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Spatial partitioning is an important method in the geographic field. It uses a set of specific constraints or criteria to divide a finite geographic area into a lot of non-overlapping subareas (Wang et al., 2014). A delimiting service area is one of the most interesting topics in spatial partitioning, as nearly everyone needs a variety of services provided by different facilities in their daily life (Wang et al., 2018). In most cases, people choose their interested facilities by considering some factors, such as the geographic distribution, the service area, and the transportation convenience of facilities. The methodology of service area delimitation has been applied to various fields, such as the delineation of school catchment areas (Caro, 2004), market areas (Ríos-Mercado & Fernández, 2009), residential care facilities (Cheng et al., 2012), healthcare facilities (Steiner et al., 2015), and political districts (Ricca et al., 2013).
A variety of models have been proposed and developed to delineate service areas, in which the Voronoi diagram is a particularly famous one. The Voronoi diagram, named after Georgy Voronoi (Voronoi, 1908), is a method to partition space into several subareas (called Voronoi regions or Voronoi cells) from a predetermined set of points (called sites, seeds, or generators) by comparing their Euclidean distances, in which generators are abstracted as cities, hospitals, and schools, and the Voronoi regions represent their service areas. It has various names in different fields, such as Thiessen polygons in geography (Thiessen, 1911) and Dirichlet tessellation in mathematics (Dirichlet, 1850). In order to solve more real-world problems, many derived Voronoi diagrams have been developed, such as the weighted Voronoi diagram (Aurenhammer & Edelsbrunner, 1984; Mu, 2010; Gong et al., 2012), the centroidal Voronoi diagram (Du et al., 1999), the Voronoi treemap (Tian et al., 2015; Tian, 2021), and the city Voronoi diagram (Görke & Wolff, 2005).
A weighted Voronoi diagram can be divided into four major categories. They are the additively weighted Voronoi diagram, the multiplicatively weighted Voronoi diagram (MWVD), the additively weighted power Voronoi diagram, and the compoundly weighted Voronoi diagram (Okabe et al., 2000). The MWVD has a wider range of applications in service area delimitation than the other three weighted Voronoi diagrams for many reasons. First, each generator is located inside its own multiplicatively weighted Voronoi region. Second, all of the available space must be divided up by a set of generators, that is, there is no unallocated space in an WWVD. Third, it supports the notion of area-weight proportionality, that is, a generator with a larger weight dominates a larger multiplicatively weighted Voronoi region. Fourth, it puts distance and weight in a ratio relationship leading to that the changing of data units will not alter the result of the MWVD construction. Fifth, and the most important, several spatial interaction models can be expressed by the MWVDs, such as the Reilly model (Reilly, 1929), Converse model (Converse, 1949), and Huff model (Huff, 1964), which make it more attractive for service area delimitation. Mu (2004) classified the applications of the MWVD into four periods: (1) Early prototypes from the1800s to the 1940s. (2) Application in market and urban analysis from 1950s to 1970s. In this period, the MWVD was used as a mathematical and geometric solution in market and urban analysis (Huff & Jenks, 1967; Boots, 1975). (3) Parallel development in computer geometry and GIS from the 1980s to the 1990s. During this time, the MWVD began to be integrated in the GIS environment (Vincent & Daly, 1990). (4) From algorithm to implementation (1990s and beyond). In this period, the MWVD was widely used as a model to solve problems, such as Zhang et al (2018) introduced, to analyze ecosystem services coverage, and Mu and Wang (2006) utilized it to study spatial patterns of urban hierarchy in the United States.