Optimizing the starting point in a precedence constrained routing problem with complicated travel cost functions

We study the optimization of the initial state, route (a permutation of indices), and track in an extremal problem connected with visiting a finite system of megalopolises subject to precedence constraints where the travel cost functions may depend on the set of (pending) tasks. This problem statement is exemplified by the task to dismantle a system of radiating elements in case of emergency, such as the Chernobyl or Fukushima nuclear disasters. We propose a solution based on a parallel algorithm, which was implemented on the Uran supercomputer. It consists of a two-stage procedure: stage one determines the value (extremum) function over the set of all possible initial states and finds its minimum and also the point where it is achieved. This point is viewed as a base of the optimal process, which is constructed at stage two. Thus, optimization of the starting point for the route through megalopolises, connected with conducting certain internal tasks there, is an important element of the solution. To this end, we employ the apparatus of the broadly understood dynamic programming with elements of parallel structure during the construction of Bellman function layers.