Dynamic programming in the generalized bottleneck problem and the start point optimization

We consider one non-additive routing problem, which is a generalization of the well-known “bottleneck problem”. The parameter is assumed to be a positive number, the degree of which determines the weight of the corresponding stage of the displacement system. By varying the parameter, it is possible to make the initial or, on the contrary, the final stages of displacement dominant. The variant of aggregation of values with the above-mentioned weights corresponds to the ideological formulation of the “bottleneck problem”, but opens the possibility of investigating new versions of routing problems with constraints. It is assumed, however, that the statement of the problem is complicated by the dependence of values on the list of tasks and includes restrictions in the form of precedence conditions. In addition, in the interest of optimization, an arbitrary choice of the initial state from a given a priori set is allowed. For the construction, the apparatus of widely understood dynamic programming is used. The possibility of realizing a global extremum with any degree of accuracy under conditions when the set of possible initial states is not finite is investigated.