Minimizing maximum lateness in two-stage projects by tropical optimization

We are considering a two-stage optimal scheduling problem, which involves two similar projects with the same starting times for workers and the same deadlines for tasks. It is required that the starting times for workers and deadlines for tasks should be optimal for the first-stage project and, under this condition, also for the second-stage project. Optimality is measured with respect to the maximal lateness (or maximal delay) of tasks, which has to be minimized. We represent this problem as a problem of tropical pseudoquadratic optimization and show how the existing methods of tropical optimization and tropical linear algebra yield a full and explicit solution for this problem.Comment: 29 page

Generalizations of Bounds on the Index of Convergence to Weighted Digraphs

We study sequences of optimal walks of a growing length, in weighted digraphs, or equivalently, sequences of entries of max-algebraic matrix powers with growing exponents. It is known that these sequences are eventually periodic when the digraphs are strongly connected. The transient of such periodicity depends, in general, both on the size of digraph and on the magnitude of the weights. In this paper, we show that some bounds on the indices of periodicity of (unweighted) digraphs, such as the bounds of Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, apply to the weights of optimal walks when one of their ends is a critical node.Comment: 17 pages, 3 figure

Optimal assignments with supervisions

International audienc

A bound for the rank-one transient of inhomogeneous matrix products in special case

summary:We consider inhomogeneous matrix products over max-plus algebra, where the matrices in the product satisfy certain assumptions under which the matrix products of sufficient length are rank-one, as it was shown in [6] (Shue, Anderson, Dey 1998). We establish a bound on the transient after which any product of matrices whose length exceeds that bound becomes rank-one

ON THE MARKOV CHAIN TREE THEOREM IN THE MAX ALGEBRA

A max-algebraic analogue of the Markov Chain Tree Theorem is presented, and its connections with the classical Markov Chain Tree Theorem and the max-algebraic spectral theory are investigated