Districting Modeling with Exact Contiguity Constraints

Abstract

A classic problem in planning is districting, which aims to partition a given area into a specified number of subareas according to required criteria. Size, compactness, and contiguity are among the most frequently used districting criteria. While size and compactness may be interpreted differently in different contexts, contiguity is an unambiguous topological property. A district is said to be contiguous if all locations in it are ‘connected’—that is, one can travel between any two locations in the district without leaving it. This paper introduces a new integer-programming-based approach to districting modeling, which enforced contiguity constraints independently of any other criteria that might be additionally imposed. Three experimental models are presented, and tested with sample data on the forty-eight conterminous US states. A major implication of this paper is that the exact formulation of a contiguity requirement allows planners to address diverse sets of districting criteria.

New Models for Commercial Territory Design

“…A survey on general territory design problems (TDPs) can be found in Kalcsics et al (2005). Two important applications of territory design are political districting (Bozkaya et al 2003;Fleischmann and Paraschis 1988;Garfinkel and Nemhauser 1970;Hess et al 1965;Hojati 1996;Mehrotra et al 1998;Ricca and Simeone 2008;Shirabe 2009) and sales territory design (Drexl and Haase 1999;Hess and Samuels 1971;Marlin 1981;Zoltners and Sinha 2005). The characteristics of the problem addressed in this paper as detailed later, make it different to those studied in the work listed previously however.…”

“…On the other hand, in territory design problems, models dealing with connectivity constraints are usually approached through heuristics, as reviewed in Kalcsics et al (2005), although a few works do provide optimal solutions, for example Garfinkel and Nemhauser (1970) and Shirabe (2009). The former studied a districting problem with 39 BUs and seven territories, while the latter proposed a solution method to a similar problem using 48 basic units to a variable number of territories.…”

“…The former studied a districting problem with 39 BUs and seven territories, while the latter proposed a solution method to a similar problem using 48 basic units to a variable number of territories. The method proposed in Shirabe (2009) was proved tractable only for a small number of territories.…”

Location Science

“…where f ij is the flow from basic unit i to j and f c k ;j D 0, 8 j 2 N.c k / (Shirabe 2009). A simpler approach is to require that each district is a subtree of a shortest path tree T .c k / rooted at the district center c k , where the edge lengths typically correspond to road distances or are all assumed to be 1.…”

“…Assume that an approximate neighborhood graph has been computed using the approach in Haugland et al (2007). Using the contiguity constraints of Shirabe (2009), the resulting districting plans for cmp ud . / and cmp wd 2 .…”