Application of Location Models for Assessing Spatial Coverage

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Location modeling is a method of determining the optimal locations of different facilities that have crucial importance for human health and wellbeing, like emergency or healthcare services. First location models were developed in 1970th, and since that time, scientists have worked hard to extend and improve them. This paper aims to describe the variety of location models and give a detailed overview of how these models are usually applied.

Location models for emergency service facilities are usually based on two methods. The minimization method is aimed at minimizing the number of facilities, and the maximization method is aimed at maximizing population coverage. Choosing between these two strategies depends on the interests of decision-makers, geographical limitations, and numerous risk factors. Today, location models are widely used to meet the population coverage demands, when a maximum travel time or distance criterion is set in advance (Li et al. 2011; Yin & Mu 2012). Location Set Covering Problem (LSCP) is the primary covering model developed by Toregas et al. (1971). It is a deterministic model that relies on the minimization method to find the best sites for emergency service stations. Here, the minimization method implies reducing the driving distance and time.

Over time, scientists have expanded the primary model to apply it in unusual situations. Murray et al. (2010) proposed to use implicit and explicit LSCP for detecting where to locate emergency services (Schilling 1979). LSCP defines the number of facilities needed to cover all demand points. However, an insignificant amount of resources may limit its application (Aktas et al. 2013). To overcome the limitations of the primary LSCP, Church & Reveille (1974) developed a Maximum Coverage Location Problem (MCLP). It is a deterministic model that maximizes population coverage with a limited number of facilities. Both LSCP and MCLP use a linear programming optimization method (Marianov & Serra 2002; Yin & Mu 2012). Researchers see these two deterministic covering models as the basis for emergency services location modeling.

However, these models aim at maximizing the coverage by the emergency services only. To overcome this flaw, scientists developed new models for backup coverage by observing if the brigades are busy due to high demand. Daskin & Stern (1981) proposed a Modified Maximal Covering Location Model (MMCLM). In addition to the primary objective, maximizing the covered population, they introduced another objective – maximizing the multiple coverages of the demand points.

Later, Hogan & ReVelle (1986) proposed two versions of MMCLM, known as BACOP1 (Backup Coverage Problem 1) and BACOP2 (Backup Coverage Problem 2). The BACOP1 intends each demand point to have priority coverage, which is not essential for many location problems. Considering this fact, Hogan & ReVelle (1986) developed a BACOP2 model using the occasional Branch and Bound (BB) method, which maximizes the population that gets priority and secondary coverage. While backup covering models require a multi-objective strategy, the BACOP2 model is applicable when there are fewer obstacles.

When the restraint is too hard, the analysis of the probabilities is required. Considering travel time limitations, Gendreau et al. (1997) introduced a Double Standard Model (DSM). Their model maximizes the population coverage multiple times, managing two different travel time restrictions, namely r1 and r2, where r1 is less than r2. Although the backup covering models and the DSM are widely recognized, they are still deterministic covering models.

Considering the chances that stations will be busy and their reliability, scientists have developed probabilistic covering models (Daskin1983; ReVelle & Hogan 1989a; Goldberg & Paz 1991; Beraldi & Ruszczynski 2002). They widened the covering models to predict real-world problems through probability distributions of random variables, estimating a set of expected future scenarios with the unknown parameters (Owen & Daskin, 1998). The Maximum Expected Covering Location Problem (MEXCLP), developed by Daskin (1983), is the most popular probabilistic model. It maximizes the coverage by using a measuring parameter (q) to describe the probability that at least one station is free to serve the calls from any object.

Then, to ease the hard assumption of MEXCLP, ReVelle & Hogan (1989a) developed a Maximal Availability Location Problem (MALP). Their model denotes each object with the probability indicator α. ReVelle & Hogan (1989b) also created a Probabilistic Location Set Covering Problem (PLSCP). The PLSCP uses an average server busy fraction (qi) and a service reliability factor (α) to determine the demand points. In PLSCP, stations are located in a unique way to maximize the probability that the emergency service will be available within a particular distance. Several other models were developed by widening MEXCLP and PLSCP. These are MOFLEET (Bianchi & Church 1988), AMEXCLP (Batta et al. 1989), and TIMEXCLP (Repede & Bernardo 1994). Nonetheless, most scientists who work with emergency service location modeling prefer MEXCLP.

The hypercube queuing method, typically determined by a linear programming model, also provides the coverage of demand points when the station is busy responding to an incident. Larson (1974, 1975) was first to apply the hypercube queuing model in emergency service location modeling. This model considered the crowding of the system by calculating the steady-state busy fractions of servers in a network. It was developed as an alternative method to obtain a better estimation of expected population coverage due to the lack of an accurate estimation in MEXCLP. Nowadays, many researchers extended the hypercube queuing model (Saydam & Aytug, 2003; Galvo et al. 2005; Takeda et al. 2007; Iannoni & Morabito 2007; Iannoni et al. 2008). In particular, Saydam & Aytug (2003) incorporated the hypercube queuing model into a linear programming model to define the server-location busyness probability.

While most models are solving single-period location problems, some researchers developed dynamic multi-period models, as demands vary both temporally and spatially (Gunawardene 1982; Gendreau et al. 2001; Rajagopalan et al. 2008; Zhang et al. 2008). In particular, Gunawardene (1982) addressed the single-period population coverage problem by widening the LSCP. Health geographers Gendreau et al. (2001) and Rajagopalan et al. (2008) developed dynamic allocation models, namely Dynamic Double Standard Model (DDSM) and Dynamically Available Coverage Location (DACL). These models solve real-time location problems by using minimization and maximization methods.

The models discussed above have so far regarded the demands of emergency points only. But if the demand comes from a line or an area, it may be only partially covered. The issue of spatial representation and geographical reality is presented in numerous studies (Church 1999; Church & Murray 2008; Murray et al. 2010). Notably, Murray et al. (2010) created the method of implicit and explicit covering models. This concept can be applied to improve spatial representation and proposes modifications of the existing covering models, namely LSCP-Implicit, LSCP-Explicit, BCLP-Implicit, MCLP-Explicit, MCLP-Explicit, and BCLP-Explicit. These modifications ease the hard assumption of the traditional models and consider area-based demands. Besides, the application of meta-heuristic methods has gained more momentum recently. These are tabu search (Rajagopalan et al. 2008; Basar et al. 2011), genetic algorithms (Jia et al. 2007; Yang et al. 2007), simulated annealing (Syam & Cote 2010), and ant colony optimizations (Doerner et al. 2005; Liu et al. 2006).

Thus, location models were described, and a detailed overview of their application was provided. Location models have long been used to find optimal locations for emergency and healthcare facilities. Most researchers have developed location models using the maximizing population coverage method and the method of minimizing the number of facilities. Researchers have applied a wide range of techniques to solve the location problems, including multi-objective strategy, probabilistic approach, and hypercube queuing.

Reference List

Li, X, Zhao, Z, Zhu, X & Wyatt, T 2011, ‘Covering models and optimization techniques for emergency response facility location and planning: a review’, Mathematical Methods of Operations Research , vol. 74, no. 3, pp. 281-310.

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