Benek Gursoy, Buket (2013) Stability and Spectral Properties in the Max Algebra with Applications in Ranking Schemes. PhD thesis, National University of Ireland Maynooth.
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Abstract
This thesis is concerned with the correspondence between the max algebra and nonnegative linear algebra. It is motivated by the PerronFrobenius theory as a powerful tool in ranking applications. Throughout the thesis, we consider maxalgebraic versions of some standard results of nonnegative linear algeb ra. We are specifically interested in the spectral and stability properties of nonnegative matrices. We see that many wellknown theorems in this context extend to the max algebra. We also consider how we can relate these results to ranking applications in decision making problems. In particular, we focus on deriving ranking schemes for the Analytic Hierarchy Process (AHP). We start by describing fundamental concepts that will be used throughout the thesis after a general introduction. We also state wellknown results in both nonnegative linear algebra and the max algebra. We are next interested in the characterisation of the spectral properties of mat rix polynomials. We analyse their relation to multistep difference equations. We show how results for matrix polynomials in the conventional algebra carry over naturally to the maxalgebraic setting. We also consider an extension of the socalled Fundamental Theorem of Demography to the max algebra. Using the concept of a multigraph, we prove that a number of inequalities related to the spectral radius of a matrix polynomial are also true for its largest max eigenvalue. We are next concerned with the asymptotic stability of nonnegative matrices in the context of dynamical systems. We are motivated by the relation of Pmatrices and positive stability of nonnegative matrices. We discuss how equivalent conditions connected with this relation echo similar results over the max algebra. Moreover, we consider extensions of the properties of sets of Pmatrices to the max algebra. In this direction, we highlight the central role of the max version of the generalised spectral radius. We then focus on ranking applications in multicriteria decision making prob lems. In particular, we consider the Analytic Hierarchy Process (AHP) which is a method to deal with these types of problems. We analyse the classical Eigenvalue Method (EM) for the AHP and its maxalgebraic version for the single criterion case. We discuss how to treat multiple criteria within the maxalgebraic framework. We address this generalisation by considering the multicriteria AHP as a multiobjective optimisation problem. We consider three approaches within the framework of multiobjective optimisation, and use the optimal solution to provide an overall ranking scheme in each case. We also study the problem of constructing a ranking scheme using a combi natorial approach. We are inspired by the socalled Matrix Tree Theorem for Markov Chains. It connects the spectral theory of nonnegative matrices with directed spanning trees. We prove that a similar relation can be established over the max algebra. We consider its possible applications to decision making problems. Finally, we conclude with a summary of our results and suggestions for future extensions of these.
Item Type:  Thesis (PhD) 

Keywords:  Stability and Spectral Properties; Max Algebra; Ranking Schemes; 
Academic Unit:  Faculty of Science and Engineering > Research Institutes > Hamilton Institute 
Item ID:  4327 
Depositing User:  IR eTheses 
Date Deposited:  19 Apr 2013 13:42 
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