A model of “nonadditive” routing problem where the costs depend on the set of pending tasks

We consider a generalization of the bottleneck (minimax) routing problem. The problem is to successively visit a number of megalopolises, complicated by precedence of constraints imposed on the order of megalopolises visited and the fact that the cost functions (of movement between megalopolises and of interior tasks) may explicitly depend on the list of tasks that are not completed at the present time. The process of movement is considered to be a sequence of steps, which include the exterior movement to the respective megalopolis and the following completion of (essentially interior) jobs connected with the megalopolis. The quality of the whole process is represented by the maximum cost of steps it consists of; the problem is to minimize the mentioned criterion (which yields a minimax problem, usually referred to as a “bottleneck problem”). Optimal solutions, in the form of a route-track pair (a track, or trajectory, conforms to a specific instance of a tour over the megalopolises, which are numbered in accordance with the route; the latter is defined by the transposition of indices), are constructed through a “nonstandard” variant of the dynamic programming method, which allows to avoid the process of constructing of all the values of the Bellman function whenever precedence constraints are present.