Catastrophe Effects and Optimal Extensions of Transportation Flows in the Developing Urban System: A Review

Abstract

This paper deals with description of all possible minimal cost extensions of transportation flows in the developing urban system. The enumeration of such extensions is based on competitive exclusion behavioral rules for suppliers and demanders connected by means of minimal cost solution of the classical Linear Programming Transportation Problem: in such a way the complete set of all topological structures for all possible optimal extensions of transportation network is constructed in the form of all maximal tries without cycles with exactly m + n — 1 basic sells (where m is a number of suppliers and n is a number of demanders). It is important to underline that there exist only finite number of such trees. The catastrophe effect analysis in this enumeration set is based on the polyhedral form of general sensitivity analysis for classical minimal cost transportation problem: for each preset topological structure of the minimal cost flow there is a polyhedral cone K in the space of supply-demand and the polyhedral wedge W in the space of transportation costs, such that the choice of arbitrary supply-demand within the cone K and the choice of arbitrary set of transportation costs within the wedge W will give the existing minimal cost flow with a preset topological structure. The finite numbers of Cartesian products K xW represents the domains of structural stability of minimal cost flows with preset topological structures. Eventually the vector method of potentials is elaborated for the construction of the domains of structural stability K xW. As a practical numeric example the expanding set of bounded Christaller-Loesh and Beckman-McPherson Central Place System are considered in detail.