an efficient solution procedure using Lagrangian relaxation. Results ... problem where the second level facilities provide backup service to the primary provider.
Annals of Operations Research, 18 (1989) 141-154
THE CAPACITATED MAXIMAL COVERING LOCATION WITH BACKUP SERVICE
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PROBLEM
Hasan PIRKUL and David SCHILLING College of Business, The Ohio State University, Columbus, Ohio 43210, U.S.A.
Abstract The maximal covering location problem has been shown to be a useful tool in siting emergency services. In this paper we expand the model along two dimensions - workload capacities on facilities and the allocation of multiple levels of backup or prioritized service for all demand points. In emergency service facility location decisions such as ambulance sitting, when all of a facility's resources are needed to meet each call for service and the demand cannot be queued, the need for a backup unit may be required. This need is especially significant in areas of high demand. These areas also will often result in excessive workload for some facilities. Effective siting decisions, therefore, must address both the need for a backup response facility for each demand point and a reasonable limit on each facility's workload. In this paper, we develop a model which captures these concerns as well as present an efficient solution procedure using Lagrangian relaxation. Results of extensive computational experiments are presented to demonstrate the viability of the approach.
1. Introduction T h e c o n c e p t of c o v e r a g e has p r o v e d to be a useful a n d intuitively a p p e a l i n g m e a s u r e of p e r f o r m a n c e for facility siting decisions w h e r e a m i n i m u m t h r e s h o l d of service is desired ( T o r e g a s a n d ReVelle, 1972 [24] and C h u r c h a n d ReVelle, 1974 [4]). This is o f t e n the case with services that are e m e r g e n c y related: fire p r o t e c t i o n and a m b u l a n c e d i s p a t c h i n g are p r i m e examples. As e v i d e n c e d b y the c o n s i d e r a b l e research efforts d i r e c t e d t o w a r d it, the m a x i m a l c o v e r i n g l o c a t i o n p r o b l e m is a p a r t i c u l a r l y p o t e n t c o v e r a g e f r a m e w o r k for these situations. T h e r e are, however, several aspects o f the p r o b l e m that still challenge researchers. T h e s e p e r t a i n in p a r t to the n a t u r e of e m e r g e n c y services d e m a n d ; a d e m a n d for which q u e u e i n g is a highly u n d e s i r a b l e option. W h e n a call for service occurs, the nearest a m b u l a n c e is d i s p a t c h e d a n d that facility's resources are fully utilized. S h o u l d a n o t h e r call arise for that a m b u l a n c e b e f o r e it r e t u r n s to d u t y , a n o t h e r facility a n d its resources m u s t be b r o u g h t to bear. T h i s b a c k u p facility is unlikely to b e as close to the incident as the p r i m a r y a m b u l a n c e , so t h a t service will be s o m e w h a t d e g r a d e d . T h e n e e d for b a c k u p service is p a r t i c u l a r l y a c u t e in regions o f high d e m a n d . T h e s e areas also will o f t e n result in excessive w o r k l o a d s o n s o m e facilities which can also d e g r a d e service quality. U n d e r these c o n d i t i o n s , there© J.C. Baltzer A.G. Scientific Publishing Company
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fore, effective siting decisions should address both the need for backup facilities for each demand point and as well as a reasonable limit on each facility's workload. Designing systems with backup service has been addressed rather sporadically in the literature. The earliest work is that of Berlin (1972) [2] in his location/ simulation approach to ambulance siting. His sequential approach involved first applying the location set covering problem (Toregas and ReVelle, 1972, [24]) to determine the minimum number and location of ambulance dispatching sites. The second phase used a simulation model to estimate ambulance utilization. Daskin and Stern (1981) [7] modify the location set covering problem with a hierarchical multiobjective form which seeks to minimize the number of facilities and maximize the number of times a demand point is covered after the primary provider. Ruefli and Storbeck (1982) [23] formulate a hierarchical maximal covering problem where the second level facilities provide backup service to the primary provider. Hogan and ReVelle (1986) [15] provide a brief review of backup service research as well as a multiobjective model formulation which trades off primary and backup coverage. Their approach, which implicitly attempts to capture finite facility capacities, is unique in that backup coverage can be provided to some nodes before all nodes have been assigned primary coverage. Pirkul and Schilling (1988) [21] consider the allocation of primary and secondary service from capacitated facilities, but in the context of minimizing average distance rather than maximizing coverage. Finally, Weaver and Church (1985) [25] take a different tack with the Vector Assignment p-Median Model. In their model, a fixed fraction of each node's demand is allocated among the facilities on the basis of their proximity, e.g. eighty percent of a node's demand is assigned t~ the closest facility, ten percent to the next closest, and so forth. While all of these approaches provide useful tools for designing systems with backup protection, none have explicitly addressed the typically concomittent problem of finite facility capacity. Workload limits or capacity restraints are often present in facility location •decisions whether one is siting warehouse, airports or fire stations. Incorporating them into siting models, however, while not difficult to formulate, typically pushes the computational complexity of the problem to the limits of tractability. Research efforts, therefore, often focus on improving the efficiency of solution procedures. Relatively few researchers have examined the capacitated version of the maximal covering location problem. As Current and Storbeck (1986) [6] indicate, "a basic underlying assumption of the location coveting models formulated to date, is that the facilities being sited are uncapacitated". They proceed to present a capacitated version of the maximal covering location problem (MCLP). No solution method was demonstrated, however, although several were suggested. Chung et al. (1983) [3] do provide a heuristic for solution of a similar capacitated MCLP with the additional restriction of binary assignment variables. Computational experience was presented for relatively small problems, however.
H. Pirkul, D. Schilling / Maximal covering with backup
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Pirkul and Schilling (1987) [20] develop an alternative formulation where all demand is both assigned a facility and contributes to that facility's workload. They also demonstrate an effective solution procedure. We present below a model formulation as well as a solution procedure for the maximal covering location problem where each facility is capacitated and primary and backup service is provided to each demand point. In the next section, the formulation of the model will be discussed. The solution procedure will be outlined in section 3 followed by a presentation of computational experience with the procedure in section 4. Conclusions are discussed in section 5.
2. Model development Stated verbally, the model seeks to maximize the amount of covered demand given a fixed number of facilities such that all demand is assigned to a facility; primary and backup service are provided from separate facilities; and total service by each facility does not exceed some specified service capacity. Mathematically the model appears as follows: PROBLEM - I P ZIp = M a x i m i z e ~
Y'.(cija P ,xij+ P P c ibj a ibx i b j)
(1)
iEl j~J
Subject to:
(2)
EYj
