Clarification on the Use of Linear Programming and GIS for Land-Use Modelling

A recent article in this journal by E. Chuvieco demonstrated the potential of integrating optimization methods (speci® cally, linear programming) with geographical information systems (GIS) through the use of a small example in land-use planning. The bene® ts attainable from the combined use of these technologies are, in our opinion, beyond question. However, several misconceptions regarding the assumptions and use of linear programming (LP) models, as well as the interpretation of results obtained from such models, require clari® cation. 1. Linear programming and GIS The use of applied optimization models has provided immeasurable bene® ts in a vast number of areas. In particular, linear programming (LP) models have been utilized to make more-informed decisions in nearly all aspects of our society, including industry, government and social services (Gass 1975). A recent paper (Chuvieco 1993) described the bene® ts that could be achieved by combining LP models with geographical information systems (GIS) in the context of land-use planning. The integration of these two technologies, as described by Chuvieco, follows a threestage approach: (1) a GIS is utilized to assemble data for the aggregate land-use alternatives; (2) the data are used as input to an LP model which determines the optimal land-use pattern as measured by the selected objective function and subject to all requisite constraints; and (3) the GIS is again employed to invoke spatial considerations and other applicable criteria to disaggregate the LP optimal solution into a detailed land-use proposal. Merging of the technologies of LP and GIS as outlined by Chuvieco is, we believe, a signi® cant contribution to both ® elds; the former in terms of the expanded arena of applications of LP methodology, and the latter due to the impact of GIS in both the preand post-processing of the large quantities of data typically present in modern land-use decision-making scenarios. As correctly noted by Chuvieco, the alternative to such an aggregation/disaggregation approach greatly expands the size and complexity of the optimization model by incorporating binary (or 0-1) decision variables specifying the land-use decision for each unit within the planning area. The motivation behind this paper is the need to correct several misinterpretations in Chuvieco’s discussion regarding LP. Left unaddressed, these misconceptions tend to propagate within the scienti® c literature. Issues requiring clari® cation include 1365± 8816/97 $12 0́0 Ñ 1997 Taylor & Francis Ltd. J. L . Arthur and D. J. Nalle 398 both the conceptual foundations of LP models and the interpretation of the results obtained in the solution of such models. 2. Foundations of linear programming LP deals with the optimization (i.e., minimization or maximization) of a linear objective function in one or more decision variables which must satisfy a set of m (m>0) linear inequalities and/or equalities. Letting the vector x= (x1 , x2 ,. . .,xn) 3⁄4 represent the (n Ö 1) vector of decision variables, the model can be expressed as