Constructing And Selecting Adjacency Constraints

Abstract

Maintaining spatial integrity is an important concern in both the tactical and operational levels of forestry planning. Spatial relationships are typically represented by adjacency constraints. The number of needed adjacency constraints for even a small number of planning units, if not kept to a minimum, may be too large to include in a mathematical programming formulation. Several approaches have been developed to “minimize” the number of adjacency constraints used. These approaches involve either constraint subset selection or constraint aggregation. We demonstrate that with constraint aggregation the theoretical minimum of necessary adjacency constraints is one. However, the range of coefficients of one aggregated adjacency constraint is impractical for actual application. As an alternative, we explore the approach of identifying a minimal subset of a class of structural adjacency constraints. As a part of this approach, we develop a two-stage procedure to identify and fine tune a minimal subset of constraints for representing adjacency conditions so that there is no loss of spatial detail. Concise mathematical formulations are presented for each stage. This process is easy to implement and yields a relatively small number of “tight” adjacency constraints.

Résumé

Il est important de maintenir l’intégrité spatiale tant au niveau tactique qu’au niveau opérationel du planning forestier. En general, on répresente les rélations spatiales par des contraintes adjacentes. Même pour un petit nombre d’unités de planning, le nombre de contraintes adjacentes risque d’être trop large pour rajouter à une formulation mathématique de programmation. Plusieurs approches ont été développées pour “minimiser” le nombre de contraintes adjacentes utilisées. Ces approches comprennent soit des contraintes de sous-ensembles soit des contraintes d’agrégat. Nous prouvons qu’avec des contraintes d’agrégat le minimum théorique de contraintes adjacentes nécessaires est de un. Cependent, l’échelle de coefficients d’une contrainte adjacente agrégée n’est pas pratique pour les applications actuelles. Comme solution alternative, nous explorons une approche d’identification d’un sous-ensemble minime d’une classe de contraintes adjacentes structurelles. Dans le but de notre approche, nous développons un procédé à deux étapes pour identifier et perfectionner un sous-ensemble minime de contraintes pour représenter les conditions adjacentes afin de ne perdre aucun détail spatial. Des formules mathématiques précises sont présentées pour chaque étape. Le procéde est facile a mettre en pratique et résulte en un nombre relativement petit de contraintes adjacentes “serrées".

Key words::

• Forest planning
• adjacency constraints
• constraint aggregation
• integer programming and linear programming.

Mots-clés::

• Planning forestier
• contraintes adjacentes
• contraintes d’agregates
• programmation de nombre entiers
• programmation linéaire.

Additional information

Notes on contributors

Alan T. Murray

Alan Murray is a Research Fellow in the Department of Geography at the University of California, Santa Barbara. Dr. Murray received his Ph.D. degree in Geography and his M.A. degree in Statistics and Operations Research from the University of California, Sant|a Barbara. Dr. Murray’s research interests include environmental and land use management, optimization methods, and spatial decision support and geographical information system applications. His research articles have appeared or will appear in Journal of Urban Planning and Development, OR Spektrum, Canadian Journal of Forest Research, and Forest Science.

Richard L. Church

Richard Church is Professor of Geography at the University of California, Santa Barbara. He received his Ph.D. degree from the Johns Hopkins University in 1974 and has also taught at the University of Tennessee. Dr. Church has published in a wide variety of areas including logistics, location theory, environmental management, and public service delivery. His papers have appeared in Geographical Analysis, Transportation Science, Journal of Regional Science, Transportation Research, European Journal of Operations Research, Journal of the Operational Research Society, Annals of Operations Research, as well as others.