Martin Gavalec

Solvability and unique solvability of max–min fuzzy equations☆

Abstract The problem of solvability and the problem of unique solvability of a fuzzy relation equation in an arbitrary max–min algebra are considered and corresponding necessary and sufficient conditions are presented. The results allow to solve both problems by an O (mnp) algorithm, where m,n,p are the dimensions of the corresponding relations in the equation. The existence of the greatest and of the least solution is also considered.

Computing matrix period in max-min algebra

Periodicity of matrix powers in max-min algebra is studied. The period of a matrix A is shown to be the least common multiple of the periods of at most n non-trivial strongly connected components in some threshold digraphs of A. An O(n3) algorithm for computing the period is described.

Computing orbit period in max-min algebra

Periodicity of vector orbits in max–min algebra is studied. It is proved that computing the coordinate-orbit period is NP-hard, while the orbit period can be computed in O(n4) time. A related problem of maximum sequence period is shown to be NP-complete.

Strong regularity of matrices in general max–min algebra☆

Abstract The problem of the strong regularity of a square matrix in a general max–min algebra is considered and a necessary and sufficient condition using the trapezoidal property is described. The results are valid without any restrictions on the underlying max–min algebra, concerning the density, or the boundedness. Previous results on this topic are special cases of the theorems presented in this paper.

Decision Making and Optimization

Well, someone can decide by themselves what they want to do and need to do but sometimes, that kind of person will need some decision making and optimization references. People with open minded will always try to seek for the new things and information from many sources. On the contrary, people with closed mind will always think that they can do it by their principals. So, what kind of person are you?

An O( n 2 ) algorithm for maximum cycle mean of Monge matrices in max-algebra

An O(n2) algorithm is described for computing the maximum cycle mean (eigenvalue) for n × n matrices, A = (aij) fulfilling Monge property, aij + akl ≤ ail + akj for any i < k, j < l. The algorithm computes the value λ(A) = max(ai1i2) + +ai2j3 + ... + aikj1/k over all cyclic permutations (i1, i2,..., ik) of subsets of the set {1,2,...,n). A similar result is presented for matrices with inverse Monge property. The standard algorithm for the general case works in O(n3) time.

Eigenspace structure of a max-prod fuzzy matrix

The eigenvectors of a fuzzy matrix correspond to steady states of a complex discrete-events system, characterized by the given transition matrix and fuzzy state vectors. The descriptions of the eigenspace (the set of all eigenvectors) for matrices in the max-min, max-ukasiewicz or max-drast fuzzy algebra have been presented in previous papers. The eigenspace of a fuzzy matrix in the max-prod algebra is investigated in this paper. First, necessary and sufficient conditions are shown under which the eigenspace restricted to increasing eigenvectors of a given matrix is non-empty, and the structure of the increasing eigenspace is described. Then, using simultaneous row and column permutations of the matrix, the complete characterization of the whole eigenspace structure of a given fuzzy matrix is shown. The details for matrices of order 3 are only presented. The method works analogously for square matrices of higher orders, with rapidly increasing complexity of the formulas.

Duality of optimization problems with generalized fuzzy relation equation and inequality constraints

Abstract A generalization of fuzzy relation equations and inequalities is introduced. An explicit method for solving optimization problems, the feasible set of which is described by a system of generalized fuzzy relation equations and inequalities, is considered. The objective function of the problems is equal to the maximum of continuous increasing functions of one variable. A duality concept for such problems is proposed. The problems are considered in a finitely dimensional space.

Computing an Eigenvector of a Monge Matrix in Max-Plus Algebra

The problem of finding one eigenvector of a given Monge matrix A in a max-plus algebra is considered. For a general matrix, the problem can be solved in O(n3) time by computing one column of the corresponding metric matrix Δ(Aλ), where λ is the eigenvalue of A. An algorithm is presented, which computes an eigenvector of a Monge matrix in O(n2) time.

The general trapezoidal algorithm for strongly regular max-min matrices

Abstract The problem of the strong regularity for square matrices over a general max–min algebra is considered. An O( n 2 log n ) algorithm for recognition of the strong regularity of a given n × n matrix is proposed. The algorithm works without any restrictions on the underlying max–min algebra, concerning the density, or the boundedness.