Simultaneous Location of Trauma Centers and Helicopters for Emergency Medical Service Planning

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1 Simultaneous Location of Trauma Centers and Helicopters for Emergency Medical Service Planning Soo-Haeng Co Hoon Jang Taesik Lee Jon Turner Tepper Scool of Business, Carnegie Mellon University, Pittsburg, PA, Department of Industrial & Systems Engineering, Korea Advanced Institute of Science and Tecnology, Daejeon, Korea, Te Paul Merage Scool of Business, University of California, Irvine, CA, Abstract: Tis paper studies te problem of simultaneously locating trauma centers and elicopters. Te standard approac to locating elicopters involves te use of elicopter busy fractions to model te random availability of elicopters. However, busy fractions cannot be estimated a priori in our problem because te demand for eac elicopter cannot be determined until te trauma center locations are selected. To overcome tis callenge, we endogenize te computation of busy fractions witin an optimization problem. Te resulting formulation as non-convex bilinear terms in te objective, for wic we develop an integrated metod tat iteratively solves a sequence of problem relaxations and restrictions. Specifically, we devise a specialized algoritm, called te Sifting Quadratic Envelopes algoritm, tat 1) generates tigter outer-approximations tan linear McCormick envelopes, and 2) outperforms a Benders-like cut generation sceme. We apply our integrated metod to te design of a nationwide trauma care system in Korea. By running a trace-based simulation on a full year of patient data, we find tat te solutions generated by our model outperform several bencmark euristics by up to 20%, as measured by an industry-standard metric: te proportion of patients successfully transported to a care facility witin one our. Our results ave elped te Korean government to plan its nationwide trauma care system. More generally, our metod can be applied to a class of optimization problems tat aim to find te locations of bot fixed and mobile servers wen service needs to be carried out witin a certain time tresold. Subject Classification: Healt care: ambulance service, ospitals. Programming: integer: algoritms: Benders/decomposition. Simulation: applications Area of Review: Policy Modeling and Public Sector OR We tank our collaborators from an earlier study (Kim et al. 2011) commissioned by te Korean Ministry of Healt and Welfare, wic elped us deepen our understanding of EMS practice. Special tanks go to Dr. Yoon Kim, principal investigator of tat study, for allowing us to use te trauma patient data for tis paper and for giving us te opportunity to contribute troug our work to te establisment of te new trauma care system in Korea. We tank te area editor, Pinar Keskinocak, as well as te anonymous associate editor and tree referees, wose comments decidedly improved our manuscript. Many tanks also to tose tat provided us wit useful feedback at INFORMS, MSOM, and POMS conferences, as well as at te Soutern California OR/OM Day at UCLA. An earlier version of tis paper was awarded te 2012 Lave-Weill Prize for te best unpublised paper on problem solving from Carnegie Mellon University. Tis researc was partially supported by Basic Science Researc Program troug te National Researc Foundation of Korea (NRF) funded by te Ministry of Education, Science and Tecnology ( ).

2 1 Introduction Trauma is a body wound or sock produced by sudden pysical injury, as from violence or accident, wic may lead to te deat of a patient if proper care is not administered in a timely fasion. Trauma is te sixt leading cause of deat worldwide and te leading cause of deat in te U.S. for tose under age 44 (Centers for Disease Control and Prevention (CDC) 2013). Trauma is a serious public ealt problem wit significant social and economic costs. Providing proper care to trauma patients requires seamless ealt care delivery operations. Wen a trauma case occurs, critical care paramedics are quickly dispatced to te scene eiter by ground ambulance or elicopter. Tey provide first aid to stabilize te patient at te scene, and ten transport im/er to a trauma center. Because any delay in transporting a patient to a trauma center can severely affect is/er survival rate, a general rule of tumb is tat an appropriate clinical intervention sould be provided witin an our from te moment of an injury incident (e.g., CDC 2012). Many countries, including Canada, Germany and Israel, ave reported significant improvements in major injury care from designating dedicated trauma care centers (Peleg et al. 2004). A trauma center is a type of ospital tat as resources and equipment needed to elp care for severely-injured patients. In te U.S., trauma centers are classified as Level I (te igest level of care) to Level IV. Te CDC estimates a 25% reduction in deats for severely-injured patients wo receive care at a Level I trauma center rater tan at a non-trauma center. Our paper studies te design of a nationwide trauma care system. Specifically, our paper is part of a broader study commissioned by te Korean Ministry of Healt and Welfare (KMHW) to make recommendations for establising a nationwide trauma care system in Korea. In te broader study (Kim et al. 2011), a group of experts conducted researc on a variety of issues related to trauma care, including infrastructure, uman resources, governance, financing, and quality control. Our paper expands te infrastructure part of tis broader study, and provides a quantitative model and analysis using oneyear nationwide data of 190,193 trauma cases. Our results ave elped te Korean government outline its initial plan in Te Korean government is currently refining its initial plan by incorporating feedback from relevant communities. Te first trauma center in Korea is sceduled to open in 2014, and te government is planning to open an additional 15 trauma centers across te country. Te results developed in tis paper can also support future decision-making of te government as it rolls out its final plan. Altoug we generate numerical results based on te data from Korea, te models and metods developed ere could be applied to te design of trauma care systems in oter countries or regions. More generally, tey could also be applied to a class of optimization problems tat aim to find te locations of bot fixed servers (e.g., trauma centers, ospitals, wareouses) and mobile servers (e.g., elicopters, ambulances, trucks) wen service needs to be carried out witin a certain time tresold. Our objective is to find te optimal locations of trauma centers and elicopters tat maximizes te effective coverage of trauma care. Ideally, we would like to always transport every citizen in te country to an available trauma center in under an our. However, due to limited budgets, tis is not practically possible even in te U.S., nearly 45 million people do not ave access to a Level I or II 1

3 trauma center witin one our s transportation distance (CDC 2012). At te request of te KMHW, we explore trauma care system designs wit various numbers of trauma centers and elicopters, assuming tat a sufficiently ig number of ground ambulances are deployed. Wen tere are a limited number of trauma centers, elicopters play te important role of expanding geograpic coverage by transporting patients from rural areas to trauma centers. Tus, in designing a nationwide trauma care system, it is important to simultaneously locate trauma centers and elicopters. Te problems of locating only one type of resource (i.e., eiter trauma centers or elicopters) belong to well-known classes of optimization problems. Te trauma center location problem can be formulated as a capacitated maximal covering location problem (e.g., Pirkul and Scilling 1991), and te elicopter location problem is structurally similar to a probabilistic ambulance location problem (e.g., Daskin 1983). However, our problem poses a unique callenge because te locations of trauma centers affect te demand for elicopters, and vice versa. Tis dependency is particularly problematic if we explicitly model te probabilistic availability of elicopters. Specifically, in te probabilistic location problem (e.g., ReVelle and Hogan 1989, Borras and Pastor 2002), it is common to model te probabilistic nature of ambulance availability using a busy fraction of an ambulance (or a elicopter, in our case). Tis fraction is usually estimated as a ratio of te workload of an ambulance at a given location (e.g., te daily number of service requests times average service time per patient) to te available service ours of an ambulance. In our model, owever, suc busy fractions cannot be estimated a priori because te demand for elicopters at eac given location cannot be determined until after te trauma centers are cosen. To address tis callenge, we endogenize te computation of busy fractions witin an optimization problem, and formulate te problem as a mixed-integer nonlinear program (MINLP) wit te objective of maximizing te expected (approximate) number of trauma patients tat can be successfully transported witin an our. However, due to te inerent dependency described above, our MINLP formulation as non-convex bilinear terms wic present serious computational callenges. Fortunately, we are able to exploit problem-specific structure to develop an integrated metod tat iteratively solves a sequence of problem relaxations and restrictions, tereby establising bounds on our model s objective. Specifically, we devise a specialized metod, called te Sifting Quadratic Envelopes (SQE) algoritm, tat creates and sifts quadratic envelopes at eac iteration. We sow tat SQE generates tigter outer-approximations tan classical linear McCormick envelopes (McCormick 1976, Floudas and Pardalos 2012), and also outperforms a cut generation sceme based on Generalized Benders Decomposition (GBD) (e.g., Geoffrion 1972). Te use of SQE allows us to get witin 6% of optimality in most problem instances on te scale required in our problem setting. Tis is significant, especially considering tat te leading global solver BARON (c.f. Tawarmalani and Sainidis 2005) acieves only 21% of optimality in te same allotted time. As a point of comparison, we also develop two simple euristics tat are motivated by existing metods in te literature. Te first no-congestion euristic is modeled after Branas and ReVelle (2001) wo also consider a joint location problem of trauma centers and elicopters as in our paper, 2

4 but ignore random availability of elicopters. Te second decoupled euristic solves for te locations of trauma centers and elicopters sequentially, ignoring te dependency between tese two resources. We ave found tat te best trauma center locations differ significantly across te tree approaces, especially wen a large number of elicopters are available. We ten compare te performance of te location solutions generated by our integrated approac wit tose from te two euristics by carrying out a trace-based simulation using one year of patient data. Our simulation treats te locations of trauma centers and elicopters as given, and sequentially processes te times and locations of patient calls from te input stream. We simulate te real-time processes of transporting eac patient to a trauma center, and categorize eac patient as successful ( 60 min transport to an under-capacity trauma center) or unsuccessful. Our results sow tat our integrated approac outperforms te two euristics significantly. For example, wen using 10 trauma centers and 20 elicopters, we acieve a 12% (resp., 14%) larger number of successful transports using our integrated approac tan te nocongestion euristic (resp., te decoupled euristic), wic translates into a potential 23,000 (resp., 26,000) additional lives saved per year. Te rest of tis paper is organized as follows. In 2, we review related literature. In 3, we describe te problem. In 4, we present our solution approaces. In 5, we describe our data and simulation model, and present te results of applying our metods to a trauma center design problem in Korea. We conclude our paper in 6. 2 Related Literature Our paper is related to te literature on location problems in ealt care delivery systems. Due to te large volume of literature in tis area, we review only te most related papers, and refer readers to Owen and Daskin (1998), Berman and Krass (2002), Brotcorne et al. (2003), Daskin and Dean (2004), ReVelle and Eiselt (2005), and Li et al. (2011) for a more compreensive review. Toregas et al. (1971) and Curc and ReVelle (1974) are among te first researcers to study Emergency Medical Services (EMS) veicle location problems. Toregas et al. (1971) study te Location Set Covering Problem (LSCP) tat identifies te minimum number of facilities (or ambulances) and teir locations wic cover all demand points witin a certain distance. Curc and ReVelle (1974) propose te Maximal Covering Location Problem (MCLP), wic locates a fixed number of facilities so as to maximize te amount of demand tat is covered by at least one facility. Many variations and extensions ave followed tese early studies: for example, Scilling et al. (1979) develop a model of locating multiple types of veicles suc as basic and advanced life support ambulances; Hogan and ReVelle (1986) and Gendreau et al. (1997) consider double coverage models in wic all demands must be covered by ambulances located at most r 2 minutes away, and in addition, a certain proportion of te demand must also be at most r 1 minutes away. Wile tese models deal wit deterministic location problems, anoter stream of researc takes into account randomness in te availability of ambulances. Tis randomness is usually modeled as te busy fraction of an ambulance, i.e., te probability tat an ambulance is unavailable to respond to a service 3

5 request immediately. Assuming a uniform busy fraction for all ambulances, Capman and Wite (1974) formulate a probabilistic version of te LSCP, and Daskin (1983) proposes te Maximum Expected Covering Location Problem (MEXCLP) tat maximizes te expected value of coverage witin a time standard. ReVelle and Hogan (1989) formulate a cance-constrained program, called te Maximum Availability Location Problem (MALP), wic positions ambulances so as to maximize te demand covered witin a time standard wit a given probability. Tey also propose a metod to estimate busy fractions tat are specific to eac demand region. Several probabilistic location models ave extended tese models, including Ball and Lin (1993), Marianov and ReVelle (1994, 1996), Borras and Pastor (2002), and Sorensen and Curc (2010). All tese models require te estimation of busy fractions. However, as ReVelle and Hogan (1989) rigtly point out, busy fractions are difficult to estimate because tese values are an output of te model and cannot be known a priori, i.e. before knowing te locations of ambulances (Brotcorne et al. 2003). 1 Tis issue is even more serious in our problem of jointly locating trauma centers and elicopters because, unlike all previous models reviewed above, te busy fraction of a elicopter depends on te locations of bot trauma centers and elicopters, wic are not given a priori. To te best of our knowledge, Branas and ReVelle (2001) is te only paper tat considers a joint location problem of trauma centers and elicopters as in our paper. Branas and ReVelle model tis as a deterministic location problem, and formulate te problem as a mixed-integer linear program. Tey could not attain solutions witin a reasonable amount of time using CPLEX directly, so tey developed an iterative euristic tat identifies te best locations of elicopters, olding te locations of trauma centers fixed, and ten finds te best locations of trauma centers, olding te locations of elicopters fixed, and so on. Similar to Branas and ReVelle (2001), tis paper considers a joint location problem of trauma centers and elicopters, but takes a fundamentally different approac in bot model and solution metod. First, our model explicitly models te capacity constraints of trauma centers, wic are critical to ensure tat trauma centers do not become over-crowded. Second, our model takes into account te randomness in te availability of elicopters as in te second stream of researc reviewed above. To address te issues regarding te estimation of busy fractions discussed above, we endogenize te computation of elicopter site-specific busy fractions witin an optimization problem. According to ReVelle and Hogan (1989), It sould be noted tat te busy fractions used ere are not specific for a particular site. Te use of [ambulance] site-specific busy fractions, rater tan [demand] area-specific busy fractions, would certainly be preferable but suc a formulation is not undertaken ere for two reasons. First, suc site-specific busy fractions cannot be obtained witout knowledge of te positions of all oter servers, and tese positions are only known as an output of te model, not in advance. Second, te constraints tat follow from suc information are of a form wic 1 Descriptive queueing models suc as Larson (1975) and Burwell et al. (1992) can estimate busy fractions under fairly realistic assumptions. However, as Marianov and ReVelle (1994) point out, suc descriptive queueing models usually fix te locations of ambulances a priori. Tere are recent developments in queueing-based location models (see, e.g., Berman and Krass (2002), Aboolian et al. (2008), Zang et al. (2010) and references terein), but as Berman and Krass (2002) point out, one invariably as to make simplifying assumptions and approximations to render te model tractable. 4

6 requires an integer programming code capable of solving large zero-one problems witout special structure. As ReVelle and Hogan (1989) predicted, te use of site-specific busy fractions witin an optimization model requires us to solve a large-scale MINLP wit a specialized algoritm. Indeed, our optimization model is a complex mixed-integer nonlinear program tat determines te allocation of patient demand to trauma centers (using eiter ground ambulance or elicopter), as well as te locations of trauma centers and elicopters. To solve tis program, we develop a novel metod tat iteratively solves a sequence of problem relaxations and restrictions. Our metod exploits te specific structure of te problem formulation to tigten bounds by systematically pusing te solution toward a global optimum. tis way, our metod teoretically guarantees convergence to an optimal solution, wereas te euristic developed by Branas and ReVelle (2001) does not. In addition, we validate our optimization model s location solution by conducting a trace-based simulation using one year of nationwide patient data from Korea. Our use of simulation for validation is in line wit Goldberg et al. (1990), Repede and Bernardo (1994), Borras and Pastor (2002), and Sorensen and Curc (2010). Lastly, we note tat our model and solution metod may be used in oter applications tat require te simultaneous location of fixed and mobile servers wen service needs to be carried out witin a certain time tresold. For example, in te transportation literature, te location-routing problem (see Nagy and Sali 2007, as well as Prodon and Prins 2014, for extensive reviews) involves (a) coosing a number of depot locations from wic to sip products, (b) assigning one or more trucks to eac depot, and (c) finding a sort route (tour) for eac truck tat starts at a depot, makes deliveries to one or more customers in sequence, and returns to te same depot. If we consider te special case were eac truck makes a delivery to only one customer, ten we can compare tis to our problem, and describe ow extensions of tis problem more closely relate to ours. Notably, location-routing problems usually do not consider transportation delays, since eac truck only makes one tour, and unanticipated demands do not occur in te planning orizon. However, if customers order te product at random times and need te product urgently (e.g., pizza delivery from a cain of pizza outlets) ten, as in our model, it is appropriate to minimize transportation-delay-induced congestion, and suc an objective will naturally ave bilinear (or more generally, if multiple trucks are stationed at eac depot, nonlinear) terms tat can be tackled by our Sifting Quadratic Envelopes algoritm. 2 Altoug tese are te main structural properties tat we need to use SQE, we point out tat our model and SQE are particularly important for problems were te busy fractions for te mobile servers are ard to estimate a priori, suc as wen fixed servers also need to be located and te mobile servers must stop at one fixed server along teir route. Tis is because, in tis case, route lengts depend on were te fixed servers are located, and so utilization and tus busy fractions depend directly on te route lengts. 3 2 Specifically, we need bot te arrival rate and workload assigned to eac mobile server to be linear functions of te same set of variables. However, as we will see from our model in 4, tis occurs quite naturally. 3 In a location-routing problem, depots are usually facilities tat ouse trucks, and tey are analogous to te eliports in our model were te elicopters are stationed; tere is no analog to our trauma centers in te canonical location-routing In 5

7 3 Problem Description Consider te problem of locating k trauma centers and m elicopters (i.e., air ambulances) to serve ī demand regions. We call te station were a elicopter is based a eliport. Tere are j ( k) candidate sites for trauma centers and candidate eliports. A eliport can eiter be on te roof of an open trauma center or at a separate location (e.g., an airport) tat permits elicopter take-off and landing. We index demand regions by i I = {1, 2,..., ī}, eliports by H = {1, 2,..., }, and trauma centers by j J = {1, 2,..., j}. More specifically, we assume witout loss of generality tat eliports 1 troug j are located on te rooftops of trauma centers 1 troug j, wile eliports j + 1 troug are not co-located at trauma center sites. Eac demand region i as expected demand rate λ i, and eac trauma center j as te fixed capacity of treating up to c j patients per unit of time. A trauma patient from any demand region can be transported to a trauma center eiter by a ground ambulance (ereinafter, in sort, ambulance) or by a elicopter. A patient is geograpically covered if s/e can be transported to an open trauma center witin 60 minutes. To define sets of tese patients and teir transportation modes, we convert travel times (e.g., 60 minutes) into distances between locations as follows (see Figure 1 for illustration). Let d r i denote te road distance between te center of demand region i and its nearest ambulance station, d r ij denote te road distance between te center of demand region i and trauma center j, d ij denote te Euclidean distance between te center of demand region i and trauma center j, and d i denote te Euclidean distance between eliport and te center of demand region i. Demand region i is geograpically covered by an ambulance if tere exists a trauma center j wit d r i + dr ij d ground, or by a elicopter if tere exists a eliport and trauma center pair (, j) suc tat d i + d ij d air. 4 In collaboration wit practitioners, we ave set d ground = 46 km and d air = 120 km, considering te average time spent in eac step of operations (a1-a3) and (1-5), respectively, as follows: Ambulance: (a1) drive to patient location i from te nearest station (d r i /(50/60) minutes), were te average ambulance speed of 50 (km/) is used; (a2) load te patient into te ambulance (5 minutes); and (a3) drive from patient location i to trauma center j (d r ij /(50/60) minutes). Helicopter: (1) take off at eliport (6 minutes); (2) fly from eliport to patient location i (d i /(180/60) minutes), were te average elicopter speed of 180 (km/) is used; (3) load te patient into te elicopter (8 minutes); (4) fly from patient location i to trauma center j (d ij /(180/60) minutes); and (5) land and and-off te patient to te trauma center (6 minutes). We assume tat a elicopter at eliport is used to cover demand region i only wen d r i + dr ij > d ground. Moreover, we assume tat eac patient is transported (as necessary) to a place were a elicopter can land (e.g., an elementary scool, a farming area, etc.) wile a elicopter is en route to te patient; tus, maneuvering to te exact pick-up location does not impose any additional delay. problem. In tis sense, altoug it is tempting to consider depots as fixed servers, tey are not. Tus, strictly speaking, te canonical location-routing problem locates only mobile servers, and not fixed servers. 4 Alternatively, we can define d r i as te average road distance between eac istorical patient in demand region i and is/er nearest ambulance station, and define d r ij similarly. Geograpic coverage is qualitatively uncanged under tis 6

8 (a) trauma center j (b) trauma center j r d ij d ij d i patient location i r d i eliport Nearest ambulance station to i patient location i Figure 1: (a) ambulance coverage: d r i + dr ij d ground; and (b) elicopter coverage: d i + d ij d air, excluding te area covered by ambulance (note: and j can be co-located). We now define te following sets of geograpically covered patients and teir transportation modes: F G = {(i, j) i I, j J, and d r i + dr ij d ground}: all feasible ambulance routes (i, j) in wic a patient from region i can be transported to trauma center j witin 60 minutes; Fi G = {j J d r i + dr ij d ground for fixed i}: te subset of trauma center sites to wic a patient from demand region i can be transported by ambulance witin 60 minutes; Fj G = {i I d r i + dr ij d ground for fixed j}: te subset of demand regions wose patients can be transported by ambulance to trauma center j witin 60 minutes; F = {(, i, j) H, i I, j J, d r i +dr ij > d ground and d i +d ij d air }: all feasible elicopter routes (, i, j) in wic a elicopter from eliport can transport a patient from region i to trauma center j witin 60 minutes, excluding any routes tat are close enoug to be witin te ambulance coverage area; F = {(i, j) d r i + dr ij > d ground and d i + d ij d air for fixed, were i I, j J}: te subset of pairs of demand region i and trauma center site j to wic a elicopter originating from eliport can transport a patient from demand region i to trauma center j witin 60 minutes; F i = {(j, ) d r i + dr ij > d ground and d i + d ij d air for fixed i, were j J, H}: te subset of pairs of trauma center j and eliport tat can be used to transport a patient from demand region i by air (using te route i j) witin 60 minutes; F j = {(, i) d r i + dr ij > d ground and d i + d ij d air for fixed j, were i I, H}: te subset of pairs of eliport and demand region i tat can be used to transport a patient by air to trauma center j (using te route i j) witin 60 minutes. Geograpic coverage is a feasibility criterion, and does not take into account te possibility tat patients may be delayed due to congestion. For example, a patient must wait for a elicopter if te nearest elicopter is already transporting anoter patient. Some proportion of patients wo are geograpically covered will not receive timely service in expectation. To take into account tis congestion effect, we define expected covered demand as te expected number of patients tat will be transported to an open trauma center in under 60 minutes, witout incurring any delays in transportation. Our objective is to find te locations of k trauma centers and m elicopters tat maximize te expected covered demand witin a time standard of 60 minutes, considering randomness in te availalternative definition. 7

9 ability of elicopters. Starting wit Daskin (1983), tis objective is commonly used in te literature for ambulance location problems (see, e.g., of Berman and Krass 2002, Sorensen and Curc 2010, and references terein). Like ambulances, elicopters in our problem are complimentary to trauma centers since eac patient needs bot transportation and a care facility. However, unlike ambulances, it is interesting to note tat elicopters also act as substitutes for trauma centers by allowing fewer trauma centers to serve a larger coverage area. To model te availability of elicopters, we compute te average service time τ ij for a elicopter to fly te circuit i j, including steps (1)-(5) described above and te following additional steps: (6) travel from trauma center j to eliport (were travel time = d j /(180/60) minutes if j or 0 oterwise), and (7) land and refuel at eliport (5 minutes). During tis service time, a elicopter is busy, and tus it is not available to serve any oter patients. As discussed earlier in 1, tis problem is callenging due to many reasons in particular, interdependency in demands for trauma centers and elicopters, and inerent uncertainties. To build a tractable model to support te decision-making of te KMHW, we make a set of assumptions as follows. First, we do not consider randomness in te availability of ambulances. 5 Tis assumption allows us to ignore te availability and routing details of tousands of ambulances across te country. In fact, te average time it takes for an ambulance to reac a patient location from te moment of a patient call is 10 minutes in major cities in Korea, suggesting tat te availability of ambulances is not a serious concern in Korea. We note, owever, tat we could also consider random availability of ambulances by modeling it in te exact same way as we model random availability of elicopters. Second, in our base model, we assume tat eac eliport can ave at most one elicopter. We give up little wit tis assumption because, in Korea, 38 candidate ospitals for trauma centers can operate at most one elicopter, wereas 16 separate eliports migt be able to operate more tan one elicopter. Moreover, elicopters primarily assist rural patients in more sparsely populated areas, and so, assuming elicopters are a scarce resource, tis suggests tat tey sould naturally be more spread out rater tan clustered togeter. Indeed, elicopters are scarce in our case: in our numerical study, we ave many more candidate eliports (54) tan elicopters (between 5 to 25). For completeness, owever, Online Appendix A describes ow our metod can be generalized to te case were more tan one elicopter is allowed at eac eliport. Moreover, Online Appendix B formulates multi-period and multi-scenario extensions of our model. Tird, to make long-term decisions of were to locate trauma centers and elicopters, our optimization model abstracts away from te detailed real-time decision-making processes used in practice. For example, in reality, a central operator for EMS (suc as service in te United States) keeps track of eac elicopter s availability at eac point in time, and dispatces eiter an ambulance or a elicopter, depending on wic option would provide faster service given te system state. Te central operator also monitors te availability of beds in eac trauma center, and may divert a patient to a farter away, yet less congested, trauma center. To test te performance of te location solutions 5 Branas and Revelle (2001) also make te same assumption, saying: Tis is a consideration tat is bot realistic and advantageous in analyzing state [Maryland] trauma systems because te number of ambulance depots at te state level is proibitively large and only a relatively small percentage of ambulance transports are devoted to severe trauma. 8

10 our optimization models generate, we use a simulation tat captures tese real-time decision-making processes. We present details of our simulation model and test results in 5. 4 Optimization Models and Solution Metods In tis section, we model and solve our problem as described in 3. In 4.1, we present an integrated model and outline a general sceme tat iteratively solves a sequence of problem relaxations and restrictions to sequentially find tigter bounds for te integrated model. In 4.2, we describe two specific solution metods tat tigten te optimality gap in te integrated approac: te Sifting Quadratic Envelopes (SQE) metod and a metod based on Generalized Benders Decomposition (GBD). Finally, in 4.3, as a point of comparison, we develop euristic metods tat build on existing approaces from te literature. 4.1 Integrated Model and Approac In our integrated approac, we endogenize te computation of busy fractions witin an optimization problem, and formulate te problem as a mixed-integer nonlinear program (MINLP). We explicitly model te allocation of patient demands to trauma centers as well as to ambulances and elicopters. As noted earlier, tis allocation represents te long-run average allocation rater tan te real-time allocation (wic we simulate later in 5). Consequently, our mat program as four principal decision variables: binary variables y j tat indicate weter or not a trauma center sould be opened at site j; binary variables x tat indicate weter or not a elicopter is stationed at eliport ; continuous variables s G ij tat represent te (expected) number of patients per unit time to transport from demand region i to trauma center j by ambulance; and continuous variables s ij tat represent te (expected) number of patients per unit time to transport from demand region i to trauma center j using a elicopter originating from eliport. We also define te following four quantities, wic are auxiliary decision variables in our mat program: λ G = i I,j F G i s G ij; (1) λ = s ij H; (2) (i,j) F λ j = s G ij + s ij j J; (3) i Fj G (,i) F j r = τ ij s ij H, (4) (i,j) F were λ G in (1) represents te total number of patients tat we plan to transport by ground ambulance (across all demand regions and all trauma centers); λ in (2) represents te total number of patients tat we plan to transport using eliport ; λ j in (3) represents te total number of patients tat we plan to transport to trauma center j; and r in (4) is te workload assigned to eliport, wic can be explained as follows. Eac patient assigned to eliport uses some elicopter time. Specifically, 9

11 a patient flown from demand region i to trauma center j using a elicopter originating from eliport causes a elicopter to be in service for τ ij units of time, i.e. te time it takes to fly te circuit i j, plus loading, unloading, and cleanup. Te total workload generated by all patients flying from i to j using a elicopter from is τ ij s ij, wic is unitless because τ ij is measured in units of time/patient wile s ij is measured in units of patients/time. Te workload assigned to eliport is simply te sum of te workloads from all patients tat te plan assigns to eliport. 6 Table 1 summarizes our notation. Symbol Definition k Number of trauma centers to be located m Number of elicopters to be located i Index for demand regions; i I = {1, 2,..., ī} j Index for eliports; H = {1, 2,..., } Index for trauma centers; j J = {1, 2,..., j} λ i Expected demand for region i c j Capacity of trauma center j d r i Road distance between te center of demand region i and its nearest ambulance station d r ij Road distance between te center of demand region i and trauma center j d ij Euclidean distance between te center of demand region i and trauma center j d j Euclidean distance between trauma center j and eliport d i Euclidean distance between eliport and te center of demand region i d ground (d air ) Maximum distance tat can be covered by an ambulance (a elicopter) τ ij Average service time for a elicopter to fly te circuit i j F G, Fi G, F j G Sets of patients tat are covered by ambulances (see 3 for teir precise definitions) F, F, F i, F j Sets of patients tat are covered by elicopters (see 3 for teir precise definitions) y j Variable: Equals 1 if a trauma center is opened at site j, or oterwise equals 0 x Variable: Equals 1 if a elicopter is stationed at eliport, or oterwise equals 0 s G ij Variable: Number of patients per unit time to transport from i to j by ambulance s ij Variable: Number of patients per unit time to transport from i to j by elicopter λ G Variable: Total number of patients per unit time to be transported by ambulance λ Variable: Total number of patients per unit time to be transported by elicopter λ j Variable: Total number of patients per unit time to be transported to trauma center j Variable: Workload assigned to eliport r Table 1: Summary of Notation Ideally, we would like to maximize te expected number of patients tat are transported and begin to receive care at a trauma center by te 60-minute tresold. Suc an objective would simultaneously incorporate te congestion at bot eliports and trauma centers. However, te expression for suc an objective involves convolutions of random variables, and is too complex to work wit. Instead, we maximize te expected number of patients tat are transported witout delay, and use a constraint to 6 In practice, tere may be times wen a elicopter can fly directly to pick up its next patient witout returning to its ome eliport. Altoug suc call-to-call travel does not affect te eliports located on te roofs of trauma centers (wic comprise 38 out of 54 candidate eliports), it may sorten te service time of a elicopter located in a eliport separate from a trauma center. Berman and Vasudeva (2005) ave proposed an approximate approac to model suc call-to-call travel. However, if we follow teir approac, our objective becomes igly nonlinear and does not yield a tractable solution. Wen we retain our existing location solutions and add call-to-call travel to our simulation, we find te percentage of successful patients increases by 1%-3%, and tat our main results presented in 5.3 remain valid. 10

12 ensure sufficient capacity exists at eac trauma center, so tat patients rarely need to wait for a trauma center bed once tey get tere. To write down an expression for our objective, we note tat wen eliport is open (i.e. x = 1), eliport can be considered as a single-server queue wit te elicopter as te server, arrival rate λ, mean service time τ = r /λ, and utilization r (were workload and utilization are equivalent in a single-server queue). Ten, te probability tat an arriving patient finds eliport busy is equal to te eliport s utilization r under te following two assumptions: (i) a patient will wait for a elicopter as needed (te eliport queue backlogs demands; it is not a loss system), 7 and (ii) patient arrivals are Poisson. 8 In oter words, r is te site-specific busy fraction for eliport, and it is endogenously-computed, since it depends on te decision variables {s ij } (see (4)). Tus, te total number of patients tat we expect to be transported (by elicopter or ambulance) witout any delay is: (1 r )λ + λ G, (5) H were, since we assume tere are ample ambulances, all λ G ambulance-transported patients are transported witout delay. Our mat program maximizes (5), wic is a proxy for te expected number of patients transported witin 60 minutes. 9 Our objective is consistent wit te so-called expected covered demand objective commonly used in te literature starting from Daskin (1983) (see, e.g., of Berman and Krass 2002, Sorensen and Curc 2010, and references terein). However, we make te important distinction tat we determine eliport-specific busy fractions r endogenously instead of estimating busy fractions exogenously tat are specific to eac demand region i (see our earlier discussion in 2). To ensure tat trauma center congestion is kept in ceck, we pre-compute an appropriate value for te effective capacity c j of eac trauma center j. We briefly describe ow we derive an appropriate value for c j from a probabilistic constraint, wile presenting details in Online Appendix C. By following Marianov and Serra (1998) and Berman and Krass (2002) wo use an M/M/k queueing model to approximate te flow of patients troug a trauma center, we can express te probabilistic constraint Prob[waiting time at trauma center j ω] ξ as ρ j ρ ω,ξ j, were ρ j is te total workload assigned to trauma center j and ρ ω,ξ j is a constant tat depends on ω and ξ. Moreover, by defining µ j as te service rate of eac server at trauma center j, we can rewrite te constraint ρ j ρ ω,ξ j we define te effective capacity of trauma center j as c j = µ j ρ ω,ξ j as λ j µ j ρ ω,ξ j. Finally,, and impose a capacity constraint of 7 Having patients queue for service is in line wit Ball and Lin (1993). Oters (e.g., see Borras and Pastor 2002) assume tat patients do not wait for ambulances and find alternate (private) modes of transportation. Our assumption seems reasonable because in our problem elicopters transport only tose patients wo are far away from a trauma center and cannot be reaced by ambulance witin 60 min. 8 Poisson arrivals is a common assumption in te literature, made for tractability as a first-order approximation of complex systems (e.g., see Berman and Krass 2002, Zang et al. 2010) even in cases wen Markovian assumptions may not old in a strict sense. 9 In fact, our objective function is a conservative underestimate for te number of patients transported witin 60 minutes. Tis follows from te fact tat te probability a patient experiences no delay is less tan or equal to te probability tat a patient s delay is small enoug tat s/e can be transported to a trauma center witin 60 minutes. 11

13 te form λ j c j y j, wic limits te number of patients served by trauma center j to c j wen it is open, or to zero wen it is closed. Putting all tis togeter, we write our full MINLP model as follows: (P ) max λ G + H(1 r )λ s.t. (1)-(4) r x H (6) y j k (7) j J x m (8) H j F G i s G ij + s ij λ i i I (9) (,j) F i λ j c j y j j J (10) x j y j j J (11) s G ij 0 (i, j) F G ; s ij 0 (, i, j) F (12) y j {0, 1} j J; x {0, 1} H. (13) Constraint (6) ensures tat te busy fraction (or utilization) of eliport, r, sould be less tan or equal to 1; in oter words, eliports sould not be overloaded. Constraints (7) and (8) ensure tat at most k trauma centers are opened and at most m elicopters are stationed across all eliports. Constraint (9) says we cannot plan to serve more people from region i tan te expected demand λ i from tat region, and constraint (10) is our capacity constraint tat keeps congestion at trauma center j under control. Constraint (11) makes sure tat wen a trauma center is closed, so is te eliport on its roof (recall tat te set H( J) is indexed suc tat eliport j is on te roof of trauma center j). Finally, constraint (12) makes sure tat te number of patients served by all transportation modes must be nonnegative, and constraint (13) makes sure tat eac trauma center is eiter open or closed, and eac eliport is assigned eiter one elicopter or no elicopter. Note tat, taken togeter, constraints (2), (4), and (6) enforce te condition tat no demands are allocated to closed eliports (i.e., x = 0 s ij = 0 (i, j) F λ = 0). Tere are a few ways tat our model differs from muc of te existing literature. Instead of pregrouping demand regions into districts and assuming tat eac district is served by a pool of elicopters, we make no suc assumptions (so called districting assumptions in te literature), and allow te mat program to determine te assignment of patients to eliports troug decision variables s ij. As discussed in Borras and Pastor (2002), demand-area-specific busy fractions witin a district can eiter be server-independent or server-dependent, wic boils down to weter servers witin a district are modeled as independent single-server queues or one multi-server queue, respectively. In te body of our paper, we assume tat eac eliport is its own single-server queue. However, if we allow multiple 12

14 elicopters per eliport, we can model server dependence, as described in Online Appendix A. Moreover, in our model, because busy fractions are endogenous, we also ave anoter type of dependence, wic spans across eliports (analogous to dependence spanning across districts). To illustrate tis point, imagine tat a demand region can be served by two eliports, 1 (nearby) and 2 (furter away). Initially, it is optimal to direct patients to 1. However, as utilization r 1 rises, congestion at 1 increases, and te mat program begins to direct patients to 2 (wic increases r 2 ). Consequently, demands get balanced across eliports 1 and 2, wic is mediated by te fact tat te busy fractions r 1 and r 2 are linked troug decision variables s ij. Te cief computational difficulty in solving te MINLP problem (P ) is te set of non-convex bilinear terms λ r tat appear in te objective. As oters (c.f. Floudas and Pardalos 2012) ave reported, bilinear terms can be notoriously callenging to cope wit. In our case, tese bilinear terms are embedded in a generalized facility location problem tat models bot trauma center and eliport locations, as well as te routing of elicopters and ambulances. Te resulting problem is significantly more difficult to solve tan te canonical facility location problem wit linear objective, wic itself is ard (c.f. Owen and Daskin 1998). Fortunately, we are able to exploit problem-specific structure to find solutions to (P ) tat are near-optimal and significantly outperform our bencmark euristics. We find solutions of (P ) by iteratively solving a sequence of problem relaxations and restrictions, all of wic are convex optimization problems and can be solved using a Mixed Integer Quadratic Programming (MIQP) solver suc as CPLEX. Tis turns out to be a more computationally efficient approac tan using a general global optimization solver suc as BARON, wic as te ability to cope wit non-convexities but doesn t exploit te problem-specific structure as well as our specialized metods. Our general sceme works as follows. Since we are maximizing te objective, any relaxation yields a valid upper bound, wile any restriction produces a lower bound. At eac point in time, we can compute an optimality gap by taking te difference between te best (lowest) upper bound and te best (igest) lower bound found tus far, and use tis gap to determine weter to continue iterating or stop. In te following, we first introduce some relaxations of (P ) in and ten a restriction of (P ) in In 4.2, we describe two metods tat we use to reduce te optimality gap Relaxations First, we relax (P ) by using McCormick envelopes (McCormick 1976, Floudas and Pardalos 2012) to linearly outer-approximate te bilinear λ r terms. A Mixed Integer Linear Program (MILP) relaxation of (P ) based on McCormick envelopes is: (P McCormick ) max λ G + H λ H w s.t. (1)-(4), (6)-(13) w λ MAX r + r MAX λ λ MAX r MAX H (14) w 0 H. (15) 13

15 Let us explain ow we ave obtained (P McCormick ). Te McCormick envelopes are derived using known constants λ MIN, λ MAX, r MIN and r MAX, wic are lower and upper bounds on te allocated demand rate and workload, respectively. Specifically, te McCormick envelope for te bilinear expression w = λ r is: w λ MAX w λ MIN w λ MAX w λ MIN r + r MAX r + r MIN r + r MIN r + r MAX λ λ MAX r MAX (16) λ λ MIN r MIN (17) λ λ MAX r MIN (18) λ λ MIN r MAX. (19) Since our objective will try to make w as small as possible, it is only te lower bounds for w tat are needed in our formulation. Tus we include only (14) and (15) in (P McCormick ), were (14) is te collection of constraints of type (16) for all eliports, and (15) was obtained by substituting λ MIN = r MIN = 0 into (17) for all eliports. Te values for λ MAX and r MAX are instance-specific; for example, we can define r MAX = 1 as te maximum utilization at eliport and λ MAX = i I:(i,j) F λ i as te total demand from regions near eliport. In general, te relaxation (P McCormick ) can be tigtened by using smaller bounds λ MAX and r MAX, wic we derive in Online Appendix D. However, it turns out tat te relaxation (P McCormick ) is quite weak, regardless of te bounds we coose. To derive a significantly tigter relaxation, we exploit te fact tat te variables λ = i,j s ij and r = i,j τ ijs ij are bot defined as linear combinations of s ij variables. In particular, we can interpret τ = r /λ as te mean service time of eliport ; tat is, te amount of time it takes a elicopter stationed at eliport to fly i j, averaged over all pick-up and drop-off points (i, j). Defining τ MAX = max i,j τ ij and τ MIN = min i,j τ ij as te maximum and minimum mean service times respectively, we derive a quadratic outer-approximation by sandwicing te bilinear term λ r as follows: τ MIN r /λ τ MAX τ MIN λ 2 λ r τ MAX λ 2. Te lower bound for λ r gives us te following Mixed Integer Quadratic Program (MIQP) relaxation of (P ): (P SQE M ) max λg + λ H H s.t. (1)-(4), (6)-(13). τ MIN λ 2 To compare te quadratic envelope used in (P SQE M ) wit te linear McCormick envelope used in (P McCormick ), we first introduce some notation and ten compare bot envelopes using an example. Treating te mean service time τ as known and fixed, we let r (λ τ ) = τ λ and w (λ τ ) = r (λ τ )λ = τ λ 2 denote te workload r and te quantity w, respectively, as a function of λ. Given a problem instance of (P ) wit τ MIN = 60 minutes and τ MAX = 140 minutes, we know tat te optimal value for τ will be in te range [60, 140]. Let us assume, for purposes of illustration, tat te optimal τ is midway between its bounds; i.e., τ = 100 minutes. Figure 2 plots w (λ 100) (dotted curve), wic is sandwiced by te quadratic envelope between w (λ τ MIN 14 ) and w (λ τ MAX ) (two

16 solid curves). Te linear McCormick envelope for τ = 100 minutes (two dased lines) is computed using (16)-(19) and r (λ τ ) as follows. Te bottom line comes from (16), and is te McCormick lower bound for w, assuming τ = 100 is fixed (wic is tigter tan w 0 from (17)). Te top line is te McCormick upper bound for w, assuming τ = 100 is fixed, and is derived from (18) (wic in tis example is tigter tan (19)). Recall tat te lower bound for w, not te upper bound, is important in our formulation because our objective will try to make w as small as possible. As Figure 2 sows, for low values of λ te quadratic envelope is tigter (iger), wereas te linear McCormick envelope is tigter for ig values of λ (te lowest solid curve crosses te lowest dased line at λ = per minute). Note tat we used λ MAX described in Online Appendix D. = 1/(2τ MIN ) = 1/120, wic comes from te tigtened bounds w In order, reading from top to bottom along te rigt-and-side of λ s domain w (λ τ MMM ) w = λ MMM r (λ τ ) w (λ τ ) Λ w = λ MMM r λ τ +r MMM λ λ MMM r MMM w (λ τ MMM ) Figure 2: Comparison of te quadratic envelope used in (P SQE M ) wit te linear McCormick envelope used in (P McCormick ), plotted over te range λ [0, λ MAX = 1/120]. In our computational experiments, we ave observed tat formulation (P SQE M ) solves faster tan (P McCormick ), suggesting tat te quadratic envelope is usually tigter tan te McCormick envelope, possibly owing to te fact tat te continuous relaxation of our problem spreads demands across many eliports, causing λ to be on te low side, were te quadratic envelope is tigter as depicted in Figure 2. We conjectured tat we could even do better by enforcing bot te quadratic and McCormick envelopes, as in te following (PM GBD) formulation (were w 0 is redundant): (PM GBD ) max λ G + λ w H H s.t. (1)-(4), (6)-(13) w τ MIN λ 2 H w λ MAX w 0 H. r + r MAX λ λ MAX r MAX H However, tis turns out to be a bad idea because CPLEX is better at andling te quadratic objective 15

17 of (P SQE ) tan te quadratic constraints in (P GBD); see Online Appendix E for computational details. M Terefore, in 4.1.2, we will use (P SQE M M ) as our master problem Restriction Solving te master problem (P SQE M ) gives us a feasible solution to (P ), since all of (P ) s constraints are present in (P SQE M ). We evaluate te quality of tis solution using te true objective from (P ), i.e., λ G + λ λ r. Te optimality gap of tis solution is te difference between te optimal value of (P SQE M (λ r τ MIN ) and te true value of tis solution; i.e., (λg + λ τ MIN λ 2 ) (λg + λ λ r ) = λ 2 ). Notice tat since r /λ τ MIN, te optimality gap will always be nonnegative. Altoug we could use te feasible solution from (P SQE M ) directly, we can often find better feasible solutions to (P ) by re-optimizing over a subset of te decision variables. Specifically, from a feasible solution from (P SQE M ), we fix te set of open trauma centers {y j}, te locations of elicopters {x }, te demand for elicopters {λ }, and te demand for ambulances ({s G ij }, λg ). Ten, we ignore te elicopter routing pattern {s ij } suggested by te master problem and re-assign elicopter-transported patients across eliports and trauma centers, wit te goal of sifting te workload {r } to te eliports tat are under-utilized. Tat is, given te fixed values for te set of master problem variables Θ = {{y j }, {x }, λ G, {λ }, {s G ij } }, we solve a restriction of (P ) to optimize over te remaining variables {{s ij }, {r }}. 10 Wit te variables Θ fixed, constraints (1), (7), (8), (11), and (13) in (P ) can be ignored because any feasible master problem solution already satisfied tese constraints. For brevity, define te constants a j = c j y j i sg ij j and b i = λ i j sg ij i, wic depend only on problem data and te fixed variables Θ. Our restriction of (P ), wic we call our subproblem, optimizes over te variables {{r }, {s ij }} and is defined as te following Linear Program (LP): (P Θ S ) min H λ r s.t. (,i) F j s ij a j j J (j,) F i s ij b i i I (i,j) F s ij = λ H (i,j) F τ ij s ij r = 0 H r x H s ij 0 (, i, j) F. Te optimality gap between te master problem and subproblem is still measured as (λ G + H λ H τ MIN λ 2 ) (λg + λ λ r ) = (λ r τ MIN λ 2 ), (20) H H H 10 Tecnically, since λ j depends on s ij, it is also re-optimized wen we solve (P Θ S ). However, since λ j does not appear directly in (P Θ S ), it is clearer to omit λ j from te set of remaining variables. 16

18 but now te (smaller) {r } values come from te subproblem instead of te master problem, and as a result te gap is reduced. 4.2 Solution Metods for Tigtening te Relaxation Once we ave a feasible solution to (P ) and a corresponding optimality gap, te next question is: Can we make any inferences from te incumbent solution or its dual tat allow us to tigten te master problem s relaxation and tereby reduce te optimality gap? A tigter relaxation of (P SQE M ) yields not only a tigter (i.e., lower) upper bound for (P ), but also wen Θ is fixed to a solution tat is closer to te true optimum, te subproblem (PS Θ ) finds better (i.e., iger) lower bounds for (P ). We ave studied two metods to tigten te master problem s relaxation: a metod tat relies on sifting quadratic envelope boundaries ( 4.2.1), and a Benders-like cut generation sceme ( 4.2.2) Sifting Quadratic Envelopes (SQE) Algoritm Te first metod tat we use to tigten te master problem uses one or more quadratic envelopes for eac eliport. We use binary variables to control wic envelope is active at eac candidate solution, and re-define te envelope boundaries at eac iteration in an attempt to tigten te relaxation. We first describe a naïve implementation of our approac, and ten discuss modifications tat we ave made for computational tractability. From (PS Θ), we ave a feasible solution to (P ), wic we denote Ψ = Θ {{r }, {s ij }} = { } {y j }, {x }, λ G, {λ }, {λ j }, {s G ij }, {r }, {s ij }. Te two crucial variables tat describe te performance of eliport are its allocated demand λ and its workload r. Terefore, for eac eliport, we use te point (λ, r ) R 2 from te feasible solution Ψ to update te quadratic envelope boundaries for eliport. Let τ = r /λ be te mean service time of candidate solution (λ, r ) and τ = r /λ be te mean service time of te optimal solution (λ, r ) of (P ). For eac eliport, our goal is to pick a mean service time τ F IX tat underestimates te optimal mean service time τ as closely as possible. Tis will allow us to closely underestimate te bilinear term w = λ r using a convex quadratic function of λ. Define two functions of λ ; namely, r (λ τ F IX as τ τ F IX ) = τ F IX, it follows by definition tat r (λ τ F IX λ and w (λ τ F IX ) = λ r (λ τ F IX ) r and w (λ τ F IX ) = τ F IX λ 2. As long ) w. Tat is, for points (λ, r ) were r r (λ τ F IX ) = λ τ F IX, te convex quadratic function w (λ τ F IX ) = τ F IX λ 2 provides a valid underestimate for w. Since, in general, we know only tat τ [τ MIN, τ MAX ] before solving our problem (P ), te best coice for τ F IX tat is always guaranteed to underestimate te optimal mean service time is τ MIN construct (P SQE M. Tis is wy, in 4.1.1, we used te underestimate τ F IX = τ MIN to ) by replacing te bilinear terms w = λ r in te objective of (P ) wit teir quadratic relaxations w (λ τ MIN ) = τ MIN λ 2. We will now describe ow we can use oter estimates for τ F IX tat lead to tigter relaxations. Notice tat since r = τ λ and τ [τ MIN defined by τ MIN, τ MAX ], all feasible (λ, r )-points must lie in te cone λ and λ r τ MAX λ, as sown in Figure 3 (e.g., consider r (λ τ,1 ) = τ MIN λ in Figure 3(a)). Moreover, we can subdivide tis cone into m slices of equal size r (λ τ,2 ) = τ MAX by splitting te domain of τ into te m subdomains [τ,1, τ,2 ], [τ,2, τ,3 ],..., [τ,m, τ,m +1], were τ,n = τ MIN + ((n 1)/m )(τ MAX τ MIN ) for n = 1, 2,..., m + 1. Wen te solution (λ, r ) is in 17

19 te nt subdomain (i.e., in te slice defined by τ,n λ r τ,n+1 λ ), we can use τ F IX = τ,n as an underestimate for te true mean service time τ at te point (λ, r ). To keep track of wic subdomain eac eliport s mean service time is in, we use binary variables; i.e. we let x n = 1 if eliport s mean service time is in te subdomain [τ,n, τ,n+1 ], or x n = 0 oterwise. Moreover, we use te constraint n x n = 1 to ensure tat eac eliport s mean service time falls in exactly one subdomain (and wen τ is on te boundary of two subdomains, we count eliport s mean service time as being in only one of te neigboring subdomains). Suc a setup allows us to activate te lower bound r r (λ τ,n ) wenever te solution (λ, r ) to (P SQE M ) is in te nt subdomain, tereby replacing te objective term w = λ r of (P ) wit te quadratic underestimate w (λ τ,n ) = τ,n λ 2. Te full formulation of (P SQE M ), wic as slices indexed by te sets N = {1,..., m }, is as follows: (P SQE M2 ) max λg + λ τ,n λ 2 n H H,n N s.t. (1)-(4), (6)-(13) λ = n N λ n, r = n N r n H 0 λ n λ MAX x n, 0 r n r MAX x n H, n N τ,n λ n r n τ,n+1 λ n x n = 1 n N H, n N H x n {0, 1} H, n N. In addition to te variables and constraints of (P SQE M ), te formulation (P SQE M2 ) includes te binary variables {x n } tat define subdomain membersip, new continuous variables {λ n } and {r n }, and several logical constraints tat link tese quantities. Te added constraints make sure tat wen x n = 1, ten λ n and r n are equal to te allocated demand and workload of eliport (i.e., λ n = λ and r n = r ), wereas wen x n = 0, ten λ n = r n = 0. Terefore, for eac eliport, only one conic slice τ,n λ n r n τ,n +1λ n is ever active at a time (corresponding to te n wit x n = 1). As we subdivide te domain of eac τ into finer slices, our relaxation (P SQE M2 ) becomes tigter. Moreover, as te number of slices m for all eliports, te area of eac slice collapses to zero and te optimal value of (P SQE M2 ) converges to te optimal value of (P ); tat is, in teory we can approximate (P ) to any arbitrary precision by simply slicing te subdomains of τ finely enoug. However, as te number of slices m increases, te problem (P SQE M2 ) becomes muc arder to solve due to te larger number of binary variables {x n }. As a result, we abandon te idea of using equally-spaced slices, and instead use a more efficient metod to decide were to slice eac cone τ MIN λ r τ MAX λ, tereby producing a tigt relaxation of (P ) using only a few slices. Our Sifting Quadratic Envelopes (SQE) algoritm begins wit only one slice defined for eac eliport; i.e., m = 1, τ,1 = τ MIN, and τ,2 = τ MAX. At eac iteration and for eac eliport, te algoritm subdivides one (judiciously cosen) slice into two. Te general idea is tat we sould focus 18

20 (a) r r (λ τ,2 ) Iteration 1 (b) r r (λ τ,2 ) Iteration 1 (c) r r (λ τ,3 ) Iteration 2 (λ, r ) r (λ τ,1 ) (λ, r ) r (λ τ NNN ) r (λ τ,1 ) (λ, r ) r (λ τ,2 ) r (λ τ,1 ) λ λ λ (d) r Iteration 2 r (λ τ,3 ) r (λ τ NNN ) r (λ τ,2 ) (λ, r ) r (λ τ,1 ) (e) r Iteration 3 r (λ τ,4 ) r (λ τ,3 ) (λ, r ) r (λ τ,2 ) r (λ τ,1 ) (f) r Iteration 3 r (λ τ,4 ) r (λ τ NNN ) (λ, r ) r (λ τ,3 ) r (λ τ,2 ) r (λ τ,1 ) λ λ λ (g) r Iteration 4 r (λ τ,4 ) r (λ τ,3 ) r (λ τ,2 ) (λ, r ) r (λ τ,1 ) () r Iteration 4 r (λ τ,4 ) r (λ τ,3 ) r (λ τ,2 ) r (λ, r ) (λ τ NNN ) r (λ τ,1 ) (i) r r (λ τ,3 ) (λ, r ) Iteration 5 r (λ τ,2 ) r (λ τ,1 ) λ Figure 3: Illustration of te Sifting Quadratic Envelopes algoritm λ λ our attention on narrowly refining te partition of τ s domain in areas tat are likely to be close to te (P )-optimal τ SQE. Terefore, at eac iteration, we solve (PM2 ) and (P S Θ ) to get te (P )-feasible solution (λ, r ), wic lies in a particular slice, say te slice defined by τ,n λ τ τ,n +1λ for some n. Next, we subdivide tis slice into two unevenly-sized slices. Because we care about producing underestimates for τ rater tan overestimates, we cut te slice in two along te ray r (λ τ NEW ) = τ NEW λ, were τ NEW = (τ + τ,n )/2 is te average of te true mean service time τ = r /λ at te point (λ, r ) and te slope τ,n of te lower boundary of tat slice (wic can be interpreted as te previous underestimate of te mean service time at te point (λ, r )). Note tat, altoug we coose were to split eac slice euristically, our coices are motivated by efficiency and ave no bearing on te correctness of our approac, or on our ability to generate valid quadratic envelopes. Moreover, to keep te total number of slices small, wenever we suspect tat te (P )-optimal (λ, r ) as a low likeliood of being in a particular slice, we merge tat slice wit te slice below it. From one iteration to te next, we may add or delete slices, but we never use more tan a small number of slices per eliport, wic keeps te computational complexity of (P SQE M2 ) in ceck. It turns out tat in practice, keeping just 3 slice boundaries below and 1 slice boundary above te current solution (λ, r ) results in a good trade-off between precision and tractability. Because tis procedure causes te boundaries of te subdomains of τ to sift at eac iteration, wic in turn define te quadratic envelope boundaries 19

21 w (λ τ,n ) = τ,n λ 2, n = 1, 2,..., m, we call tis metod te Sifting Quadratic Envelopes (SQE) algoritm. At eac iteration of te SQE algoritm, we proceed as follows: (1) solve te master problem (P SQE M2 ) to get a good solution to (P ); (2) solve te subproblem (PS Θ ) to improve tis solution; (3) evaluate te quality of te improved solution by computing its optimality gap using Equation (20); and (4) terminate if a time limit or optimality gap tresold is reaced, oterwise sift te quadratic envelope boundaries in (P SQE M2 ) and continue. Figure 3 illustrates te progression of te SQE algoritm on a ypotetical example. For eac iteration, we plot te subdomains of [τ MIN, τ MAX ] for a single eliport as conic slices in te (λ, r )-plane. (Altoug we only describe wat appens at a single eliport, te conic slices are adjusted for all eliports at eac iteration of our algoritm.) Grapically, we make use of te fact tat, for any fixed value of τ, te function r (λ τ ) = τ λ defines a ray from te origin wit slope τ. Tus, we begin wit a single slice bounded by r (λ τ,1 ) and r (λ τ,2 ), were τ,1 = τ MIN and τ,2 = τ MAX (Figure 3(a)). At Iteration 1, we solve (P SQE M2 ) and (P S Θ ), and use te solution (λ, r ) to compute te mean service time τ = r /λ (tis τ is te slope of te dotted ray in Figure 3(b)). Next, we split te subdomain of [τ,1, τ,2 ] into two slices by introducing a new slice boundary r (λ τ NEW ), were τ NEW = (τ + τ,1 )/2 is te average of te actual mean service time τ at te point (λ, r ) and te previous underestimate τ,1 (Figure 3(b)). At Iteration 2, after re-labelling te slice boundaries τ NEW and τ,2 as τ,2 and τ,3 respectively, we ten re-solve (P SQE M2 ) and (P S Θ) to get a new point (λ, r ) and its associated mean service time τ = r /λ. Assuming te point (λ, r ) is in te top slice (Figure 3(c)), we proceed by splitting te top slice in two. We do tis by creating a new slice boundary wit slope τ NEW = (τ + τ,2 )/2, i.e. a slope tat is midway between te actual mean service time τ and te previous underestimate τ,2 (Figure 3(d)). Once again, we re-label te slice boundaries and re-solve (P SQE M2 ) and (P S Θ). At Iteration 3, assume te point (λ, r ) also falls into te topmost slice (Figure 3(e)). As in te previous iteration, we split te top slice in two by defining a new region boundary wit slope τ NEW = (τ + τ,3 )/2 tat is midway between τ = r /λ and te underestimate defined by te lower boundary of tat slice, τ,3 (Figure 3(f)). But before te next iteration, we also delete one region boundary to keep te problem size manageable. Motivated by our desire to generate underestimates, we keep up to tree region boundaries below te incumbent point (λ, r ), and only one region boundary above. Specifically, we delete te region boundary defined by r (λ τ,2 ), i.e. te lowest region boundary tat can be deleted. Note tat we must keep te original region boundaries r (λ τ MIN ) and r (λ τ MAX ), since te solution (λ, r ) may ave a mean service time τ = r /λ tat falls anywere in te full domain [τ MIN, τ MAX ]. At Iteration 4, wit te slice boundaries re-labelled, we re-solve (P SQE M2 ) and (P Θ S ) once again. Assuming te new solution (λ, r ) now falls into te bottommost slice (Figure 3(g)), we split te bottommost slice in two by introducing te slice boundary τ NEW = (τ + τ,1 )/2 and delete all but one slice boundary tat lies above te point (λ, r ); i.e. we delete r (λ τ,2 ) and r (λ τ,3 ) (Figure 3()). Finally, Iteration 5 begins wit two slices for eliport, as sown (Figure 3(i)). Te algoritm continues until eiter te optimality gap is reduced below a desired tresold or a time limit as been reaced. Pseudocode for te SQE algoritm 20

22 can be found in Online Appendix F Algoritm Based on Generalized Benders Decomposition Te second metod tat we use to tigten te master problem is based on Generalized Benders Decomposition (GBD) (e.g., Benders 1962, Geoffrion 1972). GBD is a tecnique tat can be used to solve a complex mat program by structurally decomposing it into a master problem and one or more subproblems. Te subproblems dual solutions are used to infer one or more cuts tat are ten added to te master problem to make its formulation tigter. GBD iterates back and fort between solving te master problem and subproblems until a provably near-optimal solution to te full problem is found. It is wort pointing out tat wen te mat program being decomposed is nonlinear, special care must be taken to implement GBD to make sure tat Benders cuts do not inadvertently cut off te optimal solution; see, e.g., Geromel and Belloni (1986) and Sainidis and Grossman (1991). In our case, tis special care requires us to add binary variables to our formulation for eac cut generated. At eac iteration t, we augment te master problem (PM GBD ) wit a Benders cut of te form z B t ( ), were z is te objective of te master problem. For completeness, te full master problem used in te Benders decomposition takes te form: (P GBD M2 ) max z s.t. (1)-(4), (6)-(13) z λ G + H λ H w z B t ( ) t = 1..nCuts w w λ MAX w 0 H. τ MIN λ 2 H r + r MAX λ λ MAX r MAX H Te Benders subproblem is te restriction of (P ) wit te variables in te set } Θ = {{y j }, {x }, λ G, {λ }, {s G ij } fixed to te master problem solution. Tis is exactly te previouslyintroduced linear program (PS Θ) from Its dual at } eac iteration t is te linear program (DΘ S ) tat optimizes over te variables {{α t j }, {βt i}, { t }, {γt } (derived in Online Appendix G). Define: B t (Θ) = λ G + j α t ja j + i β t ib i + t λ + max(λ, γ t )x. (21) Lemma 1 At eac iteration t, z B t (Θ) is a Benders optimality cut. To implement te max(λ, γ t )x expressions in te Benders cut z B t (Θ), we introduce binary variables. For details of tis implementation and te proof of Lemma 1, see Online Appendix G Computational Results To test te SQE and GBD-based algoritms described in and 4.2.2, respectively, we ran various instances wile varying te number of trauma centers k and te number of elicopters m. We conducted 21

23 all of our computational experiments on a Dell Precision T5500 workstation wit Intel Xeon CPU 3.33GHz (6 cores) and 12.0 GB RAM, running Windows 7, 64 bit. We used CPLEX 12.3 and AMPL to solve te MIQP and MILP formulations of our problem instances. We ave found tat: (1) wen using SQE, te (P SQE M ) formulation outperforms te (P GBD M ) formulation, and (2) SQE outperforms te GBD-based algoritm. Performance varied by instance (k, m), wit te optimality gap of te SQE metod after 18 ours being in te range of 1.64% to 9.34%. For example, for (k, m) = (10, 15), we reaced a gap of 7.61%, wereas for (k, m) = (14, 25), we reaced a gap of 2.8%. Our SQE metod is substantially faster tan te global solver BARON, wic, at te 18-our mark as a gap of 21.9% and 13.6% for (k, m) = (10, 15) and (14, 25), respectively. Furter details of our computational experiments are in Online Appendix E. Altoug striving for a low optimality gap and solution time is teoretically justified, wen it comes to measuring te real-world performance of a particular solution, one of te best ways is to run te location solutions troug a simulation model. For tis reason, in 5 we use simulation to compare te performance of te solutions tat we get using SQE wit te solutions from two euristics tat we present next. 4.3 Bencmark Heuristics We now introduce two bencmark euristics tat were inspired by existing metods in te literature. We call te first te no-congestion euristic, and te second te decoupled euristic. Te no-congestion euristic is inspired by te model of Branas and ReVelle (2001). As described in 2, Branas and ReVelle (2001) also study a joint location problem of trauma centers and elicopters, but model te problem as a deterministic location problem witout taking into account te random availability of elicopters. To isolate te effect of elicopter congestion, we construct a Mixed Integer Linear Program (MILP) relaxation from (P ) by simply dropping te bilinear λ r terms from its objective: (P NC ) max λ G + H λ s.t. (1)-(4), (6)-(13). Problem (P NC ) maximizes te number of patients served, assuming eliports do not get congested. Te no-congestion euristic is simpler to implement (in terms of computation) tan our original problem (P ) because te non-convex bilinear terms λ r do not appear in te objective. As we sall see in 5.3, ignoring elicopter congestion as a detrimental impact on overall performance. (Note tat (P NC ) is not exactly te same as te formulation of Branas and ReVelle (2001) because, most notably, (P NC ) additionally models te capacity constraints of trauma centers.) In contrast, te decoupled euristic models congestion, but locates trauma centers and elicopters in sequence, allowing us to test ow important it is for trauma centers and elicopters to be located simultaneously. Tis euristic as practical relevance, since one of te planning metods te KMHW considered was to determine te locations of trauma centers first, wile postponing te decision of were to locate elicopters. Recall from our discussion in 1 tat bot of te single-resource allocation problems 22

24 (i.e., eiter trauma centers or elicopters) belong to well-known classes of optimization problems: trauma centers can be located using a capacitated version of te maximal covering location problem (e.g., Pirkul and Scilling 1991), and elicopters can be located using a probabilistic ambulance location formulation (e.g., Daskin 1983). Te latter metod estimates a elicopter busy fraction, wic can only be computed after te trauma center locations are fixed. Tus, our decoupled euristic solves te problem sequentially as follows. First, we solve a capacitated variant of te maximal covering location problem, assuming tat a elicopter is stationed at every eliport. Te solution from tis problem yields te trauma center locations as well as an allocation of patient demands to trauma centers. Second, after we estimate a elicopter busy fraction from te solution of te first step, we solve a variant of te maximum expected covering location problem (Daskin 1983) to establis te eliport locations. decoupled euristic are presented in Online Appendix H. 5 Application Furter details of te In tis section, we apply our model and solution metods to te design of a nationwide trauma care system in Korea. In 5.1, we briefly describe te data used for our analysis. Ten, in 5.2, we present a trace-based simulation model, wic takes te location solution of an optimization model as input, and simulates te arrival and service processes of trauma patients. Finally, in 5.3, we present te location solutions from te different approaces described in 4 on te map of Korea, and compare teir performance, as measured by our simulation. 5.1 Data We use te following two data sets tat were produced as part of te broader study (Kim et al. 2011) commissioned by te KMHW: (1) demand-side data: one year s wort of nationwide trauma patient calls, including te times and locations of incidents; and (2) supply-side data: te number and location of candidate trauma centers and eliports. [1] Demand-Side Data: Estimating te demand-side data involves many practical callenges suc as fragmented data sources and a lack of clinical information to measure injury severity scores. Below we briefly describe te main data issues, and refer te reader to Kim et al. (2011) for furter details. Estimates of te total annual number of trauma patients come from two data sources: te National Emergency Department Information System (NEDIS), and te National Healt Insurance (NHI). Injuryrelated Emergency Department (ED) visits were identified by teir diagnosis code, yielding a total of 1,223,750 cases. Among tese, only tose cases wit an Excess Mortality Ratio-Based Injury Severity Score (EMR-ISS) iger tan 15 were classified as trauma cases, yielding a total of 190,193 trauma cases. Te NEDIS and NHI data sets, wile useful for computing accurate total patient volumes, lack te fine granularity tat we need to model patient arrivals. For te specific locations and times of trauma incidents, we consulted te nationwide data of emergency telepone calls. Tis data set includes fieldtriage records of wic 80,300 were classified as trauma cases. Additionally, from te NEDIS data we identified 32,630 trauma patients wo were self-transported or transferred from oter local ospitals; 23

25 for tese patients, te locations and times of teir incidents were assumed to be tose of teir ED visits. For te remaining 77,263 (= 190, , , 630) trauma patients, we assigned teir locations and times by subsampling from te 112,930 (= 80, , 630) trauma patients, wile taking care to matc te regional demand rates in te original data. 11 See Figure 4(a) for te geograpical distribution of trauma patients in Korea. For our tests, we split te year into two alves: January-June (90,265 trauma cases) and July- December (99,928 trauma cases). Tis allowed us to test our metods bot in-sample (e.g., by optimizing te trauma center and eliport locations using January-June data and ten evaluating te performance of tis solution wit a simulation using January-June data) and out-of-sample (e.g., optimizing wit January-June data and evaluating wit July-December data). We report only te in-sample results from te January-June data set ere, and include te out-of-sample results, wic are qualitatively similar but serve as a robustness ceck, in Online Appendix I. Finally, we aggregated patient demand by geograpic area to keep te size of our optimization models manageable. After consultation wit practitioners, we used a 25km 25km grid to subdivide Korea into 204 (= ī) demand regions. Te optimization models use te aggregate demand rates λ i for eac region i I = {1, 2,..., ī}, were all patient demands in a region are assumed to come from its center. As a precautionary measure, we also solved our optimization models wit oter grid sizes (e.g., 15km 15km and 20km 20km) and found tat solutions were similar i.e., a reasonable amount of data aggregation seems to ave only a marginal impact on solution quality. 12 (a) (b) (c) Figure 4: Geograpical distribution of: (a) trauma cases in Korea, (b) candidate trauma centers, and (c) candidate eliports (excluding te eliports co-located at trauma centers) [2] Supply-Side Data: Te KMHW provided us wit a list of 38 (= j) candidate trauma center sites, all of wic are existing ospitals tat expressed interest in dedicating resources specifically for trauma patients. Tese ospitals are general ospitals wit operating emergency departments, most of wic 11 Alternatively, we could ave used te locations and times of te ED visits of tese 77,263 patients. After muc debate among medical and field professionals, Kim et al. (2011) concluded tat te sampling approac described ere captured reality better tan tis alterative approac. 12 We ave also conducted simulation experiments wit bot aggregated and unaggregated data, and tested solutions by increasing or decreasing regional demands by x% (were x = ±5, ±10, and ±15). Our results are robust against tese canges. 24

26 are associated wit medical scools. Figure 4(b) sows te locations of tese candidate trauma centers. To limit congestion at trauma centers, te effective capacity of eac center, c j, is set to 50 patients per day. More details on tis derivation can be found in Online Appendix C. For candidate eliports, we use 16 tat te Korean National Emergency Management Agency (NEMA) is currently operating; see Figure 4(c) for teir locations. Recall tat elicopters can also be stationed at open trauma centers. So, te total number of candidate eliport locations,, is 54 (= ). Currently, te NEMA operates elicopters for fire-figting and rescue as well as for EMS missions, so its elicopters are not specifically designed for transporting trauma patients. Tus, te KMHW was interested in exploring te optimal deployment of new EMS-dedicated elicopters. For tis purpose, we vary te total number of available elicopters, m, in our study. 5.2 Simulation Model To bencmark te performance of te location solutions generated by our integrated (SQE) metod from 4.1 and 4.2 wit tose generated by te euristics from 4.3, we use a trace-based simulation. Taking te locations of trauma centers and elicopters as given, we sequentially process eac patient call from te istorical data set. We simulate te real-time processes of serving eac patient according to te flowcart sown in Figure 5, and categorize eac patient as successful or unsuccessful. Successful patients are tose tat get transported to an available (under-capacity) trauma center witin 60 minutes. After processing all patient calls, we measure te proportion of te nation s trauma patients tat are served successfully. Te following features of our simulation, wic are abstracted away in our optimization model and euristics, capture te real-time decision-making processes in practice: Helicopter Assignment: We keep track of eac elicopter s availability at eac point in time. Wile a elicopter is transporting a patient to a trauma center or returning to its ome eliport, it is temporarily unavailable for serving anoter patient. If more tan one patient must wait for a elicopter, patients get served according to te first-in-first-out rule. Helicopter Diversion: We monitor te availability of beds at eac trauma center. If a trauma center does not ave an available bed to admit a new patient, we divert elicopters to te nearest trauma center tat as an available bed. 13 Multiple Resources: Wen multiple resources are available to serve a patient (e.g., multiple trauma centers tat operate under teir capacities, multiple elicopters/ambulances), we always transport a patient to te nearest available (under-capacity) trauma center in te fastest way possible. For furter details of our simulation, most notably te decision points marked A, B, C, and D in Figure 5, see Online Appendix J. 13 We ave obtained similar results wen implementing alternative diversion policies based on te number of waiting patients or boarding patients. 25

27 Patient i arrival <A> Is patient i outside te coverage area? Yes OUT OF COVERAGE (FAILURE) Yes No Are all trauma centers over capacity? <B> Yes Is te nearest trauma center j^ witin ambulance coverage? No No Is te nearest undercapacity trauma center j * witin ambulance coverage? Yes Transport patient i to j * by ambulance GROUND SUCCESS Ambulance <D> Wic veicle can transport patient i to j^ faster? Helicopter <C> No Is j * witin elicopter coverage? No Transport patient i to j * by ambulance GROUND FAILURE Transport patient i to j^ by ambulance GROUND FAILURE Transport patient i to j^ by elicopter HELICOPTER FAILURE <D> Yes Wic veicle can transport patient i to j * faster? Helicopter Ambulance Transport patient i to j * by ambulance Transport patient i to j * by elicopter GROUND FAILURE Is te arrival time of patient i at j * 60min? Yes No HELICOPTER SUCCESS HELICOPTER FAILURE Figure 5: Simulation flowcart 5.3 Results We tested te following tree approaces: te integrated approac (presented in 4.1 and 4.2), and te no-congestion and decoupled euristics (presented in 4.3). Te KMHW was interested in exploring different numbers of trauma centers (k) and elicopters (m), since te budget for te trauma care system ad not yet been determined. Below, we report test results for k = 10, 12, 14, and m = 5, 10, 15, 20, 25. Let TkHm denote te test case wit k trauma centers and m elicopters. First, we examine te location solutions of te trauma centers computed under te tree different approaces. Figure 6 plots te test cases T10H5, T10H15, and T10H25 as we increase te number of elicopters from 5 to 25 olding te number of trauma centers fixed at 10. From tis figure, we can make te following two important observations. First, te trauma center locations depend on te 26

28 number of available elicopters, implying tat it is important to consider te locations of trauma centers in conjunction wit elicopter transportation. Second, te trauma center locations differ significantly across te tree approaces, especially wen a large number of elicopters are used. For example, in te case of T10H25, bot te no-congestion euristic and te decoupled euristic place five trauma centers at different locations, as compared to te integrated approac. Te integrated approac tends to locate trauma centers in ig demand regions (darker areas in te figure), wereas te two euristics tend to spread trauma centers more broadly across te country. Tis is because te two euristics ignore elicopter congestion wen locating trauma centers, and simply maximize te total geograpic coverage. Tis tends to result in trauma centers tat are placed in areas tat make eavy use of elicopters and ligt use of ambulances. On te oter and, te integrated approac overcomes tis sortcoming by accounting for elicopter congestion wen coosing trauma center sites. T10H5 T10H15 T10H25 Decoupled No-Congestion Integrated For eac approac (decoupled, no-congestion, and integrated), indicates te sites cosen for T10H5; indicates te sites cosen for T10H15 but not for T10H5; indicates te sites cosen for T10H25 but not for T10H15 Figure 6: Trauma center locations determined from te tree approaces for T10H5, T10H15 and T10H25 (Note: Helicopter locations are omitted to avoid clutter.) Next, given te locations of trauma centers and elicopters determined from eac of te tree approaces, we examine te percentage of successful transports as measured by our simulation. As seen in Figure 7, wen te number of elicopters is very low (e.g., TkH5), te performance is about te same across all tree approaces. Tis is expected because wit a small number of elicopters most patients are transported by ambulances, wic are modeled rougly in te same manner in all tree approaces. As te number of elicopters increases (i.e., m increases), we observe tat te integrated 27

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