughou he week, depending on he day of week, and even he ime of day. We inroduce an incremenal search algorihm o solve he model and evaluae he effeciveness of our model wihin he framework of an experimenal design. Background The goal of emergency medical services (EMS) is o reduce moraliy, disabiliy, and suffering in persons [9, 8]. EMS adminisraors and managers ofen face he difficul ask of locaing a limied number of ambulances in a way ha will yield he bes service o a consiuen populaion. Typically, calls originaing from a populaion cener are assumed o be covered if hey can be reached wihin a ime hreshold. This noion of coverage has been widely acceped and is wrien ino he EMS Ac of 973, which requires ha in urban areas 95 percen of requess be reached in 0 minues, and in rural areas, calls should be reached in 30 minues or less [3]. The sudy of locaion models for ambulance locaion has undergone wo disinc phases of evoluion. During he firs phase mosly deerminisic models emerged and probabilisic and more realisic models emerged during he second. Undoubedly advances in hardware, commercially available solvers (e.g., CPLEX [9]), and mea-heurisics have propelled researchers o develop increasingly more realisic and sophisicaed models [2, 8, 4, 6, 8, 22, 38]. Ineresed readers can race he earlier advances in reviews by Shilling, The Universiy of Norh Carolina a Charloe, Business Informaion Sysems and Operaions Managemen, Deparmen, Charloe, NC 28223, USA, {hrajagop, saydam}@uncc.edu 2 The Universiy of Norh Carolina a Charloe, Compuer Science Deparmen, Charloe, NC 28223, USA, xiao@uncc.edu

2 622 H. K. Rajagopalan e al. Jayaraman, and Barkhi [39], Owen and Daskin [3], and find recen and projeced developmens in he laes review by Brocorne, Lapore and Seme [9]. Probabilisic models acknowledge he possibiliy ha a given ambulance may no be available when i is called. These ypes of models provide a way o model uncerainy by eiher using a queuing framework or via a mahemaical programming approach. The models based on a mahemaical programming approach employ simplifying assumpions such as all unis operae independenly, ye have he same busy probabiliy. For example server independence and sysem-wide server busy probabiliy is a common assumpion used in Daskin s maximum expeced coverage locaion problem (MEXCLP) [2] and ReVelle and Hogan s maximum availabiliy locaion problem (MALP) [35]. In heir second MALP formulaion (MALPII), ReVelle and Hogan allow server busy probabiliies o be differen in various neighborhoods, secors of a given region bu no locaion specific. The Probabilisic Locaion Se Covering Problem (PLSCP) formulaed by ReVelle and Hogan [36] minimized he number of servers needed o guaranee coverage for a ciy. This model like MALPII uses neighborhood server busy probabiliies. Laer Marianov and ReVelle [26] exended PLSCP using he assumpion of neighborhood probabiliies in MALPII o formulae Queuing Probabilisic Locaion Se Covering Problem (Q-PLSCP). They model each neighborhood as a muli-server loss sysem and calculae he neighborhood busy probabiliies a priori and hen use i as an inpu ino he sysem. True probabilisic models are based on spaially disribued queuing heory [23] or simulaion [4] and hey are by definiion descripive. Typically, hey are employed o evaluae he vehicle busy probabiliies and oher performance merics of a given allocaion of ambulances. Larson s hypercube model [23, 24, 35] represens he mos noable milesone for approaches using a queuing framework. The hypercube model and is various exensions have been found paricularly useful in deermining performance of EMS sysems [4, 0,, 3, 8, 23, 25, 34, 37, 40]. However, hypercube is compuaionally expensive. For m servers he number of simulaneous equaions o solve would be 2 m. For flee sizes of 0 or more his approach would be compuaionally impossible o solve wih he presen day compuers. To solve his problem Larson developed an approximaion o he hypercube problem [24] which would require soluion only m simulaneous nonlinear equaions for m servers. One of he assumpions used in Larson s approximaion is ha service imes are exponenially disribued and idenical for all servers, independen of he cusomers hey are serving. Jarvis generalized Larson s approximaion for loss sysems (zero queue) by allowing service ime disribuions o be of a general ype and may depend on boh server and cusomer [2]. The lieraure of EMS models has been iled owards he sraegic decision making, where EMS managers make decisions on he locaion of ambulances or ambulance sies over a longer period of ime. These models eiher focused on finding ou he minimum number of ambulances needed o adequaely cover an area or he maximum coverage ha could be obained given a se of ambulances. Boh kinds of models were complimenary and serve differen purposes. Minimizaion models were used on deciding he size of he ambulance flee while maximizaion models were used o give an esimae of how good he sysem could perform over a period of ime. Since a long erm perspecive was aken flucuaions in demand in any given day were overlooked insead selec peak demand periods were used as an esimae for demand. However, in realiy, demand is no saic bu flucuaes hroughou he week, day of week, and even hour by hour wihin a day [8, 32].

3 A muliperiod expeced covering locaion model for dynamic 623 Redeploymen models however look a operaional level decisions managers make daily or even hour by hour in an aemp o relocae ambulances in response o demand flucuaions by ime and space. There have been wo earlier papers on relocaion in he EMS lieraure [6, 33]. Repede and Bernardo [33] exended MEXCLP for muliple ime inervals o capure he emporal variaions in demand and uni busy probabiliies, hence, called heir model TIMEXCLP. Their applicaion of TIMEXCLP o Louisville, Kenucky resuled an increase of coverage while he average response ime decreased by 36%. The mos recen and comprehensive dynamic relocaion model is developed Gendreau e al. [6]. The objecive of heir dynamic double sandard model a ime (DDSM ) is o maximize backup coverage while minimizing relocaion coss. There are several imporan consideraions incorporaed ino his model. While he primary objecive is o maximize he proporion of calls covered by a leas wo vehicles wihin a disance hreshold, he model penalizes repeaed relocaion of he same vehicle, long round rips, and long rips. The model s inpu parameers are updaed each ime a call is received and DDSM is solved. To solve his complex model, paricularly a shor ime inervals, Gendreau, Lapore and Seme developed a fas abu search heurisic implemened on eigh parallel Sun Ulra worksaions. Using real daa from he Island of Monreal, heir ess showed ha he algorihm was able o generae new redeploymen sraegies for 95% of all cases. Furhermore, comparisons wih CPLEX generaed exac soluions showed ha he wors case deparure from opimaliy was only 2%. Apar from hese sudies [6, 33], here has been very lile work done concerning he periodic relocaion (redeploymen) of ambulances in an environmen where demand and he locaion and quaniy of available ambulances are changing. Brocorne, LaPore and Seme predic ha curren and fuure advances in his field are likely o be in probabilisic locaion models, dynamic redeploymen models, and fas heurisics designed o solve generally large scale problem insances [9]. In his paper, drawing from earlier developmens by Daskin [2] and Marianov and ReVelle [27] we formulae a new model which ries o deermine he minimum number of ambulances ha mee or exceed a predeermined expeced coverage requiremen for dynamic redeploymen of ambulances in response o changing demand. Similar o Galvão, Chiyoshi and Morabio [4] we increase he realism of our model by compuing server specific busy probabiliies using Jarvis hypercube approximaion [2]. We develop a an incremenal search algorihm o find he minimum number of ambulances which calls an implemenaion of reacive abu search [5] o deermine he bes possible ambulance locaions for a given flee size. We es our algorihm on a se of muli-period es problems wihin an experimenal design framework. This paper is organized as follows. Secion 2 presens he dynamic expeced coverage problem. Secion 3 deails he search algorihm. Secion 4 presens he numerical experimens and conclusions and suggesions for fuure work are discussed in Secion The model We formulae he dynamic expeced coverage locaion (DECL) model o deploy minimum number of ambulances o guaranee a sysem wide coverage under dynamic demand condiions. Le be he index of ime inervals from o T, x i,j be he number of servers

4 624 H. K. Rajagopalan e al. (ambulances) locaed in node i a ime inerval, h j, be he fracion of demand a node j a ime inerval, m be he oal number of servers available for deploymen a ime inerval, n be he number of nodes in he sysem, and c be he minimum expeced coverage requiremen a ime. Also le, p i, be he busy probabiliy of a server a node i a ime inerval, ρ be he average sysem busy probabiliy a ime inerval. P 0 be he probabiliy of having all servers free, P m be he probabiliy of having all servers busy in an M/M/m/0- loss sysem, and Q(m, ρ, j) be he Q facor for Jarvis algorihm which adjuss he probabiliies for server cooperaion in he model. Le, a ij, y ij, Subjec o: if node j is covered by i servers during ime inerval = 0 if no if node j is covered by server a node i during ime inerval = 0 if no n m Q j = i = T n x i = i= Minimize Z = n m aij, xi, yij, j, i= i= ( Z p, i ) h y ( p ) p c j, ij,, () i, (3) n i, l= l, (2) x i, m (4) i= x i,, 0 andineger i (5) ij, { 0,} y (6) Objecive funcion () minimizes he oal number of ambulances over muliple periods and consrain (2) couns he number of ambulances ha cover each node. Consrain (3) ensures ha he sysem wide expeced coverage for a given ime inerval mees or exceeds he required coverage, c. In consrain (3) we uilize he Q facor from Jarvis hypercube approximaion algorihm o adjus for server cooperaion and compue ambulance specific busy probabiliies, and he resuling expeced coverage. The minimum coverage requiremen is likely o be he same for all periods whereas he demand volume and locaions are likely o be differen. Consrain (4) ensures ha he oal number of ambulances in he sysem can never exceed he oal number of ambulances available for he relaively shorer ime inervals. For example, he hour before he rush period of wo hours and he subsequen hree hours end o exhibi differen call volumes (low, high, medium) and call locaions (e.g., suburbs, major roads or highways, downown).

5 A muliperiod expeced covering locaion model for dynamic Search algorihm There have been various aemps o idenify near-opimal soluions for locaion problems hrough he use of mea-heurisic search mehods [28, 29] and more recenly, evoluionary algorihms [2, 6-8, 4-6, 20, 37, 38]. The models wih consan busy probabiliy can be solved by ineger linear programming bu hey may sill ake oo much ime. Ineger linear programming solvers using he laes simplex, barrier, and branch and bound implemenaions guaranee opimum soluion and on average are fas bu in wors case scenarios can run as an exponenial ime algorihm []. Furher, all inpus mus be known a priori, before he algorihm sars running. I is no possible o use linear programming echniques when he parameers mus be calculaed a run ime. This is he case when he models require individual busy probabiliies for servers. Since hese values are dependen on server locaion, hey have o be calculaed a run ime. As shown by Galvão, Chiyoshi and Morabio he coefficiens in he consrain se (3) depend on he values for he decision variables x i, and Z, herefore, can only be solved by a heurisic algorihm. Therefore, i is absoluely necessary o develop effecive and efficien heurisic search echnique o guaranee good or near opimal soluions in a very shor compuing ime. Mos models in he lieraure ha are solved using mea-heurisics have one objecive such as o find he maximum coverage wih a given se of servers. The model developed in his paper, unlike ohers in he lieraure, has wo search objecives: () o find he minimum number of servers, and (2) given he number of servers, find he opimal or near opimal locaions ha mee or exceed coverage or availabiliy requiremens. Therefore, we canno direcly benefi from he experiences of mea-heurisics applied in he general domain. To solve for hese wo inerdependen objecives we develop a search heurisic which we call he Incremenal Search Algorihm (ISA). ISA has a hierarchical srucure where he firs objecive of finding he minimum number of servers can be viewed as he main search (ouer) loop, whereas he second objecive is nesed as an inner search loop. The main search (ouer) loop incremens or decremens he flee size based on he resuls from he inner loop (mees or does no mee he average requiremens.) For he inner loop of he ISA we implemen a variaion of Reacive Tabu Search (RTS) [5] in which we embed Jarvis algorihm o compue server specific workloads and coverage saisics. For he ISA we use a one dimensional daa srucure (an array) of size m * + where m is he number of response unis (servers) and is he number of ime inervals. The array sars wih he coverage value and he server locaions for ime inerval and hen coninues wih he coverage value and locaion of he servers for all ime inervals. The ISA works in he following way. A ieraion 0, we solve for m = λ /( ρ µ ) where m is he number of servers (flee size) a ime, λ is he arrival rae a ime, ρ is he average busy probabiliy of servers. The value of m is he iniial size of he search vecor. We hen randomly generae server locaions and evaluae he hard coverage consrain using he Jarvis algorihm wihin equaions. We hen use a RTS o find he se of locaions which gives us he bes coverage. If he bes coverage given by RTS is less han he required coverage, hen we increase he number of servers and coninue RTS wih he new se of servers and coninue his process unil he coverage consrain i me. If he coverage for he iniial number of servers is more han he required coverage afer RTS hen we

6 626 H. K. Rajagopalan e al. reduce he number of servers by one and coninue RTS unil he coverage drops below he required coverage, in which case he previous soluion is he required number of servers. As saed earlier, deermining he opimal locaions in hese np-hard problem domains is compuaionally challenging. We chose o implemen a variaion of he TS called Reacive Tabu Search (RTS) [5] o obain opimal or near opimal resuls. Tabu Search (TS) is a mea-heurisic search mehod developed by Glover and Laguna [7]. A unique feaure of TS is is use of a memory, or lis. Once a soluion is enered ino a TS memory, hey are hen abu, or disallowed, for some ime. The idea is o resric he algorihm from visiing he same soluion more han once in a given ime period. The exploraion pressure in TS is is abiliy o accep a worse soluion as i progresses hrough is search. Tabu search has been successfully applied o various problem domains [30], including covering locaion models [5, 6]. In RTS he memory size is deermined hrough feedback during he search. All pairs are sored in he long erm memory. Iniial abu size is se o one. When a previously visied configuraion in he long erm memory reappears abu size increases o include ha configuraion and if a configuraion in he abu lis is no repeaed for 2m ieraions, where m is he curren number of servers, i is removed from he abu lis. The basic operaion in RTS or any oher abu search involves relocaing an ambulance from a given node (i) o anoher node (j) where node j is he bes locaion in he neighborhood. The pair (i,j) become abu as long as he abu is effecive. Neighborhood for his sudy is he eigh nodes immediaely surrounding he seleced node. For a given flee size, he firs ambulance is seleced for he basic operaion which deermines he bes node in he neighborhood o relocae his vehicle. The bes node is seleced based on he bes coverage calculaed by equaion (3). This makes up he firs ieraion. Nex, he second ambulance is seleced for he basic neighborhood search and he process repeas unil he las ambulance is seleced; hen, he firs ambulance is seleced again. Throughou his search, he size of he abu lis changes according o he exploraion or exploiaion pressure needed. The sopping rule for his implemenaion of RTS is 00 ieraions. This number was deermined afer running a se of sample problems for a long period of ime (000 ieraions and more) which showed ha he incremenal gains afer 00 ieraions were negligible. The se of locaions during he 00 ieraions which resuled in he maximum coverage is sored. This soluion which is made up of he flee size, ambulance locaions, and he resuling coverage is passed o he main algorihm, ISA. The major overheads for his search process are () searching for he minimum number of servers and (2) using Jarvis algorihm o calculae he coverage for every possible relocaion while searching for he bes locaions for he curren flee size The look-ahead procedure (LAP) reduces he compuaion ime by addressing he above wo problems. ISA using RTS wih LAP follows he seps described below. ISA sars wih he same iniial soluion and uses RTS wih LAP (RTSLAP) insead of he normal RTS o calculae he bes locaions for a given flee size. During he neighborhood searches in RTSLAP insead of using Jarvis algorihm i uses a sysem wide average busy probabiliy expeced coverage compuaion. Therefore, RTSLAP unlike RTS does no use Jarvis algorihm for each ambulance relocaion and hus reduces he compuaion ime. Only a he end of 00 ieraions of he RTSLAP Jarvis algorihm is called o compue more accuraely ambulance specific busy probabiliies and he resuling coverage. This soluion which is made up of he flee size, ambulance locaions, and he resuling coverage is passed o he main algorihm, ISA.

7 A muliperiod expeced covering locaion model for dynamic 627 Once a soluion is reached which saisfies he coverage consrains, his is reaed as a good inermediae soluion. However, he prescribed ambulance locaions may be subopimal since RTSLAP uses an average busy probabiliy while searching for ambulance locaions. We view his as a good iniial soluion for ISA and re-solve he problem using full ISA wih RTS. RTSLAP procedure brings he ISA algorihm much closer o number of servers required o saisfy he coverage consrain faser han earlier algorihm wihou LAP. The algorihm for he enire search heurisic is available upon reques from he corresponding auhor. 4. Experimens and resuls We es our model, DECL, via an experimenal design framework which allows objecive analyses and saisically significan resuls, whenever applicable. The model is esed on wo imporan merics: () qualiy of soluions, and (2) CPU imes. The qualiy of soluion is measured by a comprehensive discree even simulaion model. Tha is, he soluion prescribed by he ISA is fed o he simulaion model which hen produces he rue coverage saisics along wih individual server busy probabiliies. The difference beween he coverage me or exceeded by our model for a given minimum flee size and he coverage calculaed via he simulaion model is he error margin for our model. If any of he soluions prescribed by our model does no mee he coverage requiremens when compued by he simulaion model hen we declare ha soluion as subopimal or unaccepable. The second meric is simply he CPU ime for he ISA. The experimens are conduced on a Dell PC Penium IV 2.4 MHz wih 52 MB RAM. he ISA is coded in Java (jdk.4). We generaed weny es problems (0 each for 64 and 256 demand nodes). We consider a problem wih wo disinc periods of demand paerns. During he firs ime period he demand is somewha evenly (uniformly) disribued across he region and during he second ime period he demand is non-uniformly disribued, reflecing he shifs in call volume due o relocaion of populaion from residenial areas of he region o employmen and perhaps educaion ceners (Figure ) Call Volume Y Coord X Coord 30 Figure. Demand disribuion for = 2.

8 628 H. K. Rajagopalan e al. The resuls from he 20 problems solved using ISA-RTS and ISA-RTSLAP are shown in Table. For each problem, he flee size for = and = 2, corresponding expeced coverage (%) compued via he simulaion model are shown. Simulaed coverages from all runs show ha he model along wih wo soluion approaches me or exceeded he required coverage of 95% wih minimum flee sizes. Table 2 shows he run ime values. No. of Nodes Problem Number ISA-RTS ISA-RTSLAP Z Expeced Expeced Z Cov. (%) Cov. (%)

9 A muliperiod expeced covering locaion model for dynamic 629 Table 2 summarizes he CPU imes. As anicipaed ineresing resul is ha he Look Ahead Procedure (LAP) reduces he run ime significanly wihou compromising from soluion qualiy. On average he ISA wih LAP runs 2.05 o2.33 imes faser ha ISA wihou LAP. More imporan is ha a larger size problems (256 node) LAP gives higher savings (2.33 imes faser) as compared o smaller size (64 node problems, run 2.05 imes faser). This means ha when we sar solving even larger and larger problems using LAP will give us greaer savings. No. of Nodes ISA-RTS ISA-RTSLAP , (22.48) 0.00, (28.93) , (82.45) , (52.7) Table 2. CPU imes in seconds (average, sd. dev.) 5. Conclusions In his paper we formulaed a new model for dynamic environmens wih he objecive of deploying he minimum number of ambulances required o mee mandaed coverage requiremens. Our model uses he expeced coverage concep developed by Daskin [2] and exended by Galvão e al. [4]. Similar o heir approach, we enhance he model s realism by compuing server specific busy probabiliies using Jarvis hypercube approximaion [2]. We developed an incremenal search algorihm (ISA) o solve he new model. In he ISA we implemened a reacive abu search [5] o deermine he ambulance locaions and chose o embed Jarvis hypercube approximaion algorihm in our heurisic o compue ambulance specific busy probabiliies. To minimize he compuaional burden of he combined (nesed) algorihms, we developed a look ahead procedure (LAP). We compared he soluion qualiy in expeced coverage using a comprehensive discree even simulaion model. Our preliminary findings show ha he new model produces high qualiy soluions while he ISA wih he LAP is shown o solve problems wih 256 nodes on average less han 0 minues versus wihou he LAP which akes an average of over 2 minues. Comparisons wih he 64 node problems sugges ha compuaional gains increase as he problem size grows. Undoubedly our model can be solved by anoher mea-heurisic such as geneic algorihms which could be he scope of a fuure comparaive sudy. References [] Ahuja, R.K., Magnani, T.L., and Orlin, J.B., Nework Flows Theory, Algorihms and Applicaions. 993, New Jersey: Prenice Hall. [2] Ayug, H. and Saydam, C., Solving large-scale maximum expeced covering locaion problems by geneic algorihms: A comparaive sudy. European Journal of Operaional Research, : p [3] Ball, M.O. and Lin, L.F., A reliabiliy model applied o emergency service vehicle locaion. Operaions Research, : p

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