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International Journal of Geographical Information Science iFirst, 2011, 1–26

Spatio-temporal location modeling in a 3D indoor environment: the case of AEDs as emergency medical devices

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Thi Hong Diep Dao, Yuhong Zhou, Jean-Claude Thill* and Eric Delmelle Department of Geography and Earth Sciences, University of North Carolina at Charlotte, Charlotte, NC 28223, USA (Received 4 January 2011; final version received 10 June 2011) This research innovatively extends optimal emergency facility location models to the interior space of multistory buildings with an integrated spatial-temporal framework. We present the case of deployed emergency medical devices known as automated external defibrillators (AEDs), which serve to treat sudden cardiac arrest on-site within the first few critical minutes of the event. AEDs have become a critical element of basic life support services in many public buildings. The proposed framework is based on the concept of discrete time windows to capture the time-dependence of potential demand and stochastically model the detection time component of impedance as a function of space–time distribution of demand. Different optimization objectives minimizing sudden cardiac arrest outcome consequences (e.g., brain damage or death) as a function of suffering time are formulated and solved. The first model is the multiple-time-window maximal covering location problem model which optimizes the placement of AEDs by maximizing the covered demand over all time periods. The second is the multiple-time-window p-Median model which places AEDs to maximize the expected value of prevented death or permanent brain impairment in case of defibrillation treatment over multiple time windows. Both models are implemented through tight coupling of commercial geographic information system software and a linear programming solver. The models are novel in two primary respects, namely location modeling in 3D microscale spaces and the integration of spatial and temporal considerations. Innovative visualization techniques for AED indoor location and coverage are also presented. Keywords: location-allocation modeling; 3D visualization; microscale spaces; spatio-temporal; socio-technical systems

1.

Introduction

Microscale spaces of the built environment, such as squares, courtyards, interior space of buildings, rooms, passageways, and others, have been one of the frontiers of research in geographical sciences for the past decade or so. Certainly, these spaces can be rather private and therefore quite inaccessible to researchers. More critically, however, techniques of representation of microscale spaces and of modeling of activities carried out within these spaces have been scarce until recently (Thill et al. 2011). Concerns for emergency management and response following the September 2001 attacks have been instrumental in raising interest towards geospatial modeling of microscale environments, especially building interiors (e.g., Kwan and Lee 2005). In this article, we look at extending

*Corresponding author. Email: [email protected] ISSN 1365-8816 print/ISSN 1362-3087 online © 2011 Taylor & Francis http://dx.doi.org/10.1080/13658816.2011.597753 http://www.tandfonline.com


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optimal facility location problems in two key respects. First we explicitly recognize the three-dimensionality of the interior space of multistory buildings, which circumvents the reductionist flaws inherent to collapsing space to two dimensions. Second, we closely integrate various spatial and temporal considerations in the analytical formulation of the problem. In particular, we study the case of the optimal placement of automated external defibrillators (AEDs), as a prominent example of deployed emergency devices. The proposed approach can readily be adapted to other indoor emergency devices, such as emergency phones, fire alarms, extinguishers, and others. At the notable exception of Arriola et al. (2005), optimal facility location models have stayed away from the complex reality of indoor spaces, which has degraded the applicability of these models for indoor facility planning purposes. Designed as a critical element of basic life support (BLS) services, AEDs permit the on-site treatment of persons in sudden cardiac arrest (SCA) by laypersons. SCA is a major life-threatening concern to a significant portion of the population because of its unexpected occurrence, the absence of warning signs, and because of the need for prompt BLS assistance to increase the chance of survival and avoid permanent injuries (Caffrey et al. 2002, Culley et al. 2004). According to U.S. Center for Disease Control statistics, it is the leading cause of death in the United States, accounting for an estimated number of 462,000 deaths each year, 74% of them out of hospital (Zheng et al. 2002); only 5–7% of SCA victims survive because treatment is not provided fast enough (Cobbe et al. 1996, NHLBI 2009, Weisfeldt et al. 2010), and 1 out of 10 survivors suffers from moderate to severe neurological impairments (Cobbe et al. 1996). The prevalence of SCA death throughout developed countries is equally alarming (Mehra 2007). Unlike a heart attack, where the heart muscle is injured due to blocked blood flow, SCA is often caused by an ‘electrical accident’ in the heart known as ventricular fibrillation, which causes the heart to stop altogether or to quiver in a lethal rhythm. Whereas SCA victims often have very limited ability to perform actions by themselves after collapsing, paramedics typically consume a significant amount of vital minutes to reach the victim. Consequently, not only the development of better therapeutic approaches for SCA treatment is a primary concern of emergency health professionals, but also the availability of on-site BLS service in the immediate aftermath of an SCA event. AEDs were first introduced in 1979 (Diack et al. 1979) allowing less skilled and trained personnel to provide early defibrillation. Functionally, an AED contains an internal microprocessor-based system capable of analyzing heart rhythm and detecting ventricular fibrillation, diagnosing the shockable rhythms, and releasing an electric charge to treat a patient (Weaver et al. 1989, Varon et al. 1999). Contrary to manual or fixed defibrillation units, use of modern AEDs requires no clinical skill and laypeople are allowed to respond to emergencies very effectively. Time is of the essence for the treatment of SCA, as reported by Das and Zipes (2003). This has prompted many communities and organizations such as the American Heart Association and the Sudden Cardiac Arrest Foundation to establish programs to deploy or guide the deployment of AEDs as public-access units in places such as shopping malls, airports, community centers, schools, universities, and any other location where a sizeable population may congregate. Indoor spaces are particularly challenging for the effective deployment of AEDs because of the limited visibility that may delay the discovery of a SCA victim, but also because of the greater navigational difficulties to find an AED station within confined spaces. The placement of indoor AEDs should recognize the typical sequence of events that happen from the time of cardiac arrest to resuscitation and the time frame associated with this sequence. This sequence is schematically represented in


International Journal of Geographical Information Science

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Figure 1. Furthermore, the temporal variation of the distribution of the population at risk inside buildings (henceforth referred to as ‘demand’) should also be a consideration for the efficient deployment of AEDs. In this article, we present and implement two planning models of optimal placement of AEDs inside multistory buildings for the purpose of optimizing the coverage of the dynamically changing population of users of these spaces. The proposed approach accounts for building layout in 3D network space and for indoor population distributions that are temporally dynamic. The two models follow related, yet different, placement considerations: in the case of the maximum covering location problem (MCLP), the objective is to maximize the total population being covered within a certain time limit by AEDs, while overall response time is minimized in the p-median model. Because the optimization is carried out in an indoor environment, a proper model for 3D indoor networks consisting of rooms, corridors, stairs/elevators, and exit doors is necessary. Travel time impedances being used for determining the facility coverage of a site are calculated based on the 3D network distances. Assuming certain conditions for the assisting individuals to reach an AED station, the coverage area for each defibrillator can be set based on the time needed to detect the SCA victim, to travel from the victim’s location to the AED, to optionally wait for the arrival of trained staff, and finally to travel back with the AED equipment to the victim (Figure 1). For both models, a multi-objective decision-making scheme is devised to take into consideration multiple time windows within which network travel time and indoor population distributions vary. If the fluctuation of spatial and temporal demand can be estimated, a multi-period location problem can be formulated as a single aggregated weighted model, where the weights reflect the importance of the demand at each time period. The proposed models innovate by reflecting probabilistic detection time of SCA victims and by formulating the AED placement problem in terms of the expected consequences of SCA (e.g., brain damage or death) in relation to the time lapsed between the victim’s collapse and defibrillation.

Figure 1. Typical sequence of events to rescue an indoor sudden cardiac arrest (SCA) victim.

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The article is organized as follows: The relevant literature on location modeling of emergency medical services (EMSs) and AEDs in particular are briefly reviewed in Section 2. Principles of 3D network representation of indoor spaces are introduced in Section 3. The spatio-temporal coverage models of indoor AED placement derived from the conventional MCLP and p-median models are then presented in Section 4. The model implementation process is described in Section 5, while a case study is developed and the results are presented and discussed in Section 6. Conclusions are drawn at the end of the article. 2. Locating emergency medical services Although research on AED placement is relatively new, a fertile body of literature can be found on location modeling for the purpose of EMSs (see, e.g., ReVelle et al. 1977, Eaton et al. 1985, as well as Marianov and ReVelle 1995). EMS location problems are generally concerned with the placement of facilities to (1) minimize response travel time to accident locations (p-median, see Hakimi 1964), (2) minimize necessary EMS infrastructure while keeping a certain level of coverage (set covering location problem, see Toregas et al. 1971), or (3) maximize emergency coverage (MCLP, see Church and ReVelle 1974). Many refinements have been made to the early operational models. For instance, a number of recent models depart from deterministic formulation and have concentrated on probabilistic approaches that reflect emergency service availability, since coverage may be lost when an emergency service is dispatched to a call (Brotcorne et al. 2003). Daskin (1983) formulates a maximum expected covering location problem to explicitly integrate the availability of a mobile facility when demand for service arises, with stochastic availability. To better model response time and probabilities of survival, Erkut et al. (2008) incorporate survival functions into existing coverage models applied to out-of-hospital cardiac arrests. Their efforts address the limits imposed by the existing coverage models when applied to EMS vehicle location, since they cannot discriminate between different response times. The probability of survival is modeled as a function of distance and pretravel delay. Arriola et al. (2005) departs from mainstream research in optimal facility location modeling by modeling the location of facilities in a multistory building with a static p-median objective. The authors particularly study the effect of elevators on optimal locations and present discrete and continuous versions of the problem. Static location models assume fixed and invariant demand. Because demand is likely to change over time, either existing facilities will need to be relocated (mobile facilities), or new facilities will be added sequentially to meet changing demand. Dynamic models have proved particularly useful to update mobile facility positions throughout the day as a function of demand (e.g., Repede and Bernardo 1994, Brotcorne et al. 2003). In another class of problems, facility planning is seen as a long-term effort involving multiple time periods for which demand is estimated or predicted stochastically; facilities are located in a sequential manner or at once with anticipation of future demands (e.g., Wesolowsky 1973, Rajagopalan and Saydam 2009). When placing indoor emergency devices such as AEDs, there is a recognized need to mindfully select their locations to reduce response time (e.g., Becker et al. 1998, Rauner and Bajmoczy 2003, Atkins 2010). High incidence of cardiac arrests can easily be mapped using a geographic information system (GIS) and can serve as a starting point to optimize the placement of AEDs (Lerner et al. 2005, Warden et al. 2007). In a study of spatial concentrations of SCAs in Copenhagen, Denmark, Folke et al. (2009) use simple



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geospatial techniques to evaluate and compare the effectiveness of alternative AED placement strategies. Retrospective analysis suggests that SCAs generally tend to occur in areas of great population movement patterns and influenced by the epidemiology of the population. Rauner and Bajmoczy (2003) proposed an integer model based on cost effectiveness analysis to determine the necessary number of AEDs among three Austrian regions. Spatial and temporal aspects of AED usage are not explicitly recognized in this model, although they are commonly seen as critical in assessing the actual effectiveness of AED placement or ensuring a relatively complete coverage of the area population. In practice, AEDs are located more on the basis of empirical experience and knowledge than scientific modeling. As part of its On-site AED Program, the Sudden Cardiac Arrest Foundation has compiled a number of recommendations for placement and utilization of public-access AEDs. The following considerations are recommended for optimal placement of the devices (SCAF 2010):

• ‘Is it unlikely that the local EMS system would be able to reliably achieve a “call-toshock” interval of 5 minutes or less at this site?

• Has an SCA incident occurred at this site in the past 5 years and has the • • • •

demographics of the population served at this site remained relatively constant? Does this location have an at-risk population? Is this location considered a higher risk location? Can an active, hands-on medical director be identified for this location? Does this location have personnel willing and able to respond to cardiac emergencies to provide CPR and defibrillation?’

Myers and Mohite (2009) apply a set covering location model to optimally place AEDs in a university community, using a 4-minute travel time threshold for service delivery. Although the importance of each building was weighted by its population, travel times were empirically recorded by actual walking time between buildings and the model was restricted to an outdoor 2D environment. Mandel and Becker (1996) propose a multiobjective integer program where the two objectives are the overall survival and equity of survival rates to determine which BLS companies serving a certain territorial jurisdiction should be equipped with an AED. 3. Indoor 3D transportation network modeling Modeling the optimal placement of emergency devices and possible movement paths inside buildings requires a network representation of spaces that can be traversed through the buildings. Different approaches to such representation have been proposed in the literature (e.g., Graf and Winter 2003, Lee 2004, Kwan and Lee 2005, Lee and Kwan 2005, Pu and Zlatanova 2005, Lee 2007, Li et al. 2010). We use here an object-oriented 2.5D data modeling approach proposed by Mandloi and Thill (2010) according to which a multilevel building is conceptualized as a stack of separate compartments representing each floor of the building. Each floor compartment is further subdivided into room and corridor entities that form the network connecting all the rooms and entrance/exit points on a floor. Conventionally, a room consists in spatially delimited space that may be occupied by non-ephemeral population, whereas a corridor is a space of transit between rooms and devoid of stable population. Movement between floor compartments is enabled by an access system of staircases and elevators; access points represent exterior doors through which movement between indoor and outdoor spaces is enabled. Nodes represent


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rooms, arc junctions, or access points. Hence the elemental granularity of the model is the room. Corridors are represented by arcs (such as the medial axis of the polygonal corridor) stored in a line feature class. The access system formed of stairs, elevators, and doors is modeled as node features through which corridor arcs are connected. Thus, the staircase and the elevator locations on each floor can be represented as node features in a node feature class. Effectively, the network of each floor is placed in a separate group and connectivity between each group is allowed only at defined points corresponding to elevator shafts and stairwells. Thus, 3D connectivity required to represent a traversable indoor network is achieved and the indoor network created on these principles becomes fully routable once proper impedances are assigned to all the arcs of the network (Mandloi and Thill 2010). It should be noted that this data model possesses all the functionality required for facility location modeling. Some other models designed for real-time emergency response (e.g., Lee 2007) exhibit additional properties, such as 3D topological capability, that are superfluous here. Travel time impedance associated with each network arc is estimated as a function of the arc length and travel speed on the arc. For instance, for a corridor arc, the travel impedance value tc can be calculated as tc =

dc sc

(1)

where d c is the distance traveled along the corridor and sc is the walking speed along the corridor. The impedance of moving up (tsu )) or down (tsd ) between floors by stairs can be modeled as tsu = mθ

ds sus

and

tsd = mθ

ds sds

(2)

where m is the number of floors starting from one, d s refers to the vertical distance between two neighboring floors, sus (sds ) is the walking speed on stairs upward (downward, respectively), and θ is a parameter to be specified. The impedance of moving up or down explicitly accounts for the discomfort experienced when the individual must travel between a large number of floors. For moving up or down using an elevator, the impedance can be estimated as te = c +

de se

(3)

in which c is the elevator waiting time, d e refers to the vertical distance between two vertically neighboring elevators, and se is the moving speed of elevators. Parameterization of Equations (1)–(3) is discussed in more detail in Mandloi and Thill (2010). 4. Indoor 3D facility location modeling In this section, we first discuss the principles underpinning the integration of space and time for the modeling of optimal indoor location of emergency medical devices. We then proceed with presenting the mathematical formulations of the two models discussed in the article, namely the multiple-time-window maximum covering location problem (MTWMCLP) and the multiple-time-window p-median (MTW-P-Median) problem.


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4.1. Handling time and space For the efficient placement of emergency devices within indoor spaces, spatial and temporal considerations play a critical role in several respects. First, because SCA strikes without warning and because of the incapacitating nature of SCA, therapeutic intervention will happen only after the victim’s collapse has been detected by a third person. The detection time hinges upon the number of individuals in the vicinity of the victim at the time of the collapse and upon the spatial pattern of activities they are involved in, both of which vary with the time of the day, the day of the week, and so on. Furthermore, the magnitude of the population at risk and its distribution throughout a building is time dependent. Finally, the ease and speed of movement across a building (travel impedances), from the site of an SCA event and potential AED stations, depends on the level of usage of the building, which can be expected to be time dependent. The threefold space–time dependencies can be modeled using discrete time windows defined over a typical 24-hour period. The idea behind the spatio-temporal 3D facility location-allocation modeling proposed in this article is based on the concept of time windows: each time window represents a time period of a typical day during which travel impedances, as well as demand population and distribution, are temporally and spatially invariant. Essentially, the 3D facility location models divide each typical 24-hour day into a number N of time windows. This allows the models to capture variations in demand population and distribution (e.g., population is higher during working hours than during evening hours), in network movement flows throughout the day (e.g., elevator waiting time is longer during working hours than night time), and consequentially in the time it takes for an SCA victim to be detected by people in his/her surrounding. These three aspects are now discussed further. Demand distribution is the distribution of room-based aggregated demands over the whole space of a building. Demands distributed inside rooms are often uneven, depending on the use of each room. For example, consider an academic building on a university campus: if a room is mainly occupied by staff, a single person is likely to be the occupant during daytime only. However, if it is designed as a lecture hall or auditorium, there are a significant number of people inside the room during certain time periods, and none at other times. Network travel time is defined as the time required for a person to travel from one location to another location within the building based on the 3D indoor transportation network. Network travel time (TN) from a demand node i (i ∈ I) to an AED station j (or vice versa) within a time window n (n ∈ N) is denoted by TNijn . Network travel time between a certain demand node and a certain AED station may be characterized by different corridor movement speeds and elevator waiting times in different time windows. For the sake of simplification, this study considers variation in elevator waiting time but assumes that corridor congestion is not a meaningful issue and therefore speed of moving along network corridors is the same in every time window. An SCA victim is unconscious after collapsing and is unable to call for help. Survival then relies only on the chance of being quickly detected and helped by surrounding people. This issue becomes vital for indoor environment because the visibility is dramatically limited between floors and between rooms. The detection time is the expected time to detect a victim stricken by SCA at demand node i. It is denoted by TOin for a given time window n and determined as follows. Let us consider the hypothetical occurrence of an SCA event at node i of a certain building in time window n. Several situations should be recognized in determining the timeliness of the event’s detection, and, if so, who may discover the victim. The case where at least one other person is in the same space as the victim at the time of the SCA event


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is rather trivial; the SCA event is then definitely observed so that TOin = 0. If nobody else than the victim is at node i, detection will occur only in the event that someone else in the building journeys to node i either deliberately or by chance. For the purpose of modeling the detection time, given the inherent compartmentalization of buildings on the basis of floors, it is assumed that such journey to node i happens only from nodes on the same floor as node i, or from the floors directly above or below it. Thus, if the victim is alone on these floors, detection time is set to an arbitrarily large default value (TOin = Tn (∞)). In all other cases, detection time is modeled stochastically as a function of indoor demand distribution and network travel time in time window n as detailed below. Each person on the same floor as node i, or one floor above or one floor below, during time window n has a chance of witnessing the event or of being first to discover the victim. This probability depends on the number and distribution of people in the building at that time. Specifically, for each node k = i, the probability of detection by someone at node k can be expressed as follows: d c Pikn Pikn = Pin

(4a)

d It is the product of the probability Pin that the victim is discovered before he/she is c that discovery is made by an expected to pass away and of the conditional probability Pikn individual at node k. The former probability is a direct function of the number of occupants of the three nearest floors of the building during the said time window. Its probability is captured through a Weibull equation with parameters β 1 and β 2 ≥ 01 : ⎛ ⎞ d Pin = 1 − exp ⎝−β1 dl1 n − β2 dl2 n ⎠ (4b) l1 =i

l2

where dl1 n is the number of individuals at demand node l1 on the same floor as node i within the time window n, and dl2 n is the number of individuals at demand node l2 on the floor above or below. If the victim is discovered, the probability that the discovery is made by a person at node k = i is a function of the aggregated count of potential witnesses at k and of the travel time from k to i, in relation to those measures aggregated over all demands on the three nearest floors of the building during the same time window. Formally, this is expressed in accordance with the principles of spatial interaction (Hägerstrand 1953, Roy and Thill 2004): ⎞ ⎛ ⎧ dl n dl n ⎪ dkn β 2 ⎪ 1 2 ⎠ if (k, i) same floor ⎝ ⎪ ⎪ α α + α ⎪ ⎨ TNikn TN β TN 1 il il n n 1 2 l1 =i l2 c ⎞ = β (4c) Pikn ⎛ 2 ⎪ d ⎪ kn d d β ⎪ β1 l1 n 2 l2 n ⎠ ⎪ ⎝ ⎪ if (k, i) above or below floor ⎩ TN α α + α TN β TN 1 ikn il il2 n n 1 l =i l 1

2

where dkn is the number of individuals at demand node k within the time window n, TNikn is the network travel time from a witness k to victim i for time window n, and α (>0) is α . The higher α, the greater the impact the exponent for the distance decay function 1 TNikn of travel time on the probability of detection by witness k. In our case study, following the gravity hypothesis of spatial interaction, we conventionally set α = 2. The expected detection time for a potential victim at demand node i within the time window n, TOin , is then formulated as

International Journal of Geographical Information Science TOin =

d Tn (∞) (Pikn TNikn ) + 1 − Pin

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(5)


k

where TNikn is the network travel time between demand nodes i and k, and Tn (∞) is set to a large default value. It is estimated by the weighted average of detection times over all nodes k in the building, where the weight is given by the expected probability Pikn that a victim at d of node i will be detected by a person from node k and the expected probability 1 − Pin no detection. In case of no detection during time window n, default detection time Tn (∞) is set to be the time that will lapse until the next time window with full occupancy of the building. Despite the fact that today’s AEDs are designed for use by the general public even without prior training, some individuals are reluctant to do so and prefer relying on trained personnel. When AEDs are installed inside a building, it is often the case that some designated building occupants are trained for AED usage. If an SCA victim helper requires assistance, he or she can call up one of these AED-trained individuals. In case none of the trained staff is currently available at the location where the SCA event occurred (i.e., either at AED station or at victim’s location), it may take some time for other trained personnel to arrive on site. Waiting time within time window n, TWn , is a user-defined parameter that quantifies the time needed to get a willing individual to operate the AED. The analyst can modify this parameter to reflect different assumptions: for example, if it is assumed that everybody is willing and able to operate an AED, TWn is set to 0. SCA victim suffering time is given by the estimated time that has lapsed between the victim’s collapse and the initiation of defibrillation. Due to the spatio-temporal variation in population distribution and in network travel impedance, victim suffering time is expected to be different for different time windows. Consistently with the depiction in Figure 1, during time window n, the suffering time for a potential victim located at site i who receives treatment with an AED stationed at site j, Tijn , is the sum of detection time, waiting time, and time to travel on the network forth and back from demand node i to the AED station j: Tijn = TOin + TWn + 2TNijn

(6)

4.2. Multiple-time-window maximal covering location problem model The traditional MCLP model is modified to capture the spatial-temporal variation of demand and of network travel impedances. We propose a multiple-time-window maximal covering location problem (MTW-MCLP) model which optimizes the location of a predefined number of emergency devices by maximizing the covered demand, aggregated over all building rooms, and summed over all time periods. Demand is said to be covered if suffering time is less than a preset threshold. 4.2.1. Notations Indices: i index for demand nodes j index for candidate AED stations n index for time windows

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Sets: I set of demand nodes J set of candidate AED stations N set of time windows NEin set of locations for which facilities can cover demand node i within a predefined threshold of suffering time during time window n: NEin {j|Tijn ≤ D}


Parameters: wn weight factor for time window n din total demand at demand node i during time window n Tijn suffering time (Equation (6)) p predefined number of AEDs to locate D suffering time threshold beyond which a demand node is deemed not covered Decision variables: Xin = 1 if demand node i during time window n is covered, 0 otherwise Yj = 1 if an AED is located at location j, 0 otherwise 4.2.2. MTW-MCLP model

Maximize

n∈N

wn

din Xin

(7)

i∈I

subject to

Yj = p

(8)

j∈J

Yj − Xin ≥ 0

∀i, n

(9)

j∈NEin

Xin , Yj binary

(10)

where the objective function in Equation (7) maximizes the total covered demand weighted by time span wn of window n. Constraint (8) ensures that p AEDs are placed in the building; constraint (9) ensures that a demand node i can only be covered if one or more than one emergency devices are within acceptable reach time of demand node i. Finally, constraint (10) imposes binary restrictions on variables Xin and Yj . Since SCA is a rare event, it is assumed that the probability of two simultaneous events is null, that an AED is available for use at the designated station, and therefore travel times to AED stations other than the closest one need not be considered. 4.3. Multiple-time-window p-median model Because the service rendered by a facility is modeled in its simplest form as a binary variable (covered or not covered) in the maximal covering location problem, this problem



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can be construed as a coarse representation of EMS provision. The p-median problem enhances this approach with a continuous service variable. The conventional p-median model locates p facilities so as to minimize the total weighted travel impedance from every demand node to their closest facility. However, in the case of AEDs, demand-weighted distances are often not attractive or pragmatic measures for optimal placement. Ideally, the consequences of delay in defibrillation treatment would be taken into consideration. Consequences can be summarized to permanent and irreversible damage to the brain and death of the victim. The analysis of Das and Zipes (2003) indicates that a linear relationship exists between victim suffering time and the likelihood of irreversible brain injury/damage or death. For each time window, the probability of an SCA victim facing irreversible brain damage or death is therefore modeled as a linear function of victim suffering time, Tijn , as shown in Figure 2. The solid line depicts the probability of death against Tijn while the gray line is the probability of suffering permanent brain damage against Tijn . Research has shown that irreversible brain damage becomes a statistical certainty after 5 minutes of suffering time, while the victim faces certain death about 10 minutes after cardiac arrest (Das and Zipes 2003). Given these assumptions, the slope b of the probability function of irreversible brain damage when suffering time is under 5 minutes is 0.2, while that of the probability function of death is d = 0.1 for suffering time being under 10 minutes. As a result, the expected monetary value of irreversible brain injury and/or loss of life due to SCA can be expressed as a function of the time lapsed until defibrillation and used as a decision criterion instead of distance (or time). Therefore, a suitable decision criterion is the expected value of prevented death or permanent brain impairment in case of defibrillation treatment. This criterion is a function of suffering time Tijn . Generalizing research reported by Das and Zipes (2003), if Tijn is d ∗ , its value is 0; this is particularly the case when the probability Pin of greater than Tijn ∗ discovery of the victim is 0 (detection time TOin is set to a very large quantity). Between Tijn ∗∗ and Tijn , given that brain injury is certain, it is the product of the likelihood that death has not occurred yet (1 –dTijn ) and of the residual value of life (Pd – Pb ). Finally, if Tijn is less ∗ , it factors in the value of survival with brain impairment bTijn (1 − dTijn )(Pd − Pb ) than Tijn and the value of survival without brain impairment (1 − bTijn )(1 − dTijn )Pd . Hence, for a victim at node i covered by an AED stationed at j within a time window n, the criterion denoted by Vijn is formulated as

Vijn

bT (1 − dT )(P − P ) + (1 − bT )(1 − dT )P ijn ijn d b ijn ijn d = (1 − dTijn )(Pd − Pb ) 0

∗ if 0 ≤ Tijn < Tijn ∗ ∗∗ Tijn ≤ Tijn < Tijn ∗∗ if Tijn ≥ Tijn

(11)

Figure 2. Modeling the relationship between sudden cardiac arrest victim suffering time and probabilities of suffering irreversible brain damage and death.

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where Pb refers to the monetary value of irreversible brain injury, and Pd is the monetary value of life. It can be seen as an extension of Erkut et al.’s (2008) survival probability function.


4.3.1. Notations Indices: i index for demand nodes j index for candidate AED locations n index for time windows Sets: I set of demand nodes J set of candidate locations N set of time windows Parameters: wn weight factor for each time window din total demand at demand node i during time window n Vijn total monetary value of irreversible brain injury and/or loss of life due to SCA (Equation (11)) Pd monetary value of life Pb monetary value of irreversible brain injury d probability slope of death when suffering time is less than 10 minutes b probability slope of irreversible brain damage when suffering time is less than 5 minutes p predefined number of AEDs to locate Decision variables: Xijn = 1 if demand node i is covered by an AED stationed at j during time window n, 0 otherwise The MTW-P-Median model is mathematically expressed as

Maximize

⎡

wn ⎣

n∈N

⎤ din Vijn Xijn ⎦

(12)

i∈I j∈J

subject to

Xijn = 1 ∀i, n

(13)

j∈J

j∈J

Yj = p

(14)

International Journal of Geographical Information Science Xijn ≤ Yj

∀i, j, n


Xijn , Y j binary

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(15) (16)

where the objective Equation (12) is to maximize the total monetized value of irreversible brain injury and/or loss of life that can be prevented, using multiple-time-window criteria and weighting each component by the time duration wn of window n; constraint (13) ensures every demand node is covered; constraint (14) restricts the number of open facilities to p; and constraint (15) ensures that a station serves a demand node if and only if an AED is effectively placed at this station. Finally, constraint (16) imposes binary restrictions on the decision variables. As for the MTW-MCLP, since SCA is a rare event, it is assumed that the probability of two simultaneous events is null and that an AED is available for use at the designated station. Therefore travel times to AED stations other than the closest need not be considered. In the first approximation, the demand at node i can be estimated by occupant capacity and computed as ρ in Ai , where ρ in is the population density at demand node i during time window n and Ai is the space area of node i. This approach is followed to estimate daytime demand in the scenarios described below. 5. Implementation The MTW-MCLP and MTW-P-Median models of indoor AED facility location are implemented in a tightly coupled system integrating ArcGISTM VBA and LINGOTM , a GIS and commercial solver, respectively (Figure 3). These models are part of the 3DCityNet suite of

Figure 3. Spatio-temporal indoor 3D facility location modeling framework.


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Figure 4. User interface for solving the MTW-P-Median problem.

spatial analysis applications built using the 3D network data model described in Section 3. ArcGISTM is used for spatial data preparation, spatial analysis, and visualization. The data preparation process involves building a 3D transportation network and specifying locations of AED candidate stations. In this research, spatial queries and network analysis (i.e., shortest route calculation) in ArcMapTM are the main spatial analysis functions used to calculate necessary elements of LINGOTM inputs (e.g., matrices comprising the coverage provided by each AED, the potential victim suffering time, Tijn , the estimated total monetary value of irreversible brain injury and/or loss of life due to SCA, Vijn ). Within the tightly coupled integrated framework, LINGOTM computes the optimal solution and the optimized AED stations. The MTW-MCLP and MTW-P-Median models are integer linear programs solved by a branch-and-bound algorithm. Innovative visualization techniques are implemented in ArcSceneTM , primarily to represent the coverage of each located AED in each time window. A user interface (Figure 4) is designed to help decision-makers easily set up parameters and scenarios, such as the number of time windows, the time span associated with each time window, and the preset number of AEDs to be placed. Users are also required to specify information related to the input data, such as workspace directory, transportation network layer (network built in ArcGIS), covering layer (AEDs), and demand node layer. A transportation network is specified for visualization purposes. 6. Case study analysis 6.1. Scenario setup A five-story academic building on a college campus is chosen to test our model. The locations of the potential demand nodes (placed at room centroids) and of the candidate AED


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Building Exit

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Floor Exit Stairs/Elevator Demand Node Candidate AED Station and its index Travelling Network CAD Floor Plan

Figure 5. Indoor 3D transportation network for the building under study.

stations (placed along corridors) along with the 3D transportation network within the building is constructed as shown in Figure 5. The test building contains 316 demand nodes; 43 candidate sites are considered for AED placement. Each candidate site is assigned a two- or three-digit label in Figure 5, with the first digit identifying the floor. For the purpose of this implementation, Equations (1)–(3) are parameterized as follows using empirical values derived from local knowledge: θ = 1.5, sc = sus = sds = 3 miles/hour, se = 150 feet/minute, c = 25 seconds during day time or 15 seconds during night time. The Weibull d (Equation (4c)) is here parameterized function used to specify the detection probability Pin with β 1 = 0.1 and β 2 = 0.05. Three time windows covering a 24-hour period are defined for the case study:

• Window 1, W1: [08:00–18:00] • Window 2, W2: [18:00–22:00] • Window 3, W3: [22:00–08:00]. The selection of these time spans is mainly based on common sense for the presence of population in an academic working environment. The first window represents normal working hours with expected daytime population, including faculty, staff, and students. The second window covers evening hours, with significantly reduced population on average compared to the daytime window. In addition, during this time span, a dramatic increase in population density is expected at selected spots, due to evening events such as movie shows or group gatherings. The third window (night time) is associated with sparsely distributed population throughout the building. More generally, it is possible to create alternative scenarios capturing temporal variations in population distributions as well as regular and irregular activities during a day using a combination of up to 24 time windows. Given the specification of time windows, default detection time Tn (∞) can be set as follows: Tn (∞) =

1 wn + wm 2 m>n

where W1 is busiest time window

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⎧1 ⎪ ⎪ ⎪ 2 W1 + W2 + W3 ⎪ ⎨ 1 = W2 + W3 ⎪2 ⎪ ⎪ ⎪ ⎩ 1 W3 2

if n = 1 if n = 2

W1 = 10 hours:day [8 AM−6 PM] where

if n = 3

W2 = 4 hours:evening [6 PM−10 PM] W3 = 10 hours:night [10 PM−8 AM] (17)

For this case study, two scenarios, labeled S1 and S2, are designed to test the performance of our models. Each scenario has the same time window setting, as mentioned above. The scenarios differ from each other in terms of demand distribution and network travel time. The assumed demand distributions of each time window are summarized for the two scenarios in Table 1, in which the index i denotes demand nodes. They are also visualized in Figure 6a and b for scenarios S1 and S2, respectively. The two scenarios are characterized by exactly the same demand distributions in periods W1 and W3. The demand distributions differ however during the evening time period W2. Under S1, the demand distribution during W2 is uniformly reduced by 80% of W1’s distribution, leading to the overall demand in the building dropping from 992 to 149 people. For S2, the reduction in demand during window W2 is assumed not to be uniform throughout the building. It is assumed that during time window W2 under scenario S2, there is a movie show attracting 391 people in the auditorium R250 on the second floor, a lecture attended by 100 students in R439 on the fourth floor, and a group gathering of 50 individuals in R140 on the first floor. Accordingly, during W2, demand reaches 707 in scenario S2. These ‘hot spots’ in population concentration are expected to make an observable impact on optimal AED solutions.

6.2. Results visualization and analysis Figure 7 is a legend cue with graphic symbols and colors used in the rest of the article for visualization of solutions. Square dots in light gray are used to show non-occupied demand nodes with zero population during a given time window, while spheres represent nodes with nonzero demand. The gray spheres represent demand nodes with nonzero demand Table 1. Scenario

S1

Demand distribution for time windows under different scenarios. Time window 1 (W1) 08:00–18:00

di1 = 992

Time window 2 (W2) 18:00–22:00

di2 = 0.2 × di1

i

S2

di2 = 149

Time window 3 (W3) 22:00–08:00 ⎧ 1 if i = C101, C322, R409, ⎪ ⎨ R227 di3 = 2 if i = RB23 ⎪ ⎩ 0 otherwise di3 = 6

i

i

i

i

⎧ ⎧ 391 if i = R250 1 if i = C101, C322, R409, ⎪ ⎪ ⎨ ⎨ 50 if i = R140 R227 d = di1 = 992 di2 = if i = R439 i3 ⎪ ⎪ i ⎩ 100 ⎩ 2 if i = RB23 0.2 × di1 otherwise 0 otherwise di2 = 707 di3 = 6


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(a) Demand (people) 1–2 3–5 6–10 11–20 21–50 50–100 101–200 201–391


(b) Time window 1 Time window 2 Time window 3

Figure 6. Demand distribution of three time windows under scenarios (a) S1 and (b) S2.

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Zero demand node Non-covered demand node Covered demand node (Suffering time