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

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

Switched max-plus linear-dual inequalities for makespan minimization: the case study of an industrial bakery shop

In this paper, an industrial bakery shop is modeled by switched max-plus linear-dual inequalities (SLDIs). SLDIs are timed discrete event systems suitable for describing flow shops with time-window constraints and switching operating modes, where each mode corresponds to a job type. We consider the scheduling problem of minimizing the makespan of the shop, and we show that the application of methods based on the max-plus algebra leads to a faster solution compared to standard techniques. The results of the paper are general, in the sense that they can be applied to any permutation flow shop with time-window constraints.Comment: conference, 3 figures. Some typos in the Appendix and Introduction correcte

Topological distances and geometry over the symmetrized Omega algebra

[EN] The aim of this paper is to study some topological distances properties, semidendrites and convexity on th symmetrized omega algebra. Furthermore, some properties and exponents on the symmetrized omega algebra are introduced.Alqahtani, M.; Özel, C.; Zekraoui, H. (2020). Topological distances and geometry over the symmetrized Omega algebra. Applied General Topology. 21(2):247-264. https://doi.org/10.4995/agt.2020.13049OJS247264212A. C. F. Bueno, On the exponential function of right circulant matrices, International Journal of Mathematics and Scientific Computing 3, no. 2 (2013).L. Hörmander, Notions of convexity, Progress in Mathematics 127, Birkh¨auser, Boston- Basel-Berlin (1994).S. Khalid Nauman, C. Ozel and H. Zekraoui, Abstract Omega algebra that subsumes min and max plus algebras, Turkish Journal of Mathematics and Computer Science 11 (2019) 1-10.G. L. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: a brief introduction, Journal of Mathematical Sciences 140, no. 3 (2007), 426-444. https://doi.org/10.1007/s10958-007-0450-5D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161, American Mathematical Society, 2015. https://doi.org/10.1090/gsm/161C. Ozel, A. Piekosz, E. Wajch and H. Zekraoui, The minimizing vector theorem in symmetrized max-plus algebra, Journal of Convex Analysis 26, no. 2 (2019), 661-686.J.-E. Pin, Tropical semirings, Idempotency (Bristol, 1994), 50-69, Publ. Newton Inst., vol. 11, Cambridge Univ. Press, Cambridge, 1998. https://doi.org/10.1017/CBO9780511662508.004I. Simon, Recognizable sets with multiplicities in the tropical semiring, in: Mathematical Foundations of Computer Science (Carlsbad, 1988), Lecture Notes in Computer Science, vol. 324, Springer, Berlin, 1988, pp. 107-120. https://doi.org/10.1007/BFb001713

The level set method for the two-sided eigenproblem

We consider the max-plus analogue of the eigenproblem for matrix pencils Ax=lambda Bx. We show that the spectrum of (A,B) (i.e., the set of possible values of lambda), which is a finite union of intervals, can be computed in pseudo-polynomial number of operations, by a (pseudo-polynomial) number of calls to an oracle that computes the value of a mean payoff game. The proof relies on the introduction of a spectral function, which we interpret in terms of the least Chebyshev distance between Ax and lambda Bx. The spectrum is obtained as the zero level set of this function.Comment: 34 pages, 4 figures. Changes with respect to the previous version: we explain relation to mean-payoff games and discrete event systems, and show that the reconstruction of spectrum is pseudopolynomia

Tropical linear algebra with the Lukasiewicz T-norm

The max-Lukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics "a+b"=max(a,b) and "ab"=max(0,a+b-1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Lukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Lukasiewicz semiring.Comment: 27 page