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Integrated fire and ambulance siting: A deterministic model
~ ) Pergamon
Socio-Econ. Plann. Sci. Vol. 29, No. 4, pp. 261-271, 1995
0038-0121(95)00014-3
Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-0121/95 $9.50+0.00
Integrated Fire and Ambulance Siting: A Deterministic Model C H A R L E S REVELLE and S T E P H A N I E S N Y D E R The Johns Hopkins University, Department of Geography and Environmental Engineering, Baltimore, M D 21218, U.S.A.
Abstract--The model presented in this paper is developed for an integrated emergency system in which ambulance and fire service deployment are simultaneously considered. The model extends and blends the Maximal Covering Location Problem (MCLP) for ambulance siting and the Facility Location-Equipment Emplacement Technique (FLEET) for fire service placement. The objective of the new formulation is to maximize and trade-off both ambulance and fire coverage subject to constraints on the total number of vehicles and stations of both types that can be sited. We introduce a new concept in integrated coverage that permits free-standing ambulance stations to be sited in addition to fire stations. Typically, ambulances are located at fire stations and, sometimes, hospitals; that is, eligible sites are driven by prior choices that did not include consideration of ambulance service. Our results, based on three sample problems, suggest that, for a given budget level for facility construction, it may be possible for ambulance coverage to be increased with little or no loss in fire coverage. This condition pertains if ambulance stations are freed of the requirement to be sited only at fire stations.
INTRODUCTION A significant amount of research has focused, and continues to be focused, on the development of models for coverage availability in emergency service systems [19]. Coverage in this sense may be defined as the positioning of a vehicle to respond to a call within a specified time or distance standard. Availability refers to the actual ability of a vehicle or unit to respond to a call within the standard. Time or distance standards are widely used to ensure that emergency services can be successfully delivered [15]. A number of increasingly more sophisticated emergency service models, which utilize coverage or availability as the measure of deployment effectiveness, have been developed in the past two decades. Many of these models are reviewed and compared by ReVelle [12]. The basic models began with the formulation of Toregas et al. [22] and Toregas and ReVelle [21] who developed the Location Set Covering Problem (LSCP). Their formulation, which applied to ambulance coverage, sought to minimize the number of facilities required to cover all demand nodes within a specified time or distance standard. Church and ReVelle [3] and White and Case [23] extended the LSCP to deal with the situation in which the number of vehicles available to position was less than the number needed to cover all demand nodes. That formulation sought to cover the maximum population within the specified distance standard given a limited number of vehicles or servers available. The formulation is referred to as the Maximal Covering Location Problem (MCLP). Subsequent coverage models have dealt with locating hierarchical services, back-up services and probabilistic or stochastic services. The model proposed in this paper represents a new class of emergency service coverage model. Here, we consider three types of emergency servers: ambulances, fire engines and fire trucks. In all prior works, the siting of emergency medical services and fire services were separate and distinct issues. Indeed, it was assumed in the siting of ambulances that the fire houses (or, sometimes, hospitals, if these were possible home bases) were already in place. The fire houses had been sited, presumably historically, or, perhaps, by solution of a prior model which optimized fire service placement. The siting of ambulances was assumed to be a secondary activity with the primary activity being the siting of fire stations. With some exceptions ambulance siting has not often been thought of as the priority activity. Of course, there is historical reason for this. Fire suppres261
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Charles ReVelle and Stephanie Snyder
sion/protection evolved prior to mobile emergency medical services, while fire stations already occupied positions in the location landscape when ambulance service was first becoming available. The reason, then, for this apparent precedence of fire-over-ambulance coverage is one of simple economics. The capital costs of building free-standing ambulance stations would thus place an additional burden on municipal budgets. As a consequence, free-standing ambulance stations are the exception rather than the rule. ReVelle [13] argued, however, that since ambulance crew operating costs: (1) represent the principal financial outlay in providing coverage; and (2) are constant across ambulance location sites, free-standing ambulance stations should be taken as a priority issue in the location literature. As a consequence of the manner in which actual ambulance siting decisions were made, modelers treated ambulance and fire services sequentially. They were thus never accounted for in the same model--as they are in this investigation. A second, and parallel feature of this study is that ambulances are not constrained to be located only at fire stations. We thus propose a model in which free-standing ambulance stations are an eligible element of the siting decisions for emergency services. We show, for two hypothetical examples, that ambulance coverage can be increased without a change in investment, and with little or no loss of fire coverage as ambulances are allowed to go free of fire stations.
L I T E R A T U R E REVIEW The past two decades have seen the development of a wealth of emergency service coverage models [13]. In this regard, the emergency service coverage models reviewed here share several general characteristics. First, facilities and demand nodes are considered to be concentrated at defined nodes of a network. Secondly, servers can respond to demands only by traveling along pre-defined arcs of the network. Finally, distance or time standards are defined as the criteria against which the success of emergency deployment systems is measured. Success is the ability to reach calls within a time standard so that services can be promptly provided. As noted previously, early emergency service coverage models dealt with either ambulance or fire coverage, but not both. One shortcoming of the simplest of these models, the LSCP [21, 22], is the prohibitive expense require to cover all demand nodes. In addition, the LSCP gives no consideration to call frequency or to the demand node population. In theory, nodes with greater populations are likely to have greater call frequencies and require more frequent service. The LSCP was thus soon superseded by more sophisticated models which, in turn, have served to generate even more realistic problem formulations. To deal with the limitations of the LSCP, a new model, referred to as the MCLP, was developed [3, 23]. The objective of this siting model is to maximize the number of people (or calls) who are provided ambulance coverage within a specified standard, subject to a constraint on the number of ambulances that can be sited. Their reinterpretation of the ambulance coverage problem provided rational ambulance coverage in the face of limited resources, although this formulation has limitations as well. One crucial limitation of the MCLP is that it does not allow for differentiated servers. Fire protection systems, in most cases, consist of service by two separate types of fire vehicles and crews, each of which has its own distinctive functions and to which different distance standards apply. One of the fire vehicle types is the engine company that provides pumper service, while the other is the truck company that provides ladder rescue service. The M C L P does not directly provide for the several types of servers as is required for fire service. A more recent emergency service coverage model, the F L E E T (Facility Location/Equipment Emplacement Technique [18]) model, was developed to help achieve coverage of demands by siting two types of servers. In particular, the goal of F L E E T is to site engine and truck companies in such a way as to maximize the population (or calls) having at least one engine company located within the engine company standard and at least one truck company within its own distance standard, subject to limits on the number of both types of vehicles available and on the number of stations that can be located. A limiting assumption of these earlier coverage models is that servers, once placed, would always
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be available to respond to calls. That is, the idea of congestion in the system is not addressed. In other words, these models simply locate servers without guaranteeing availability. Recognition of this limitation has lead to the development of a new generation of coverage models in which redundant, or back-up, coverage is sought within the distance standard. Daskin and Stern [7] and Berlin [1], for example, developed a redundant coverage model for ambulance services. The goal of their model is to site a fixed number of ambulances in such a manner that all demand nodes have primary coverage within an ambulance distance standard while maximizing the number of redundant servers operating in the system. A demand node would have a redundant server if a second ambulance was located within its distance standard, in addition to the first-sited ambulance which was providing primary coverage. Hogan and ReVelle [8] formulated a redundant coverage model in which first coverage is again required, but only a specified number of additional coverers for each demand node is sought. Additional coverage is population-weighted in this, and other models, in an effort to site additional servers where they might be needed most; i.e. where the population and call frequency are the greatest. Hogan and ReVelle [8] structured a further formulation in which primary coverage was no longer a requirement, but rather a goal to be traded-off against population-weighted back-up coverage in a multiobjective model. The preceding coverage models all share the assumption that the relevant system is deterministic. However, the underlying system is, in fact, a stochastic one in which the availability of a server to respond to a call is the crucial issue. The redundant coverage models have addressed this issue by seeking to site additional ambulances that can respond to calls in the event that the primary ambulance is unavailable. However, there has been little or no accounting of the percentage or fraction of the time that an ambulance can actually respond. This condition has facilitated generation of probabilistic models better able to account for a server's "busy fraction." Chapman and White [2] were the first to formulate such a probabilistic coverage model. They constructed a probabilistic location set covering model in which the busy fraction is estimated from the local rate of calls and call durations given an initial estimate of the number of servers in the system. Using this estimated busy fraction, a reliability constraint is written, for each demand node, that specifies the minimum fraction of the time a server must be available within the distance standard. The current line of research in probabilistic coverage models expands upon the work of Chapman and White [2] as well as that of Daskin [5]. Daskin [5] formulated a model in which the expected population that could be covered by a fixed number of ambulances is maximized. Recently, research by ReVelle and Hogan [16] has concentrated on deriving more realistic, sector-specific (vs system-wide) estimates of the busy fraction. Despite the progress made in modeling emergency service systems with the LSCP, MCLP, FLEET, and subsequent models, important issues remain. As noted previously, critical among these is the integration of fire and ambulance services, i.e. the simultaneous siting of both types of services. The model proposed here thus addresses the issue of service integration, with particular focus on the issue of allowing ambulances to be sited at other than fire stations; that is, to occupy free-standing ambulance stations.
MODEL FORMULATION In many cities, the placement and even operation of ambulance services is the responsibility of the fire department. Ambulances are typically housed at, and despatched from, fire stations. The fire stations, for their part, are typically located by professional judgment in such a manner that fire events, not medical emergencies, are best covered. Ambulances are then located secondarily at the fire stations--which have likely been located without due regard to where medical emergencies arise. Nevertheless, these stations often contain sufficient space for an ambulance vehicle. This practice may not be optimal in terms of providing emergency medical services, but ifa fire station is already sited, it is far cheaper to locate an ambulance there since the infra-structure is in place and capital costs are sunk. In light of these circumstances, we here explore a new concept, a scenario in which ambulances
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Charles ReVelle and Stephanie Snyder
are allowed to be sited at free-standing ambulance stations as well as at existing fire stations. The integrated emergency services problem is modelled as the following multiple-objective integer program, which we term the Fire and Ambulance Service Technique (FAST):
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where i, I = index and set of demand nodes, j, J = index and set of potential facility sites, ai = population at demand node i, y,~= {1 if demand node i has at least one ambulance stationed within the ambulance distance or time standard; 0 otherwise}, y~ = {1 if demand node i has at least one fire engine stationed within the engine standard and at least one fire truck stationed within the truck standard; 0 otherwise}, x~ = {1 if an ambulance is stationed at j; 0 otherwise}, x T = {1 if a fire truck is stationed at j; 0 otherwise}, X~= {1 if a fire engine is stationed at j; 0 otherwise}, ~j={1 if a fire station is located at j; 0 otherwise}, uj = {1 if an ambulance station is located at j; 0 otherwise}, p A = the number of ambulances to be sited, p E+ T = the number of engines and trucks to be positioned, S A = distance standard for ambulances, S E = distance standard for engines, S T = distance standard for trucks,
Integrated fire and ambulance siting
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N A = {j Idji ~< SA}; the set of all potential facility sites that are eligible to provide node i with ambulance service, N,.r = {j [dji ~< s'r}; the set of all potential facility sites that are eligible to provide node i with truck service, N E = {jldjg ~< SE}; the set of all potential facility sites that are eligible to provide node i with engine service, B = budget, expressed in fire station equivalents, available to build and maintain fire and ambulance stations; and b = fraction of the cost of a fire station that an ambulance station incurs. The first objective maximizes population, or ambulance call frequency that achieves ambulance coverage, while the second objective maximizes population, or fire call frequency that achieves fire coverage. Constraints (2) stipulate that each demand node will not have ambulance coverage unless at least one ambulance is sited within the ambulance distance standard of the node. Constraints (3) and (4) ensure that a node will not have fire coverage unless there is at least one truck company positioned within the truck company distance standard and at least one engine company positioned within the engine company standard. Constraints (5) and (6) ensure that a truck and/or engine will not be located at a site unless a fire station has already been positioned at that site. An ambulance, fire engine and fire truck are all permitted to be located at a fire station. No issues of station capacity are considered except that no more than one of each server type is allowed at a node. Ambulances are permitted to be located at fire or ambulance stations, but no fire companies may be located at ambulance stations. Constraints (7) ensure that an ambulance will not be sited at a node unless either a fire or ambulance station exists at that node. Constraints (8) and (9) limit the number of ambulances, fire engines, and trucks that can be sited, respectively. Constraints (10) ensure that, at most, either a fire station or an ambulance station, but not both, can be located at a node. Constraint (11) ensures that the number of fire and ambulance stations located is equal to a predetermined level.
COMPUTATIONAL RESULTS AND DISCUSSION Two data sets were utilized to evaluate the proposed model. The first data set consisted of the 55-node test network of Kroll [9], while the second was a 66-node set from Serra [20]. All computations were carried out on a Micro-Vax 4000 workstation using the mixed-integer software package, CPLEX. The results from a number of runs on these data sets will be reported in this section, with the outcomes from the Kroll network discussed first. The following distance standards were utilized on the 55-node set: 60 distance units for ambulance coverage, 40 distance units for truck coverage and 30 distance units for engine coverage. In the real world, the Insurance Services Office (ISO) has set maximum distance standards for fire trucks at 1 mile and for fire engines at 3/4 mile for high value areas. The distance standards utilized in this research were set with these standards in mind, with the engine company standard being 3/4 of that for truck companies. No mandatory response time or distance has been set by the ISO. The standards are used "to grade" cities and towns as to the quality of their fire coverage. We set the ambulance standard at a greater distance than that for fire protection coverage under the realistic assumption that fewer ambulances than fire companies would be sited in a region. The idea throughout the runs was to maintain a constant facilities total budget level for the combined system of fire and ambulance services, and to investigate how ambulance and fire coverage might vary as the number of free-standard ambulance stations was allowed to increase while the number of fire stations decreased. By constant facilities budget level, we are referring only to the cost of building and maintaining those facilities that house the ambulances and fire companies. The cost of building and maintaining an ambulance station was assumed to be one-half that of a fire station. Therefore, to maintain a constant budget level, every time the number of ambulance stations to site was increased by two, the number of fire stations to site was decreased by one. In the first set of runs, the number of ambulances to site in the system was set at 12, while the
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Charles ReVelle and Stephanie Snyder
total number of fire engines and fire trucks was fixed at 16. The number of free-standing ambulance stations and the number of fire stations was incrementally varied and examined in the following (ambulance station, fire station) combinations: (4, 10), (6, 9) and (8, 8). Thus, for any run, the total number, or sum, of fire stations to be located was set to a number Z and the total number of ambulance stations was set to a number U, where Z + 1/2U = B. To maintain the constant facility budget level, the values of Z and U, from constraints (11), were varied as the problem was re-solved for various combinations of the two values that satisfied the constraint. Note that it is necessary for the following two expressions to be met in order for the problem to remain feasible:
(13)
E Zj ~ (1/2)p E+T J
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J
Alternatively stated, these two bounds must be considered in determining eligible combinations of Z and U for use in subsequent problem analyses. Since this is a two-objective model that seeks to maximize both fire and ambulance coverages a trade-off curve is needed. Since the weighting of multiple objectives [4] preserves the integer friendliness of the constraint set [14], we chose to produce the trade-off curve in this manner. Weights were thus attached to each objective and incrementally varied between 0 and 1 as the problem was re-solved. This helped generate the needed trade-off curve. A separate trade-off curve was output for each combination of free-standing ambulance stations and fire stations, namely: (4, 10), (6, 9) and (8, 8). In Fig. 1, we display the entire set of trade-off curves for a constant budget level, divided in various ways between ambulance dispatching stations and fire stations. The results here indicate that as ambulances are allowed to go free of fire stations, i.e. to be located at their own dedicated ambulance stations, emergency medical coverage can increase with little loss in fire coverage. A reasonable increase in ambulance coverage appears to come between 4240 4200-
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AMBULANCE COVERAGE (POPULATION) Fig. 1. Trade-off between population with ambulance coverage and population with fire coverage (A = number of ambulance stations and F = number of fire stations), (12 ambulance units, 16 fire units), (Kroll Network [9]).
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the first two trade-off curves. In the curve closest to the origin, the number of ambulance stations is four. In the next curve out from the origin, the number of ambulance stations has increased from four to six. At the same time, the number of fire stations decreases from 10 to 9 in order to maintain a constant budget level. The ambulance coverage increases from 5510 to 5760 people between the first and second curves from the origin. Thus, one could gain 4.5% in ambulance coverage with only a 1.4% loss in fire coverage by increasing the number of ambulance stations to six in the second curve out from the origin. Further still, increasing the number of ambulance stations to eight results in a 6.5% increase in ambulance coverage with only a 2.8% decrease in fire coverage. Clearly, this model is producing viable alternatives to be considered in emergency service planning. Depending on how the decision-maker weighs ambulance and fire coverage, the model appears able to generate solutions superior to those that restrict ambulance location to fire stations. Many of the problem scenarios of interest here solved exactly as relaxed LPs, with the remainder requiring minimal branching and bounding (less than 50 nodes). A sense of the spatial configuration of the generated solutions is given in Fig. 2 where a single solution is displayed graphically on the Kroll network. The plain circles represent the fire stations while the boxed circles locate the ambulance stations. The location of vehicles at facilities has also been indicated. Figure 2 corresponds to the situation in which there are 10 fire stations and four ambulance stations to site, and fire coverage is heavily weighted over ambulance coverage. In general, the first stations are clustered around the higher population nodes, as would be expected. Engines and trucks are located together at a node at six out of the 10 stations, though they are not required to be. They may be located separately, as long as they are within their respective distance standards so as to provide coverage to demand nodes. Figure 2 corresponds to a point on the trade-off curves in which fire coverage has a much larger weight relative to ambulance coverage. Solutions for which the reverse is true display a much more dispersed configuration of both station types. This is likely due to the fact that the distance standard for ambulances is greater than that for engines and trucks, and, therefore, ambulances can cover more distance and nodes (within their standards) than fire companies and spread out farther in providing coverage. As a side note, the top most points on each trade-off curve actually correspond to the F L E E T solution for each scenario. Recall that the F L E E T model maximizes the population that has at least one engine and at least one truck within their respective distance standards, subject to a budget limitation. For the FAST model, if fire service were weighted at 1.00 and ambulance coverage at 0.00, the solution would correspond exactly with the F L E E T solution. Another set of runs was performed with the same 55-node network, in which more fire vehicles were added to the system. The number of fire vehicles was increased from 16 to 20, while all other conditions remained as in the previous set of runs. With the increased number of fire vehicles in the system, and the same number of stations, all stations became occupied by fire vehicles when fire coverage was maximized. That is, in all runs reported here, a fire station never functioned solely as an ambulance depot. As in the previous set of runs, a constant budget level was maintained, as resources were divided between the various combinations of fire and ambulance stations. A separate trade-off curve was again developed for each combination of stations. The entire set of curves is shown in Fig. 3. In the trade-off curve closest to the origin, there were 0 ambulance stations and 12 fire stations in the system. In going to the second curve out from the origin, one of the original 12 fire stations is replaced by two ambulance stations, maintaining a constant budget level. By allowing a replacement of a fire station with two ambulance stations, ambulance coverage increased from 4953 people to 5410 people between the top-most points of the two trade-off curves closest to the origin, an increase of approx. 9%. With this increase in ambulance coverage came a 0.9% decrease of people with fire coverage. Similarly, in examining the coverage differences between the top most points on the second and third trade-off curves from the origin, there is a 3.9% increase in ambulance coverage for a 1.3% decrease in fire coverage. Thus, as in Fig. 1, ambulance coverage seems capable of a reasonable increase with only a small decrease in fire coverage. That is, by maintaining a constant facility
Charles ReVelle and Stephanie Snyder
268
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Fig. 2. Graphical representation of one solution on the Kroll Network. budget level and replacing fire stations with free-standing ambulance stations, ambulance coverage can increase with little or no loss in fire coverage. It is likely that the changes in coverage displayed in Fig. 3 are not as great as they might be since there were more fire vehicles in the system, and, thus, not as much " r o o m to m o v e " to exploit greater levels of ambulance coverage.
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As previously noted, a second data set was utilized to generate a third set of runs. This second network is the 66-node structure of Serra [20]. A set of runs was performed under conditions similar to those in the previous set. Thus, the number of fire vehicles in the system was set to 20, with the number of ambulances set to 12. The number of ambulance stations varied between zero and four, while the number of fire stations varied between 12 and 10 in the following combinations: (0, 12), (2, 11) and (4, 10). Distance standards were set at higher levels to account for the greater network distances here. Again, separate trade-off curves were generated for different combinations of ambulance and fire stations, at a constant facility budget level. The results are displayed in Fig. 4. Again, results indicate that as ambulances are allowed to go free of fire stations, ambulance coverage can increase with only a small loss in fire coverage. For example, in comparing the top most points of the two trade-off curves closest to the origin, an increase of 23% in ambulance coverage can be achieved with only a 5.5% loss in fire coverage as one fire station is replaced by two ambulance stations (at a constant budget level). Similarly, in comparing the top most point of the middle and outer curves, ambulance coverage increases 7.6% for a 6.4% decrease in fire coverage. Thus, the greatest gain in ambulance coverage comes between the first two trade-off curves. Based on the test runs reported here, a relatively convincing argument can be made, as follows: as free-standing ambulance stations replace fire stations, at a constant facility budget level, it appears that ambulance coverage can be increased with relatively little loss in fire coverage.
FUTURE RESEARCH The current model represents a first step in integrating ambulance and fire emergency services. Future research is clearly needed. The issues of redundant coverage, back-up coverage, and probabilistic coverage must be addressed within the context of integrated emergency service planning. In addition, new measures of coverage should be developed. The model presented here, for example, assumes no congestion. Thus, it is assumed that when a call comes in, vehicles are free to respond from their stations. In fact, emergency service systems are often congested. As a result, both redundant/back-up coverage models and probabilistic models have been created to deal
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270
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AMBULANCE COVERAGE Fig. 4. Trade-off between population with ambulance coverage and population with fire coverage (A = number of ambulance stations and F = number of fire stations), (12 ambulance units, and 20 fire units), (Serra Network [20]). with congestion scenarios. Both model types can allow multiple servers of the same or multiple type at a single site--a feature not available in this first deterministic model of ours. Future research should thus seek to build both types of models (see Refs [6, 8, 16], as well as Refs [10, 11, 17]) within problem scenarios that remain responsive to evolving conditions. SUMMARY The deterministic, multi-objective model presented in this paper represents a first step in research on integrated emergency service systems. Traditionally, fire coverage takes precedence over ambulance coverage in such systems. Ambulances are generally located at fire stations that have been located without sufficient regard to medical emergencies. The reason for this precedence structure is rooted largely in history and economics. We have developed here an integrated emergency service coverage model in which ambulances are allowed to be sited at free-standing stations. We thus seek to show that ambulance coverage may benefit from a different placement of servers. Results indicate, for a series of non-contrived hypothetical examples, that with a constant facilities budget, ambulance coverage can be increased with little or no loss in fire coverage as ambulances are allowed to go free of fire stations. These findings suggest that the proposed model has the potential to enhance the process of locating emergency service vehicles and stations. Convention has largely dictated that ambulances be located at pre-existing fire stations. Since distance standards for fire vehicles are typically smaller than that for ambulances, the resulting pattern of fire station location is generally more dispersed than if ambulance coverage was explicitly considered in the placement of facilities. Thus, the existing system of stations may be sub-optimal with respect to ambulance service needs. Our illustrations of the value of allowing ambulance stations to be sited free of fire stations may be conservative. The reason for this is that in the model, fire stations are always optimally located. But, in a real urban situation, it is doubtful that fire stations would be so positioned. Thus, the additional coverage value from allowing free-standing ambulance stations might be somewhat greater than that found here.
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O b v i o u s l y , the s i t u a t i o n described in this p a p e r , viz. a city has no pre-existing e m e r g e n c y service facilities a n d is trying to design their entire e m e r g e n c y service system, will n o t often be encountered. M o s t , if n o t all, cities have some stage o f e m e r g e n c y service n e t w o r k in place. Thus, the utility o f o u r m o d e l m a y be m o r e in d e t e r m i n i n g where, a n d w h a t type o f a d d i t i o n a l emergency service vehicles a n d stations, to a d d to a system. This situation c o u l d arise, for example, as a city grows a n d requires m o r e emergency service coverage a n d a larger budget. A d d i t i o n a l l y , the m o d e l c o u l d p r o v i d e v a l u a b l e i n f o r m a t i o n as to h o w to place o r r e a r r a n g e vehicles at existing stations in the system. W e believe, further, t h a t o u r a p p r o a c h can p r o v i d e useful i n f o r m a t i o n to a city m a n a g e r w h o is faced with a d d i n g , rearranging, or redesigning a city's emergency service response system. T h e m o d e l was f o r m u l a t e d as an integer p r o g r a m t h a t solved m o s t p r o b l e m s exactly as relaxed LPs w i t h o u t a n y b r a n c h a n d b o u n d nodes. R e m a i n i n g p r o b l e m s all solved in fewer t h a n 50 nodes. A s suggested earlier, future research is p l a n n e d in the areas o f r e d u n d a n t coverage a n d p r o b a b i l i s t i c i n t e g r a t e d e m e r g e n c y service models. REFERENCES 1. G. Berlin. Facility location and vehicle allocation for an emergency service. Ph.D. Dissertation, The Johns Hopkins University, Baltimore, MD (1972). 2. S. Chapman and J. White. Probabilistic formulations of emergency service facilities location problems. O R S A / T I M S Conf., San Juan, Puerto Rico (1974). 3. R. Church and C. ReVelle. The maximal covering location problem. Papers Regl Sci. Assoc. 32, 101-118 (1974). 4. J. Cohon. Multiobjective Programming and Planning. Academic Press Inc., San Diego, CA (1978). 5. M. Daskin. The maximal expected covering location model: formulation, properties and heuristic solution. Transport. Sci. 17, 48-70 (1983). 6. M. Daskin, K. Hogan and C. ReVelle. Integration of multiple excess, back-up and expected covering models. Environ. Plann. B15, 15-35 (1988). 7. M. Daskin and E. Stern. A multi-objective set covering model for EMS vehicle deployment. Transport. Sci. 15, 137-152 (1981). 8. K. Hogan and C. ReVelle. Concepts and applications of back-up coverage. Magmt Sci. 32, 1434-1444 (1986). 9. P. Kroll. The multiple-depot, multiple-tour, and multiple-stop delivery problem. Ph.D. Dissertation, The Johns Hopkins University, Baltimore, MD (1988). 10. V. Marianov and C. ReVelle. The standard response fire protection siting problem. INFOR 29, 116-129 (1991). 11. V. Marianov and C. ReVelle. A probabilistic fire-protection siting model with joint vehicle reliability requirements. Papers Regl Sci.: The J. RSA1 71, 217-241 (1992). 12. C. ReVelle. Review, extension and prediction in emergency service siting models. Fur. J. Operl Res. 40, 58-65 (1989). 13. C. ReVelle. Siting ambulances and fire companies: new tools for planners. J. Amer. Plann. Assoc. 57, 471-484 (1991). 14. C. ReVelle. Facility siting and integer-friendly programming. Eur. J. Operl Res. 65, 147-158 (1993). 15. C. ReVelle, D. Bigman, D. Schilling, J. Cohon and R. Church. Facility location: a review of context-free and EMS models. Hlth Serv. Res. 129-146 (1977). 16. C. ReVelle and K. Hogan. The maximum availability location problem. Transport. Sci. 23, 192-200 (1989). 17. C. ReVelle and V. Marianov. A probabilistic FLEET model with individual vehicle reliability requirements. Eur. J. Operl Res. 53, 93-105 (1991). 18. D, Schilling, D. Elzinga, J. Cohon, R. Church and C. ReVclle. The Team/FLEET models for simultaneous facility and equipment siting. Transport. Sci. 13, 163 175 (1979). 19. D. Schilling, V. Jayaraman and R. Barkhi. A review of covering problems in facility location. Locat. Sci. 1, 25-55 (1993). 20. D. Serra. Location and districting of hierarchical facilities. Ph.D. Dissertation, The Johns Hopkins University, Baltimore, MD (1989). 21. C. Toregas and C. ReVelle. Binary logic solutions to a class of location problems. Geogr. Anal. 5, 145-158 (1973). 22. C. Toregas, R. Swain, C. ReVelle and L. Bergman. The location of emergency service facilities. Opers Res. 19, 1363-1373 (1971). 23. J. White and K. Case. On covering problems and the central facility location problem. Geogr. Anal. 2, 30-42 (1974).
Socio-Econ. Plann. Sci. Vol. 29, No. 4, pp. 261-271, 1995
0038-0121(95)00014-3
Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-0121/95 $9.50+0.00
Integrated Fire and Ambulance Siting: A Deterministic Model C H A R L E S REVELLE and S T E P H A N I E S N Y D E R The Johns Hopkins University, Department of Geography and Environmental Engineering, Baltimore, M D 21218, U.S.A.
Abstract--The model presented in this paper is developed for an integrated emergency system in which ambulance and fire service deployment are simultaneously considered. The model extends and blends the Maximal Covering Location Problem (MCLP) for ambulance siting and the Facility Location-Equipment Emplacement Technique (FLEET) for fire service placement. The objective of the new formulation is to maximize and trade-off both ambulance and fire coverage subject to constraints on the total number of vehicles and stations of both types that can be sited. We introduce a new concept in integrated coverage that permits free-standing ambulance stations to be sited in addition to fire stations. Typically, ambulances are located at fire stations and, sometimes, hospitals; that is, eligible sites are driven by prior choices that did not include consideration of ambulance service. Our results, based on three sample problems, suggest that, for a given budget level for facility construction, it may be possible for ambulance coverage to be increased with little or no loss in fire coverage. This condition pertains if ambulance stations are freed of the requirement to be sited only at fire stations.
INTRODUCTION A significant amount of research has focused, and continues to be focused, on the development of models for coverage availability in emergency service systems [19]. Coverage in this sense may be defined as the positioning of a vehicle to respond to a call within a specified time or distance standard. Availability refers to the actual ability of a vehicle or unit to respond to a call within the standard. Time or distance standards are widely used to ensure that emergency services can be successfully delivered [15]. A number of increasingly more sophisticated emergency service models, which utilize coverage or availability as the measure of deployment effectiveness, have been developed in the past two decades. Many of these models are reviewed and compared by ReVelle [12]. The basic models began with the formulation of Toregas et al. [22] and Toregas and ReVelle [21] who developed the Location Set Covering Problem (LSCP). Their formulation, which applied to ambulance coverage, sought to minimize the number of facilities required to cover all demand nodes within a specified time or distance standard. Church and ReVelle [3] and White and Case [23] extended the LSCP to deal with the situation in which the number of vehicles available to position was less than the number needed to cover all demand nodes. That formulation sought to cover the maximum population within the specified distance standard given a limited number of vehicles or servers available. The formulation is referred to as the Maximal Covering Location Problem (MCLP). Subsequent coverage models have dealt with locating hierarchical services, back-up services and probabilistic or stochastic services. The model proposed in this paper represents a new class of emergency service coverage model. Here, we consider three types of emergency servers: ambulances, fire engines and fire trucks. In all prior works, the siting of emergency medical services and fire services were separate and distinct issues. Indeed, it was assumed in the siting of ambulances that the fire houses (or, sometimes, hospitals, if these were possible home bases) were already in place. The fire houses had been sited, presumably historically, or, perhaps, by solution of a prior model which optimized fire service placement. The siting of ambulances was assumed to be a secondary activity with the primary activity being the siting of fire stations. With some exceptions ambulance siting has not often been thought of as the priority activity. Of course, there is historical reason for this. Fire suppres261
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Charles ReVelle and Stephanie Snyder
sion/protection evolved prior to mobile emergency medical services, while fire stations already occupied positions in the location landscape when ambulance service was first becoming available. The reason, then, for this apparent precedence of fire-over-ambulance coverage is one of simple economics. The capital costs of building free-standing ambulance stations would thus place an additional burden on municipal budgets. As a consequence, free-standing ambulance stations are the exception rather than the rule. ReVelle [13] argued, however, that since ambulance crew operating costs: (1) represent the principal financial outlay in providing coverage; and (2) are constant across ambulance location sites, free-standing ambulance stations should be taken as a priority issue in the location literature. As a consequence of the manner in which actual ambulance siting decisions were made, modelers treated ambulance and fire services sequentially. They were thus never accounted for in the same model--as they are in this investigation. A second, and parallel feature of this study is that ambulances are not constrained to be located only at fire stations. We thus propose a model in which free-standing ambulance stations are an eligible element of the siting decisions for emergency services. We show, for two hypothetical examples, that ambulance coverage can be increased without a change in investment, and with little or no loss of fire coverage as ambulances are allowed to go free of fire stations.
L I T E R A T U R E REVIEW The past two decades have seen the development of a wealth of emergency service coverage models [13]. In this regard, the emergency service coverage models reviewed here share several general characteristics. First, facilities and demand nodes are considered to be concentrated at defined nodes of a network. Secondly, servers can respond to demands only by traveling along pre-defined arcs of the network. Finally, distance or time standards are defined as the criteria against which the success of emergency deployment systems is measured. Success is the ability to reach calls within a time standard so that services can be promptly provided. As noted previously, early emergency service coverage models dealt with either ambulance or fire coverage, but not both. One shortcoming of the simplest of these models, the LSCP [21, 22], is the prohibitive expense require to cover all demand nodes. In addition, the LSCP gives no consideration to call frequency or to the demand node population. In theory, nodes with greater populations are likely to have greater call frequencies and require more frequent service. The LSCP was thus soon superseded by more sophisticated models which, in turn, have served to generate even more realistic problem formulations. To deal with the limitations of the LSCP, a new model, referred to as the MCLP, was developed [3, 23]. The objective of this siting model is to maximize the number of people (or calls) who are provided ambulance coverage within a specified standard, subject to a constraint on the number of ambulances that can be sited. Their reinterpretation of the ambulance coverage problem provided rational ambulance coverage in the face of limited resources, although this formulation has limitations as well. One crucial limitation of the MCLP is that it does not allow for differentiated servers. Fire protection systems, in most cases, consist of service by two separate types of fire vehicles and crews, each of which has its own distinctive functions and to which different distance standards apply. One of the fire vehicle types is the engine company that provides pumper service, while the other is the truck company that provides ladder rescue service. The M C L P does not directly provide for the several types of servers as is required for fire service. A more recent emergency service coverage model, the F L E E T (Facility Location/Equipment Emplacement Technique [18]) model, was developed to help achieve coverage of demands by siting two types of servers. In particular, the goal of F L E E T is to site engine and truck companies in such a way as to maximize the population (or calls) having at least one engine company located within the engine company standard and at least one truck company within its own distance standard, subject to limits on the number of both types of vehicles available and on the number of stations that can be located. A limiting assumption of these earlier coverage models is that servers, once placed, would always
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be available to respond to calls. That is, the idea of congestion in the system is not addressed. In other words, these models simply locate servers without guaranteeing availability. Recognition of this limitation has lead to the development of a new generation of coverage models in which redundant, or back-up, coverage is sought within the distance standard. Daskin and Stern [7] and Berlin [1], for example, developed a redundant coverage model for ambulance services. The goal of their model is to site a fixed number of ambulances in such a manner that all demand nodes have primary coverage within an ambulance distance standard while maximizing the number of redundant servers operating in the system. A demand node would have a redundant server if a second ambulance was located within its distance standard, in addition to the first-sited ambulance which was providing primary coverage. Hogan and ReVelle [8] formulated a redundant coverage model in which first coverage is again required, but only a specified number of additional coverers for each demand node is sought. Additional coverage is population-weighted in this, and other models, in an effort to site additional servers where they might be needed most; i.e. where the population and call frequency are the greatest. Hogan and ReVelle [8] structured a further formulation in which primary coverage was no longer a requirement, but rather a goal to be traded-off against population-weighted back-up coverage in a multiobjective model. The preceding coverage models all share the assumption that the relevant system is deterministic. However, the underlying system is, in fact, a stochastic one in which the availability of a server to respond to a call is the crucial issue. The redundant coverage models have addressed this issue by seeking to site additional ambulances that can respond to calls in the event that the primary ambulance is unavailable. However, there has been little or no accounting of the percentage or fraction of the time that an ambulance can actually respond. This condition has facilitated generation of probabilistic models better able to account for a server's "busy fraction." Chapman and White [2] were the first to formulate such a probabilistic coverage model. They constructed a probabilistic location set covering model in which the busy fraction is estimated from the local rate of calls and call durations given an initial estimate of the number of servers in the system. Using this estimated busy fraction, a reliability constraint is written, for each demand node, that specifies the minimum fraction of the time a server must be available within the distance standard. The current line of research in probabilistic coverage models expands upon the work of Chapman and White [2] as well as that of Daskin [5]. Daskin [5] formulated a model in which the expected population that could be covered by a fixed number of ambulances is maximized. Recently, research by ReVelle and Hogan [16] has concentrated on deriving more realistic, sector-specific (vs system-wide) estimates of the busy fraction. Despite the progress made in modeling emergency service systems with the LSCP, MCLP, FLEET, and subsequent models, important issues remain. As noted previously, critical among these is the integration of fire and ambulance services, i.e. the simultaneous siting of both types of services. The model proposed here thus addresses the issue of service integration, with particular focus on the issue of allowing ambulances to be sited at other than fire stations; that is, to occupy free-standing ambulance stations.
MODEL FORMULATION In many cities, the placement and even operation of ambulance services is the responsibility of the fire department. Ambulances are typically housed at, and despatched from, fire stations. The fire stations, for their part, are typically located by professional judgment in such a manner that fire events, not medical emergencies, are best covered. Ambulances are then located secondarily at the fire stations--which have likely been located without due regard to where medical emergencies arise. Nevertheless, these stations often contain sufficient space for an ambulance vehicle. This practice may not be optimal in terms of providing emergency medical services, but ifa fire station is already sited, it is far cheaper to locate an ambulance there since the infra-structure is in place and capital costs are sunk. In light of these circumstances, we here explore a new concept, a scenario in which ambulances
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Charles ReVelle and Stephanie Snyder
are allowed to be sited at free-standing ambulance stations as well as at existing fire stations. The integrated emergency services problem is modelled as the following multiple-objective integer program, which we term the Fire and Ambulance Service Technique (FAST):
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where i, I = index and set of demand nodes, j, J = index and set of potential facility sites, ai = population at demand node i, y,~= {1 if demand node i has at least one ambulance stationed within the ambulance distance or time standard; 0 otherwise}, y~ = {1 if demand node i has at least one fire engine stationed within the engine standard and at least one fire truck stationed within the truck standard; 0 otherwise}, x~ = {1 if an ambulance is stationed at j; 0 otherwise}, x T = {1 if a fire truck is stationed at j; 0 otherwise}, X~= {1 if a fire engine is stationed at j; 0 otherwise}, ~j={1 if a fire station is located at j; 0 otherwise}, uj = {1 if an ambulance station is located at j; 0 otherwise}, p A = the number of ambulances to be sited, p E+ T = the number of engines and trucks to be positioned, S A = distance standard for ambulances, S E = distance standard for engines, S T = distance standard for trucks,
Integrated fire and ambulance siting
265
N A = {j Idji ~< SA}; the set of all potential facility sites that are eligible to provide node i with ambulance service, N,.r = {j [dji ~< s'r}; the set of all potential facility sites that are eligible to provide node i with truck service, N E = {jldjg ~< SE}; the set of all potential facility sites that are eligible to provide node i with engine service, B = budget, expressed in fire station equivalents, available to build and maintain fire and ambulance stations; and b = fraction of the cost of a fire station that an ambulance station incurs. The first objective maximizes population, or ambulance call frequency that achieves ambulance coverage, while the second objective maximizes population, or fire call frequency that achieves fire coverage. Constraints (2) stipulate that each demand node will not have ambulance coverage unless at least one ambulance is sited within the ambulance distance standard of the node. Constraints (3) and (4) ensure that a node will not have fire coverage unless there is at least one truck company positioned within the truck company distance standard and at least one engine company positioned within the engine company standard. Constraints (5) and (6) ensure that a truck and/or engine will not be located at a site unless a fire station has already been positioned at that site. An ambulance, fire engine and fire truck are all permitted to be located at a fire station. No issues of station capacity are considered except that no more than one of each server type is allowed at a node. Ambulances are permitted to be located at fire or ambulance stations, but no fire companies may be located at ambulance stations. Constraints (7) ensure that an ambulance will not be sited at a node unless either a fire or ambulance station exists at that node. Constraints (8) and (9) limit the number of ambulances, fire engines, and trucks that can be sited, respectively. Constraints (10) ensure that, at most, either a fire station or an ambulance station, but not both, can be located at a node. Constraint (11) ensures that the number of fire and ambulance stations located is equal to a predetermined level.
COMPUTATIONAL RESULTS AND DISCUSSION Two data sets were utilized to evaluate the proposed model. The first data set consisted of the 55-node test network of Kroll [9], while the second was a 66-node set from Serra [20]. All computations were carried out on a Micro-Vax 4000 workstation using the mixed-integer software package, CPLEX. The results from a number of runs on these data sets will be reported in this section, with the outcomes from the Kroll network discussed first. The following distance standards were utilized on the 55-node set: 60 distance units for ambulance coverage, 40 distance units for truck coverage and 30 distance units for engine coverage. In the real world, the Insurance Services Office (ISO) has set maximum distance standards for fire trucks at 1 mile and for fire engines at 3/4 mile for high value areas. The distance standards utilized in this research were set with these standards in mind, with the engine company standard being 3/4 of that for truck companies. No mandatory response time or distance has been set by the ISO. The standards are used "to grade" cities and towns as to the quality of their fire coverage. We set the ambulance standard at a greater distance than that for fire protection coverage under the realistic assumption that fewer ambulances than fire companies would be sited in a region. The idea throughout the runs was to maintain a constant facilities total budget level for the combined system of fire and ambulance services, and to investigate how ambulance and fire coverage might vary as the number of free-standard ambulance stations was allowed to increase while the number of fire stations decreased. By constant facilities budget level, we are referring only to the cost of building and maintaining those facilities that house the ambulances and fire companies. The cost of building and maintaining an ambulance station was assumed to be one-half that of a fire station. Therefore, to maintain a constant budget level, every time the number of ambulance stations to site was increased by two, the number of fire stations to site was decreased by one. In the first set of runs, the number of ambulances to site in the system was set at 12, while the
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Charles ReVelle and Stephanie Snyder
total number of fire engines and fire trucks was fixed at 16. The number of free-standing ambulance stations and the number of fire stations was incrementally varied and examined in the following (ambulance station, fire station) combinations: (4, 10), (6, 9) and (8, 8). Thus, for any run, the total number, or sum, of fire stations to be located was set to a number Z and the total number of ambulance stations was set to a number U, where Z + 1/2U = B. To maintain the constant facility budget level, the values of Z and U, from constraints (11), were varied as the problem was re-solved for various combinations of the two values that satisfied the constraint. Note that it is necessary for the following two expressions to be met in order for the problem to remain feasible:
(13)
E Zj ~ (1/2)p E+T J
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J
Alternatively stated, these two bounds must be considered in determining eligible combinations of Z and U for use in subsequent problem analyses. Since this is a two-objective model that seeks to maximize both fire and ambulance coverages a trade-off curve is needed. Since the weighting of multiple objectives [4] preserves the integer friendliness of the constraint set [14], we chose to produce the trade-off curve in this manner. Weights were thus attached to each objective and incrementally varied between 0 and 1 as the problem was re-solved. This helped generate the needed trade-off curve. A separate trade-off curve was output for each combination of free-standing ambulance stations and fire stations, namely: (4, 10), (6, 9) and (8, 8). In Fig. 1, we display the entire set of trade-off curves for a constant budget level, divided in various ways between ambulance dispatching stations and fire stations. The results here indicate that as ambulances are allowed to go free of fire stations, i.e. to be located at their own dedicated ambulance stations, emergency medical coverage can increase with little loss in fire coverage. A reasonable increase in ambulance coverage appears to come between 4240 4200-
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AMBULANCE COVERAGE (POPULATION) Fig. 1. Trade-off between population with ambulance coverage and population with fire coverage (A = number of ambulance stations and F = number of fire stations), (12 ambulance units, 16 fire units), (Kroll Network [9]).
Integrated fire and ambulance siting
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the first two trade-off curves. In the curve closest to the origin, the number of ambulance stations is four. In the next curve out from the origin, the number of ambulance stations has increased from four to six. At the same time, the number of fire stations decreases from 10 to 9 in order to maintain a constant budget level. The ambulance coverage increases from 5510 to 5760 people between the first and second curves from the origin. Thus, one could gain 4.5% in ambulance coverage with only a 1.4% loss in fire coverage by increasing the number of ambulance stations to six in the second curve out from the origin. Further still, increasing the number of ambulance stations to eight results in a 6.5% increase in ambulance coverage with only a 2.8% decrease in fire coverage. Clearly, this model is producing viable alternatives to be considered in emergency service planning. Depending on how the decision-maker weighs ambulance and fire coverage, the model appears able to generate solutions superior to those that restrict ambulance location to fire stations. Many of the problem scenarios of interest here solved exactly as relaxed LPs, with the remainder requiring minimal branching and bounding (less than 50 nodes). A sense of the spatial configuration of the generated solutions is given in Fig. 2 where a single solution is displayed graphically on the Kroll network. The plain circles represent the fire stations while the boxed circles locate the ambulance stations. The location of vehicles at facilities has also been indicated. Figure 2 corresponds to the situation in which there are 10 fire stations and four ambulance stations to site, and fire coverage is heavily weighted over ambulance coverage. In general, the first stations are clustered around the higher population nodes, as would be expected. Engines and trucks are located together at a node at six out of the 10 stations, though they are not required to be. They may be located separately, as long as they are within their respective distance standards so as to provide coverage to demand nodes. Figure 2 corresponds to a point on the trade-off curves in which fire coverage has a much larger weight relative to ambulance coverage. Solutions for which the reverse is true display a much more dispersed configuration of both station types. This is likely due to the fact that the distance standard for ambulances is greater than that for engines and trucks, and, therefore, ambulances can cover more distance and nodes (within their standards) than fire companies and spread out farther in providing coverage. As a side note, the top most points on each trade-off curve actually correspond to the F L E E T solution for each scenario. Recall that the F L E E T model maximizes the population that has at least one engine and at least one truck within their respective distance standards, subject to a budget limitation. For the FAST model, if fire service were weighted at 1.00 and ambulance coverage at 0.00, the solution would correspond exactly with the F L E E T solution. Another set of runs was performed with the same 55-node network, in which more fire vehicles were added to the system. The number of fire vehicles was increased from 16 to 20, while all other conditions remained as in the previous set of runs. With the increased number of fire vehicles in the system, and the same number of stations, all stations became occupied by fire vehicles when fire coverage was maximized. That is, in all runs reported here, a fire station never functioned solely as an ambulance depot. As in the previous set of runs, a constant budget level was maintained, as resources were divided between the various combinations of fire and ambulance stations. A separate trade-off curve was again developed for each combination of stations. The entire set of curves is shown in Fig. 3. In the trade-off curve closest to the origin, there were 0 ambulance stations and 12 fire stations in the system. In going to the second curve out from the origin, one of the original 12 fire stations is replaced by two ambulance stations, maintaining a constant budget level. By allowing a replacement of a fire station with two ambulance stations, ambulance coverage increased from 4953 people to 5410 people between the top-most points of the two trade-off curves closest to the origin, an increase of approx. 9%. With this increase in ambulance coverage came a 0.9% decrease of people with fire coverage. Similarly, in examining the coverage differences between the top most points on the second and third trade-off curves from the origin, there is a 3.9% increase in ambulance coverage for a 1.3% decrease in fire coverage. Thus, as in Fig. 1, ambulance coverage seems capable of a reasonable increase with only a small decrease in fire coverage. That is, by maintaining a constant facility
Charles ReVelle and Stephanie Snyder
268
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Fig. 2. Graphical representation of one solution on the Kroll Network. budget level and replacing fire stations with free-standing ambulance stations, ambulance coverage can increase with little or no loss in fire coverage. It is likely that the changes in coverage displayed in Fig. 3 are not as great as they might be since there were more fire vehicles in the system, and, thus, not as much " r o o m to m o v e " to exploit greater levels of ambulance coverage.
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As previously noted, a second data set was utilized to generate a third set of runs. This second network is the 66-node structure of Serra [20]. A set of runs was performed under conditions similar to those in the previous set. Thus, the number of fire vehicles in the system was set to 20, with the number of ambulances set to 12. The number of ambulance stations varied between zero and four, while the number of fire stations varied between 12 and 10 in the following combinations: (0, 12), (2, 11) and (4, 10). Distance standards were set at higher levels to account for the greater network distances here. Again, separate trade-off curves were generated for different combinations of ambulance and fire stations, at a constant facility budget level. The results are displayed in Fig. 4. Again, results indicate that as ambulances are allowed to go free of fire stations, ambulance coverage can increase with only a small loss in fire coverage. For example, in comparing the top most points of the two trade-off curves closest to the origin, an increase of 23% in ambulance coverage can be achieved with only a 5.5% loss in fire coverage as one fire station is replaced by two ambulance stations (at a constant budget level). Similarly, in comparing the top most point of the middle and outer curves, ambulance coverage increases 7.6% for a 6.4% decrease in fire coverage. Thus, the greatest gain in ambulance coverage comes between the first two trade-off curves. Based on the test runs reported here, a relatively convincing argument can be made, as follows: as free-standing ambulance stations replace fire stations, at a constant facility budget level, it appears that ambulance coverage can be increased with relatively little loss in fire coverage.
FUTURE RESEARCH The current model represents a first step in integrating ambulance and fire emergency services. Future research is clearly needed. The issues of redundant coverage, back-up coverage, and probabilistic coverage must be addressed within the context of integrated emergency service planning. In addition, new measures of coverage should be developed. The model presented here, for example, assumes no congestion. Thus, it is assumed that when a call comes in, vehicles are free to respond from their stations. In fact, emergency service systems are often congested. As a result, both redundant/back-up coverage models and probabilistic models have been created to deal
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270
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AMBULANCE COVERAGE Fig. 4. Trade-off between population with ambulance coverage and population with fire coverage (A = number of ambulance stations and F = number of fire stations), (12 ambulance units, and 20 fire units), (Serra Network [20]). with congestion scenarios. Both model types can allow multiple servers of the same or multiple type at a single site--a feature not available in this first deterministic model of ours. Future research should thus seek to build both types of models (see Refs [6, 8, 16], as well as Refs [10, 11, 17]) within problem scenarios that remain responsive to evolving conditions. SUMMARY The deterministic, multi-objective model presented in this paper represents a first step in research on integrated emergency service systems. Traditionally, fire coverage takes precedence over ambulance coverage in such systems. Ambulances are generally located at fire stations that have been located without sufficient regard to medical emergencies. The reason for this precedence structure is rooted largely in history and economics. We have developed here an integrated emergency service coverage model in which ambulances are allowed to be sited at free-standing stations. We thus seek to show that ambulance coverage may benefit from a different placement of servers. Results indicate, for a series of non-contrived hypothetical examples, that with a constant facilities budget, ambulance coverage can be increased with little or no loss in fire coverage as ambulances are allowed to go free of fire stations. These findings suggest that the proposed model has the potential to enhance the process of locating emergency service vehicles and stations. Convention has largely dictated that ambulances be located at pre-existing fire stations. Since distance standards for fire vehicles are typically smaller than that for ambulances, the resulting pattern of fire station location is generally more dispersed than if ambulance coverage was explicitly considered in the placement of facilities. Thus, the existing system of stations may be sub-optimal with respect to ambulance service needs. Our illustrations of the value of allowing ambulance stations to be sited free of fire stations may be conservative. The reason for this is that in the model, fire stations are always optimally located. But, in a real urban situation, it is doubtful that fire stations would be so positioned. Thus, the additional coverage value from allowing free-standing ambulance stations might be somewhat greater than that found here.
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O b v i o u s l y , the s i t u a t i o n described in this p a p e r , viz. a city has no pre-existing e m e r g e n c y service facilities a n d is trying to design their entire e m e r g e n c y service system, will n o t often be encountered. M o s t , if n o t all, cities have some stage o f e m e r g e n c y service n e t w o r k in place. Thus, the utility o f o u r m o d e l m a y be m o r e in d e t e r m i n i n g where, a n d w h a t type o f a d d i t i o n a l emergency service vehicles a n d stations, to a d d to a system. This situation c o u l d arise, for example, as a city grows a n d requires m o r e emergency service coverage a n d a larger budget. A d d i t i o n a l l y , the m o d e l c o u l d p r o v i d e v a l u a b l e i n f o r m a t i o n as to h o w to place o r r e a r r a n g e vehicles at existing stations in the system. W e believe, further, t h a t o u r a p p r o a c h can p r o v i d e useful i n f o r m a t i o n to a city m a n a g e r w h o is faced with a d d i n g , rearranging, or redesigning a city's emergency service response system. T h e m o d e l was f o r m u l a t e d as an integer p r o g r a m t h a t solved m o s t p r o b l e m s exactly as relaxed LPs w i t h o u t a n y b r a n c h a n d b o u n d nodes. R e m a i n i n g p r o b l e m s all solved in fewer t h a n 50 nodes. A s suggested earlier, future research is p l a n n e d in the areas o f r e d u n d a n t coverage a n d p r o b a b i l i s t i c i n t e g r a t e d e m e r g e n c y service models. REFERENCES 1. G. Berlin. Facility location and vehicle allocation for an emergency service. Ph.D. Dissertation, The Johns Hopkins University, Baltimore, MD (1972). 2. S. Chapman and J. White. Probabilistic formulations of emergency service facilities location problems. O R S A / T I M S Conf., San Juan, Puerto Rico (1974). 3. R. Church and C. ReVelle. The maximal covering location problem. Papers Regl Sci. Assoc. 32, 101-118 (1974). 4. J. Cohon. Multiobjective Programming and Planning. Academic Press Inc., San Diego, CA (1978). 5. M. Daskin. The maximal expected covering location model: formulation, properties and heuristic solution. Transport. Sci. 17, 48-70 (1983). 6. M. Daskin, K. Hogan and C. ReVelle. Integration of multiple excess, back-up and expected covering models. Environ. Plann. B15, 15-35 (1988). 7. M. Daskin and E. Stern. A multi-objective set covering model for EMS vehicle deployment. Transport. Sci. 15, 137-152 (1981). 8. K. Hogan and C. ReVelle. Concepts and applications of back-up coverage. Magmt Sci. 32, 1434-1444 (1986). 9. P. Kroll. The multiple-depot, multiple-tour, and multiple-stop delivery problem. Ph.D. Dissertation, The Johns Hopkins University, Baltimore, MD (1988). 10. V. Marianov and C. ReVelle. The standard response fire protection siting problem. INFOR 29, 116-129 (1991). 11. V. Marianov and C. ReVelle. A probabilistic fire-protection siting model with joint vehicle reliability requirements. Papers Regl Sci.: The J. RSA1 71, 217-241 (1992). 12. C. ReVelle. Review, extension and prediction in emergency service siting models. Fur. J. Operl Res. 40, 58-65 (1989). 13. C. ReVelle. Siting ambulances and fire companies: new tools for planners. J. Amer. Plann. Assoc. 57, 471-484 (1991). 14. C. ReVelle. Facility siting and integer-friendly programming. Eur. J. Operl Res. 65, 147-158 (1993). 15. C. ReVelle, D. Bigman, D. Schilling, J. Cohon and R. Church. Facility location: a review of context-free and EMS models. Hlth Serv. Res. 129-146 (1977). 16. C. ReVelle and K. Hogan. The maximum availability location problem. Transport. Sci. 23, 192-200 (1989). 17. C. ReVelle and V. Marianov. A probabilistic FLEET model with individual vehicle reliability requirements. Eur. J. Operl Res. 53, 93-105 (1991). 18. D, Schilling, D. Elzinga, J. Cohon, R. Church and C. ReVclle. The Team/FLEET models for simultaneous facility and equipment siting. Transport. Sci. 13, 163 175 (1979). 19. D. Schilling, V. Jayaraman and R. Barkhi. A review of covering problems in facility location. Locat. Sci. 1, 25-55 (1993). 20. D. Serra. Location and districting of hierarchical facilities. Ph.D. Dissertation, The Johns Hopkins University, Baltimore, MD (1989). 21. C. Toregas and C. ReVelle. Binary logic solutions to a class of location problems. Geogr. Anal. 5, 145-158 (1973). 22. C. Toregas, R. Swain, C. ReVelle and L. Bergman. The location of emergency service facilities. Opers Res. 19, 1363-1373 (1971). 23. J. White and K. Case. On covering problems and the central facility location problem. Geogr. Anal. 2, 30-42 (1974).