Tree cutting algorithm for single commodity spatial market price equilibrium problems

Abstract

This thesis involves the study of an algorithm for single commodity spatial market price equilibrium problems. The conventional approach is the conversion to the related nonlinear programming problem. When it is the multicommodity problem, the integrability assumption about supply and demand functions is needed to induce a so-called quasi-welfare function. The presented algorithm works with a linear objective function which represents the total cost of transportation, while it considers a excess supply function in its derivation of the desired excess supply at each node. And it proceeds to find an equilibrium solution while it implicitly cuts some basic arcs when certain conditions are met. This approach has advantages in that it stops with an exact solution within a finite number of iterations, and the integrability assumption is not needed when it is extended to the case of multicommodity problem. This tree cutting algorithm is basically a parametric programming technique, which is a natural extension of the study of Srinivasan and Thompson [23]. The operation which finds an adjacent dual basis provides the linear network problem with a direction so that it can be solved by the parametric network simplex method. It is shown that the tree cutting algorithm with adjustment routine converges to an optimal solution in a finite number of iterations. All the proofs are presented for the case of linear excess supply functions. Computational results indicates that the suggested algorithm converges quite well for the linear case. Finally, the comments about the problems which may arise in applying the algorithm to more extended problems are suggested.