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A hybrid fleet model for emergency medical service system design
Sot. Sci. Med. Vol. 26, No. I, pp. 163-171, Printed in Great Britain. All rights reserved
1988 Copyright
0277-9536188 163.00 + 0.00 % 1988 Pergamon Journals Ltd
A HYBRID FLEET MODEL FOR EMERGENCY MEDICAL SERVICE SYSTEM DESIGN GEOFFREY BIANCHI’ and RICHARD L. CHURCH* ‘General Research Corp., Santa Barbara, CA 93160 and 2University of California at Santa Barbara, Santa Barbara, CA 93106, U.S.A.
Abstract-Covering models have been used to locate emergency services such as ambulance and fire protection systems. As an example, in the late seventies, an analysis of the Baltimore, Maryland fire protection system was conducted with the development and use of a covering model called the Facility Location and Equipment Emplacement Technique (FLEET). The FLEET model combined the location of fire stations with the allocation of primary and special service equipment to the stations. Further. in a recent study of Austin, Texas the ambulance system was restructured based on the use of a covering model. Covering models have also been extended to handle some of the special circumstances involved in emergency service systems. One example is the maximal expected coverage problem (MEXCLP). This paper presents a new covering model which utilizes both the special coverage structure of the MEXCLP and the simultaneous station location and equipment allocation of the FLEET model. Optimal solutions are found using linear and integer programming. Results of the model applied to several planning data sets (including a form of the Austin, Texas planning problem) demonstrate that more concentrated ambulance allocation patterns exist which may lead to easier dispatching, reduced facility costs, and better crew load balancing with little or no loss of service coverage. Tradeoff curves are presented which show that significant reductions in the number of dispatching sites (keeping the number of ambulances constant) can be made without any major changes in service level. Key words-ambulance
modeling, emergency services, expected coverage, location-allocation
INTRODLMXION
In the past ten years many context-free location models have been developed that can be utilized in the location of emergency services. Many of these models can be classified as covering models in that they utilize a maximal service distance or time standard. A demand is considered ‘covered’ as long as a facility has been placed at a location which is within the maximal distance or time standard of that demand point. The first location model to utilize the ‘covering’ concept was the Location Set Covering Problem (LSCP). The LSCP identifies the minimum number and location of facilities which insures that no demand point will be farther than the maximal service distance (or time) from a facility. Since the development of the LSCP by Toregas and ReVelle [l] in 1972, many context-free covering models have been defined. Many of the recently developed models are either based on the LSCP or are based on the concept of ‘maximizing coverage’ which was first introduced in the Maximal Covering Location Problem (MCLP) [2]. The MCLP deals with locating a fixed number of facilities in order to maximize the amount of demand provided coverage. Both LSCP and MCLP models have been applied to a number of planning problems, including fire station location [3-71, ambulance location [S-11], and health clinic location [12]. The use of the MCLP to analyze ambulance service in Austin, Texas received awards from both The Institute of Management Sciences [S] and The Association of American Geographers. Even in a 1977 paper, ReVelle cites that over 100 cities have requested information on the LSCP model for use in fire station location analysis (31. 163
The use of covering models in analyzing emergency service systems has produced a new surge of model development representing new model forms addressing certain properties not included in the earlier covering models. Several of these properties are discussed in Church and Roberts [3]. Almost all covering models utilized in ambulance deployment modeling locate ambulances rather than simultaneously locate stations and allocate ambulances to these sites. Ambulance stations are then assumed to be placed wherever ambulances are sited. Given that there is no constraint on cost included for stations, solutions tend to be dispersed with one ambulance per site. The cost of developing an ambulance system is a function of both the number of ambulances and the number of stations. Costs can be reduced by employing stations which house several ambulances or share facilities with other services such as fire protection. Such deployment patterns (i.e. multiple units assigned to a station) can be analyzed only when explicitly included in the model. Such patterns may cost less, be easier to administer and yield better load balance between crews than more dispersed patterns and at the same time provide essentially the same level of service as completely dispersed patterns. It is important to analyze such solutions because they in fact may be superior. Further, approaches that do handle station location and ambulance allocation as a two step decision making process cannot capture the dependent nature between the two decisions. Both processes must be resolved simultaneously in the same model. The objective of this paper is to present a hybrid covering model which simultaneously locates stations
164
GEOFFREY BIANCHIand
and ahocates ambulances. The model is based in part on the Facility Location and Equipment Emplacement Technique (FLEET) model that Schilling et al. [5] developed for locating fire stations and allocating equipment, as well as, on the Maximal Expected Coverage Location Problem (MEXCLP) of Daskin [14] that was developed for problems where vehicle availability needs to be considered. A general integer programming approach is discussed and computational results are given. Several tradeoff curves are presented which show that significant reductions in the number of dispatching sites (keeping the number of ambulances constant) can be made without major changes in expected coverage for the two planning problems presented in this paper. The results of this paper suggest that station location and ambulance allocation need to be modeled as simultaneous decisions and that many different robust alternatives can be generated from such a model. This type of covering model will ultimately lead to better decision making in emergency service delivery problems. BACKGROUND Many of the extensions to covering models for emergency medical services (EMS) are based on the MCLP [2]. The MCLP seeks to find the location of a fixed number of facilities on a network such that the amount of population covered (or served) within a prespecified maximal service distance (or time) is maximized. Several solution methods have been employed to solve maximal covering problems: heuristics [2], linear programming with branch and bound [2], lagrangian relaxation [15], and dual ascent [15]. Utilizing the MCLP one can generate a tradeoff curve which shows the maximum possible coverage as a function of the number of facilities located. The smallest number of facilities at which 100% coverage is achieved represents a solution of the (LSCP). MCLP solutions with less than 100% coverage utilize fewer resources than LSCP solutions. Church and ReVelle have shown that substantially high values of coverage can be achieved with significantly fewer facilities than what is required for total coverage [2]. Advancements in covering models have been developed along four major areas. The first area involves the calculation of coverage utilizing different measures such as the type of demand (e.g. coverage of the elderly population, or coverage of those areas with the highest frequencies of injuries due to assault). Utilizing different coverage measures was first modeled as a multiobjective coverage model by Schilling [ 161.Eaton et al. [8] give an excellent discussion of the use of eight different criteria used in covering for ambulance deployment in Austin. A second major area involves the expansion of the definition of covering to include varying coverage values as a function of distance. Several model forms have been developed by Church and Roberts [13]. A third major area deals with the simultaneous location of stations and equipment. Schilling et al. [5], in analyzing a fire station location problem for the City of Baltimore, combined the processes of fire station location and equipment allocation into one model called FLEET. FLEET was used to analyze a
RICHARDL. CHURCH fire station location problem where two types of equipment (primary service and special service equipment) were being positioned. This particular model is significantly different from earlier covering models in that it handled equipment and facilities simultaneously. Usually in fire protection systems, equipment is in use only a small fraction of the time and is available (not busy) when most calls for service are placed. The FLEET model was formulated on the assumption that a vehicle is essentially always available (and not busy). However, in ambulance operations, equipment can frequently be busy or not available when called. Therefore, it is important to address vehicle availability in any ambulance location model. Good service is achieved only when an available unit (currently not busy) can respond within the desired maximal service time to a call. Equipment availability has been introduced into covering problems with three different approaches. The first approach is one that is comprised of two steps. The first step utilizes a covering model to select station locations. The second step involves the use of a simulation or queuing model to allocate ambulances to the dispatching stations in order to insure coverage with a high probability. This approach was first suggested by Berlin and Liebman [9]. Although the recent ambulance location study of Eaton er al. utilized primarily the maximal covering location problem to locate-allocate ambulances, they utilized this two step process to an extent to insure that ambulance patterns could reasonably respond to calls when ambulance availability was taken into account. The second approach involves extending covering models to include the notion of backup coverage. A demand area is typically considered covered if it can be reached by at least one ambulance within the maximal service distance. Should that ambulance be busy, the demand area is not ‘actually’ covered unless a second ambulance exists that can also reach the demand within the covering distance. The second ambulance can provide ‘backup’ coverage in case the first is busy. One type of ‘backup’ coverage model would be to minimize the number of ambulances needed to cover all demand, and then locate that fixed number in such a way that all are covered once and as many as possible are covered twice (that is backup covered). This is simply a restatement of the Hierarchical Set Covering Problem of Daskin and Stern [lo]. Hogan and ReVelle [ 171 have also generated several types of backup coverage models utilizing both the location set covering problem and the maximal covering location problem. Further, Charnes and Storbeck [18] have also utilized concepts of backup coverage between two types of responding equipment. The third approach to extending coverage models to consider equipment availability is to incorporate equipment availability directly into the definition of coverage. This was accomplished by Daskin [14] in the definition of the MEXCLP. The MEXCLP objective is defined in terms that an emergency vehicle is not always available (with a given probability) when called for service. Daskin has demonstrated an application of the MEXCLP to EMS system design [l 11 utilizing the Austin, Texas planning problem that was
A
hybrid FLEET model
analyzed by Eaton et al. [8]. The MEXCLP model has been used with linear programming to optimally solve several planning problems by Daskin [l 11. Daskin has also developed a heuristic to solve the MEXCLP [ 141. In analyzing the location of ambulances by a covering model two important properties must be included. First, equipment availability must be included in the definition of coverage whenever the probability of being unavailable is significant (which is true for most urban ambulance systems). This can be approached by either the expected coverage format of Daskin or the backup coverage format of Hogan and ReVelle [17]. Second, it is important to analyze both equipment allocation and station location decisions simultaneously. For this reason, a hybrid model incorporating the concepts of MEXCLP and FLEET was developed. This model maximizes expected coverage within a maximal service distance (or time) while locating a fixed number of stations and allocating a given number of ambulances. We will designate this model as a Multiplecover, One-unit FLEET problem (MOFLEET). The multiple-cover refers to the expected coverage objective where expected coverage is a function of the number of times a demand is covered. The one-unit specifies that it utilizes only one type of unit (ambulances) compared to the original two equipment FLEET model.
THE MOFLEET MODEL
MOFLEET seeks to find the location of a fixed number of stations and a given number of ambulances on a network such that the amount of population within the service standard is maximized. The MOFLEET model utilizes a multiple coverage goal or expected coverage objective which makes it sensitive to the conditions of an ambulance not always being available when called upon. MOFLEET defined on a network of nodes and arcs can be formulated as follows: MOFLEET:
Min xX(1 i k
ST. c x,+ 2 y,a&.f, IEN
- U)U~- I a,y,
for each
i
k=l
xx,= E Xz,=P x, G Cz,
for each j
x, > 0, integer z, and
(4)
for each j
y,k = 0, 1 for each
i, j,
and k
(5) (6)
where: x, = the number of ambulances located at node j 1 if a station is located at site j z, = 1 0 if otherwise C = the maximum number of ambulances that can be located at a station
Yik =
165
1 if demand node i is not covered by k ambulances 0 if otherwise
a, = the population to be served at demand node i = the number of stations to locate E = the number of ambulances to allocate II = the probability of a randomly selected ambulance being unavailable Mp = the maximum number of ambulances necessary to provide complete coverage S, = the maximal service distance for demand i I = the set of demand nodes J = the set of facility sites N,= {jEJld,< Si} d, = the shortest distance or travel time from node i to j. P
The objective function seeks to minimize the number of people not covered within the maximal service distance, Si, where coverage is a function of the number of ambulances that can respond to a demand zone within the specified maximal service distance. The constraints of form (1) are used to determine whether an area i is covered up to h4p times; when Xj is equal to one or greater then there are one or more ambulances that can provide coverage to the demand zone and Yik = 0. If there is not a station or an ambulance in the set Ni (i.e. Zxj = 0) then demand zone i does not have an ambulance that can respond within the service criteria and Yik = 1. Constraint 2 establishes that exactly E emergency vehicles will be used and the third constraint maintains that exactly P facility locations will be used. Constraints of type (4) limit the allocation of ambulances to a site j that has a facility located at j. Constraints (5) and (6) complete the constraint set with the integer conditions on the x, y, and z variables. The MOFLEET model is essentially the generalization of the MEXCLP model with the added feature of locating stations to house emergency equipment. When E = P, then the MOFLEET model is essentially equivalent to the MEXCLP model. However, when E > P, then there will be multiple unit stations and the two models are not equivalent. The MOFLEET model is an important extension to the MEXCLP in that one should not only be concerned with how many units are being located, but also with how many dispatching points are used. The costs and ease of operation when units are clustered as compared to a completely dispersed set of units could outweigh the additional benefits of complete dispersal in many cases. In order to support this argument, one would want to generate a tradeoff of expected coverage versus the number of stations. Integer programming was utilized to obtain optimal solutions to the MOFLEET problem. To solve MOFLEET with integer programming the programming system required the specification of the Yik and Zj variables as O/l integers, and the Xj variables as continuous integers. We used the Linear INteractive Discrete Optimizer (LINDO) programming system [ 191. LINDO was executed on a VAX 1I/750 under UNIX to generate optimal integer solutions to MOFLEET. LINDO accepts program formulations in the IBM Mathematical Programming System
GEOFFREY BIANCHIand RICHARDL.
166
(MPS) format. A FORTRAN program creates the file of MPS statements that specifies the model formulation which is then read by LINDO. Additionally, LINDO permits the specification of O/l integer variables directly by marking their start and stop point in the input file. The LINDO program uses the method of branch and bound to resolve fractional decision variables into O/l integers. An alternative heuristic solution approach has also been developed as a part of this research and is described elsewhere
WI. MODELRESULTS Two test networks are used to illustrate the value of the MOFLEET modeling approach. The first spatial data set is the 55 node network of Swain [21]. The second data set is a 33 node network of census tract centroids of Austin, Texas. The Austin data contains actual calls for emergency medical service during a five month period [1 11. An important input of the MOFLEET model is the parameter u, the probability that a randomly selected ambulance is unavailable when called for service. An emergency vehicle is considered unavailable when it is serving another emergency call or the ambulance cannot be deployed (i.e. it is in repair). Daskin [11] gives an excellent discussion of the model parameter utilized in the maximum expected covering problem. ReVelle [22] has, in a recent paper, dealt with the estimation and application of the probability of spatially varying demand for designing emergency service systems within the context of the maximum expected coverage problem. The parameter 11may be thought of as the ratio of two terms, the expected EMS workload per day, w, to the number of deployed ambulances, E. As the number of ambulances, E, is increased (given a value for w), the probability that a randomly selected ambulance is unavailable is decreased. In this research, the value used for w is equal to 0.90 for the 33 node network and 1.50 for the 55 node network (the same as those used by Daskin [1 1] for comparisons). The actual value of w using the Austin data set is lower than 0.90 (approx. 0.73), therefore, the degree of ambulance unavailability is slightly overestimated. In addition, the probability of an ambulance being unavailable is the same for all units in the fleet and it is also independent of the ambulance
Table Number of units 5 5 5 5 5 4 4 4 4 3 3 3
Prob. of an unavailable vehicle 0.18 0.18 0.18 0.18 0.18 0.225 0.225 0.225 0.225 0.3 0.3 0.3
Number of locations 5 4 3 2 I 4 3 2 1 3 2 1
I.
CHURCH
locations. Since workload can vary spatially, the probability of ambulance unavailability being the same for all units over all locations represents an assumption that could be relaxed in future modeling. This relaxation can be based on ReVelle’s method to estimate spatially varying probabilities of vehicle unavailability [22]. Tables 1 and 2 present the optimal MOFLEET solutions for both the 33 node and the 55 node networks. All results in this paper were found using a maximal service distance of S = 10.0 (same as that used by Daskin (111). For each MOFLEET solution generated, the tables indicate the amount of demand covered at least once by a station-ambulance combination and the expected coverage. Computer CPU time in hours is also presented. The LINDO system (written in FORTRAN 77) was utilized with the Unix 4.3 BSD operating system. This system is relatively slow, however, one should expect substantial time saving by utilizing a more efficient branch and bound system in a better operating environment. Since operating costs were $2.5 per hour, the actual costs of generating most of the solutions were relatively small. All solutions in this study were generated for less than $350. Table 1 presents optimal solutions of MOFLEET on the 33 node Austin, Texas data set. Solutions were identified for allocating 5, 4, and 3 ambulances over a range of locations. The j-unit, 4-station solution, as well as, the 4-unit, 4-station and the 3-unit, 3-station results represent optimal solutions to the MEXCLP model. The j-unit, 4-station solution indicates that higher coverage can be achieved by locating 2 ambulances at node 25 instead of having the ambulance allocated to site 17 as indicated by the j-station, j-unit solution. By moving a unit to node 4 and thus eliminating station 11, the 3 station solution results in a loss of total expected coverage of only 2.25% as compared to the j-station solution. Similarly, the 4and 3-unit solutions indicate that the number of stations may be reduced with a minimal loss of coverage. In addition, the average CPU usage for the 5, 4, and 3 ambulance MOFLEET problems with LINDO was 0.033, 0.045, and 0.017 hours, respectively. Table 2 presents the optimal solutions found by LINDO on the 55 node network. Solutions were identified for allocating 10, 7, and 5 ambulances. This table indicates that clustered solutions (i.e. reducing
33 Node-MOFLEEToetimal solutions
Locations
Number of demands covered at least once
Percent covered at least once
Expected number of demands covered
Percent expected coverage
CPU time (hr)
4, II. 17.25.31 4, 11,25’,31 4’. 25’. 31 11’,25’ 4J 4,11,25,31 4’. 25, 31 llz,25’ 4’ 4, II,25 1 I’, 25 4’
4249 4249 4141 3896 2989 4249 4141 3896 2989 4042 3896 2989
100.00 100.00 97.46 91.69 70.35 100.00 97.46 91.69 70.35 95.13 91.69 70.35
4111.87 4111.87 4016.31 3848.63 2988.43 3914.79 3829.16 3716.50 2981.34 3460.03 3357.35 2908.30
96.77 96.79 94.52 90.58 70.33 92.13 90.12 87.47 70.17 El .43 79.01 68.45
0.036 0.030 0.051 0.021 0.027 0.022 0.104 0.024 0.028 0.012 0.022 0.018
A hybrid FLEET Table 2. 55 Node-MOFLEET
Number of units IO
Prob. of an unavailable vehicle
Number of locations
0.15
IO
0.15
9
0.15
8
0.15 0.15 0.2143 0.2143 0.2143 0.2143 0.2143 0.2143 0.3 0.3 0.3 0.3 0.3
6 6 4
4 3 2
Locations 3.7, IO, 13, 17, 20 22, 27, 36, 55 32, IO, 13, 17, 20 22.27.36. 55 17i, 2d, 27.34.36 422, 53,55 172. 21. 27,34. 36’. 422, 55 2’, 17’. 2 I, 27.36’. 55 4. IO, 13. 17,275 30,36 42, 10. 17.27.34, 36 IO, 17’. 36, 38.42’ 17*, 36,38,42j 22I, 362, 42’ 22’, 42’ 7, IO. 22,34.42 3,223 36, 422 4’. 22, 36 22=, 42’ 45
167
mddel
optimal
solutions
Number of demands covered at least once
Percent covered at least once
Expected number of demands covered
Percent expected coverage
CPU time (hr)t
6350
99.22
6177.38
96.52
0.05
6350
99.22
6169.10
96.39
0.81
6380
99.69
6155.15
96.17
13.45
6300
98.44
6131.91
95.81
47.00
6250 6190 6140 6040 5810 5480 5020 5600 5570 5480 5020 4250
97.66 96.72 95.94 94.38 90.78 85.62 78.44 87.50 87.03 85.62 78.44 66.41
6073.48 5771.05 5739.06 5665.16 5563.21 5386. I8 5006.44 5118.97 5093.26 5014.85 4851.52 4239.66
94.89 90.17 89.67 88.52 86.92 84.16 78.22 79.98 79.58 78.36 75.80 66.24
0.02 I.19 7.06 23.29 34.24 4.65 0.02 0.16 1.50 2.07 0.27
l
lIP halted prior to completion. tVAX I l/750 utilizing LINDO (written in FORTRAN 77) under Unix 4.3 BSD. This is a relatively slow system. No special procedures were taken to reduce time. One can expect substantial time savings by utilizing a more efficient branch and bound code and an efficient machine. Operating costs are approximately $2.5 per hour.
the number of stations) can be generated by MOFLEET without substantial reductions in coverage. For example, the optimal placement of 10 units at 6 locations covers only 1.64% less than the optimal placement of 10 units at 10 locations. Another important item of information in this table is the computer processing time. A considerable amount of time was spent to identify optimal integer solutions. The average CPU usage for the set of 10, 7, and 5 units allocated was 15.33, 11.74, and 0.80 hr, respectively. The tables show the reduction in coverage as the deployment pattern for ambulances goes from a dispersed pattern to a clustered pattern. These coverage values indicate that high levels of coverage can be maintained while at the same time achieving a significant degree of system centralization. By reducing the quantity and spatial distribution of system components, real-time system monitoring and control can be performed with greater ease. Figure 1 shows the deployment pattern generated by the LINDO system on the S-node network for allocating 10 units over IO facility sites. The circular regions represent the theoretical coverage areas of the ambulances located at the designated stations. This pattern represents the optimal solution to the MEXCLP because the solution is allowed to be dispersed (i.e. there is no constraint or penalty for adding a station). This solution also represents the only MEXCLP deployment pattern on the S-node network which provides at least 95% coverage. MOFLEET, however, identifies solutions in which 10 units may be allocated to 9, 8, or 7 stations while maintaining at least 95% c’overage. In fact, MOFLEET identified a 10 unit, 6 facility solution illustrated in Fig. 2 which is short of 95% coverage by only 0.11%. It is also important tlo recognize that Fig. 2 demonstrates the value of the MOFLEET modeling approach. That is that high levels of coverage can be maintained while
reducing the number of stations to be operated and managed on a daily basis. The reduction in the number of required stations to meet demand needs can provide for further ease and efficiency of system management. Figure 3 presents two solutions to the MOFLEET mode1 on the 33 node Austin. Texas network. The 4-station and 5-ambulance solution is represented graphically by squares on the map. This solution also represents the optimal pattern of distributing 5 ambulances with no consideration for station location (i.e. optimal MEXCLP solution). By reducing the number of stations by one. a more centralized distribution of stations and ambulances can be realized as illustrated by the solution in Fig. 3 represented by triangles. In this case, a station is eliminated from census tract 11 and the ambulance is reallocated to tract 4. The loss of the station results in a loss of total expected coverage of only 2.27%. Although coverage falls below 95% to 94.52%, the solution demonstrates that the number of stations may be reduced without seriously reducing the emergency system performance. The MOFLEET problem can be defined for any maxima1 service distance and a number of facilities P to be located. By solving the model over a range of facilities, a tradeoff curve can be described which illustrates the effectiveness of increasing or decreasing the number of facilities located. Figure 4 presents a set of cost-effectiveness curves that describe demand covered as a function of the number of stations located, P, when E ambulances are allocated. The results of LINDO in Table 1 for the 33 node data are used to outline the tradeoff curves of Fig. 4a. This family of curves can be used to find the best coverage of demand attainable given a maximum number of available ambulances and stations. Similarly, LINDO results for the 55 node data set are used to describe the tradeoff curves of
GEOFFREYBIANCHIand RICHARDL.
CHURCH
Fig. 1. MOFLEET solution for 55 node problem with Fig. 4h. The ‘flatness’ of the curves shown in Fig. 4b emphasizes the importance of the MOFLEET model. For instance, in the case of allocating 10 units over a set of station locations, the MOFLEET model results indicate a small decrease in coverage with a reduction in the number of stations located. For example a 40% reduction in stations (i.e. 10 down to
10
stations.
6) resulted in only a 1.64% loss of total expected coverage. SUMMARY
We have presented a hybrid model called MOFLEET which simultaneously locates stations and
A hybrid FLEET model
169
with 2 Units m
63
With 2 Units
4D
Fig. 2. MOFLEET solution for 55 node problem with 6 stations
allocates equipment to maximize expected coverage. The MOFLEET model provides decision makers and managers of EMS systems with a powerful locational analysis technique. The sample results represented by the 55 node and 33 node configuration maps and the
tradeoff curves indicated that high levels of coverage can be maintained while reducing the number of system facilities that require operation and management. Therefore, location decisions based on management and operating costs can be analyzed with
170
_
GEOFFREY
BIANCHIand RICHARDL. CHURCH
_=
./
n
=3 site/5 ambulance MOFLEET configuration
lJ
=4 site/5 ambulance MOFLEET configuration number in symbol represents number of ambulances allocated at that location.
Fig. 3. Comparison
of 3- and 4-station MOFLEET solutions for Austin. Texas.
simulation models are used to deal with the changes in the spatial distribution of EMS demand over time. However, by extending MOFLEET into a multiple period planning model one can address issues associated with crew allocation. For example, the avail-
respect to a desired number of locations and operating units at the locations. The MOFLEET problem can be easily reformulated to handle equipment and crew allocation in a multiple time period framework. In general,
(4
100 8 E ?i 0 B zl%
3 S
(b)
100
5-units 90
p
90
60
z 6
60
70
z tj
70
60
B ;
60
lo-units
50
50 1
2
3
4
Number of Stations
5
6
1
2
3
4
5
6
7
6
9
Number of Stations
Fig. 4. (a) Cost-effectiveness curves for the 33 node data set with service distance = 10.0. (b) Costeffectiveness curves for the 55 node data set with service distance = 10.0.
10
A hybrid FLEET model
ability of crews during peak versus off-peak times and the balancing of work loads can be addressed within the multiple time period context. In addition, one can optimize tradeoffs between demand covered versus the number of stations, ambulances, and emergency service crews utilized.
10.
11. 12.
REFERENCES
1. Toregas C. and ReVelle C. Optimal location under time or distance constraints. Socio-Econ. Plating Sci. 28, 133, 1972. 2. Church R. L. and ReVelle C. The maximal covering location problem. Paper Regl Sci. Ass. 32, 101, 1974. 3. ReVelle C., Bigman D., Schilling D., Cohon J. and Church R. L. Facility location: a review of context-free and EMS models. kh Serv. Res. 12, 129, 1977. 4. Plane D. R. and Hendrick F. E. Mathematical nrogramming and the location of fire companies for- the Denver fire department. Mgmr Sci. Rep. Ser. 74-79, 1974. 5. Schilling D., Elzinga D. J., Cohon J., Church R. L. and ReVelle C. The TEAM/FLEET models for simultaneous facility and equipment siting. Transport. Sci. 13, 163, 1979. 6. Schilling D., ReVelle C., Cohon J. and Elzinga D. Some models for fire protection locational decisions. Eur. J. Opns Res. 5, 1-7, 1980. 7. Kolesar P. and Walker W. E. An algorithm for the dynamic relocation of fire companies. Opns Res. 22,249, 1974. 8. Eaton D., Daskin M. S., Simmons D., Bulloch B. and Jansma G. Determining emergency medical service vehicle deployment in Austin, Texas. Interfaces 15, 96, 1985. 9. Berlin G. N. and Liebman J. C. Mathematical analysis
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1988 Copyright
0277-9536188 163.00 + 0.00 % 1988 Pergamon Journals Ltd
A HYBRID FLEET MODEL FOR EMERGENCY MEDICAL SERVICE SYSTEM DESIGN GEOFFREY BIANCHI’ and RICHARD L. CHURCH* ‘General Research Corp., Santa Barbara, CA 93160 and 2University of California at Santa Barbara, Santa Barbara, CA 93106, U.S.A.
Abstract-Covering models have been used to locate emergency services such as ambulance and fire protection systems. As an example, in the late seventies, an analysis of the Baltimore, Maryland fire protection system was conducted with the development and use of a covering model called the Facility Location and Equipment Emplacement Technique (FLEET). The FLEET model combined the location of fire stations with the allocation of primary and special service equipment to the stations. Further. in a recent study of Austin, Texas the ambulance system was restructured based on the use of a covering model. Covering models have also been extended to handle some of the special circumstances involved in emergency service systems. One example is the maximal expected coverage problem (MEXCLP). This paper presents a new covering model which utilizes both the special coverage structure of the MEXCLP and the simultaneous station location and equipment allocation of the FLEET model. Optimal solutions are found using linear and integer programming. Results of the model applied to several planning data sets (including a form of the Austin, Texas planning problem) demonstrate that more concentrated ambulance allocation patterns exist which may lead to easier dispatching, reduced facility costs, and better crew load balancing with little or no loss of service coverage. Tradeoff curves are presented which show that significant reductions in the number of dispatching sites (keeping the number of ambulances constant) can be made without any major changes in service level. Key words-ambulance
modeling, emergency services, expected coverage, location-allocation
INTRODLMXION
In the past ten years many context-free location models have been developed that can be utilized in the location of emergency services. Many of these models can be classified as covering models in that they utilize a maximal service distance or time standard. A demand is considered ‘covered’ as long as a facility has been placed at a location which is within the maximal distance or time standard of that demand point. The first location model to utilize the ‘covering’ concept was the Location Set Covering Problem (LSCP). The LSCP identifies the minimum number and location of facilities which insures that no demand point will be farther than the maximal service distance (or time) from a facility. Since the development of the LSCP by Toregas and ReVelle [l] in 1972, many context-free covering models have been defined. Many of the recently developed models are either based on the LSCP or are based on the concept of ‘maximizing coverage’ which was first introduced in the Maximal Covering Location Problem (MCLP) [2]. The MCLP deals with locating a fixed number of facilities in order to maximize the amount of demand provided coverage. Both LSCP and MCLP models have been applied to a number of planning problems, including fire station location [3-71, ambulance location [S-11], and health clinic location [12]. The use of the MCLP to analyze ambulance service in Austin, Texas received awards from both The Institute of Management Sciences [S] and The Association of American Geographers. Even in a 1977 paper, ReVelle cites that over 100 cities have requested information on the LSCP model for use in fire station location analysis (31. 163
The use of covering models in analyzing emergency service systems has produced a new surge of model development representing new model forms addressing certain properties not included in the earlier covering models. Several of these properties are discussed in Church and Roberts [3]. Almost all covering models utilized in ambulance deployment modeling locate ambulances rather than simultaneously locate stations and allocate ambulances to these sites. Ambulance stations are then assumed to be placed wherever ambulances are sited. Given that there is no constraint on cost included for stations, solutions tend to be dispersed with one ambulance per site. The cost of developing an ambulance system is a function of both the number of ambulances and the number of stations. Costs can be reduced by employing stations which house several ambulances or share facilities with other services such as fire protection. Such deployment patterns (i.e. multiple units assigned to a station) can be analyzed only when explicitly included in the model. Such patterns may cost less, be easier to administer and yield better load balance between crews than more dispersed patterns and at the same time provide essentially the same level of service as completely dispersed patterns. It is important to analyze such solutions because they in fact may be superior. Further, approaches that do handle station location and ambulance allocation as a two step decision making process cannot capture the dependent nature between the two decisions. Both processes must be resolved simultaneously in the same model. The objective of this paper is to present a hybrid covering model which simultaneously locates stations
164
GEOFFREY BIANCHIand
and ahocates ambulances. The model is based in part on the Facility Location and Equipment Emplacement Technique (FLEET) model that Schilling et al. [5] developed for locating fire stations and allocating equipment, as well as, on the Maximal Expected Coverage Location Problem (MEXCLP) of Daskin [14] that was developed for problems where vehicle availability needs to be considered. A general integer programming approach is discussed and computational results are given. Several tradeoff curves are presented which show that significant reductions in the number of dispatching sites (keeping the number of ambulances constant) can be made without major changes in expected coverage for the two planning problems presented in this paper. The results of this paper suggest that station location and ambulance allocation need to be modeled as simultaneous decisions and that many different robust alternatives can be generated from such a model. This type of covering model will ultimately lead to better decision making in emergency service delivery problems. BACKGROUND Many of the extensions to covering models for emergency medical services (EMS) are based on the MCLP [2]. The MCLP seeks to find the location of a fixed number of facilities on a network such that the amount of population covered (or served) within a prespecified maximal service distance (or time) is maximized. Several solution methods have been employed to solve maximal covering problems: heuristics [2], linear programming with branch and bound [2], lagrangian relaxation [15], and dual ascent [15]. Utilizing the MCLP one can generate a tradeoff curve which shows the maximum possible coverage as a function of the number of facilities located. The smallest number of facilities at which 100% coverage is achieved represents a solution of the (LSCP). MCLP solutions with less than 100% coverage utilize fewer resources than LSCP solutions. Church and ReVelle have shown that substantially high values of coverage can be achieved with significantly fewer facilities than what is required for total coverage [2]. Advancements in covering models have been developed along four major areas. The first area involves the calculation of coverage utilizing different measures such as the type of demand (e.g. coverage of the elderly population, or coverage of those areas with the highest frequencies of injuries due to assault). Utilizing different coverage measures was first modeled as a multiobjective coverage model by Schilling [ 161.Eaton et al. [8] give an excellent discussion of the use of eight different criteria used in covering for ambulance deployment in Austin. A second major area involves the expansion of the definition of covering to include varying coverage values as a function of distance. Several model forms have been developed by Church and Roberts [13]. A third major area deals with the simultaneous location of stations and equipment. Schilling et al. [5], in analyzing a fire station location problem for the City of Baltimore, combined the processes of fire station location and equipment allocation into one model called FLEET. FLEET was used to analyze a
RICHARDL. CHURCH fire station location problem where two types of equipment (primary service and special service equipment) were being positioned. This particular model is significantly different from earlier covering models in that it handled equipment and facilities simultaneously. Usually in fire protection systems, equipment is in use only a small fraction of the time and is available (not busy) when most calls for service are placed. The FLEET model was formulated on the assumption that a vehicle is essentially always available (and not busy). However, in ambulance operations, equipment can frequently be busy or not available when called. Therefore, it is important to address vehicle availability in any ambulance location model. Good service is achieved only when an available unit (currently not busy) can respond within the desired maximal service time to a call. Equipment availability has been introduced into covering problems with three different approaches. The first approach is one that is comprised of two steps. The first step utilizes a covering model to select station locations. The second step involves the use of a simulation or queuing model to allocate ambulances to the dispatching stations in order to insure coverage with a high probability. This approach was first suggested by Berlin and Liebman [9]. Although the recent ambulance location study of Eaton er al. utilized primarily the maximal covering location problem to locate-allocate ambulances, they utilized this two step process to an extent to insure that ambulance patterns could reasonably respond to calls when ambulance availability was taken into account. The second approach involves extending covering models to include the notion of backup coverage. A demand area is typically considered covered if it can be reached by at least one ambulance within the maximal service distance. Should that ambulance be busy, the demand area is not ‘actually’ covered unless a second ambulance exists that can also reach the demand within the covering distance. The second ambulance can provide ‘backup’ coverage in case the first is busy. One type of ‘backup’ coverage model would be to minimize the number of ambulances needed to cover all demand, and then locate that fixed number in such a way that all are covered once and as many as possible are covered twice (that is backup covered). This is simply a restatement of the Hierarchical Set Covering Problem of Daskin and Stern [lo]. Hogan and ReVelle [ 171 have also generated several types of backup coverage models utilizing both the location set covering problem and the maximal covering location problem. Further, Charnes and Storbeck [18] have also utilized concepts of backup coverage between two types of responding equipment. The third approach to extending coverage models to consider equipment availability is to incorporate equipment availability directly into the definition of coverage. This was accomplished by Daskin [14] in the definition of the MEXCLP. The MEXCLP objective is defined in terms that an emergency vehicle is not always available (with a given probability) when called for service. Daskin has demonstrated an application of the MEXCLP to EMS system design [l 11 utilizing the Austin, Texas planning problem that was
A
hybrid FLEET model
analyzed by Eaton et al. [8]. The MEXCLP model has been used with linear programming to optimally solve several planning problems by Daskin [l 11. Daskin has also developed a heuristic to solve the MEXCLP [ 141. In analyzing the location of ambulances by a covering model two important properties must be included. First, equipment availability must be included in the definition of coverage whenever the probability of being unavailable is significant (which is true for most urban ambulance systems). This can be approached by either the expected coverage format of Daskin or the backup coverage format of Hogan and ReVelle [17]. Second, it is important to analyze both equipment allocation and station location decisions simultaneously. For this reason, a hybrid model incorporating the concepts of MEXCLP and FLEET was developed. This model maximizes expected coverage within a maximal service distance (or time) while locating a fixed number of stations and allocating a given number of ambulances. We will designate this model as a Multiplecover, One-unit FLEET problem (MOFLEET). The multiple-cover refers to the expected coverage objective where expected coverage is a function of the number of times a demand is covered. The one-unit specifies that it utilizes only one type of unit (ambulances) compared to the original two equipment FLEET model.
THE MOFLEET MODEL
MOFLEET seeks to find the location of a fixed number of stations and a given number of ambulances on a network such that the amount of population within the service standard is maximized. The MOFLEET model utilizes a multiple coverage goal or expected coverage objective which makes it sensitive to the conditions of an ambulance not always being available when called upon. MOFLEET defined on a network of nodes and arcs can be formulated as follows: MOFLEET:
Min xX(1 i k
ST. c x,+ 2 y,a&.f, IEN
- U)U~- I a,y,
for each
i
k=l
xx,= E Xz,=P x, G Cz,
for each j
x, > 0, integer z, and
(4)
for each j
y,k = 0, 1 for each
i, j,
and k
(5) (6)
where: x, = the number of ambulances located at node j 1 if a station is located at site j z, = 1 0 if otherwise C = the maximum number of ambulances that can be located at a station
Yik =
165
1 if demand node i is not covered by k ambulances 0 if otherwise
a, = the population to be served at demand node i = the number of stations to locate E = the number of ambulances to allocate II = the probability of a randomly selected ambulance being unavailable Mp = the maximum number of ambulances necessary to provide complete coverage S, = the maximal service distance for demand i I = the set of demand nodes J = the set of facility sites N,= {jEJld,< Si} d, = the shortest distance or travel time from node i to j. P
The objective function seeks to minimize the number of people not covered within the maximal service distance, Si, where coverage is a function of the number of ambulances that can respond to a demand zone within the specified maximal service distance. The constraints of form (1) are used to determine whether an area i is covered up to h4p times; when Xj is equal to one or greater then there are one or more ambulances that can provide coverage to the demand zone and Yik = 0. If there is not a station or an ambulance in the set Ni (i.e. Zxj = 0) then demand zone i does not have an ambulance that can respond within the service criteria and Yik = 1. Constraint 2 establishes that exactly E emergency vehicles will be used and the third constraint maintains that exactly P facility locations will be used. Constraints of type (4) limit the allocation of ambulances to a site j that has a facility located at j. Constraints (5) and (6) complete the constraint set with the integer conditions on the x, y, and z variables. The MOFLEET model is essentially the generalization of the MEXCLP model with the added feature of locating stations to house emergency equipment. When E = P, then the MOFLEET model is essentially equivalent to the MEXCLP model. However, when E > P, then there will be multiple unit stations and the two models are not equivalent. The MOFLEET model is an important extension to the MEXCLP in that one should not only be concerned with how many units are being located, but also with how many dispatching points are used. The costs and ease of operation when units are clustered as compared to a completely dispersed set of units could outweigh the additional benefits of complete dispersal in many cases. In order to support this argument, one would want to generate a tradeoff of expected coverage versus the number of stations. Integer programming was utilized to obtain optimal solutions to the MOFLEET problem. To solve MOFLEET with integer programming the programming system required the specification of the Yik and Zj variables as O/l integers, and the Xj variables as continuous integers. We used the Linear INteractive Discrete Optimizer (LINDO) programming system [ 191. LINDO was executed on a VAX 1I/750 under UNIX to generate optimal integer solutions to MOFLEET. LINDO accepts program formulations in the IBM Mathematical Programming System
GEOFFREY BIANCHIand RICHARDL.
166
(MPS) format. A FORTRAN program creates the file of MPS statements that specifies the model formulation which is then read by LINDO. Additionally, LINDO permits the specification of O/l integer variables directly by marking their start and stop point in the input file. The LINDO program uses the method of branch and bound to resolve fractional decision variables into O/l integers. An alternative heuristic solution approach has also been developed as a part of this research and is described elsewhere
WI. MODELRESULTS Two test networks are used to illustrate the value of the MOFLEET modeling approach. The first spatial data set is the 55 node network of Swain [21]. The second data set is a 33 node network of census tract centroids of Austin, Texas. The Austin data contains actual calls for emergency medical service during a five month period [1 11. An important input of the MOFLEET model is the parameter u, the probability that a randomly selected ambulance is unavailable when called for service. An emergency vehicle is considered unavailable when it is serving another emergency call or the ambulance cannot be deployed (i.e. it is in repair). Daskin [11] gives an excellent discussion of the model parameter utilized in the maximum expected covering problem. ReVelle [22] has, in a recent paper, dealt with the estimation and application of the probability of spatially varying demand for designing emergency service systems within the context of the maximum expected coverage problem. The parameter 11may be thought of as the ratio of two terms, the expected EMS workload per day, w, to the number of deployed ambulances, E. As the number of ambulances, E, is increased (given a value for w), the probability that a randomly selected ambulance is unavailable is decreased. In this research, the value used for w is equal to 0.90 for the 33 node network and 1.50 for the 55 node network (the same as those used by Daskin [1 1] for comparisons). The actual value of w using the Austin data set is lower than 0.90 (approx. 0.73), therefore, the degree of ambulance unavailability is slightly overestimated. In addition, the probability of an ambulance being unavailable is the same for all units in the fleet and it is also independent of the ambulance
Table Number of units 5 5 5 5 5 4 4 4 4 3 3 3
Prob. of an unavailable vehicle 0.18 0.18 0.18 0.18 0.18 0.225 0.225 0.225 0.225 0.3 0.3 0.3
Number of locations 5 4 3 2 I 4 3 2 1 3 2 1
I.
CHURCH
locations. Since workload can vary spatially, the probability of ambulance unavailability being the same for all units over all locations represents an assumption that could be relaxed in future modeling. This relaxation can be based on ReVelle’s method to estimate spatially varying probabilities of vehicle unavailability [22]. Tables 1 and 2 present the optimal MOFLEET solutions for both the 33 node and the 55 node networks. All results in this paper were found using a maximal service distance of S = 10.0 (same as that used by Daskin (111). For each MOFLEET solution generated, the tables indicate the amount of demand covered at least once by a station-ambulance combination and the expected coverage. Computer CPU time in hours is also presented. The LINDO system (written in FORTRAN 77) was utilized with the Unix 4.3 BSD operating system. This system is relatively slow, however, one should expect substantial time saving by utilizing a more efficient branch and bound system in a better operating environment. Since operating costs were $2.5 per hour, the actual costs of generating most of the solutions were relatively small. All solutions in this study were generated for less than $350. Table 1 presents optimal solutions of MOFLEET on the 33 node Austin, Texas data set. Solutions were identified for allocating 5, 4, and 3 ambulances over a range of locations. The j-unit, 4-station solution, as well as, the 4-unit, 4-station and the 3-unit, 3-station results represent optimal solutions to the MEXCLP model. The j-unit, 4-station solution indicates that higher coverage can be achieved by locating 2 ambulances at node 25 instead of having the ambulance allocated to site 17 as indicated by the j-station, j-unit solution. By moving a unit to node 4 and thus eliminating station 11, the 3 station solution results in a loss of total expected coverage of only 2.25% as compared to the j-station solution. Similarly, the 4and 3-unit solutions indicate that the number of stations may be reduced with a minimal loss of coverage. In addition, the average CPU usage for the 5, 4, and 3 ambulance MOFLEET problems with LINDO was 0.033, 0.045, and 0.017 hours, respectively. Table 2 presents the optimal solutions found by LINDO on the 55 node network. Solutions were identified for allocating 10, 7, and 5 ambulances. This table indicates that clustered solutions (i.e. reducing
33 Node-MOFLEEToetimal solutions
Locations
Number of demands covered at least once
Percent covered at least once
Expected number of demands covered
Percent expected coverage
CPU time (hr)
4, II. 17.25.31 4, 11,25’,31 4’. 25’. 31 11’,25’ 4J 4,11,25,31 4’. 25, 31 llz,25’ 4’ 4, II,25 1 I’, 25 4’
4249 4249 4141 3896 2989 4249 4141 3896 2989 4042 3896 2989
100.00 100.00 97.46 91.69 70.35 100.00 97.46 91.69 70.35 95.13 91.69 70.35
4111.87 4111.87 4016.31 3848.63 2988.43 3914.79 3829.16 3716.50 2981.34 3460.03 3357.35 2908.30
96.77 96.79 94.52 90.58 70.33 92.13 90.12 87.47 70.17 El .43 79.01 68.45
0.036 0.030 0.051 0.021 0.027 0.022 0.104 0.024 0.028 0.012 0.022 0.018
A hybrid FLEET Table 2. 55 Node-MOFLEET
Number of units IO
Prob. of an unavailable vehicle
Number of locations
0.15
IO
0.15
9
0.15
8
0.15 0.15 0.2143 0.2143 0.2143 0.2143 0.2143 0.2143 0.3 0.3 0.3 0.3 0.3
6 6 4
4 3 2
Locations 3.7, IO, 13, 17, 20 22, 27, 36, 55 32, IO, 13, 17, 20 22.27.36. 55 17i, 2d, 27.34.36 422, 53,55 172. 21. 27,34. 36’. 422, 55 2’, 17’. 2 I, 27.36’. 55 4. IO, 13. 17,275 30,36 42, 10. 17.27.34, 36 IO, 17’. 36, 38.42’ 17*, 36,38,42j 22I, 362, 42’ 22’, 42’ 7, IO. 22,34.42 3,223 36, 422 4’. 22, 36 22=, 42’ 45
167
mddel
optimal
solutions
Number of demands covered at least once
Percent covered at least once
Expected number of demands covered
Percent expected coverage
CPU time (hr)t
6350
99.22
6177.38
96.52
0.05
6350
99.22
6169.10
96.39
0.81
6380
99.69
6155.15
96.17
13.45
6300
98.44
6131.91
95.81
47.00
6250 6190 6140 6040 5810 5480 5020 5600 5570 5480 5020 4250
97.66 96.72 95.94 94.38 90.78 85.62 78.44 87.50 87.03 85.62 78.44 66.41
6073.48 5771.05 5739.06 5665.16 5563.21 5386. I8 5006.44 5118.97 5093.26 5014.85 4851.52 4239.66
94.89 90.17 89.67 88.52 86.92 84.16 78.22 79.98 79.58 78.36 75.80 66.24
0.02 I.19 7.06 23.29 34.24 4.65 0.02 0.16 1.50 2.07 0.27
l
lIP halted prior to completion. tVAX I l/750 utilizing LINDO (written in FORTRAN 77) under Unix 4.3 BSD. This is a relatively slow system. No special procedures were taken to reduce time. One can expect substantial time savings by utilizing a more efficient branch and bound code and an efficient machine. Operating costs are approximately $2.5 per hour.
the number of stations) can be generated by MOFLEET without substantial reductions in coverage. For example, the optimal placement of 10 units at 6 locations covers only 1.64% less than the optimal placement of 10 units at 10 locations. Another important item of information in this table is the computer processing time. A considerable amount of time was spent to identify optimal integer solutions. The average CPU usage for the set of 10, 7, and 5 units allocated was 15.33, 11.74, and 0.80 hr, respectively. The tables show the reduction in coverage as the deployment pattern for ambulances goes from a dispersed pattern to a clustered pattern. These coverage values indicate that high levels of coverage can be maintained while at the same time achieving a significant degree of system centralization. By reducing the quantity and spatial distribution of system components, real-time system monitoring and control can be performed with greater ease. Figure 1 shows the deployment pattern generated by the LINDO system on the S-node network for allocating 10 units over IO facility sites. The circular regions represent the theoretical coverage areas of the ambulances located at the designated stations. This pattern represents the optimal solution to the MEXCLP because the solution is allowed to be dispersed (i.e. there is no constraint or penalty for adding a station). This solution also represents the only MEXCLP deployment pattern on the S-node network which provides at least 95% coverage. MOFLEET, however, identifies solutions in which 10 units may be allocated to 9, 8, or 7 stations while maintaining at least 95% c’overage. In fact, MOFLEET identified a 10 unit, 6 facility solution illustrated in Fig. 2 which is short of 95% coverage by only 0.11%. It is also important tlo recognize that Fig. 2 demonstrates the value of the MOFLEET modeling approach. That is that high levels of coverage can be maintained while
reducing the number of stations to be operated and managed on a daily basis. The reduction in the number of required stations to meet demand needs can provide for further ease and efficiency of system management. Figure 3 presents two solutions to the MOFLEET mode1 on the 33 node Austin. Texas network. The 4-station and 5-ambulance solution is represented graphically by squares on the map. This solution also represents the optimal pattern of distributing 5 ambulances with no consideration for station location (i.e. optimal MEXCLP solution). By reducing the number of stations by one. a more centralized distribution of stations and ambulances can be realized as illustrated by the solution in Fig. 3 represented by triangles. In this case, a station is eliminated from census tract 11 and the ambulance is reallocated to tract 4. The loss of the station results in a loss of total expected coverage of only 2.27%. Although coverage falls below 95% to 94.52%, the solution demonstrates that the number of stations may be reduced without seriously reducing the emergency system performance. The MOFLEET problem can be defined for any maxima1 service distance and a number of facilities P to be located. By solving the model over a range of facilities, a tradeoff curve can be described which illustrates the effectiveness of increasing or decreasing the number of facilities located. Figure 4 presents a set of cost-effectiveness curves that describe demand covered as a function of the number of stations located, P, when E ambulances are allocated. The results of LINDO in Table 1 for the 33 node data are used to outline the tradeoff curves of Fig. 4a. This family of curves can be used to find the best coverage of demand attainable given a maximum number of available ambulances and stations. Similarly, LINDO results for the 55 node data set are used to describe the tradeoff curves of
GEOFFREYBIANCHIand RICHARDL.
CHURCH
Fig. 1. MOFLEET solution for 55 node problem with Fig. 4h. The ‘flatness’ of the curves shown in Fig. 4b emphasizes the importance of the MOFLEET model. For instance, in the case of allocating 10 units over a set of station locations, the MOFLEET model results indicate a small decrease in coverage with a reduction in the number of stations located. For example a 40% reduction in stations (i.e. 10 down to
10
stations.
6) resulted in only a 1.64% loss of total expected coverage. SUMMARY
We have presented a hybrid model called MOFLEET which simultaneously locates stations and
A hybrid FLEET model
169
with 2 Units m
63
With 2 Units
4D
Fig. 2. MOFLEET solution for 55 node problem with 6 stations
allocates equipment to maximize expected coverage. The MOFLEET model provides decision makers and managers of EMS systems with a powerful locational analysis technique. The sample results represented by the 55 node and 33 node configuration maps and the
tradeoff curves indicated that high levels of coverage can be maintained while reducing the number of system facilities that require operation and management. Therefore, location decisions based on management and operating costs can be analyzed with
170
_
GEOFFREY
BIANCHIand RICHARDL. CHURCH
_=
./
n
=3 site/5 ambulance MOFLEET configuration
lJ
=4 site/5 ambulance MOFLEET configuration number in symbol represents number of ambulances allocated at that location.
Fig. 3. Comparison
of 3- and 4-station MOFLEET solutions for Austin. Texas.
simulation models are used to deal with the changes in the spatial distribution of EMS demand over time. However, by extending MOFLEET into a multiple period planning model one can address issues associated with crew allocation. For example, the avail-
respect to a desired number of locations and operating units at the locations. The MOFLEET problem can be easily reformulated to handle equipment and crew allocation in a multiple time period framework. In general,
(4
100 8 E ?i 0 B zl%
3 S
(b)
100
5-units 90
p
90
60
z 6
60
70
z tj
70
60
B ;
60
lo-units
50
50 1
2
3
4
Number of Stations
5
6
1
2
3
4
5
6
7
6
9
Number of Stations
Fig. 4. (a) Cost-effectiveness curves for the 33 node data set with service distance = 10.0. (b) Costeffectiveness curves for the 55 node data set with service distance = 10.0.
10
A hybrid FLEET model
ability of crews during peak versus off-peak times and the balancing of work loads can be addressed within the multiple time period context. In addition, one can optimize tradeoffs between demand covered versus the number of stations, ambulances, and emergency service crews utilized.
10.
11. 12.
REFERENCES
1. Toregas C. and ReVelle C. Optimal location under time or distance constraints. Socio-Econ. Plating Sci. 28, 133, 1972. 2. Church R. L. and ReVelle C. The maximal covering location problem. Paper Regl Sci. Ass. 32, 101, 1974. 3. ReVelle C., Bigman D., Schilling D., Cohon J. and Church R. L. Facility location: a review of context-free and EMS models. kh Serv. Res. 12, 129, 1977. 4. Plane D. R. and Hendrick F. E. Mathematical nrogramming and the location of fire companies for- the Denver fire department. Mgmr Sci. Rep. Ser. 74-79, 1974. 5. Schilling D., Elzinga D. J., Cohon J., Church R. L. and ReVelle C. The TEAM/FLEET models for simultaneous facility and equipment siting. Transport. Sci. 13, 163, 1979. 6. Schilling D., ReVelle C., Cohon J. and Elzinga D. Some models for fire protection locational decisions. Eur. J. Opns Res. 5, 1-7, 1980. 7. Kolesar P. and Walker W. E. An algorithm for the dynamic relocation of fire companies. Opns Res. 22,249, 1974. 8. Eaton D., Daskin M. S., Simmons D., Bulloch B. and Jansma G. Determining emergency medical service vehicle deployment in Austin, Texas. Interfaces 15, 96, 1985. 9. Berlin G. N. and Liebman J. C. Mathematical analysis
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