The Max‐P‐Regions Problem

In this paper, we introduce a new spatially constrained clustering problem called the max‐‐regions problem. It involves the clustering of a set of geographic areas into the maximum number of homogeneous regions such that the value of a spatially extensive regional attribute is above a predefined threshold value. We formulate the max‐‐regions problem as a mixed integer programming (MIP) problem, and propose a heuristic solution.

[1]  Felipe Caro,et al.  School redistricting: embedding GIS tools with integer programming , 2004, J. Oper. Res. Soc..

[2]  W. Macmillan,et al.  Redistricting in a GIS environment: An optimisation algorithm using switching-points , 2001, J. Geogr. Syst..

[3]  J. Byfuglien,et al.  Region-Building — a Comparison of Methods , 1973 .

[4]  F. Glover.  HEURISTICS FOR INTEGER PROGRAMMING USING SURROGATE CONSTRAINTS , 1977 .

[5]  Richard L. Church,et al.  The p-Regions Problem. p-区域问题 , 2011 .

[6]  Robert Haining,et al.  Regionalisation Tools for the Exploratory Spatial Analysis of Health Data , 1997 .

[7]  John C. Gower,et al.  Maximum Split Clustering Under Connectivity Constraints , 2003, J. Classif..

[8]  Bruno Simeone,et al.  Local search algorithms for political districting , 2008, Eur. J. Oper. Res..

[9]  Raúl Ramos Lobo,et al.  Supervised regionalization methods: A survey , 2006 .

[10]  Peter Haggett,et al.  On the Efficiency of Alternative Aggregations in Region-Building Problems , 1970 .

[11]  Peter J. Taylor,et al.  Some implications of the spatial organization of elections , 1973 .

[12]  Stan Openshaw,et al.  A geographical solution to scale and aggregation problems in region-building, partitioning and spatial modelling , 1977 .

[13]  Christophe Perruchet,et al.  Constrained agglomerative hierarchical classification , 1983, Pattern Recognit..

[14]  J. William Ahwood,et al.  CLASSIFICATION , 1931, Foundations of Familiar Language.

[15]  Wang Yin Chai,et al.  A Multiobjective Hybrid Metaheuristic Approach for GIS-based Spatial Zoning Model , 2004, J. Math. Model. Algorithms.

[16]  Alan T. Murray,et al.  Integrating attribute and space characteristics in choropleth display and spatial data mining , 2000, Int. J. Geogr. Inf. Sci..

[17]  K E Rosing,et al.  Optimal Clustering , 1986 .

[18]  Roger Sammons.  A simplistic approach to the redistricting problem , 1978 .

[19]  Pip Forer.  ELEMENTS OF SPATIAL STRUCTURE: A QUANTITATIVE APPROACH , 1977 .

[20]  S. Nagel.  Simplified Bipartisan Computer Redistricting , 1965 .

[21]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[22]  J D Thoreson,et al.  Computers in behavioral science. Legislative districting by computer simulation. , 1967, Behavioral science.

[23]  A. Zoltners,et al.  Sales Territory Alignment: A Review and Model , 1983 .

[24]  W. Vickrey.  On the Prevention of Gerrymandering , 1961 .

[25]  John M. Liittschwager,et al.  Legislative districting by computer , 1969 .

[26]  Stan Openshaw,et al.  Optimal Zoning Systems for Spatial Interaction Models , 1977 .

[27]  Bruno Simeone,et al.  A spanning tree heuristic for regional clustering , 1995 .

[28]  S Openshaw,et al.  Algorithms for Reengineering 1991 Census Geography , 1995, Environment & planning A.

[29]  Mark E. T. Horn,et al.  Solution Techniques for Large Regional Partitioning Problems , 2010 .

[30]  David Rossiter,et al.  Program GROUP: The Identification of All Possible Solutions to a Constituency-Delimitation Problem , 1981 .

[31]  A. Ferligoj,et al.  Clustering with relational constraint , 1982 .

[32]  W. Macmillan.  Optimization modelling in a GIS framework:The problem of political redistricting. , 1994 .

[33]  Corina da Costa Freitas,et al.  Efficient regionalization techniques for socio‐economic geographical units using minimum spanning trees , 2006, Int. J. Geogr. Inf. Sci..

[34]  Michelle H Browdy.  Simulated Annealing: An Improved Computer Model for Political Redistricting , 1990 .

[35]  W. Bunge,et al.  Gerrymandering, Geography, and Grouping , 1966 .

[36]  C. Margules,et al.  An Adjacency Constraint in Agglomerative Hierarchical Classifications of Geographic Data , 1985 .

[37]  Lawrence D. Bodin.  A DISTRICTING EXPERIMENT WITH A CLUSTERING ALGORITHM , 1973 .

[38]  J Moshman,et al.  A REDISTRICTING ALGORITHM APPLIED TO GEOGRAPHIC REORGANIZATION OF CIRCUIT COURTS , 1973, Annals of the New York Academy of Sciences.

[39]  A. D. Gordon.  A survey of constrained classification , 1996 .

[40]  Justin C. Williams.  POLITICAL REDISTRICTING: A REVIEW , 2005 .

[41]  David Martin,et al.  The Application of Zone-Design Methodology in the 2001 UK Census , 2001 .

[42]  R. Webster,et al.  COMPUTER‐BASED SOIL MAPPING OF SMALL AREAS FROM SAMPLE DATA , 1972 .

[43]  John M. Liittschwager.  THE IOWA REDISTRICTING SYSTEM , 1973 .

[44]  Rajan Batta,et al.  A simulated annealing approach to police district design , 2002, Comput. Oper. Res..

[45]  Gilbert Laporte,et al.  Solving a home-care districting problem in an urban setting , 2003, J. Oper. Res. Soc..

[46]  M Keane,et al.  The Size of the Region-Building Problem , 1975 .

[47]  G. W. Milligan,et al.  An examination of procedures for determining the number of clusters in a data set , 1985 .

[48]  Cort J. Willmott,et al.  Spatial statistics and models , 1984 .

[49]  B. Simeone,et al.  Clustering on trees , 1997 .

[50]  R. Keith Semple,et al.  Classification in Human Geography , 1984 .

[51]  Gilbert Laporte,et al.  A tabu search heuristic and adaptive memory procedure for political districting , 2003, Eur. J. Oper. Res..

[52]  P. Legendre,et al.  Developments in Numerical Ecology , 1988 .

[53]  Elena Fernández,et al.  A reactive GRASP for a commercial territory design problem with multiple balancing requirements , 2009, Comput. Oper. Res..

[54]  Fionn Murtagh,et al.  Contiguity-constrained clustering for image analysis , 1992, Pattern Recognit. Lett..

[55]  Takeo Yamada,et al.  A mini–max spanning forest approach to the political districting problem , 2009, Int. J. Syst. Sci..

[56]  Manfred M. Fischer,et al.  Regional taxonomy: A comparison of some hierarchic and non-hierarchic strategies , 1980 .

[57]  Richard L. Church,et al.  Contiguity Constraints for Single‐Region Site Search Problems , 2010 .

[58]  A Gordon,et al.  Classification, 2nd Edition , 1999 .

[59]  Fionn Murtagh,et al.  A Survey of Algorithms for Contiguity-Constrained Clustering and Related Problems , 1985, Comput. J..

[60]  Philip M. Lankford.  Regionalization: Theory and Alternative Algorithms , 2010 .