A Zero-One Programming Model for Contiguous Land Acquisition

Abstract

The land acquisition problem is a spatial partitioning problem that involves selecting multiple parcels to be acquired for a particular land use. Three selection criteria are considered: total cost, total area, and spatial contiguity. Achieving contiguity or connectivity has been problematic in previous exact methods for land acquisition. Here we present a new zero-one programming model that enforces necessa y and sufficient conditions for achieving contiguity in discrete cell landscapes, independent of other spatial attributes such as compactness. Computational experience with several demonstration problems is reported, and results and extensions are discussed.Spatial partitioning problems, in which the landscape is divided into distinct regions or zones, are ubiquitous in geography and planning. They have appeared as "regionalization" problems (Nutenko 1970), "region building" problems (Cliff and Haggett 1970; Keane 1975), "districting" problems (Horn 1995; Williams 1995), "clustering" problems (Rosing and ReVelle 1986), "land acquisition/ allocation" problems (Wright, ReVelle, and Cohon 1983; Gilbert et al. 1985; Diamond and Wright 1991; Cova and Church ZOOO), and "reserve design" problems (Williams and ReVelle 1996). In addition, facility siting problems such as the p-median problem (ReVelle and Swain 1970), as well as the delineation of market areas in central place theory (Dacey 1965), also involve partitioning space into a set of distinct regions.In this paper we address a particular type of spatial partitioning problem, the land acquisition problem, which can be described in the following way. We are given a two-dimensional landscape that is represented as a set of n discrete parcels or cells. The landscape is to be partitioned into two regions by assigning each cell to one region or the other. One of the regions contains those cells selected for acquisition (for a particular land use, for example, green space) and the other region contains the remaining (unselected) cells.The land acquisition problem was introduced as an optimization problem by Wright, ReVelle, and Cohon (1983) who formulated a zero-one programming model with three objectives: minimize the total cost of selected cells; maximize total area; and maximize compactness. The last objective was achieved by minimizing the total external border length of selected cells. The authors employed multiobjective linear programming together with a branch-and-bound routine to find all pareto-optimal or Justin C. Williams is associate research professor of geography and environmental engineering at Johns Hopkins University.

Area aggregation in map generalisation by mixed-integer programming

“…The largest instance that we could solve optimally contained 15 nodes with the approach of Williams (2002) and 30 nodes with the approach of Shirabe (2005). However, for the special cases that compactness is either neglected or expressed by c shortest-path , we found a MIP of linear size.…”

“…To express this requirement by means of linear constraints is a nontrivial task. Williams (2002) and Shirabe (2005) have found different solutions for this in the context of spatial allocation problems. Their models allow constraints to be defined that forbid non-contiguous aggregates but do not exclude any contiguous aggregate.…”

Zone design of specific sizes using adaptive additively weighted Voronoi diagrams

“…If N units are to generate M zones, that is, if M < N, there are around M N partitions without imposing size and contiguity constraints (Williams 2002). On the one hand, even where the M value is small, the number of solutions will grow exponentially as N increases; and, on the other hand, when the zones must be connected, there is no general formula to determine the total number of solutions (possible partitions).…”

“…The following models have been developed within this line of research: integer linear programming (IP or ILP), which has been largely unsuccessful due to the difficulty of explicitly formalizing the contiguity constraint in algebraic terms; and mixed linear programming (MIP), which, from a computational point of view, is only efficient in problems involving sizes that are reduced, both in the number of basic units and in the number of zones to be generated. The most important MIP models are those created by Zoltners and Sinha (1983), applied to sales zone design; Cova and Church (2000) and Williams (2002), applied to land-use allocation or terrain acquisition; Shirabe (2005), used for generic zoning problems; and, finally, Solis et al (2009) and Rios-Mercado and Fernandez (2009), which is applied in designing commercial zones.…”

“…This process is carried out by clustering the units into a smaller number M of zones in order to create a new spatial structure that reflects a specific set of criteria. The issue of spatial partitioning is present in a wide spectrum of applications, as evinced by the literature on demarcating political-electoral districts (Horn 1995, Ricca and Simeone 1997e, Mehrotra et al 1998; delimitating socio-economic zones under a particular jurisdiction or administrative control (Openshaw and Alvanides 2001, Alvanides et al 2002, Martin 2003; delimitating market areas or sales 'territories' (Zoltners and Sinha 1983, Rios-Mercado and Fernandez 2009; delimitating school districts (Caro et al 2004, Ahmadi 2006; demarcating zones for delivering mobile domiciliary services, such as planning salt apportionment operations on roads (Muyldermans et al 2002) and demarcating police districts (D'Amico et al 2002); delimitating zones that receive energy resources (electrical power) (Bergey et al 2003, Tiede andStrobl 2006); and finally, delimitating zones for land-use allocation and/or land acquisition and apportionment (Cova and Church 2000, Aerts and Heuvelink 2002, Williams 2002.…”