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Review, extension and prediction in emergency service siting models
58
European Journal of Operational Research 40 (1989) 58-69 North-Holland
Theory and Methodology
Review, extension and prediction in emergency service siting models Charles R e V E L L E
Program in Systems Analysis and Economics for Public Decision Making, Department of Geography and Environmental Engineering. The Johns Hopkins University, Baltimore, MD 21218, USA
Abstract: This is a review of a selected set of location papers and a research agenda. The location models reviewed fall into three categories: (1) basic deterministic covering models, (2) deterministic models which consider the value of additional covering servers, and (3) probabilistic models which allow randomness in server availability. Contributions in each of these categories are compared in a sequence which highlights the intellectual evolution of the models. Several new models are structured using the new formulation tools that are described. Several additional models whose development is predicted are described verbally.
Keywords: Location, reliability, public service, linear programming, integer programming
1. Introduction Since the 1960s, the field of location has moved in many directions, providing to researchers a vast set of theoretical and practical problems. These problems furnish the mathematician with theorems to prove, offer the mathematical programmer the opportunity to develop algorithms, and provide planners the capability to examine alternative solutions to practical problems. Because of the wide opportunities location problems present, the field of location or siting has attracted an array of researchers from numerous disciplines. This paper is a review of a segment of the location field, namely, urban public facility location. An attempt will be made here at synthesis of a set of models in this area. As a consequence, no claim is offered for a comprehensive treatment of urban public facility models. The reader interested in a fairly comprehensive review of urban public
Received September 1987; revised May 1988
facility location is referred to ReVelle (1987). Here, a small subset of the literature is selected for review, for integration, for extension, and for prediction. In short, this is both a review and the laying out of a research agenda with the first few steps on that agenda being undertaken in this paper. All of the models we consider here are covering models. Their goal is to provide 'coverage' to demand areas. A demand area is considered to be covered if a facility or vehicle is available to serve the demand area within some mandated distance or time standard. That is, coverage is achieved if the server at the facility can reach the demand area within the standard, or conversely, if individuals from the demand area can reach the facility within the time standard. The proximate presence of the server is the criterion which we use to measure whether or not a demand area is covered. It is surprising to note that such a simple concept can generate so much literature and so many models, but the richness of the covering notion will still not be fully fathomed here.
0377-2217/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
C. ReVelle / Emergency service siting models In Section 2, we review three basic deterministic location covering models and their uses; we also note one basic model that has yet to be structured. In Section 3, we show how criticism of these models has led to a second generation of siting models, as yet incomplete and awaiting further developments and extensions. In Section 4, this second generation is criticized in turn and a third generation of models, correcting the deficiencies of the second is discussed. This third generation of location models is still in a formative stage; much growth is anticipated. Indeed, in Section 5, we begin to fill in some of the blank spots in the first and second generation models and predict research directions in the third generation.
2. Basic deterministic covering models The location set covering problem (LSCP) seeks to position the least number of facilities needed to cover all points of demand within S distance or time units. The LSCP was an early statement of the emergency facilities location problem in mathematical programming form. It had also been stated in the early 1970s in virtually the same form as that above by a health planner with no knowledge of mathematical programming, suggesting that the problem statement as offered by mathematical programmers closely conformed to real needs. In mathematics, the problem is stated as Minimize
z = ~ xj jEJ
subject to
E
Xj>/]
Vi E I ,
jEN,
xj=0, 1
v J,
where index and set of demands; index and set of eligible sites for facilities; 1, 0; 1 if a facility placed at j; 0, otherwise; the shortest distance (or time) from site j to demand point i; = the m a x i m u m distance (or time) that a demand point can be from its nearest facility;
i,I= j, J=
X.I dji S
59
and N,
= ( j [ dji ~< S } = the set of facility sites eligi-
ble to serve demand point i, by virtue of being within S of i. The objective is minimization of the number of facilities required. The constraints require coverage of each demand point i; such coverage is achieved by the presence of at least one facility within S distance units of travel from the demand point i. The problem can be solved easily as a linear programming problem in which the integer variables are simply required to be non-negative. An occasional cut constraint (about 5% of runs) may be required in the circumstance where a non-integer number of facilities results from solution of the linear programming problem. See Toregas et al. (1971) for details. The problem can also be solved by the method of reductions. See Toregas and ReVelle (1973). A derivative model of the LSCP is the p-center problem. The p-center problem seeks the locations of the p facilities which minimizes the m a x i m u m distance separating any demand point and its nearest facility. A solution is obtained by solving a sequence of location set covering problems in which the m a x i m u m distance is successively reduced by the smallest unit of measurement of distance. The total of facilities required may be p in number in each of a set of successive solutions as distance is decreased. When a value of S is reached at which the number of facilities increases to p + 1, the preceding value of distance is the smallest m a x i m u m distance for which p facilities are feasible. Any smaller m a x i m u m distance requires a greater number of facilities. This method of solution is described by Minieka (1970) and by Christofides and Viola (1971). The location set covering problem required coverage of all demand points, no matter how low their population, how distant, or how great their need for service. Recognizing that the resources required could be excessive to cover all points of demand, no matter how small or remote, Church and ReVelle (1974) and White and Case (1974) framed a new problem that did not required the coverage of all nodes. Church and ReVelle called it the maximal covering location problem (MCLP); White and Case referred to it as the partial covering problem. We will use the former designation here.
60
C. ReVelle / Emergency service siting models
The problem begins with a statement of sources, in this case p facilities, and asks question: " H o w can the p facilities be best ployed where the goal is coverage within S of m a x i m u m possible population within S ? " In mathematics, the problem is stated as Maximize
rethe dethe
z = ~ aiy i i,E l
subject to
y,<~ ~_, xj
Viii,
tance standards. Other objectives included in a multi-objective formulation of the FLEET model were m a x i m u m coverage of fire frequency, maxim u m coverage of property value, m a x i m u m coverage of population at risk (as measured by the product of fire frequency and population for each demand point), m a x i m u m coverage of property at risk (formed in a similar fashion), and so on. The m a x i m u m population coverage version can be stated mathematically as z = ~ aiy ~
Maximize
i~l
j~N i
2xj=p,
subject to
j~J
xj, yi=O, 1 qi, j, where additional notation is a i = the population at demand node i; y ~ = l , 0; it is 1 if demand i is covered by a facility within N,.; 0, otherwise; and p = the number of facilities that can be sited. All other notation is as defined earlier. Relaxed linear programming supplemented by occasional use of branch and bound was used by Church and ReVelle to provide solutions to this problem. Church and ReVelle also suggested heuristic procedures for solution as did White and Case. The most informative use of the problem statement is to develop a tradeoff curve between population covered and the number of facilities utilized. The maximal covering idea was generalized for fire protection by Schilling et al. (1979). Their model accounts for coverage by two different types of service; in addition, both the servers and the facilities which housed the servers were to be sited simultaneously. Of several model types, the most general formulation was that known as the FLEET model (for facility location, equipment emplacement technique). The FLEET model sought the locations of a limited number of engine companies (the p u m p e r brigades) and truck companies (ladder brigades) as well as the fire stations that housed them. The goal of the FLEET model was coverage of the m a x i m u m number of people by both an engine company sited within an engine c o m p a n y distance standard and a truck c o m p a n y sited within the truck c o m p a n y distance standard. That is, coverage required the simultaneous siting of both types of service within their respective dis-
j ~ NiE
Yi <~ E
x7
Viii,
v;
J,
j~ NJ
xf
V
Ex~+Eg=p E+T, ExS=p s, y,, V , xf,
= o, 1
Vi, j ,
where new notation is x?
= 1, 0; 1 if an engine c o m p a n y positioned in
x5
=
#
=
N/E =
N/T
pE+T= pS =
a fire house at site j; 0 otherwise; 1, 0; 1 if a truck c o m p a n y positioned in a fire house at site j; 0 otherwise; 1, 0; 1 if a fire station established at site j; 0 otherwise; ( J I dji ~< E ) = the sites j which can provide engine c o m p a n y coverage to point i by virtue of being within the engine company distance standard E; { j ldji ~< T} = the sites j which can provide truck c o m p a n y coverage to point i by virtue of being within the truck c o m p a n y distance standard T; number of fire companies; and number of fire stations.
In this formulation, the first two constraint types define coverage as attainable only if both one or more engine companies is sited within the engine distance standard and one or more truck companies is sited within the truck distance stan-
C. Re gelle / Emergency service siting models
dard. The third and fourth constraint types allow companies to site only at nodes that have been allocated a fire station. The fifth constraint limits the total number of companies or brigades, but does not prescribe the proportion of resources placed in engine companies as opposed to truck companies. This constraint reflects the fact that the major cost of fire companies is personnel and not equipment. The last constraint limits the total number of fire houses. Schilling et al. found relaxed linear programming plus occasional branch and bound a viable solution technique. One additional model from the fire protection arena belongs in this brief description of deterministic models. To describe the model, it is first necessary to define the term standard response. In most major US cities, the standard response to a call for fire protection service is three engines and two trucks. This brings the requisite number of firefighters to the scene. The engine companies are to be available for dispatch within an engine distance and the truck companies within a truck distance. The problem then, is one of siting engine companies, truck companies and fire stations in a way that provides the m a x i m u m population with the standard response coverage. The model is not included in this section, because it is new; it requires tools to write the model that will not be available until we discuss the methods for redundant and backup coverage to be introduced in Section 3. The models discussed above, although they originated with a set covering model applied to emergency services, have been used relatively frequently for ordinary services as well. The set covering model has been used to allocate bus stops (Gleason, 1975). The maximal covering model has been used to site clinics and other health facilities (Eaton et al., 1981). A distinctive relative of the FLEET model in which services nest hierarchically has been used to site health facilities with differing but overlapping service characteristics (Moore and ReVelle, 1982). Indeed, the use of these deterministic models for ordinary services may be equally appropriate as their use for emergency services even though emergency service was the motivation for the creation of these models. The reason for the appropriateness for use for siting ordinary facilities is that people generally go to these ordinary facili-
61
ties and they can be built and staffed large enough to deal with the expected clientele. In contrast, when these models are used for emergency services, facilities are sited only on the basis of geographical coverage and not on the basis of availability. One has to assume that the facilities are continuously available in order to justify their use for emergency services. In uncongested, low demand settings, this assumption is not unreasonable, but in congested systems where frequent calls keep an emergency vehicle on the streets sometimes up to 80-90% of the time, this assumption simply cannot be justified. Congestion in service systems, as seen in a frequent lack of availability of a server within the distance or time standard, motivated the creation of a new set of models which emphasized additional coverage beyond the first. These models, mostly developed in the last half dozen years, will be discussed next.
3. Deterministic models which focus on additional coverage
Several investigators recognized the possibility that in congested systems the first server (and possibly the only server in a particular coverage area) might not be available at the time of a call's arrival. These investigators focussed on the importance of providing additional servers beyond the first within the coverage region of a demand area. Berlin (1972) and Daskin and Stern (1981) structured a model which utilized a number of facilities greater than or equal to the minimum number required by the location set covering problem. They focussed simply on the total of redundant or additional coverers in the system beyond the first cover required by the location set covering problem. When the number of facilities was equal to the minimum, this amounted to finding the alternate optimum to the LSCP (there are many alternate optima to the LSCP) which produced the greatest number of covering servers, summed over all coverage sets. In mathematical terms: Maximize
z= ~R, iEI
C ReVelle / Emergency service siting models
62
subject to
ViEI,
E xj-Ri=l jG IV
Z
j=p,
jEJ
xj=O, 1
W,
Ri are integers
Vi,
where R~ are the number of servers beyond the first in the set N,, and all other terms and variables are as described earlier. If p is the minimum number required to cover, the solution obtained is the alternate optimum which maximizes the sum of redundant coverages. If the number of facilities is ranged from the minimum required to cover p up to some larger number, the tradeoff curve arrays redundant coverage against total number of servers. Again, relaxed linear programming provides frequent all zero-one solutions. Although this formulation was an improvement on the simple location set covering problem, it still required coverage of each and every demand node. Further, it did not in any way account for the greater importance of having additional eligible coverers for areas of high demand. The latter defect could have led to solutions which clustered servers around low population demand areas simply because they were easy to cover. To correct for the lack of attention to the magnitude of demand, Benedict (1983) and Eaton et al. (1986), both restructured the objective of the model of Berlin and of Daskin and Stern. For a given number of servers, p, in the system, they operated on the objective Maximize
emphasize the importance of at least one redundant coverer for each demand node, Hogan and ReVelle (1986) modified the location set covering problem to the following form, which they referred to as BAcoP1 (for Backup Coverage Problem 1):
z = ~ a~R,,
where a~ is, again, the population or demand at node i. This objective maximizes the product of population and the number of servers which provide additional cover to the population. Trading off p against the product of population and redundant servers provides a sense of the redundant coverage attainable by deploying additional servers. Solution method is not altered by this simple modification. True, this modification makes additional coverers more likely for high demand nodes, but redundant coverers can pile up for some demand areas and not for others. Some demand nodes may be left with only a first coverer and no backup. To
Maximize
z = ~ aiu i i~I
subject to
ui<~ Y[ x j - 1 )oN,
ViEI,
E xj=p,
j~J
Xj, Ui = O, 1
W, j,
where ut = (1, 0); it is one if the total of facilities in N, is greater than or equal to two (one or more backup servers), and zero otherwise. All other notation is as defined earlier. This model still requires first coverage of every node, an assumption we will relax in a moment. The formulation seeks to maximize the population that has at least one redundant coverer. It de-emphasizes multiple redundant coverers of a n o d e - - a l t h o u g h these can still occur, but not at the expense of first redundant coverage. As before, relaxed linear programming provides frequent zero-one solutions. Hogan and ReVelle (1986) extend BAcoP1 to the situation in which first coverage is not required, but is an objective as in the maximal covering location problem. In addition, second coverage (or higher) is also a goal. The two objectives are to maximize the population achieving first coverage, and to maximize the population achieving second (or higher) coverage. In mathematics, their BACOP2 model is Maximize
zi = E aiYi, i~l Z2 :
E aiui iEl
subject to ui+yi~
E Xj j~N,
Viii,
E xj=p,
j~J
ui <~Yi V i i i , xg, ui, y , = 0 , 1
Vi, j,
C. ReVelle / Emergency service siting models
where all notation is as previously defined. The first constraint type, in combination with the (0, 1) restrictions, says that the sum of first and second coverage is bounded by the number of coverers in N, or by two, whichever is smaller. The third constraint type says that second coverage cannot be counted until first coverage achieved. Thus, if the number of coverers in N, is one, first coverage will be counted, but second coverage will not. Hogan and ReVelle, in an uncontrived example, found that the extent of second coverage could be much improved with only marginal reductions in first coverage from its maximum level. Again, relaxed linear programming provides a straightforward and rapid solution method for the (0, 1) programming problem. Occasional use of branch and bound was found necessary. The authors point out that the BACOP models can be extended to count third and subsequent coverage variables as well. Still another model considers redundant coverage and in a strikingly different way than any of the preceding models. Storbeck (1982), borrowing a page from goal programming, defines the difference between the number of coverers in AT, and the value one as the difference between two nonnegative variables, which we call k + and k 7 . His model, in multi-objective notation, is Maximize
z1= £
ai k + ,
i~l
z 2 = -- y " a i k 7 i~I
subject to k i ~ -- k i
xa-
=
1
Viii,
j~N, xI=P, j~J
x j, k~ = 0 ,
1
k [ = integers
Vi, j,
Vi,
where k + = the coverage in excess of one for node i;
k , = the coverage less than one for node i. Objective 1 reflects maximization of the population receiving redundant coverage. Objective 2 suggests minimization of the population which is 'undercovered' Storbeck's formulation is ex-
63
tremely versatile and one of the cleverest of the additional coverer models. Missing from this section is a model which focuses on providing backup or additional coverage for fire protection systems, where coverage is achieved only by the proximate stationing of t w o d i f f e r e n t k i n d s of servers, engine companies and truck companies. This, along with other new models, will be presented in Section 5. These formulations which focus on additional coverage beyond the first are meant to apply to congested systems, and they represent an attempt to treat the probabilistic nature of service availability with deterministic models. These models have not been tested in a simulation that we know of, but certainly they add an appropriate new dimension to the basic deterministic models that preceded them. Surprisingly, they also pave the way for extensions to both the basic deterministic models they evolved from and to the more recently developed probabilistic models. We shall describe the probabilistic models next.
4. ProbabUistic models consider randomness vehicle availability,
in
The models developed in Section 3 were intended to account for the possibility of servers being busy by stationing within the time standard additional units beyond the first. While these were not probabilistic models, they did provide a new direction to emergency location modelling and new techniques to the analysis. The models are not dismissed as a passing phase but are, in fact, a robust contribution to emergency services siting. Although the deterministic additional coverage models just described are of value, it was a natural step for researchers to investigate probabilistic siting models next. By probabilistic siting model, we mean here a formulation in which the randomness is in the availability of the server. It is possible to approach models in which randomness exists in both server availability and in time of travel, time of service, etc., but for this review we are considering only randomness in availability. We choose this research line because the impact of the randomness in travel and service time is far less than the impact of randomness on availability.
C. ReVelle / Emergency service siting models
64
Surprisingly, the first probabilistic emergency model was constructed a dozen years ago, long before redundant coverage modelling had gotten off the ground. C h a p m a n and White (1974) proposed a probabilistic location set covering model in which servers were not always available. A simplified version of their model (Daskin, 1983) uses a single across-the-board estimate of server busy probability; we call that busy probability q. We can then write an expression for the probability that one or more servers in N~ is free to accept the next arriving call from demand node i. That probability is here constrained to be greater than or equal to a, a system-wide level of reliability that must be met for all nodes.
structured as follows: n/
Maximize
E ( 1 - q)(q)k
z= E i~l
laiYik
k=l
subject to n,
Z Yik ~
k=l
Z
Xj
Viii,
j~U,
2xj=p, j~J
Yik = O, 1 Vi, k, xj are integers >~ 0
Vj,
where 1 -- qZ:~ulX: >1 a
Vi ~ I.
Yik = 1, 0; it is 1 if node i is covered by at least k
The busy probability q, raised to the number of servers in N~, is the probability that a// servers in N~ are busy. This value subtracted from one is the probability that one or more servers in ~ is free, and it is this quantity that is constrained to be greater than or equal to a. This constraint has a simple linear equivalent, namely, E
xj>
l o g 0 - a) ]
l~g q
Viii,
j~N,
where [u] means the smallest integer greater than or equal to u. Appending an objective of minimizing the number of servers that are required in the system completes the formulation of the probabilistic location set covering problem. It is of interest that the variable xj can now take on any positive integer value; that is, multiple servers can be stationed at a single site. The C h a p m a n and White model was not entirely so simple as this, but did attempt to prespecify busy fractions in different sectors of the system and proposed updates of busy fraction using simulation output. Daskin (1983) recognizing the utility of the probabilistic approach of Chapman and White, incorporated the idea of the stable busy fraction into a formulation that maximized the expected value of population coverage given that p facilities are to be placed on the network. His model is one type of a probabilistic analog of the maximal covering location problem of Church and ReVelle and of White and Case. Daskin's m a x i m u m expected covering location problem (MEXCLP) is
facilities; it is 0, otherwise; x: = the number of facilities at site j (an integer >/0); and n, = number of facility sites in N,. The constraint for each demand point i says that the sum of the indices of coverage, the Yik, can rise only to the number of facilities sited within N~. The k-th component of the objective for node i, while the demand at i is absent, is ( 1 q)qk-~yik. If Yik equals one, then the probability that the demand at node i will be assigned to its k-th coverer is ( 1 - q)qk-1. Summed over all k, the total of components becomes the probability that demand i is covered. Multiplied by the dem a n d at i, the product formed is the demand times the probability of the cover. Summed over all demands, the objective becomes finally the expected demand covered. The BACOP models of H o g a n and ReVelle are very similar to this formulation. Separation of the coverage objective into separate objectives for first cover, second cover, etc., and removal of the probability weights makes the objectives look precisely like those of Hogan and ReVelle. In the Daskin model, the Y~k variables replace y~, u~, vi, etc. in Hogan and ReVelle. Finally the 'can't be covered unless' constraints such as u~ ~
C ReVelle / Emergency service siting models
relaxed linear program, with relatively high confidence of all integer solutions resulting--as Hogan and ReVelle showed with their more constrained model. The expected coverage model of Daskin preceded the Hogan and ReVelle backup models by several years, but the parallels were not noted until a presentation of the backup models in 1984 at the International Symposium on Location Decisions (Boston and Martha's Vineyard, MA). A comprehensive review of backup models is presented in Daskin et al. (1988). The original probabilistic set covering problem model using one estimate of busy fraction for the system, for all its elegance, produces results that are extremely sensitive to q. A minor change in q that causes the bracketed term to go from 0.99 to 1.01 will double the right hand sides of the above reliability constraints, and as a consequence, often double the number of facilities. This doubling of required servers is a result of the assumption of a single system wide busy fraction estimate. As a consequence, it becomes important to pursue different estimates of the busy fractions in different sectors of the planning region. The last in the sequence of probabilistic models is an enhancement of the probabilistic location set covering model of Chapman and White. ReVelle and Hogan (1986), embracing the concept of the reliability constraint, incorporated in the formulation prior knowledge of the components of the busy fraction which is used in the reliability constraint. They thereby created a reliability constraint which had built into it an explicit bound on the busy fraction. This bound guaranteed a value of the busy fraction that would support the reliability constraint. ReVelle and Hogan first define an average busy fraction for servers in the same sector of the planning area for which the reliability constraint is to be written. Let us say that one of the reliability constraints is being written for demand point i. The vehicles that can provide coverage for point i must be in N,, defined as the set of eligible coverers. The average busy fraction of these vehicles is -t " E k ~ M, f k qi
-
-
24Ej~ N Xj '
where } = the average duration of a call (hours);
65
fk = frequency of calls at demand node k (calls/ day); and 34,= the set of demand nodes within S of node i. The numerator is the daily hours of service needed in the zone around node i. The denominator is the daily hours of service available in that same zone. Hence, the ratio provides an estimate of the busy fraction in the zone within which the reliability constraint is written. The actual value of this busy fraction estimate is not calculable until the number of servers allocated to N, is known. By defining
kGM,
we can write
qi - y~j ~ x, x / ' which will allow us a nearer final constraint. If the original reliability constraint is rewritten with qi replacing q, we have
1 - q~ .... ,~, >~ a
Vi ~ l,
or, replacing qi with its estimator from above,
1-
/ Y'. x,
~,~.
The above expression has no exact inverse, no analytical solution for the number of vehicles in N,. It does, however, have a numerical inverse. That is, the number of vehicles in N; must be greater than or equal to the smallest integer which satisfies the above non-linear reliability constraint. More formally, we can write for demand point i, the following linear equivalent: y" xi>~b,, j ~ N,
where b, is the smallest positive integer satisfying
1 - ( i /Fb , )
h'>~,,
the only determinants of b, being F, and a. The value of b i is a number which simultaneously satisfies the reliability requirement a n d provides a busy fraction qi whose value supports that satisfaction.
C. ReVelle / Emergency service siting models
66
The probabilistic location set covering problem (PLSCP), as enhanced by ReVelle and Hogan with local busy fractions can thus be summarized as Minimize
z = ~ Xg j~J
suggested; (3) a frankly r a n d o m environment where the notion of backup coverage is replaced by a concern for the reliability of service presence. In the following section we will partly fill in the several blank spots we have left in this review and indicate where new developments are most likely.
subject to Vi,
j~N~ xj are integers, where bi is the smallest integer satisfying
1 - (F,/bi)b'>_- a. Two more models can be derived from this simple model. These are: (1) the a-reliable p-center problem, an analog to the original p-center problem (Mineika, 1970); and (2) the m a x i m u m reliability location problem. The a-reliable p-center problem seeks the positions of p facilities in the system which minimizes the m a x i m u m time within which service is available with a reliability. In other words, given p facilities to site we are to find the smallest maxim u m time within which service can be available a fraction of the time. Example problems and a tradeoff curve of m a x i m u m distance versus number of facilities is provided in ReVelle and Hogan (1987). The maximum reliability location problem seeks to position p facilities in such a way that the probability of service actually being available within a stated distance or time is as large as possible. A simple technique is derived in ReVelle and Hogan (1987) which utilizes solutions to the probabilistic location set covering problem to generate solutions to the m a x i m u m reliability location problem. A tradeoff curve between reliability and the number of servers is provided there, as well. Both the a-reliable p-center problem and the maxi m u m reliability location problem can be formulated and solved using the local estimation procedure for busy fractions. Both involve solving a sequence of (0, 1)-programming problems using relaxed linear programming and occasional branch and bound. This completes the review of the basic siting models that have been created to date in: (1) the deterministic environment; (2) a congested environment where concern for backup coverage is
5. Filling in the missing models and making predictions on models yet to come In Section 2, we indicated that a certain basic deterministic model had never been formulated. We call this model, which is from the fire protection area, the maximal covering standard response problem. The first coverage of a demand area, which is the focus of the FLEET model, is covered only by the first due company. The standard response to a fire in most major cities is three engine companies and two truck companies. The engine companies are to be within an engine standard response distance and the truck companies from within a truck standard response distance. Our goal is to maximize the population (or calls) which have standard response coverage; i.e., three engine companies positioned within their distance standard and two truck companies deployed within their distance standard, given limited resources in companies and fire stations. With the tools provided by the models which dealt with additional covering, we can now structure the maximal covering standard response problem. One important feature of the structure to be presented is that we should avoid requiring any level of coverage for a demand area in order to be true to the objective statement. To structure the problem, we utilize the backup coverage concepts of Hogan and ReVelle: Maximize
z = E aiwi i~1
subject to W i q - Y 7 <~
E
Viii,
XT
j ~ Ni TSR
E x? Vi l, j ~ N/ESR
UiE<~yiE,
i~I,
Xff<~X s
Vj~J,
67
C. ReVelle / Emergency service siting models
wi<~u~,
i~l,
w,~yT,
iEl,
xf<~x s
Vj~J,
and backup coverage are defined by the simulta-
where newly utilized variables and parameters are:
neous presence of both types of fire vehicles, engines and trucks. First coverage is the stationing of at least one engine company and one truck company within their respective distance standards. Second coverage is achieved by the deployment of at least two engine companies and at least two truck companies within the appropriate distance standards. In mathematics, the model, which we call FLEET BACOP, is written:
yl I"
Maximize
£
X/"E"}- E
jcJ
XT = p E + T ,
j~d
S, xS.1 = p S j~J
all variables are 0, 1,
yi E
=
Wt
Nt TSR :
Nc ESR :
(1, 0); it is 1 if point i is covered by at least one truck company within the truck standard response distance; it is 0 otherwise; (1, 0); it is 1 if point i is covered by at least one engine company within the engine company standard response distance; it is 0 otherwise; (1, 0); it is 1 if i is covered by at least two engine companies; it is 0 otherwise; (1,0); it is 1 if i is covered by two truck companies and three engine companies within their respective standard response distances; it is 0 otherwise; the set of sites from which trucks can provide the standard response to node i, which sites are within the truck standard response (TSR) distance; the set of sites from which engine companies can provide the standard response to node i, which sites are within the engine standard response distance.
All other terms are as defined previously. The formulation is fairly large but workable, and linear programming, supplemented by occasional branch and bound, should provide a solution method. Another missing formulation was listed in Section 3. Although conceptually related to the model offered immediately above, this model more formally belongs in the category of models which focus on additional coverage, primarily because the model is an analyst's idea and not a notion which arose in the fire protection sector. The model, again, is in the fire protection arena. It is analogous to the BACOP2 model of Hogan and ReVelle in that is arrays first coverage of population against backup coverage of population. The difference is that in this model both first coverage
z 1 = Y'~ a,y,, i~l Z 2 ~ £ OjUt i~I
subject to U, + Y i
x~
Vi~l,
xf
ViEI,
j~Nf U,q'-Yi<~
xT
E j~Nf
W E J,
XE ~ xSJ Vj E J, ] u i<~y, V i i i , j~J
E x,
2cJ
:p s ,
j~J
all variables are 0, 1, where y, indicates coverage by at least one engine company and by at least one truck company, and u i indicates coverage by at least two engine companies and at least two truck companies. Additionally, the sets return to the form as given originally in the F L E E T model. Linear programming and branch and bound should suffice as a general solution method. This model will provide a tradeoff curve between first coverage, the presence of both an engine and a truck within their distance standards, and second coverage, the presence of at least two engines and two trucks, within their distance standards. In addition to the new models presented above, we can predict attempts at the development of some additional probabilistic siting models, models which are proposed here for the first time, but
68
c. ReVelle / Emergency service siting models
which have n e i t h e r been s t r u c t u r e d n o r solved at the time of writing. T h e first of these m o d e l s is a p r o b a b i l i s t i c a n a l o g of the m a x i m a l covering l o c a t i o n p r o b l e m ( M C L P ) . W h e r e a s the M C L P d e p l o y s p facilities in such a w a y that the m a x i m u m p o p u l a t i o n has a server p o s i t i o n e d within S travel distance, the new model, the m a x i m u m a v a i l a b i l i t y l o c a t i o n p r o b l e m ( M A L P ) d e p l o y s its p servers so as to m a x i m i z e the p o p u l a t i o n which will have a server available within S d i s t a n c e with a reliability. T h e c o n s t r a i n t f r o m the p r o b a b i l i s t i c l o c a t i o n set covering p r o b lem that each a n d every d e m a n d p o i n t have service a v a i l a b l e within S with a reliability is n o w relaxed. All that is sought is that, given p facilities, the m a x i m u m p o p u l a t i o n achieves such levels of service. Of course, it m a y b e d e s i r a b e to a p p e n d to this m o d e l tight p r o b a b i l i s t i c c o n s t r a i n t s of coverage within a d i s t a n c e greater t h a n S, m u c h as m a n d a t o r y closeness c o n s t r a i n t s can b e a d d e d to the M C L P . T h e s e c o n d such m o d e l is the p r o b a b i l i s t i c analog of the FLEET model, which we here call the m a x i m u m a v a i l a b i l i t y FLEET model. This m o d e l w o u l d seek to d e p l o y engines, trucks a n d stations in such a w a y that the m a x i m u m p o p u l a t i o n (or calls) has a n engine available within the engine d i s t a n c e s t a n d a r d a n d a truck available within the truck d i s t a n c e s t a n d a r d with a reliability. This p r o b l e m s t a t e m e n t implies that the p r o b a b i l i t y of service being a v a i l a b l e is the p r o b a b i l i t y of two events occurring simultaneously. A n o t h e r p r o b l e m s t a t e m e n t is possible. W e c a n n o t call one version of an u n s o l v e d p r o b l e m m o r e t r a c t a b l e than a n o t h e r , b u t we c a n say that this s e c o n d version m i g h t be m o r e tractable. T h e second w a y of stating the p r o b l e m is to d e p l o y engines, trucks a n d stations in such a w a y that the m a x i m u m p o p u l a t i o n (calls) has a truck c o m p a n y available within its d i s t a n c e with fl r e l i a b i l i t y a n d an engine c o m p a n y a v a i l a b l e within its d i s t a n c e with fl reliability. Finally, a third new m o d e l w o u l d be the maxim u m a v a i l a b i l i t y s t a n d a r d response model. Here, engines, trucks, a n d stations w o u l d be p o s i t i o n e d to m a x i m i z e the p o p u l a t i o n which have the engine c o m p o n e n t of the s t a n d a r d response available a n d the truck c o m p o n e n t of the s t a n d a r d response a v a i l a b l e with a reliability. Or, the engine c o m p o n e n t a v a i l a b l e with fl reliability a n d the truck c o m p o n e n t a v a i l a b l e with /3 reliability could be used to define the criterion of merit.
Still o t h e r p r o b l e m s a n d f o r m u l a t i o n s besides these few u n d o u b t e d l y await s t a t e m e n t a n d solution, b u t for now this seems a r a t h e r full research a g e n d a to s i m p l y r o u n d out d e v e l o p m e n t s to date.
References Benedict, J. (1983), "Three hierarchical objective models which incorporate the concept of excess coverage for locating EMS vehicles or hospitals", MSc thesis, Northwestern University. Berlin, G. (1972), "Facility location and vehicle allocation for provision of an emergency service", Ph.D. thesis, The Johns Hopkins University. Christofides, N., and Viola, P. (1971), "The optimum location of multi-centers on a graph", Operational Research Quarterly 22, 145. Church, R., ReVelle, C., White, and Case (1974), "The maximal covering location problem", Papers of the Regional Science Association.
Daskin, M. (1983), "The maximal expected coveting location model: Formulation, properties and heuristic solution", Transportation Seience 17 (1), 48-70. Daskin, M., and Stern, E. (1981), "A multi-objective set covering model, for EMS vehicle deployment", Transportation Science 15, 137. Daskin, M., Hogan, K., and ReVelle, C. (1988), "Integration of multiple, excess, backup, and expected covering models", Environment and Planning 15, 15 - 35.
Eaton, D., Church, R., Bennet, V., and Namon, B. (1981), "On deployment of health resources in rural columbia", TIMS Studies in the Management Sciences 17, 331-359. Eaton, D., Hector, M., Sanchez, V., Lantigua, R., and Morgan, J. (1986), "Determining ambulance deployment in Santa Domingo, Dominican Republic", Journal of the Operational Research Society 37, 113. Gleason, J. (1975), "A set coveting approach to bus stop location", OMEGA 3 (5), 605-608. Hogan, K., and ReVelle, C. (1986), "Concepts and applications of backup coverage", Management Science 32 (11), 1434. Minieka, E. (1970), "The M-centre problem", S I A M Review 12, 138. Moore, G., and ReVelle, C. (1982), "The hierarchical service location problem", Management Science 28 (7), 775. ReVelle, C. (1987), "Urban public facility location", in: Handbooks of Urban and Regional Economics, E. Mills (ed.), North-Holland, Amsterdam. ReVelle, C., and Hogan, K. (1986), "A reliability constrained siting model with local estimates of busy fractions", Environment and Planning B15, 143. ReVelle, C., and Hogan, K. (1987), "The probahilistic location set covering problem and its derivative models", International Symposium on Location Decisions, Namur, Belgium, to appear in Annals of Operations Research. Schilling, D., Elzinga, D., Cohon, J., Church, R., and ReVelle, C. (1979), "The TEAM/FLEET models for simultaneous
C. ReVelle / Emergency service siting models facility and equipment siting", Transportation Science, 167. Storbeck, J. (1982), "Slack, natural slack, and location covering", Socio-Economic Planning Sciences 16 (3), 99. Toregas, C., Swain, R., ReVelle, C., and Bergmann, L. (1971), "The location of emergency service facilities", Operations Research 19, 1363.
69
Toregas, C., and ReVelle, C. (1973), "Binary logic solutions to a class of location problems", Geographical Analysis. White, J., and Case, K. (1974), " O n covering problems and the central facility location problem", Geographical Analysis 6 (3), 281.
European Journal of Operational Research 40 (1989) 58-69 North-Holland
Theory and Methodology
Review, extension and prediction in emergency service siting models Charles R e V E L L E
Program in Systems Analysis and Economics for Public Decision Making, Department of Geography and Environmental Engineering. The Johns Hopkins University, Baltimore, MD 21218, USA
Abstract: This is a review of a selected set of location papers and a research agenda. The location models reviewed fall into three categories: (1) basic deterministic covering models, (2) deterministic models which consider the value of additional covering servers, and (3) probabilistic models which allow randomness in server availability. Contributions in each of these categories are compared in a sequence which highlights the intellectual evolution of the models. Several new models are structured using the new formulation tools that are described. Several additional models whose development is predicted are described verbally.
Keywords: Location, reliability, public service, linear programming, integer programming
1. Introduction Since the 1960s, the field of location has moved in many directions, providing to researchers a vast set of theoretical and practical problems. These problems furnish the mathematician with theorems to prove, offer the mathematical programmer the opportunity to develop algorithms, and provide planners the capability to examine alternative solutions to practical problems. Because of the wide opportunities location problems present, the field of location or siting has attracted an array of researchers from numerous disciplines. This paper is a review of a segment of the location field, namely, urban public facility location. An attempt will be made here at synthesis of a set of models in this area. As a consequence, no claim is offered for a comprehensive treatment of urban public facility models. The reader interested in a fairly comprehensive review of urban public
Received September 1987; revised May 1988
facility location is referred to ReVelle (1987). Here, a small subset of the literature is selected for review, for integration, for extension, and for prediction. In short, this is both a review and the laying out of a research agenda with the first few steps on that agenda being undertaken in this paper. All of the models we consider here are covering models. Their goal is to provide 'coverage' to demand areas. A demand area is considered to be covered if a facility or vehicle is available to serve the demand area within some mandated distance or time standard. That is, coverage is achieved if the server at the facility can reach the demand area within the standard, or conversely, if individuals from the demand area can reach the facility within the time standard. The proximate presence of the server is the criterion which we use to measure whether or not a demand area is covered. It is surprising to note that such a simple concept can generate so much literature and so many models, but the richness of the covering notion will still not be fully fathomed here.
0377-2217/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
C. ReVelle / Emergency service siting models In Section 2, we review three basic deterministic location covering models and their uses; we also note one basic model that has yet to be structured. In Section 3, we show how criticism of these models has led to a second generation of siting models, as yet incomplete and awaiting further developments and extensions. In Section 4, this second generation is criticized in turn and a third generation of models, correcting the deficiencies of the second is discussed. This third generation of location models is still in a formative stage; much growth is anticipated. Indeed, in Section 5, we begin to fill in some of the blank spots in the first and second generation models and predict research directions in the third generation.
2. Basic deterministic covering models The location set covering problem (LSCP) seeks to position the least number of facilities needed to cover all points of demand within S distance or time units. The LSCP was an early statement of the emergency facilities location problem in mathematical programming form. It had also been stated in the early 1970s in virtually the same form as that above by a health planner with no knowledge of mathematical programming, suggesting that the problem statement as offered by mathematical programmers closely conformed to real needs. In mathematics, the problem is stated as Minimize
z = ~ xj jEJ
subject to
E
Xj>/]
Vi E I ,
jEN,
xj=0, 1
v J,
where index and set of demands; index and set of eligible sites for facilities; 1, 0; 1 if a facility placed at j; 0, otherwise; the shortest distance (or time) from site j to demand point i; = the m a x i m u m distance (or time) that a demand point can be from its nearest facility;
i,I= j, J=
X.I dji S
59
and N,
= ( j [ dji ~< S } = the set of facility sites eligi-
ble to serve demand point i, by virtue of being within S of i. The objective is minimization of the number of facilities required. The constraints require coverage of each demand point i; such coverage is achieved by the presence of at least one facility within S distance units of travel from the demand point i. The problem can be solved easily as a linear programming problem in which the integer variables are simply required to be non-negative. An occasional cut constraint (about 5% of runs) may be required in the circumstance where a non-integer number of facilities results from solution of the linear programming problem. See Toregas et al. (1971) for details. The problem can also be solved by the method of reductions. See Toregas and ReVelle (1973). A derivative model of the LSCP is the p-center problem. The p-center problem seeks the locations of the p facilities which minimizes the m a x i m u m distance separating any demand point and its nearest facility. A solution is obtained by solving a sequence of location set covering problems in which the m a x i m u m distance is successively reduced by the smallest unit of measurement of distance. The total of facilities required may be p in number in each of a set of successive solutions as distance is decreased. When a value of S is reached at which the number of facilities increases to p + 1, the preceding value of distance is the smallest m a x i m u m distance for which p facilities are feasible. Any smaller m a x i m u m distance requires a greater number of facilities. This method of solution is described by Minieka (1970) and by Christofides and Viola (1971). The location set covering problem required coverage of all demand points, no matter how low their population, how distant, or how great their need for service. Recognizing that the resources required could be excessive to cover all points of demand, no matter how small or remote, Church and ReVelle (1974) and White and Case (1974) framed a new problem that did not required the coverage of all nodes. Church and ReVelle called it the maximal covering location problem (MCLP); White and Case referred to it as the partial covering problem. We will use the former designation here.
60
C. ReVelle / Emergency service siting models
The problem begins with a statement of sources, in this case p facilities, and asks question: " H o w can the p facilities be best ployed where the goal is coverage within S of m a x i m u m possible population within S ? " In mathematics, the problem is stated as Maximize
rethe dethe
z = ~ aiy i i,E l
subject to
y,<~ ~_, xj
Viii,
tance standards. Other objectives included in a multi-objective formulation of the FLEET model were m a x i m u m coverage of fire frequency, maxim u m coverage of property value, m a x i m u m coverage of population at risk (as measured by the product of fire frequency and population for each demand point), m a x i m u m coverage of property at risk (formed in a similar fashion), and so on. The m a x i m u m population coverage version can be stated mathematically as z = ~ aiy ~
Maximize
i~l
j~N i
2xj=p,
subject to
j~J
xj, yi=O, 1 qi, j, where additional notation is a i = the population at demand node i; y ~ = l , 0; it is 1 if demand i is covered by a facility within N,.; 0, otherwise; and p = the number of facilities that can be sited. All other notation is as defined earlier. Relaxed linear programming supplemented by occasional use of branch and bound was used by Church and ReVelle to provide solutions to this problem. Church and ReVelle also suggested heuristic procedures for solution as did White and Case. The most informative use of the problem statement is to develop a tradeoff curve between population covered and the number of facilities utilized. The maximal covering idea was generalized for fire protection by Schilling et al. (1979). Their model accounts for coverage by two different types of service; in addition, both the servers and the facilities which housed the servers were to be sited simultaneously. Of several model types, the most general formulation was that known as the FLEET model (for facility location, equipment emplacement technique). The FLEET model sought the locations of a limited number of engine companies (the p u m p e r brigades) and truck companies (ladder brigades) as well as the fire stations that housed them. The goal of the FLEET model was coverage of the m a x i m u m number of people by both an engine company sited within an engine c o m p a n y distance standard and a truck c o m p a n y sited within the truck c o m p a n y distance standard. That is, coverage required the simultaneous siting of both types of service within their respective dis-
j ~ NiE
Yi <~ E
x7
Viii,
v;
J,
j~ NJ
xf
V
Ex~+Eg=p E+T, ExS=p s, y,, V , xf,
= o, 1
Vi, j ,
where new notation is x?
= 1, 0; 1 if an engine c o m p a n y positioned in
x5
=
#
=
N/E =
N/T
pE+T= pS =
a fire house at site j; 0 otherwise; 1, 0; 1 if a truck c o m p a n y positioned in a fire house at site j; 0 otherwise; 1, 0; 1 if a fire station established at site j; 0 otherwise; ( J I dji ~< E ) = the sites j which can provide engine c o m p a n y coverage to point i by virtue of being within the engine company distance standard E; { j ldji ~< T} = the sites j which can provide truck c o m p a n y coverage to point i by virtue of being within the truck c o m p a n y distance standard T; number of fire companies; and number of fire stations.
In this formulation, the first two constraint types define coverage as attainable only if both one or more engine companies is sited within the engine distance standard and one or more truck companies is sited within the truck distance stan-
C. Re gelle / Emergency service siting models
dard. The third and fourth constraint types allow companies to site only at nodes that have been allocated a fire station. The fifth constraint limits the total number of companies or brigades, but does not prescribe the proportion of resources placed in engine companies as opposed to truck companies. This constraint reflects the fact that the major cost of fire companies is personnel and not equipment. The last constraint limits the total number of fire houses. Schilling et al. found relaxed linear programming plus occasional branch and bound a viable solution technique. One additional model from the fire protection arena belongs in this brief description of deterministic models. To describe the model, it is first necessary to define the term standard response. In most major US cities, the standard response to a call for fire protection service is three engines and two trucks. This brings the requisite number of firefighters to the scene. The engine companies are to be available for dispatch within an engine distance and the truck companies within a truck distance. The problem then, is one of siting engine companies, truck companies and fire stations in a way that provides the m a x i m u m population with the standard response coverage. The model is not included in this section, because it is new; it requires tools to write the model that will not be available until we discuss the methods for redundant and backup coverage to be introduced in Section 3. The models discussed above, although they originated with a set covering model applied to emergency services, have been used relatively frequently for ordinary services as well. The set covering model has been used to allocate bus stops (Gleason, 1975). The maximal covering model has been used to site clinics and other health facilities (Eaton et al., 1981). A distinctive relative of the FLEET model in which services nest hierarchically has been used to site health facilities with differing but overlapping service characteristics (Moore and ReVelle, 1982). Indeed, the use of these deterministic models for ordinary services may be equally appropriate as their use for emergency services even though emergency service was the motivation for the creation of these models. The reason for the appropriateness for use for siting ordinary facilities is that people generally go to these ordinary facili-
61
ties and they can be built and staffed large enough to deal with the expected clientele. In contrast, when these models are used for emergency services, facilities are sited only on the basis of geographical coverage and not on the basis of availability. One has to assume that the facilities are continuously available in order to justify their use for emergency services. In uncongested, low demand settings, this assumption is not unreasonable, but in congested systems where frequent calls keep an emergency vehicle on the streets sometimes up to 80-90% of the time, this assumption simply cannot be justified. Congestion in service systems, as seen in a frequent lack of availability of a server within the distance or time standard, motivated the creation of a new set of models which emphasized additional coverage beyond the first. These models, mostly developed in the last half dozen years, will be discussed next.
3. Deterministic models which focus on additional coverage
Several investigators recognized the possibility that in congested systems the first server (and possibly the only server in a particular coverage area) might not be available at the time of a call's arrival. These investigators focussed on the importance of providing additional servers beyond the first within the coverage region of a demand area. Berlin (1972) and Daskin and Stern (1981) structured a model which utilized a number of facilities greater than or equal to the minimum number required by the location set covering problem. They focussed simply on the total of redundant or additional coverers in the system beyond the first cover required by the location set covering problem. When the number of facilities was equal to the minimum, this amounted to finding the alternate optimum to the LSCP (there are many alternate optima to the LSCP) which produced the greatest number of covering servers, summed over all coverage sets. In mathematical terms: Maximize
z= ~R, iEI
C ReVelle / Emergency service siting models
62
subject to
ViEI,
E xj-Ri=l jG IV
Z
j=p,
jEJ
xj=O, 1
W,
Ri are integers
Vi,
where R~ are the number of servers beyond the first in the set N,, and all other terms and variables are as described earlier. If p is the minimum number required to cover, the solution obtained is the alternate optimum which maximizes the sum of redundant coverages. If the number of facilities is ranged from the minimum required to cover p up to some larger number, the tradeoff curve arrays redundant coverage against total number of servers. Again, relaxed linear programming provides frequent all zero-one solutions. Although this formulation was an improvement on the simple location set covering problem, it still required coverage of each and every demand node. Further, it did not in any way account for the greater importance of having additional eligible coverers for areas of high demand. The latter defect could have led to solutions which clustered servers around low population demand areas simply because they were easy to cover. To correct for the lack of attention to the magnitude of demand, Benedict (1983) and Eaton et al. (1986), both restructured the objective of the model of Berlin and of Daskin and Stern. For a given number of servers, p, in the system, they operated on the objective Maximize
emphasize the importance of at least one redundant coverer for each demand node, Hogan and ReVelle (1986) modified the location set covering problem to the following form, which they referred to as BAcoP1 (for Backup Coverage Problem 1):
z = ~ a~R,,
where a~ is, again, the population or demand at node i. This objective maximizes the product of population and the number of servers which provide additional cover to the population. Trading off p against the product of population and redundant servers provides a sense of the redundant coverage attainable by deploying additional servers. Solution method is not altered by this simple modification. True, this modification makes additional coverers more likely for high demand nodes, but redundant coverers can pile up for some demand areas and not for others. Some demand nodes may be left with only a first coverer and no backup. To
Maximize
z = ~ aiu i i~I
subject to
ui<~ Y[ x j - 1 )oN,
ViEI,
E xj=p,
j~J
Xj, Ui = O, 1
W, j,
where ut = (1, 0); it is one if the total of facilities in N, is greater than or equal to two (one or more backup servers), and zero otherwise. All other notation is as defined earlier. This model still requires first coverage of every node, an assumption we will relax in a moment. The formulation seeks to maximize the population that has at least one redundant coverer. It de-emphasizes multiple redundant coverers of a n o d e - - a l t h o u g h these can still occur, but not at the expense of first redundant coverage. As before, relaxed linear programming provides frequent zero-one solutions. Hogan and ReVelle (1986) extend BAcoP1 to the situation in which first coverage is not required, but is an objective as in the maximal covering location problem. In addition, second coverage (or higher) is also a goal. The two objectives are to maximize the population achieving first coverage, and to maximize the population achieving second (or higher) coverage. In mathematics, their BACOP2 model is Maximize
zi = E aiYi, i~l Z2 :
E aiui iEl
subject to ui+yi~
E Xj j~N,
Viii,
E xj=p,
j~J
ui <~Yi V i i i , xg, ui, y , = 0 , 1
Vi, j,
C. ReVelle / Emergency service siting models
where all notation is as previously defined. The first constraint type, in combination with the (0, 1) restrictions, says that the sum of first and second coverage is bounded by the number of coverers in N, or by two, whichever is smaller. The third constraint type says that second coverage cannot be counted until first coverage achieved. Thus, if the number of coverers in N, is one, first coverage will be counted, but second coverage will not. Hogan and ReVelle, in an uncontrived example, found that the extent of second coverage could be much improved with only marginal reductions in first coverage from its maximum level. Again, relaxed linear programming provides a straightforward and rapid solution method for the (0, 1) programming problem. Occasional use of branch and bound was found necessary. The authors point out that the BACOP models can be extended to count third and subsequent coverage variables as well. Still another model considers redundant coverage and in a strikingly different way than any of the preceding models. Storbeck (1982), borrowing a page from goal programming, defines the difference between the number of coverers in AT, and the value one as the difference between two nonnegative variables, which we call k + and k 7 . His model, in multi-objective notation, is Maximize
z1= £
ai k + ,
i~l
z 2 = -- y " a i k 7 i~I
subject to k i ~ -- k i
xa-
=
1
Viii,
j~N, xI=P, j~J
x j, k~ = 0 ,
1
k [ = integers
Vi, j,
Vi,
where k + = the coverage in excess of one for node i;
k , = the coverage less than one for node i. Objective 1 reflects maximization of the population receiving redundant coverage. Objective 2 suggests minimization of the population which is 'undercovered' Storbeck's formulation is ex-
63
tremely versatile and one of the cleverest of the additional coverer models. Missing from this section is a model which focuses on providing backup or additional coverage for fire protection systems, where coverage is achieved only by the proximate stationing of t w o d i f f e r e n t k i n d s of servers, engine companies and truck companies. This, along with other new models, will be presented in Section 5. These formulations which focus on additional coverage beyond the first are meant to apply to congested systems, and they represent an attempt to treat the probabilistic nature of service availability with deterministic models. These models have not been tested in a simulation that we know of, but certainly they add an appropriate new dimension to the basic deterministic models that preceded them. Surprisingly, they also pave the way for extensions to both the basic deterministic models they evolved from and to the more recently developed probabilistic models. We shall describe the probabilistic models next.
4. ProbabUistic models consider randomness vehicle availability,
in
The models developed in Section 3 were intended to account for the possibility of servers being busy by stationing within the time standard additional units beyond the first. While these were not probabilistic models, they did provide a new direction to emergency location modelling and new techniques to the analysis. The models are not dismissed as a passing phase but are, in fact, a robust contribution to emergency services siting. Although the deterministic additional coverage models just described are of value, it was a natural step for researchers to investigate probabilistic siting models next. By probabilistic siting model, we mean here a formulation in which the randomness is in the availability of the server. It is possible to approach models in which randomness exists in both server availability and in time of travel, time of service, etc., but for this review we are considering only randomness in availability. We choose this research line because the impact of the randomness in travel and service time is far less than the impact of randomness on availability.
C. ReVelle / Emergency service siting models
64
Surprisingly, the first probabilistic emergency model was constructed a dozen years ago, long before redundant coverage modelling had gotten off the ground. C h a p m a n and White (1974) proposed a probabilistic location set covering model in which servers were not always available. A simplified version of their model (Daskin, 1983) uses a single across-the-board estimate of server busy probability; we call that busy probability q. We can then write an expression for the probability that one or more servers in N~ is free to accept the next arriving call from demand node i. That probability is here constrained to be greater than or equal to a, a system-wide level of reliability that must be met for all nodes.
structured as follows: n/
Maximize
E ( 1 - q)(q)k
z= E i~l
laiYik
k=l
subject to n,
Z Yik ~
k=l
Z
Xj
Viii,
j~U,
2xj=p, j~J
Yik = O, 1 Vi, k, xj are integers >~ 0
Vj,
where 1 -- qZ:~ulX: >1 a
Vi ~ I.
Yik = 1, 0; it is 1 if node i is covered by at least k
The busy probability q, raised to the number of servers in N~, is the probability that a// servers in N~ are busy. This value subtracted from one is the probability that one or more servers in ~ is free, and it is this quantity that is constrained to be greater than or equal to a. This constraint has a simple linear equivalent, namely, E
xj>
l o g 0 - a) ]
l~g q
Viii,
j~N,
where [u] means the smallest integer greater than or equal to u. Appending an objective of minimizing the number of servers that are required in the system completes the formulation of the probabilistic location set covering problem. It is of interest that the variable xj can now take on any positive integer value; that is, multiple servers can be stationed at a single site. The C h a p m a n and White model was not entirely so simple as this, but did attempt to prespecify busy fractions in different sectors of the system and proposed updates of busy fraction using simulation output. Daskin (1983) recognizing the utility of the probabilistic approach of Chapman and White, incorporated the idea of the stable busy fraction into a formulation that maximized the expected value of population coverage given that p facilities are to be placed on the network. His model is one type of a probabilistic analog of the maximal covering location problem of Church and ReVelle and of White and Case. Daskin's m a x i m u m expected covering location problem (MEXCLP) is
facilities; it is 0, otherwise; x: = the number of facilities at site j (an integer >/0); and n, = number of facility sites in N,. The constraint for each demand point i says that the sum of the indices of coverage, the Yik, can rise only to the number of facilities sited within N~. The k-th component of the objective for node i, while the demand at i is absent, is ( 1 q)qk-~yik. If Yik equals one, then the probability that the demand at node i will be assigned to its k-th coverer is ( 1 - q)qk-1. Summed over all k, the total of components becomes the probability that demand i is covered. Multiplied by the dem a n d at i, the product formed is the demand times the probability of the cover. Summed over all demands, the objective becomes finally the expected demand covered. The BACOP models of H o g a n and ReVelle are very similar to this formulation. Separation of the coverage objective into separate objectives for first cover, second cover, etc., and removal of the probability weights makes the objectives look precisely like those of Hogan and ReVelle. In the Daskin model, the Y~k variables replace y~, u~, vi, etc. in Hogan and ReVelle. Finally the 'can't be covered unless' constraints such as u~ ~
C ReVelle / Emergency service siting models
relaxed linear program, with relatively high confidence of all integer solutions resulting--as Hogan and ReVelle showed with their more constrained model. The expected coverage model of Daskin preceded the Hogan and ReVelle backup models by several years, but the parallels were not noted until a presentation of the backup models in 1984 at the International Symposium on Location Decisions (Boston and Martha's Vineyard, MA). A comprehensive review of backup models is presented in Daskin et al. (1988). The original probabilistic set covering problem model using one estimate of busy fraction for the system, for all its elegance, produces results that are extremely sensitive to q. A minor change in q that causes the bracketed term to go from 0.99 to 1.01 will double the right hand sides of the above reliability constraints, and as a consequence, often double the number of facilities. This doubling of required servers is a result of the assumption of a single system wide busy fraction estimate. As a consequence, it becomes important to pursue different estimates of the busy fractions in different sectors of the planning region. The last in the sequence of probabilistic models is an enhancement of the probabilistic location set covering model of Chapman and White. ReVelle and Hogan (1986), embracing the concept of the reliability constraint, incorporated in the formulation prior knowledge of the components of the busy fraction which is used in the reliability constraint. They thereby created a reliability constraint which had built into it an explicit bound on the busy fraction. This bound guaranteed a value of the busy fraction that would support the reliability constraint. ReVelle and Hogan first define an average busy fraction for servers in the same sector of the planning area for which the reliability constraint is to be written. Let us say that one of the reliability constraints is being written for demand point i. The vehicles that can provide coverage for point i must be in N,, defined as the set of eligible coverers. The average busy fraction of these vehicles is -t " E k ~ M, f k qi
-
-
24Ej~ N Xj '
where } = the average duration of a call (hours);
65
fk = frequency of calls at demand node k (calls/ day); and 34,= the set of demand nodes within S of node i. The numerator is the daily hours of service needed in the zone around node i. The denominator is the daily hours of service available in that same zone. Hence, the ratio provides an estimate of the busy fraction in the zone within which the reliability constraint is written. The actual value of this busy fraction estimate is not calculable until the number of servers allocated to N, is known. By defining
kGM,
we can write
qi - y~j ~ x, x / ' which will allow us a nearer final constraint. If the original reliability constraint is rewritten with qi replacing q, we have
1 - q~ .... ,~, >~ a
Vi ~ l,
or, replacing qi with its estimator from above,
1-
/ Y'. x,
~,~.
The above expression has no exact inverse, no analytical solution for the number of vehicles in N,. It does, however, have a numerical inverse. That is, the number of vehicles in N; must be greater than or equal to the smallest integer which satisfies the above non-linear reliability constraint. More formally, we can write for demand point i, the following linear equivalent: y" xi>~b,, j ~ N,
where b, is the smallest positive integer satisfying
1 - ( i /Fb , )
h'>~,,
the only determinants of b, being F, and a. The value of b i is a number which simultaneously satisfies the reliability requirement a n d provides a busy fraction qi whose value supports that satisfaction.
C. ReVelle / Emergency service siting models
66
The probabilistic location set covering problem (PLSCP), as enhanced by ReVelle and Hogan with local busy fractions can thus be summarized as Minimize
z = ~ Xg j~J
suggested; (3) a frankly r a n d o m environment where the notion of backup coverage is replaced by a concern for the reliability of service presence. In the following section we will partly fill in the several blank spots we have left in this review and indicate where new developments are most likely.
subject to Vi,
j~N~ xj are integers, where bi is the smallest integer satisfying
1 - (F,/bi)b'>_- a. Two more models can be derived from this simple model. These are: (1) the a-reliable p-center problem, an analog to the original p-center problem (Mineika, 1970); and (2) the m a x i m u m reliability location problem. The a-reliable p-center problem seeks the positions of p facilities in the system which minimizes the m a x i m u m time within which service is available with a reliability. In other words, given p facilities to site we are to find the smallest maxim u m time within which service can be available a fraction of the time. Example problems and a tradeoff curve of m a x i m u m distance versus number of facilities is provided in ReVelle and Hogan (1987). The maximum reliability location problem seeks to position p facilities in such a way that the probability of service actually being available within a stated distance or time is as large as possible. A simple technique is derived in ReVelle and Hogan (1987) which utilizes solutions to the probabilistic location set covering problem to generate solutions to the m a x i m u m reliability location problem. A tradeoff curve between reliability and the number of servers is provided there, as well. Both the a-reliable p-center problem and the maxi m u m reliability location problem can be formulated and solved using the local estimation procedure for busy fractions. Both involve solving a sequence of (0, 1)-programming problems using relaxed linear programming and occasional branch and bound. This completes the review of the basic siting models that have been created to date in: (1) the deterministic environment; (2) a congested environment where concern for backup coverage is
5. Filling in the missing models and making predictions on models yet to come In Section 2, we indicated that a certain basic deterministic model had never been formulated. We call this model, which is from the fire protection area, the maximal covering standard response problem. The first coverage of a demand area, which is the focus of the FLEET model, is covered only by the first due company. The standard response to a fire in most major cities is three engine companies and two truck companies. The engine companies are to be within an engine standard response distance and the truck companies from within a truck standard response distance. Our goal is to maximize the population (or calls) which have standard response coverage; i.e., three engine companies positioned within their distance standard and two truck companies deployed within their distance standard, given limited resources in companies and fire stations. With the tools provided by the models which dealt with additional covering, we can now structure the maximal covering standard response problem. One important feature of the structure to be presented is that we should avoid requiring any level of coverage for a demand area in order to be true to the objective statement. To structure the problem, we utilize the backup coverage concepts of Hogan and ReVelle: Maximize
z = E aiwi i~1
subject to W i q - Y 7 <~
E
Viii,
XT
j ~ Ni TSR
E x? Vi l, j ~ N/ESR
UiE<~yiE,
i~I,
Xff<~X s
Vj~J,
67
C. ReVelle / Emergency service siting models
wi<~u~,
i~l,
w,~yT,
iEl,
xf<~x s
Vj~J,
and backup coverage are defined by the simulta-
where newly utilized variables and parameters are:
neous presence of both types of fire vehicles, engines and trucks. First coverage is the stationing of at least one engine company and one truck company within their respective distance standards. Second coverage is achieved by the deployment of at least two engine companies and at least two truck companies within the appropriate distance standards. In mathematics, the model, which we call FLEET BACOP, is written:
yl I"
Maximize
£
X/"E"}- E
jcJ
XT = p E + T ,
j~d
S, xS.1 = p S j~J
all variables are 0, 1,
yi E
=
Wt
Nt TSR :
Nc ESR :
(1, 0); it is 1 if point i is covered by at least one truck company within the truck standard response distance; it is 0 otherwise; (1, 0); it is 1 if point i is covered by at least one engine company within the engine company standard response distance; it is 0 otherwise; (1, 0); it is 1 if i is covered by at least two engine companies; it is 0 otherwise; (1,0); it is 1 if i is covered by two truck companies and three engine companies within their respective standard response distances; it is 0 otherwise; the set of sites from which trucks can provide the standard response to node i, which sites are within the truck standard response (TSR) distance; the set of sites from which engine companies can provide the standard response to node i, which sites are within the engine standard response distance.
All other terms are as defined previously. The formulation is fairly large but workable, and linear programming, supplemented by occasional branch and bound, should provide a solution method. Another missing formulation was listed in Section 3. Although conceptually related to the model offered immediately above, this model more formally belongs in the category of models which focus on additional coverage, primarily because the model is an analyst's idea and not a notion which arose in the fire protection sector. The model, again, is in the fire protection arena. It is analogous to the BACOP2 model of Hogan and ReVelle in that is arrays first coverage of population against backup coverage of population. The difference is that in this model both first coverage
z 1 = Y'~ a,y,, i~l Z 2 ~ £ OjUt i~I
subject to U, + Y i
x~
Vi~l,
xf
ViEI,
j~Nf U,q'-Yi<~
xT
E j~Nf
W E J,
XE ~ xSJ Vj E J, ] u i<~y, V i i i , j~J
E x,
2cJ
:p s ,
j~J
all variables are 0, 1, where y, indicates coverage by at least one engine company and by at least one truck company, and u i indicates coverage by at least two engine companies and at least two truck companies. Additionally, the sets return to the form as given originally in the F L E E T model. Linear programming and branch and bound should suffice as a general solution method. This model will provide a tradeoff curve between first coverage, the presence of both an engine and a truck within their distance standards, and second coverage, the presence of at least two engines and two trucks, within their distance standards. In addition to the new models presented above, we can predict attempts at the development of some additional probabilistic siting models, models which are proposed here for the first time, but
68
c. ReVelle / Emergency service siting models
which have n e i t h e r been s t r u c t u r e d n o r solved at the time of writing. T h e first of these m o d e l s is a p r o b a b i l i s t i c a n a l o g of the m a x i m a l covering l o c a t i o n p r o b l e m ( M C L P ) . W h e r e a s the M C L P d e p l o y s p facilities in such a w a y that the m a x i m u m p o p u l a t i o n has a server p o s i t i o n e d within S travel distance, the new model, the m a x i m u m a v a i l a b i l i t y l o c a t i o n p r o b l e m ( M A L P ) d e p l o y s its p servers so as to m a x i m i z e the p o p u l a t i o n which will have a server available within S d i s t a n c e with a reliability. T h e c o n s t r a i n t f r o m the p r o b a b i l i s t i c l o c a t i o n set covering p r o b lem that each a n d every d e m a n d p o i n t have service a v a i l a b l e within S with a reliability is n o w relaxed. All that is sought is that, given p facilities, the m a x i m u m p o p u l a t i o n achieves such levels of service. Of course, it m a y b e d e s i r a b e to a p p e n d to this m o d e l tight p r o b a b i l i s t i c c o n s t r a i n t s of coverage within a d i s t a n c e greater t h a n S, m u c h as m a n d a t o r y closeness c o n s t r a i n t s can b e a d d e d to the M C L P . T h e s e c o n d such m o d e l is the p r o b a b i l i s t i c analog of the FLEET model, which we here call the m a x i m u m a v a i l a b i l i t y FLEET model. This m o d e l w o u l d seek to d e p l o y engines, trucks a n d stations in such a w a y that the m a x i m u m p o p u l a t i o n (or calls) has a n engine available within the engine d i s t a n c e s t a n d a r d a n d a truck available within the truck d i s t a n c e s t a n d a r d with a reliability. This p r o b l e m s t a t e m e n t implies that the p r o b a b i l i t y of service being a v a i l a b l e is the p r o b a b i l i t y of two events occurring simultaneously. A n o t h e r p r o b l e m s t a t e m e n t is possible. W e c a n n o t call one version of an u n s o l v e d p r o b l e m m o r e t r a c t a b l e than a n o t h e r , b u t we c a n say that this s e c o n d version m i g h t be m o r e tractable. T h e second w a y of stating the p r o b l e m is to d e p l o y engines, trucks a n d stations in such a w a y that the m a x i m u m p o p u l a t i o n (calls) has a truck c o m p a n y available within its d i s t a n c e with fl r e l i a b i l i t y a n d an engine c o m p a n y a v a i l a b l e within its d i s t a n c e with fl reliability. Finally, a third new m o d e l w o u l d be the maxim u m a v a i l a b i l i t y s t a n d a r d response model. Here, engines, trucks, a n d stations w o u l d be p o s i t i o n e d to m a x i m i z e the p o p u l a t i o n which have the engine c o m p o n e n t of the s t a n d a r d response available a n d the truck c o m p o n e n t of the s t a n d a r d response a v a i l a b l e with a reliability. Or, the engine c o m p o n e n t a v a i l a b l e with fl reliability a n d the truck c o m p o n e n t a v a i l a b l e with /3 reliability could be used to define the criterion of merit.
Still o t h e r p r o b l e m s a n d f o r m u l a t i o n s besides these few u n d o u b t e d l y await s t a t e m e n t a n d solution, b u t for now this seems a r a t h e r full research a g e n d a to s i m p l y r o u n d out d e v e l o p m e n t s to date.
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Eaton, D., Church, R., Bennet, V., and Namon, B. (1981), "On deployment of health resources in rural columbia", TIMS Studies in the Management Sciences 17, 331-359. Eaton, D., Hector, M., Sanchez, V., Lantigua, R., and Morgan, J. (1986), "Determining ambulance deployment in Santa Domingo, Dominican Republic", Journal of the Operational Research Society 37, 113. Gleason, J. (1975), "A set coveting approach to bus stop location", OMEGA 3 (5), 605-608. Hogan, K., and ReVelle, C. (1986), "Concepts and applications of backup coverage", Management Science 32 (11), 1434. Minieka, E. (1970), "The M-centre problem", S I A M Review 12, 138. Moore, G., and ReVelle, C. (1982), "The hierarchical service location problem", Management Science 28 (7), 775. ReVelle, C. (1987), "Urban public facility location", in: Handbooks of Urban and Regional Economics, E. Mills (ed.), North-Holland, Amsterdam. ReVelle, C., and Hogan, K. (1986), "A reliability constrained siting model with local estimates of busy fractions", Environment and Planning B15, 143. ReVelle, C., and Hogan, K. (1987), "The probahilistic location set covering problem and its derivative models", International Symposium on Location Decisions, Namur, Belgium, to appear in Annals of Operations Research. Schilling, D., Elzinga, D., Cohon, J., Church, R., and ReVelle, C. (1979), "The TEAM/FLEET models for simultaneous
C. ReVelle / Emergency service siting models facility and equipment siting", Transportation Science, 167. Storbeck, J. (1982), "Slack, natural slack, and location covering", Socio-Economic Planning Sciences 16 (3), 99. Toregas, C., Swain, R., ReVelle, C., and Bergmann, L. (1971), "The location of emergency service facilities", Operations Research 19, 1363.
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Toregas, C., and ReVelle, C. (1973), "Binary logic solutions to a class of location problems", Geographical Analysis. White, J., and Case, K. (1974), " O n covering problems and the central facility location problem", Geographical Analysis 6 (3), 281.