Tropical determinant on transportation polytope

Let ${\mathcal D}^{k,l}(m,n)$ be the set of all the integer points in the transportation polytope of $kn\times ln$ matrices with row sums $lm$ and column sums $km$. In this paper we find the sharp lower bound on the tropical determinant over the set ${\mathcal D}^{k,l}(m,n)$. This integer piecewise-linear programming problem in arbitrary dimension turns out to be equivalent to an integer non-linear (in fact, quadratic) optimization problem in dimension two. We also compute the sharp upper bound on a modification of the tropical determinant, where the maximum over all the transversals in a matrix is replaced with the minimum.Comment: 16 pages, 2 figure

On the job rotation problem

The job rotation problem (JRP) is the following: Given an $n \times n$ matrix $A$ over \Re \cup \{\ -\infty\ \}\ and $k \leq n$, find a $k \times k$ principal submatrix of $A$ whose optimal assignment problem value is maximum. No polynomial algorithm is known for solving this problem if $k$ is an input variable. We analyse JRP and present polynomial solution methods for a number of special cases

The tropical version of El Gamal Encryption

In this paper, we consider the new version of tropical cryptography protocol, i.e the tropical version of El Gamal encryption.  We follow the ideas and modify the clasical El Gamal encryption using tropical matrices and matrix power in tropical algebra. Then we also provide a toy example for the reader’s understanding.

Tropical Determinant on Transportation Polytopes

Let Dk,l(m, n)be the set of all the integer points in the transportation polytope of kn × ln matrices with row sums lm and column sums km. In this paper we find the sharp lower bound on the tropical determinant over the set Dk,l(m, n). This integer piecewise linear programming problem in arbitrary dimension turns out to be equivalent to an integer non-linear (in fact, quadratic) optimization problem in dimension two. We also compute the sharp upper bound on a modification of the tropical determinant, where the maximum over all the transversals in a matrix is replaced with the minimum

TIME COORDINATION OF SELECTED PUBLIC TRANSPORT LINES IN PROSTĚJOV

Příspěvek se zabývá modelováním synchronizace odjezdů spojů z přestupních zastávek pomocí Max-plus algebry ve Scilabu. Problém přestupu cestujících je formulován pomocí matematického aparátu Max-plus algebry. Následuje charakteristika MHD Prostějov a seznámení s operacemi Max-plus algebry, kterých je využito při modelování synchronizace linek MHD Prostějov.The paper deals with modelling of the synchronization of departures from the transfer stations using Max-plus algebra in Scilab. The problem of passengers’ transfer is formulated using mathematical Max-plus algebra. In the next part of the paper there is a characteristic of Prostějov public transport and an introduction to Max-plus algebra operations that are used for modelling of the synchronization of public transport lines in Prostějov

Network models and biproportional rounding for fair seat allocations in the UK elections

Systems for allocating seats in an election offer a number of socially and mathematically interesting problems. We discuss how to model the allocation process as a network flow problem, and propose a wide choice of objective functions and allocation schemes. Biproportional rounding, which is an instance of the network flow problem, is used in some European countries with multi-seat constituencies. We discuss its application to single seat constituencies and the inevitable consequence that seats are allocated to candidates with little local support. However, we show that variants can be selected, such as regional apportionment, to mitigate this problem. In particular, we introduce a parameter based family of methods, which we call Balanced Majority Voting, that can be tuned to meet the public's demand for local and global ``fairness''. Using data from the 2010 and 2015 UK General Elections, we study a variety of network models and implementations of biproportional rounding, and address conditions of existence and uniqueness

MODELLING SYNCHRONIZATION PUBLIC TRANSPORT LINES IN INTERCHANGE STOP

Příspěvek se zabývá modelováním synchronizace odjezdů spojů z přestupních zastávek pomocí max-plus algebry. Problém přestupu cestujících je formulován pomocí matematického aparátu max-plus algebry. Následuje charakteristika MHD Prostějov a seznámení s operacemi max-plus algebry, kterých je využito při modelování synchronizace linek MHD Prostějov.The paper deals with modelling of the synchronization of departures from the transfer stations using max-plus algebra. The problem of passengers’ transfer is formulated using mathematical max-plus algebra. In the next part of the paper there is a characteristic of Prostějov public transport and an introduction to max-plus algebra operations that are used for modelling of the synchronization of public transport lines in Prostějov

Optimal solution of assignment problem in Rojam manufacturing company

This research is about assignment problems consist to determine one of the special cases of transportation problems.The assignment problem is a of combinatorial optimization problems in the branch of optimization of or operation research in the mathematics.This research is trying to dig a lot of information by looking at the statistics of the management department with problems whose structure are identical with assignment problems in manufacturing company, which for separate jobs and the cost of assign each job to each person. This study comes out with two elements as an objective which is to minimize total cost of assignment problem in manufacturing company and to maximize the profit in manufacturing company.This study will conduct visit to a manufacturing company and the data will be collected from all employees in departments such as quality department, operation department, human resources department, marketing department and others.This study using quantitative methodology and the data will be analysed using the AMPL model to solve the assignment problem in the manufacturing company.The findings of this research are showing the vehicle assignment problem and the computational procedure, which lead to an optimal solution of the problem. It can reduce the total cost and increases the quality performance the workers

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