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.
The cyclicity index of a matrix is the cyclicity index of its critical subgraph, namely, the subgraph of the adjacency graph which consists of all cycles of the maximal average weight. The cyclicity index of a graph is the least common multiple of the cyclicity indices of all its maximal strongly connected subgraphs, and the cyclicity index of a strongly connected graph is the least common divisor of the lengths of its (directed) cycles. In this paper we obtain the characterization of linear, possibly non-surjective, transformations of tropical matrices preserving the cyclicity index. It appears that non-bijective maps with these properties exist and all maps are exhausted by transposition, renumbering of vertices, Hadamard multiplication with a matrix of a certain special structure, and certain diagonal transformation. Moreover, only diagonal transformation can be non-bijective.