Stability and Spectral Properties in the Max Algebra with Applications in Ranking Schemes

This thesis is concerned with the correspondence between the max algebra and non-negative linear algebra. It is motivated by the Perron-Frobenius theory as a powerful tool in ranking applications. Throughout the thesis, we consider max-algebraic versions of some standard results of non-negative linear algeb- ra. We are specifically interested in the spectral and stability properties of non-negative matrices. We see that many well-known 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 well-known results in both non-negative 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 multi-step difference equations. We show how results for matrix polynomials in the conventional algebra carry over naturally to the max-algebraic setting. We also consider an extension of the so-called 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 non-negative matrices in the context of dynamical systems. We are motivated by the relation of P-matrices and positive stability of non-negative 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 P-matrices 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 multi-criteria 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 max-algebraic version for the single criterion case. We discuss how to treat multiple criteria within the max-algebraic framework. We address this generalisation by considering the multi-criteria AHP as a multi-objective optimisation problem. We consider three approaches within the framework of multi-objective 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 so-called Matrix Tree Theorem for Markov Chains. It connects the spectral theory of non-negative 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

The Markov chain tree theorem and the state reduction algorithm in commutative semirings

We extend the Markov chain tree theorem to general commutative semirings, and we generalize the state reduction algorithm to commutative semifields. This leads to a new universal algorithm, whose prototype is the state reduction algorithm which computes the Markov chain tree vector of a stochastic matrix.Comment: 13 page

The Markov Chain Tree Theorem in commutative semirings and the State Reduction Algorithm in commutative semifields

We extend the Markov Chain Tree Theorem to general commutative semirings, and we generalize the State Reduction Algorithm to general commutative semifields. This leads to a new universal algorithm, whose prototype is the State Reduction Algorithm which computes the Markov chain tree vector of a stochastic matrix

The Analytic Hierarchy Process, Max Algebra and Multi-objective Optimisation

The Analytic Hierarchy Process (AHP) is widely used for decision making involving multiple criteria. Elsner and van den Driessche introduced a max-algebraic approach to the single criterion AHP. We extend this to the multi-criteria AHP, by considering multi-objective generalisations of the single objective optimisation problem solved in these earlier papers. We relate the existence of globally optimal solutions to the commutativity properties of the associated matrices; we relate min-max optimal solutions to the generalised spectral radius; and we prove that Pareto optimal solutions are guaranteed to exist.Comment: 1 figur

Stability and Spectral Properties in the Max Algebra with Applications in Ranking Schemes

This thesis is concerned with the correspondence between the max algebra and non-negative linear algebra. It is motivated by the Perron-Frobenius theory as a powerful tool in ranking applications. Throughout the thesis, we consider max-algebraic versions of some standard results of non-negative linear algeb- ra. We are specifically interested in the spectral and stability properties of non-negative matrices. We see that many well-known 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 well-known results in both non-negative 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 multi-step difference equations. We show how results for matrix polynomials in the conventional algebra carry over naturally to the max-algebraic setting. We also consider an extension of the so-called 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 non-negative matrices in the context of dynamical systems. We are motivated by the relation of P-matrices and positive stability of non-negative 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 P-matrices 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 multi-criteria 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 max-algebraic version for the single criterion case. We discuss how to treat multiple criteria within the max-algebraic framework. We address this generalisation by considering the multi-criteria AHP as a multi-objective optimisation problem. We consider three approaches within the framework of multi-objective 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 so-called Matrix Tree Theorem for Markov Chains. It connects the spectral theory of non-negative 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

Stability and Spectral Properties in the Max Algebra with Applications in Ranking Schemes

This thesis is concerned with the correspondence between the max algebra and non-negative linear algebra. It is motivated by the Perron-Frobenius theory as a powerful tool in ranking applications. Throughout the thesis, we consider max-algebraic versions of some standard results of non-negative linear algeb- ra. We are specifically interested in the spectral and stability properties of non-negative matrices. We see that many well-known 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 well-known results in both non-negative 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 multi-step difference equations. We show how results for matrix polynomials in the conventional algebra carry over naturally to the max-algebraic setting. We also consider an extension of the so-called 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 non-negative matrices in the context of dynamical systems. We are motivated by the relation of P-matrices and positive stability of non-negative 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 P-matrices 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 multi-criteria 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 max-algebraic version for the single criterion case. We discuss how to treat multiple criteria within the max-algebraic framework. We address this generalisation by considering the multi-criteria AHP as a multi-objective optimisation problem. We consider three approaches within the framework of multi-objective 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 so-called Matrix Tree Theorem for Markov Chains. It connects the spectral theory of non-negative 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

Stability and Spectral Properties in the Max Algebra with Applications in Ranking Schemes

This thesis is concerned with the correspondence between the max algebra and non-negative linear algebra. It is motivated by the Perron-Frobenius theory as a powerful tool in ranking applications. Throughout the thesis, we consider max-algebraic versions of some standard results of non-negative linear algeb- ra. We are specifically interested in the spectral and stability properties of non-negative matrices. We see that many well-known 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 well-known results in both non-negative 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 multi-step difference equations. We show how results for matrix polynomials in the conventional algebra carry over naturally to the max-algebraic setting. We also consider an extension of the so-called 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 non-negative matrices in the context of dynamical systems. We are motivated by the relation of P-matrices and positive stability of non-negative 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 P-matrices 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 multi-criteria 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 max-algebraic version for the single criterion case. We discuss how to treat multiple criteria within the max-algebraic framework. We address this generalisation by considering the multi-criteria AHP as a multi-objective optimisation problem. We consider three approaches within the framework of multi-objective 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 so-called Matrix Tree Theorem for Markov Chains. It connects the spectral theory of non-negative 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

Stability and Spectral Properties in the Max Algebra with Applications in Ranking Schemes

This thesis is concerned with the correspondence between the max algebra and non-negative linear algebra. It is motivated by the Perron-Frobenius theory as a powerful tool in ranking applications. Throughout the thesis, we consider max-algebraic versions of some standard results of non-negative linear algeb- ra. We are specifically interested in the spectral and stability properties of non-negative matrices. We see that many well-known 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 well-known results in both non-negative 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 multi-step difference equations. We show how results for matrix polynomials in the conventional algebra carry over naturally to the max-algebraic setting. We also consider an extension of the so-called 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 non-negative matrices in the context of dynamical systems. We are motivated by the relation of P-matrices and positive stability of non-negative 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 P-matrices 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 multi-criteria 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 max-algebraic version for the single criterion case. We discuss how to treat multiple criteria within the max-algebraic framework. We address this generalisation by considering the multi-criteria AHP as a multi-objective optimisation problem. We consider three approaches within the framework of multi-objective 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 so-called Matrix Tree Theorem for Markov Chains. It connects the spectral theory of non-negative 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