Environment and Planning, 1973, volume 5, pages 397-408 On the solution of regional planning models via geometric programming J J Dinkel, G A Kochenberger Department of Management Science and Organizational Behavior, The Pennsylvania State University, University Park, Pennsylvania 16802, USA YSeppala Department of Computer Science, University of Helsinki, Helsinki, Finland Received 13 February 1973 Abstract. The purpose of this paper is to demonstrate the applicability of geometric programming to regional land-use planning models. The regional mode!s are of the type where the criterion is maximum accessibility, and the model is constrained by population limits on each district. After a brief discussion of geometric programming, the relationship of total accessibility models to geometric programs is developed and a method of obtaining numerical solutions is presented. Several models are analyzed using the geometric programming approach, including a game theoretic model which is used to generate decentralized plans. 1 Introduction Geometric programming, as developed by Duffin et al (1967), has had many and varied applications to engineering design problems. Recent extensions of this theory, to include more general functions [Duffin and Peterson, forthcoming (a) and (b)], have opened up many potential applications in the area of management science. Our purpose here is to demonstrate the applicability of geometric programming to regional land-use planning models. Because of the nature of the models, we will need to employ the methods of signed geometric programming; however, certain classes retain the global properties of ordinary geometric programming. We consider regional planning models which have the form of determining the distribution of population to various districts within a region, in such a way that they optimize some criterion function. A criterion which has been proposed is that of total accessibility of the region. In general, accessibility is inversely related to the distances among the districts of the region, and the total accessibility is the sum over all districts. Thus in geometric programming terms the models take the form of maximizing a posynomial (positive polynomial) subject to posynomial constraints. Section 2 contains a brief review of geometric programming and provides the basis for studying the regional planning models. Section 3 describes the maximization of posynomials by geometric programming. Applications of these results to regional planning models are given in section 4. 2 A review of geometric programming For the purposes of this paper the term geometric programming is used to refer to the theory as developed by Duffin et al (1967). As we proceed we will need to refer to extensions of this theory; these extensions will be referred to as signed geometric programs [Duffin and Peterson, forthcoming (a) and (b)]. Geometric programming deals with posynomial functions. A posynomial is defined as n m git) = i Q n'/", i=l /=1 where Q are positive real numbers, and the a tj are arbitrary real numbers. The