AGGREGATION THEORY IN OPTIMIZATION AND MARKOVIAN THEORY: A SURVEY AND AN ANALYSIS OF AGGREGATION IN MARKOV CHAINS (CLUSTER ANALYSIS)

DAVID FRANKLIN ROGERS, Purdue University

Abstract

Aggregation theory in quantitative business analysis is surveyed in this thesis with particular emphasis on aggregation in optimization problems and Markovian theory. In general, aggregation theory is the study of reducing large, possibly intractable models. Often, aggregation involves the solution of a model by using an auxiliary model which is reduced in size and/or complexity. The main components of the aggregation process are (1) clustering--determining which elements of a model to combine into single elements and how the clustered data is defined and (2) disclustering--deriving elements of a more refined model from reduced models. Aggregation theory is the study of these four components and their relationships and is typically examined with respect to particular models. Usually regarded as a method for handling computationally intractable problems, aggregation theory is useful for any situation in which the loss of a more detailed model may be rationalized by the decrease of model size or complexity. Furthermore, aggregation theory is useful for situations in which macro results are specifically desired. The term "aggregation" is used in several disciplines and has different connotations even within particular disciplines. The purpose of this dissertation is to (1) develop a unified framework for aggregation theory by defining its elements and their relationships, (2) present an overview of previous research in aggregation theory relating to optimization problems and dynamic systems, (3) present a more detailed study of the current status of aggregation theory relating to stochastic processes that have the Markovian property, (4) develop procedures to test the success of various clustering and disclustering routines for aggregation in Markov chains, and (5) introduce and test procedures for clustering and disclustering expressly designed for these structured Markov chains. The type of Markov chains considered in this thesis may naturally arise in inventory theory, queueing theory and manpower management. They are forms of a generalized random walk and appear as bands of elements in (typically sparse) matrix representations of Markov models. For very large intractable Markov models, the proposed methods are shown to be excellent for defining aggregate models that best represent (retain information for) the original model.

Degree

Ph.D.

Subject Area

Operations research

COinS