Optimal Ambulance Location. with Random Delays and Travel Times

Transcription

1 1 Optmal Ambulance Locaton wth Random Delays and Travel Tmes Armann Ingolfsson 1, Susan Budge, Erhan Erkut Unversty of Alberta School of Busness Edmonton, Alberta, T6G 2R6 Last revson: March 2006 Abstract We descrbe an optmzaton model for ambulance locaton that maxmzes the expected system wde coverage, gven a total number of ambulances. The model measures expected coverage as the fracton of calls reached wthn a gven tme standard and consders response tme to be composed of a random delay (pror to travel to the scene) plus a random travel tme. Pre-travel delays at dspatch and actvaton stages can be sgnfcant, and models that do not account for such delays can severely overestmate the possble coverage for a gven number of ambulances and underestmate the number of ambulances needed to provde a specfed coverage level. By explctly modelng the randomness n the delays and the travel tme, we arrve at a more realstc model for ambulance locaton. In order to capture the dependence of ambulance busy fractons on the allocaton of ambulances between statons, we terate between solvng the optmzaton model and usng the approxmate hypercube model to calculate busy fractons. We llustrate the use of the model usng actual data from Edmonton. Key words: emergency servces, ambulance locaton, faclty locaton, dspatch delays 1 Correspondng author (e-mal: armann.ngolfsson@ualberta.ca)

2 2 Introducton The desgn of emergency medcal servce (EMS) systems nvolves several nterconnected strategc decsons, such as the number and locatons of ambulance statons, the number and locatons of the vehcles, and the dspatch system used. In ths paper we focus on the allocaton of vehcles to a set of (exstng or planned) ambulance statons wth known locatons. The man concern n an EMS system s the response tme to calls. The most obvous and sgnfcant component of response tme s the travel tme between the ambulance staton and the demand locaton. Almost all of the exstng operatons research lterature on ambulance locaton focuses on travel tmes, but ths s not the only component of the response tme, whch s generally defned as the tme from when a call for ambulance servce arrves untl paramedcs reach the patent. Therefore, the response tme ncludes any delays pror to the trp. Such delays can nclude tme spent on the phone obtanng the address and establshng the serousness of the call, tme spent decdng whch ambulance to dspatch, tme to contact the paramedc crew of that ambulance, and tme for the paramedc crew to reach ts ambulance and start t. Queueng delays (when no ambulances are avalable) can also occur, but they occur nfrequently. In the rare stuatons when all ambulances are busy, ncomng calls are typcally responded to usng some type of backup system, such as supervsor s vehcles or fre engnes. An overrdng ssue when desgnng an EMS system s the coverage provded, and a common performance target s to respond to (or cover) a fracton α of all calls n δ mnutes or less (for example 90% n under 9 mnutes). Our paper s motvated by the observaton that the estmated coverage depends on the way delays and travel tmes are modeled. Appendx A of the onlne supplement provdes a smple numercal example that llustrates the relevant ssues, ncludng:

3 3 Not accountng for varablty n travel tmes can result n large errors. For example, f all demand nodes are at an average travel tme of 9.01 mnutes away from the staton, then a determnstc model estmates zero coverage whle a probablstc model estmates roughly 50% coverage, assumng the response tme dstrbuton s close to beng symmetrc. Although negatve and postve errors at ndvdual demand locatons may cancel each other to some extent when computng the total expected number of covered calls, the error n ths system performance estmate can be consderable (around 40% n the example n the Appendx when the pre-trp delays are ncluded). A probablstc model s a better representaton of realty, and the use of determnstc travel tmes n ambulance locaton models ntroduces avodable errors. Ignorng pre-travel delays entrely results n large errors. When one models randomness n travel tmes, gnorng randomness n the duraton of delays causes smaller errors than gnorng delays altogether. The drecton of the change n probablty of coverage when one ncorporates randomness n delay duratons s not always the same, as llustrated n the onlne supplement. We beleve that these errors can nfluence decsons adversely when every percent counts n tryng to reach a coverage target. For nstance, n a recent proect we completed for the Cty of Edmonton, Alberta (Ingolfsson et al., 2003), current coverage was 87% and most ndvdual system desgn changes had mpacts on the order of one percentage pont or less. To be useful n such stuatons, prescrptve models must be able to dscrmnate correctly between system desgns wth coverage dfferences of one percentage pont or so.

4 4 In ths paper we ntroduce new methodology that ncorporates randomness n both pre-travel delays and travel tmes and s therefore free of the errors demonstrated n the example n the onlne supplement. Ths paper s motvated by two real-world ambulance locaton proects that we completed recently the Edmonton proect mentoned above and another conducted n St. Albert, a town of 50,000, near Edmonton. We use data from the latter study n ths secton. We have analyzed data from approxmately 6,997 EMS calls servced n over 4 years n St. Albert. Fgure 1 dsplays the emprcal dstrbuton of pre-trp delays, whch s well approxmated by a lognormal dstrbuton. The delays ranged from 20 seconds to 20 mnutes, wth an average of 175 seconds and a standard devaton of 95 seconds. Lmtng the analyss to calls classfed as heart and respratory (.e., hgh prorty) yelded almost the same mean and standard devaton. The average delay of almost 3 mnutes s a very substantal fracton of the 9-mnute response tme standard, and the varaton n the delay s too large to gnore (the standard devaton s more than 50% of the mean). Green and Kolesar (1989) report delays smlar to the ones that we are concerned wth. They found unexpected dspatch delays when valdatng a queueng model of polce patrol n New York Cty. They found that about 50% of calls experenced dspatch delays averagng about 4 mnutes. Henderson and Mason (2004) had a smlar experence. They report that for many of the calls, a large amount of tme s spent before an ambulance s dspatched to a call and dscuss the mpact that ths has on the ablty to meet the coverage goals as well as the potental to acheve a consderable mprovement n performance wth only small decreases n these pre-trp delays. Anyone that has experence wth real emergency servce systems wll be aware of the presence of such delays, and several past researchers have mentoned them (see, for example,

5 5 several of the chapters n Wlleman and Larson, 1977, and Brandeau and Larson, 1986) suggestng, n some cases, that such delays are neglgble, and n other cases that they can be ncorporated n exstng models by addng the average delay to the average travel tme. Probablty Emprcal Lognormal Pre-trp delay (sec.) Fgure 1: Emprcal cumulatve dstrbuton functon of pre-trp delays for 6,997 EMS calls servced n St. Albert, and a ftted lognormal dstrbuton. The St. Albert dataset contans multple trps to several locatons, whch allows us to analyze dstrbutons of travel tmes. Fgure 2 shows the emprcal dstrbuton of travel tmes for 352 trps from a partcular staton to the same multple-resdent demand pont. The trp tmes range from 55 seconds to 370 seconds, wth an average of 143 seconds and a standard devaton of 52 seconds. Of these 352 calls, 94 are classfed as heart and respratory. For these hgh-prorty calls the average travel tme s 126 seconds, ndcatng faster travel for hgh-prorty calls. However, the standard devaton s stll a very substantal 57 seconds. We analyzed a total of nne locatons wth multple trps and found that the standard devaton was always consderable (on average 40% of the mean). Reportng on a proect for locatng emergency vehcle bases n Tucson, Arzona, Goldberg et al. (1990a) also found substantal varaton n emprcal travel

6 6 tmes for gven base-demand zone pars. Ths varaton can be due to varablty n the effectve travel speed, or due to randomness n the locaton of the ncdent (demand aggregaton). Probablty Emprcal Lognormal Travel tme (sec.) Fgure 2: Emprcal cumulatve dstrbuton functon for travel tmes between a partcular staton and demand pont par for a total of 352 trps, together wth a ftted lognormal dstrbuton. To summarze, when analyzng response tme data, we notced that delays can be sgnfcant and hghly varable, and that travel tmes between a gven par of ponts are hghly varable. We conclude that a convoluton of the delay and travel tme dstrbutons s needed to obtan an accurate response tme dstrbuton, assumng travel tme and delay are statstcally ndependent an assumpton that s supported by the data that we worked wth. Stuatons where the travel tme and delay are dependent can be handled as well, as we wll demonstrate. We beleve that the explct modelng of the uncertanty n travel tmes s an mportant feature of ths paper. In addton, our model s ntended to overcome three lmtatons of exstng models that gnore ether delays or the randomness n delays.

7 7 Frst, models that gnore delays or randomness n delays may severely overestmate the coverage acheved wth a gven number of ambulances and, conversely, underestmate the number of ambulances needed to meet a specfed coverage obectve (see Fgures 4 and 5 n the Computatonal Experments secton). Second, for a gven number of ambulances, exstng models may prescrbe a suboptmal dstrbuton of ambulances to statons. Thrd, exstng models do not enable predcton of the consequences of reducng delays. Ths last pont s mportant because delays can be far easer and less costly to reduce than travel tmes. It mght be possble to reduce delays through smple process changes, such as dspatchng an ambulance before the serousness of the call has been establshed (thereby performng two actvtes n parallel rather than n seres), or through the ntegraton of 911 and EMS call centers (thus elmnatng hand-off tme from one call center to the other), whereas reducng travel tmes usually requres addng ambulances or statons. Our model can help compare the costs and benefts of actons to reduce delays versus actons to reduce travel tmes. Ths s valuable for decson-makers who are nterested n the least-costly way of reachng servce standards. As far as the response tme standard s concerned, 30 seconds saved are 30 seconds saved, regardless of whch component of the response tme these savngs come from. There s an extensve lterature on optmal locaton of ambulances. Yet very few papers model the randomness n travel tmes, and we know of no papers that ncorporate randomness n pretrp delays nto an optmzaton model. We consder both omssons serous mpedments to applyng optmzaton models to ambulance locaton, and we beleve our model s a frst step n overcomng these shortcomngs.

8 8 In the remander of the paper, we dscuss the relevant lterature, and then descrbe the problem data, our problem formulaton, some useful propertes of the formulaton, the results of computatonal experments, and further research that we ntend to undertake to extend and experment wth the model. Lterature There s an extensve lterature on locatng emergency servce facltes. Wlleman and Larson (1977), Swersey (1994), and Maranov and ReVelle (1995) provde revews of ths area. Berman and Krass (2001) revew the lterature on faclty locaton wth stochastc demands, much of t motvated by emergency servce applcatons. In ths secton we survey selected papers wth an emphass on those that are most relevant to our research. Past models can be usefully characterzed as prescrptve or descrptve. Ths dstncton s not perfect, because every mathematcal model of EMS operatons provdes predctons of performance, as a functon of decson varables such as the number of ambulances at each staton, and every such mathematcal model allows one to experment wth the decson varables to search for a better confguraton. All models make smplfyng assumptons, for varous reasons. At one extreme are models that make strong smplfyng assumptons n the nterest of makng t possble to fnd optmal or near-optmal confguratons for large problem nstances usng general purpose mathematcal programmng solvers. At the other extreme are models whose focus s on accurately predctng the performance for a partcular confguraton. Even though some models fall n the mddle between these two extremes, many models can be usefully classfed as ether prescrptve (where the focus s on makng optmzaton possble) or descrptve (where the focus s on accurate predcton of performance measures). Descrptve models are typcally ether analytcal queueng models or smulaton models.

9 9 Related to the dscusson of prescrptve and descrptve models s problem sze. For ambulance locaton models, the number of demand nodes and the number of statons are the prmary determnants of problem sze. Demand s typcally aggregated nto demand nodes, n part to provde a reasonable problem sze. The number of demand nodes s nfluenced by the sze of the geographc regon, the populaton, and the method used to dvde the regon nto demand nodes,.e., the demand aggregaton method. The number of statons s nfluenced by the sze of the regon, the sze of the populaton, the level of fundng, and by operatng polces (for example, f ambulances can wat on street corners for the next call, then there would be more possble statons ). Both the number of demand nodes and the number of statons wll nfluence the tme to evaluate a sngle soluton, but only the number of statons (and not the number of demand nodes) wll nfluence the sze of the soluton space for a prescrptve model. Moreover, the number of statons wll mpact the sze of the problem for a prescrptve model n a combnatoral fashon. Gven that the number of demand nodes can be manpulated va preprocessng (aggregaton) and that ths number s expected to mpact the evaluaton tme for a sngle soluton approxmately lnearly, the determnng factor for computatonal effort for a prescrptve model s the number of statons. Most of the prescrptve models use an all-or-none noton of coverage, where a demand pont s consdered covered f the closest ambulance staton s wthn some specfed maxmum dstance. The obectve of the set-coverng locaton problem (SCLP), frst formulated by Toregas et al. (1971), s to mnmze the number of statons such that all demand ponts are covered. Although ths s a bnary problem, the LP relaxaton (or the addton of a smple cuttng plane) usually generates all-nteger solutons. By changng the coverage dstance, one can generate a number of solutons wth varyng number of facltes.

10 10 Whle SCLP has been used n several locaton studes, t has a number of shortcomngs. For example, the requrement of coverng every demand pont s rather strngent and usually results n the locaton of an unreasonably hgh number of facltes. To address ths problem, Church and ReVelle (1974) extended SCLP by proposng the maxmal coverng locaton problem (MCLP) where the goal s to maxmze the proporton of the demand covered wth a fxed number of facltes. The LP relaxaton of ths bnary problem s reported to result n all-nteger solutons most of the tme. One can solve MCLP parametrcally n the number of facltes and obtan a cost-coverage tradeoff curve. Unlke SCLP, MCLP dfferentates between demand ponts based on relatve demand and t s able to trade off system coverage and resources. Hence, t s better suted for emergency servce faclty locaton than SCLP, and there are several reported applcatons. However, the classfcaton of a demand pont that s wthn a specfed dstance of a staton as covered makes the mplct assumpton that there s always a vehcle at the staton to respond to a call. Whle most emergency response systems are desgned for low utlzaton levels, n many ctes ambulances are busy a sgnfcant porton of the tme (for example, 30%). To account for the potental unavalablty of ambulances, Daskn (1983) extended MCLP by formulatng the maxmum expected coverng locaton problem (MEXCLP), whch maxmzes the expected value of populaton coverage for a fxed number of servers. MEXCLP uses a sngle, system-wde busy probablty, and computes the probablty of a subset of busy vehcles from a gven staton usng the bnomal dstrbuton. Whle the model s an nteger program wth a nonlnear obectve functon, t can be lnearzed, and nstances of realstc sze can be solved wth general-purpose nteger programmng solvers.

11 11 Revelle and Hogan (1989) also attempted to account for ambulance unavalablty by extendng MCLP n a dfferent drecton, through ther maxmum avalablty locaton problem (MALP), whch maxmzes the populaton that s covered wth α relablty. Unfortunately, ths obectve functon s nconsstent wth the expected coverage performance measure that drves most EMS systems n practce. See Erkut et al. (2006) for a crtque of MALP and related models. Although there are many prescrptve ambulance locaton models n the lterature, the four models dscussed above can be consdered the most nfluental ones on subsequent research, snce most other models are extensons of these four. Whle many of these prescrptve models can be solved to global optmalty wth reasonable effort, they suffer from smplfyng assumptons. On the other hand, descrptve models provde more realsm. The man descrptve model that s relevant for our purposes s the hypercube model developed by Larson (1974) and subsequent approxmate versons of that model (Larson, 1975 and Jarvs, 1985). Ths model allows busy fractons to vary between ambulances and can accommodate ambulances respondng to calls outsde ther assgned dstrcts. Larson (1979), and Brandeau and Larson (1986) descrbe applcatons and extensons of the hypercube model. We use an extenson of the approxmate hypercube model that allows multple servers at a staton (Budge et al., 2005). Dscrete event smulaton can be used when even greater realsm s needed (e.g., Henderson and Mason, 2000 and 2004, and Ingolfsson et al., 2003). Fnally, some authors have combned descrptve models wth optmzaton heurstcs. Both Batta et al. (1989) and Saydam and Aytug (2003) combne the approxmate hypercube model wth heurstcs, the former usng a sngle node substtuton heurstc and the latter usng a genetc algorthm.

12 12 We extend the prescrptve modelng paradgm by ncorporatng randomness n response tmes, wthout sacrfcng the ablty to use general-purpose solvers to fnd optmal solutons. All of the prescrptve coverng models that we dscussed above use determnstc (average) travel tmes. Whle delays are usually not explctly mentoned n papers dealng wth prescrptve coverage models, t s easy to ncorporate a constant (average) delay nto all coverage models by smply subtractng the delay from the specfed maxmum response tme. (For example, Eaton et al. (1985) uses MCLP wth a 5-mnute travel tme, whch may have been part of an 8-mnute response tme wth an average delay of 3 mnutes.) The assumpton made by early coverng models s that f (and only f) an ambulance s avalable wthn a specfed maxmum dstance of a demand pont, then the demand pont s covered. EMS systems typcally measure performance based on the fracton of calls responded to wthn a specfed tme standard. However, for a gven ambulance locaton and a demand pont, t s not possble to know wth certanty whether the call wll be responded to wthn the tme standard t depends on the pre-trp delay and the travel tme as well as the avalablty of the ambulance, none of whch can be predcted wth certanty. Our model does not rely solely on average travel tmes, and hence, t s not lmted by the resultng strct classfcaton of demand ponts as covered or not covered. It allows ncorporaton of randomness n pre-trp delays and travel tmes, and computes an expected coverage for each demand pont, gven the ambulance locatons. Hence, we ncrease model realsm by replacng the 0-1 consequences mpled by solutons of tradtonal coverng models for demand ponts by real numbers, whch are better estmates of the fracton of calls emanatng from dfferent demand ponts that can be reached wthn the specfed tme standard.

13 13 In the remander of ths secton, we focus on ambulance locaton models that ncorporate response tme varablty. As we mentoned above, a constant pre-trp delay can be ncorporated nto all coverng models. However, we know of no papers n the lterature that ncorporate random delays nto a prescrptve model. We are aware of three nstances where travel tme varablty was ncluded n coverng models. Maranov and ReVelle (1996) assume travel tme from staton to node s normally dstrbuted wth known mean and varance. Then they defne a node to be covered by staton f the average travel tme plus K standard devatons s less than a specfed constant. Whle they acknowledge the varablty n travel tmes, they do not use the dstrbutons drectly n the model. Ths model s more conservatve (for K > 0) than a coverage model that uses the average travel tmes only. However, t s stll a tradtonal coverng model n the sense that a demand pont s ether covered or not. Perhaps the paper that s most relevant to ours s Goldberg and Paz (1991), whch s nspred by a case study reported n Goldberg et al. (1990a) and Goldberg et al. (1990b). They formulate an emergency faclty locaton model that ncludes the probablty P that an ambulance at staton can travel to a call from demand node wthn a response tme standard. Ths quantty s used to calculate expected coverage n the obectve functon of ther optmzaton problem. Daskn (1987) models random travel tmes smlarly, but the focus of hs model s the ntegraton of locaton and routng, takng nto account that some calls may requre two vehcles to respond. Daskn s model does not account for ambulance unavalablty and s qute large, even for small networks. Goldberg and hs co-workers used an approxmaton related to the hypercube model to estmate the busy probabltes of the vehcles, and ncluded an upper bound on the number of statons. They use regresson to estmate average travel tmes as a functon of dstance along

14 14 roads of varous types, and compute the P values usng ths mean and the standard devaton of the resduals, assumng normal dstrbuton of path travel tmes. Whle the way we model expected coverage s smlar to that of Goldberg and Paz (1991), there are several dfferences between ther work and ours. Perhaps the most sgnfcant modelng dfference s the ncluson of pre-trp delays n our model. Also, we treat the calculaton of the busy probabltes for the vehcles, and the computaton of coverage probabltes for demand ponts n dfferent ways. We consder dspatch polces as gven, rather than ncludng them as decson varables. For all of these reasons, our model s more compact and tractable and we are able to solve problems of realstc sze optmally usng off-the-shelf solvers, whle Goldberg and Paz (1991) propose parwse nterchange heurstcs for ther model. Problem Data We assume that the followng data are avalable: A set S of m staton locatons, ndexed by, and a set N of n demand nodes, ndexed by. A postve arrval rate λ for each demand node. We assume that the node arrval processes are ndependent Posson processes. We denote the system wde arrval rate wth λ λ N and the fracton of the total demand comng from demand node by h λ / λ. A dspatch order for each demand node,.e., a lst of the m statons n order of preference for dspatchng to a call orgnatng from node. Parameters δ and α whch specfy the coverage obectve that calls should be responded to n at most δ tme unts wth probablty of at least α.

15 15 The probablty w that the response tme R for a call that s responded to from the th staton (n node s dspatch order) to node s less than or equal to δ tme unts. The average on-scene tme, and average tme spent travelng to and remanng at a hosptal, denoted E[ T on scene], and E[ T hosptal], respectvely. The busy fracton ρ for ambulances at staton,.e., the probablty that an ambulance at staton s not avalable to respond to calls, and correcton factors Q for each statonnode par, to approxmately account for the dependence n the busy fractons between servers. We assume that ρ (0,1) and Q > 0. The last assumpton, that the busy fractons and correcton factors are exogenous nput to the model, s obvously a lmtng one. We dscuss how to overcome ths assumpton later. The best way to calculate the probabltes w depends on the avalablty of data and the context. We now outlne three possble methods. Frst, f detaled data for a sample of ndvdual calls s avalable, then one could estmate δ w as the rato k / k, where k s the total number of calls n the sample where an ambulance from staton responded to a call from node and k δ s the number of such calls that had a response tme less than or equal to δ. Second, suppose that the dstrbuton functon H ( t ) of the travel tme T from the th staton (n node s dspatch order) to node as well as the dstrbuton functon F( t ) for the delay are avalable, and that t s reasonable to assume that the travel tme and the delay are ndependent random varables. Then one can use convoluton to calculate the probabltes,.e., δ w = H (δ x) df( x) (1) x= 0

16 16 Thrd, suppose that both travel tmes and pre-travel delays depend on call prorty, but that for a gven prorty level, these two random varables are ndependent. Addng a superscrpt p, for prorty level, to the notaton defned n the precedng paragraph, and usng p v to denote the probablty that a call from node s of prorty p, then the calculaton n (1) would be adusted as follows: δ p p p = p x= 0 w v H (δ x) df ( x) The frst method s the most general n that t requres no ndependence assumptons, but t has two lmtatons: (1) the sample sze k mght be small or even zero for some staton-node pars, even f the overall sample s large, and (2) the method s slent about how one could predct the consequences of changes to the pre-travel delay dstrbuton. The second and thrd methods requre the ndependence assumpton, but they do not suffer from the two lmtatons ust mentoned. Note that the w are condtonal probabltes they assume that the call comes from demand node and s responded to by the -th preferred staton. Hgher system congeston makes t more lkely that less preferred statons respond to calls, and ths can nduce dependence between pretravel delays and travel tmes. Our model captures such dependence by combnng the condtonal probablty, w, wth the probablty f ( x) that the -th preferred staton responds to a call from node, as shown below. We emphasze that the calculaton of w s done for all staton-node pars, before solvng the optmzaton problem that we pose n the next secton. The optmzaton model requres no

17 17 nformaton about the probablty dstrbutons of travel tmes or delays other than the probabltes w. We wll assume that the dspatch order for each node s such that: w w K w (2) 1 2 m That s, the statons are arranged n descendng order of the lkelhood of respondng to a call from node n less than δ tme unts. Although dspatchng the closest avalable unt s not always optmal (see, for example, Larson, 1979), studes such as that by Jarvs (1981) ndcate that ths polcy s generally near-optmal. Our experence wth real EMS systems ndcates that devatng from closest-avalable-unt dspatchng would be dffcult n practce. The formulaton that we present n the next secton s vald wthout ths assumpton, but the concavty property that we dscuss later requres t. Problem Formulaton and Propertes Let x be the number of ambulances located at staton, and let x be the number of ambulances at the th preferred staton for demand node. The vector ( x1, x2, K, xm ) s a permutaton of (,, K, ) x1 x2 x m, for each. Smlarly, let ρ be the busy probablty for the th most preferred staton for demand node. The optmzaton problem s: (P1) maxmze s( x) h s ( x) N subect to z( x) x = b (3) S x 0, nteger, for all S (4)

18 18 where =, for all N s ( x) f ( x) w S (5) and x ( ) 1 xu ( ) = 1 ρ ρu u= 1 f x Q, for all S, N (6) Problem (P1) maxmzes the expected coverage s(x), subect to a constrant on the total number of ambulances z(x) beng equal to b. For the moment, we assume b to be gven, but n the algorthm n the next secton, b wll become a decson varable. The system-wde coverage s(x) s a weghted combnaton of the coverages for ndvdual demand nodes, and the coverage s ( x ) for demand node s calculated n (5) by condtonng on whch staton sends an ambulance to respond to a call from node. The calculaton of the node coverage requres the dspatch probablty f ( x ), the probablty that a call from node s responded to by an ambulance from ts th preferred staton. Ths probablty s calculated, as shown n (6), as the product of the probabltes that all ambulances at the 1 more preferred statons are busy, at least one ambulance at the th preferred staton s free, and a correcton factor Q, to approxmately account for the dependence between servers. Settng the correcton factors to 1 s equvalent to assumng that the probablty of an ambulance beng busy s statstcally ndependent of the status of all other ambulances n the system.

19 19 Concavty Result Proposton 1: If w1 w2 K wm for all N, and Q and ρ are nvarant wth x (recall that these are assumed to be exogenous nput to the model) for all S, N, then the systemwde coverage s a concave functon of x. Proof: Recall that the system wde coverage s ( x ) = h s ( x ) s a convex combnaton of the coverages s ( x ) for each demand node. To prove that s(x) s concave, t suffces to prove that the coverage s ( x ) for a partcular node s concave, snce the weghts h are postve. Therefore, we assume wthout loss of generalty that there s only one demand node and we drop the demand node subscrpt n the proof to smplfy notaton. N By assumpton we have w = w 1 w 0 for all. We can express the probablty f ( x ) as: x x u xu x u f( x) = Q ( 1 ρ ) ρ u = Q ρu ρ u = g 1( x) g( x) u= 1 u= 1 u = 1 where xu g ( x) = Q ρ and g ( x ) = 1. Consequently, u = 1 u 0 m s( x) = f ( x) w = g ( x) w g ( x) w 1 S = 1 = 1 m m m = g ( x) w g ( x) w = w + g ( x) w = 0 = 1 = 1 m wth the understandng that w m + 1 = 0. The gradent of s(x) wth respect to x has the followng entres: m s = (ln ρ k ) g( x) w x k = k

20 20 The entres n the Hessan matrx H are (assumng k l ): 2 m s hkl = = (ln ρk )(ln ρl ) g( x) w x x = l k l Recallng that Q > 0, ρ (0,1) and w 0, we see that s / xk s non-negatve for all k, and 2 s / xk xl s non-postve for all k and l. Consder the quadratc form quadratc form can be expressed as: T y Hy where y s an arbtrary column vector wth m elements. Ths m m m m m T 2 y Hy = yk ylhkl = yl hll + 2 yk ylhkl k = 1 l = 1 l = 1 k = 1 l = k + 1 Substtutng the expresson for h kl we get: m m m m m T 2 2 y Hy = yl (ln ρl ) g( x) w + 2 yk yl (ln ρk )(ln ρl ) g( x) w l = 1 = l k = 1 l = k + 1 = l (7) By changng the order of summaton, the double sum n (7) can be expressed as: m m m yl (ln ρl ) g( x) w = g( x) w (ln ρl ) yl l = 1 = l = 1 l = 1 Smlarly, the trple sum n (7) can be expressed as: m m m m m y y (ln ρ )(ln ρ ) g ( x) w = g ( x) w y y (ln ρ )(ln ρ ) k l k l k l k l k = 1 l = k + 1 = l k = 1 = k + 1 l = k + 1 m 1 = g ( x) w y y (ln ρ )(ln ρ ) k l k l = 2 k = 1 l = k + 1

21 21 Substtuton n (7) results n: m 1 T 2 2 y Hy = g( x) w (ln ρ l ) yl + 2 (ln ρk )(ln ρl ) yk yl = 1 l = 1 k = 1 l = k + 1 m = g( x) w (ln ρl ) yl = 1 l = 1 2 We see that each term n the outer summaton s non-postve (because g( x) 0, w 0, and T the squared summaton s non-negatve) and therefore y Hy 0 for all y. Consequently, H s negatve sem-defnte and s(x) s concave. Q.E.D. The obectve functon n (P1) s concave and the constrants are lnear. Consequently, the contnuous relaxaton of (P1) s a convex programmng problem, and a local optmum s also global. Note that as a result of ths proposton, the coverage s ( x ) for each demand node has the followng propertes: An ncrease n the number of ambulances at any staton ncreases the coverage for each demand node. When the number of ambulances at a partcular staton s ncreased, the margnal ncrease n coverage decreases. Busy Fractons and Correcton Factors The assumpton that the busy fractons ρ and correcton factors for dependence Q are exogenous nput s not realstc, as they wll depend on the number and dstrbuton of

22 22 ambulances between statons. To overcome ths lmtaton, we propose teratng between solvng (P1) and estmatng the busy fractons and correcton factors. If all ambulances are assumed to have the same busy fracton, then a relatvely smple estmaton procedure can be used (refer to Appendx 1 for detals). If all ambulances are not assumed to have the same busy fracton, then a more complcated estmaton procedure s necessary. We use a generalzaton of the approxmate hypercube model, detaled n Budge et al. (2005), that allows for multple vehcles at a staton. Ths procedure evaluates the busy fractons ρ, the correcton factors AH Q, and the expected coverage. We wll use s ( x ) to denote the expected coverage evaluated wth the approxmate hypercube model, to dstngush t from the expected coverage s(x) as computed n formulaton (P1). In the orgnal hypercube model (Larson, 1974), servce tmes (the tme an ambulance s ted up wth a call) are assumed exponentally dstrbuted. The pre-travel delay and the travel tme are part of the servce tme and f these components are lognormally dstrbuted then the servce tmes wll be far from exponentally dstrbuted. Fortunately, one can expect the loss-verson of the approxmate hypercube model (whch we use) to be relatvely nsenstve to the shape of the servce tme dstrbuton, as argued by Jarvs (1981). The related nsenstvty property of the M/M/s/s loss system s dscussed, for example, by Gross and Harrs (1998). We propose the followng teratve algorthm to overcome the assumpton of the busy fractons and correcton factors beng exogenous nput. Step 1: Choose an ntal value for the total number of ambulances, b. Step 2: Attempt to maxmze coverage wth b ambulances, as follows:

23 23 Step 2a: Set the busy fractons correcton factors n ρ to an ntal estmate of the busy fracton, set all n Q equal to 1, and set smoothng parameter γ (0,1). 0,* x = 0. Set 1 n and choose a Step 2b: Solve (P1), usng busy fractons soluton n,* x that maxmzes s(x) subect to, convergence crteron s satsfed, go to Step 3. n ρ and correcton factors n 1,* n Q. Fnd the x x 1, S, (3), and (4). If the Step 2c: Estmate the busy fractons soluton n,* x. Set out ρ and correcton factors ρ γρ + (1 γ) ρ for all statons and n out n out Q that result from the Q γ Q + (1 γ ) Q for all staton-node pars and n n + 1. Go back to step n out n 2b. AH Step 3: Evaluate the expected coverage s ( x ) for the fnal soluton(s), usng the approxmate hypercube model. Adust the total number of ambulances b based on whether the hghest coverage among the fnal solutons s less than or greater than the target of α. When t has been determned that the current total number of ambulances s the smallest one that wll acheve the target coverage, then stop. Otherwse, return to step 2. The algorthm ncludes an outer loop, whch s a one-dmensonal search (such as bsecton search) for the smallest total number of ambulances needed to provde the requred coverage, and an nner loop, whch terates between solvng (P1) and estmatng the busy fractons and correcton factors. The expected coverage for each soluton that s returned by the algorthm s evaluated usng the approxmate hypercube model, thus avodng the smplfyng assumptons

24 24 made n formulaton (P1), namely, that the busy fractons and correcton factors are exogenous nputs. The constrants x n 1,* x 1 are added n Step 2b to prevent the allocaton of ambulances to statons from changng too much from one teraton to the next, recognzng that the busy fractons and correcton factors depend on the allocaton of ambulances to statons. The convergence crteron for the nner loop could be expressed n terms of the sequence of n,* out n,* solutons { x }, the estmated busy fractons { ρ ( x )}, or both. The nner loop algorthm s not guaranteed to converge to a unque soluton. Indeed, we have sometmes observed convergence to a cycle of two or more smlar solutons. In such cases, planners could be presented wth multple good solutons, whch could be compared n terms of the values that they gve for the coverage (as estmated by the busy fracton estmaton procedure) or for other performance measures. Goldberg et al. (1991) use a dfferent approach, where they nclude the busy fractons as decson varables and nclude a constrant n the problem formulaton that s smlar to equaton (12) n Appendx 1. An advantage of our approach s that the contnuous relaxaton of (P1) s a convex optmzaton problem, under certan assumptons, as we have shown. Goldberg et al. (1991) do not solve ther formulaton as a mathematcal program, but use specalzed heurstcs. Computatonal Experments In the nstances of (P1) that we solved, based on data from Edmonton EMS, we used determnstc travel tmes n order to solate the effect of randomness n delays. The dspatch orders satsfed assumpton (2). These nstances have 10 statons and 180 demand nodes. We were able to solve these nstances to optmalty n at most a few mnutes per nstance wth a

25 25 standard branch-and-bound algorthm that calls a nonlnear programmng algorthm to solve the contnuous relaxatons. To overcome the assumpton of the busy fractons and correcton factors beng gven exogenously, we used the algorthm descrbed n the last secton. Fgure 3 shows an example of how n ρ and out ρ evolved over 3 teratons for one problem nstance based on Edmonton data, wth the total number of ambulances equal to 16. In ths nstance, γ was set to 0.9, and 33%. n ρ and out ρ converged n about 3 teratons wth an average after convergence of about 0.6 Output Staton Busy Fracton Staton 10 Statons 1, and 4-9 Statons 2 and Iteraton Fgure 3: An example of teratng on the busy fractons ρ, where the ntal nput busy fracton was set to 0.3 for each staton, and a smoothng constant of 0.9 was used. We used the model to emprcally explore the mpact of varyng the parameters of the delay dstrbuton. Fgure 4 shows how the mnmum total number of ambulances needed to provde the specfed coverage (90% reached n 9 mnutes) changes when the mean and standard devaton of the delay dstrbuton vary. We tred values that were 0%, 50%, 100%, 125%, and 150% of the current value for the mean (2.6 mnutes) and for the standard devaton (1.3 mnutes), except for combnatons of parameters that made t mpossble to meet the coverage

26 26 goal. We wll refer to the combnaton where both the mean and the standard devaton equal ther current values as the base case. As Fgure 4 shows, the total number of ambulances needed changes consderably when the parameters of the delay dstrbuton are vared. The dramatc mpact of gnorng the delay s llustrated by comparng the case when the delay s assumed to be zero to the base case. In the former case, only 11 ambulances are needed, whle n the base case, 16 are needed. Total Number of Ambulances Zero Mean 50% Lower Mean Current Mean 25% Hgher Mean 50% Hgher Mean Base case 0 Zero St. Dev. 50% Lower St. Dev. Current St. Dev. 25% Hgher St. Dev. 50% Hgher St. Dev. Standard Devaton of Delay Fgure 4: Senstvty of the mnmum total number of ambulances needed to provde the coverage goal to the mean and standard devaton of the delay dstrbuton. Comparson of the case where the delay s assumed determnstc and equal to the current mean and the base case results n a less dramatc dfference, of course: the number of ambulances needed ncreases from 15 to 16. However, the mpact of gnorng the varablty n delays would be far greater f the mean delay were hgher. For example, f the mean delay were to ncrease by

27 27 25% (from 2.6 mnutes to 3.25 mnutes), whle the standard devaton stayed the same, then 21 ambulances would be needed to reach the coverage goal. In ths case, f the delay varablty were gnored (.e., the standard devaton s assumed to be zero), then the model predcts that only 18 ambulances would be needed to reach the coverage goal. Hence, a model that ncorporates delays but treats them as determnstc would underestmate the number of ambulances needed to provde the target coverage by (21-18)/21 = 14%. Fgure 5 gves the complementary perspectve and provdes addtonal nsght nto the mpact of the delay standard devaton. It demonstrates how the system wde coverage vares when the parameters of the delay dstrbuton are vared n the same way as for the results n Fgure 4, wth the total number of ambulances fxed at 16. From Fgure 5, we see that f the varablty n the delay s not consdered, the estmated coverage s about 92%, compared to ust over 90% f the varablty n the delay s ncorporated. When the standard devaton s ncreased 25% from the base case, the coverage drops to about 89%. The results are magnfed as the average level of the delay ncreases. These results llustrate the mportance of accountng for delays, and specfcally the randomness n the delays, n order to obtan accurate estmates of the coverage and of the resources requred to attan a specfed coverage. They also llustrate the mportance of controllng the call-takng and dspatchng processes to ensure that delays do not ncrease (but preferably, decrease).

28 28 Estmated Coverage % Lower Mean Current Mean 25% Hgher Mean 50% Hgher Mean Zero Mean Base case 0.75 Zero St. Dev. 50% Lower St. Dev. Current St. Dev. 25% Hgher St. Dev. 50% Hgher St. Dev. Standard Devaton of Delay Fgure 5: Senstvty of the system wde servce to the mean and standard devaton of the delay dstrbuton, when the total number of ambulances s fxed at 16. Dscusson Ths secton outlnes several possble avenues for further research nvolvng exploraton of the optmzaton model (P1), ts propertes, soluton approaches, and nsghts from ts applcaton. Frst we dscuss three extensons of the model that are farly straghtforward, and then we dscuss some avenues for further research. Model Extensons One can add a constrant to (P1) to ensure that the probablty that at least one ambulance s avalable s above some threshold β, as follows (assumng ndependence between ambulances): ρ β (8) x 1 S

29 29 The constrant can be lnearzed by solatng the product of the busy fractons on one sde of the nequalty and takng logarthms of both sdes, resultng n: ( ln( ρ)) x ln(1 β) (9) S Note that the coeffcents ln( ρ ) and ln(1 β ) wll be postve. Prelmnary experments usng data from Edmonton ndcated that the expected coverage target of reachng 90% of all calls n 9 mnutes or less was tghter than constrant (9) for β In addton to maxmzng the system-wde coverage, one could add constrants on the coverage for each demand node, of the form s ( x) α, for all N (10) where α s the target coverage for demand node. Ths constrant set could, for example, be used to mpose a common mnmum coverage for all demand nodes or some subset of the demand nodes. One can also add varables and constrants to decde whch statons to open and to lmt the number of ambulances at each staton. Specfcally, let y be a bnary ndcator varable for whether staton s opened; let c be the fxed cost of openng staton ; let d be the varable cost of locatng one ambulance at staton ; and let b be the maxmum number of ambulances at staton, f t s opened (f there are no such lmts, then one can set b = B for some suffcently large number B). Upon replacng constrant (3) on the total number of ambulances wth a budget constrant, the extended problem formulaton becomes:

30 30 (P2) maxmze s( x) h s ( x) N subect to ( c y + d x ) budget S (9), (4) x b y, for all S y {0,1}, for all S (11) The contnuous relaxaton of (P2) s a convex programmng problem, by Proposton 1, but (P2) s more dffcult to solve than (P1) because t has more nteger varables. Future Research Incorporaton of random delays and travel tmes may nfluence not only the total number of ambulances needed to provde a gven level of servce, but also how ambulances are dstrbuted through the system. We plan to perform experments to generate nsght nto whether ths happens and how. In order to do further computatonal testng of the model, data from a cty of smlar sze to Edmonton, but whch s aggregated nto many more (smaller) zones and has up to 40 potental locatons for ambulances wll be used. We also hope to use the model to estmate the mpact of varous changes to the operaton of an ambulance system. For example, t may be possble to reduce delays by performng actvtes n parallel rather than n seres, but such a change may ncrease ambulance workload, f t results n more false alarms. Therefore, we would lke to explore the trade-off between reducng delays and ncreasng busy fractons. Estmaton of the travel tme dstrbuton functons H ( t ) s lkely to be challengng. We are workng on developng procedures to estmate these functons, and have obtaned detaled travel

31 31 tme data from a number of ctes that we wll use to valdate such procedures. Prelmnary results are reported n Budge (2004). Although we can solve nstances of our formulaton nvolvng Edmonton data to optmalty n reasonable tme, t s concevable that problem nstances for ctes wth more statons and ambulances wll requre the development of heurstcs to generate near-optmal solutons. Conclusons We have presented an optmzaton model for allocatng a specfed number of ambulances to statons so as to maxmze system-wde expected coverage. The model dffers from prevous related work n that the varaton n pre-travel delay s consdered (n addton to the varaton n travel tme) when calculatng the coverage. Data from recent proects wth the town of St. Albert and the Cty of Edmonton ndcate that pre-travel delays are mportant and hghly varable (wth a standard devaton of about 40% of the mean). Our computatonal experments demonstrate that the ncluson of the varablty of such delays has a substantal mpact on the soluton that the model prescrbes. Our formulaton s suffcently tractable that t can be solved to global optmalty for problems wth 180 demand nodes and 10 ambulance statons wth general-purpose solvers.

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