The subject of research in the field of unmanned maritime and river navigation vessels is the new technological principles and models of unmanned navigation with high reliability and accuracy in the autonomous regimes on the specified path, with the use of satellite navigation, electronic charts and ground-based radio navigation systems. Non-crew vessels management from the shore requires the implementation of group management by the mobile objects, taking into account the update of the local maps of the shipping situation. When selecting the vessels routes, it is necessary to solve the operational problem of determining the shortest vessel route (Autonomous mobile object) at the specific navigation conditions, which directly defines the energy efficiency of the vessel in the voyage. The solution of the operational problem of selecting the shortest paths (routes) for a group of vessels moving in the direction of the specified targets with known coordinates located in a limited space is presented in the paper. A model and an optimization algorithm allowing, in comparison with existing solutions, to significantly expand the class of tasks by introducing the negative weights of individual sections of the path (graph arcs) and make the transition to solving high-dimensional problems are proposed. The key issue in the problem of automation and formation of the routes is the selection of the mathematical apparatus not only for the calculation of the shortest routes, but also for their recovery. For this purpose, a recursive optimization method based on the integer linear programming is proposed for high-dimensional tasks. The practical implementation of the proposed algorithm is demonstrated by the example of calculating a network model with a complex topology, using an iterative procedure for the program compiled in the MATLAB codes. It is shown that the computer model realized on its basis has properties of convergence of calculations and convenience of practical use. In contrast to the existing ones, the proposed model allows to remove the restrictions associated with the presence of negative weights and cycles at the network, to automate the calculations of the shortest paths in the places of branching by means of digital technologies. The correctness of the obtained solutions is confirmed, which makes it possible to use the model and algorithm as a component of the tasks tool designed to control the transport process.

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The operational task of automating the construction and routing of the network model with the known coordinates of the conditional goals set for a group of vessels to achieve them in the minimum time is solved; it makes it possible to obtain the reserves of running time necessary for saving fuel and energy, taking into account the load, the cost of cargo, transportation costs, logistics characteristics, etc. It is emphasized that in stormy weather conditions and vessel management in situations related to schedule correction, flexible operational decisions of dispatching services, made on the basis of numerical optimization methods using modern computing environments, are necessary. In this regard, the method of dynamic programming, implemented using the Bellman-Ford routing algorithm, which is supplemented by a recursive step-by-step optimization procedure that removes the limitation of the algorithm in the presence of inversely oriented edges with negative weights in the graph, is discussed in the paper. In the presence of negative weights, there are conditions for the appearance of a negative cycle in the graph, in which the practical implementation of the Bellman-Ford algorithm will become impossible due to an endless cycle of relaxations (attenuation) of the vertices weights included in this cycle. Hence, at a limited period of time for weighing all vertices (passes on all edges), the algorithm can give a knowingly false result. The proposed procedure for modifying the well-known Bellman-Ford algorithm eliminates this limitation and allows it to be used not only for estimating the shortest paths in a network containing arcs with negative weights, but also to detect negative cycles in it. The modified Bellman-Ford algorithm is implemented as a program compiled in MATLAB codes, and it is demonstrated by the example of automated construction and calculation of a network model containing both positive and negative edges (flows), using a recursive procedure of step-by-step optimization. It is shown that the proposed model, unlike the known models, eliminates the limitations caused by the presence of negative cycles in the network model, which makes it possible to automate the search for the shortest paths to conditional goals by the functional means of the MATLAB environment. The constructed computer model is simple and compact. The proposed algorithm and the recursive procedure are recommended for finding energy-efficient solutions for managing mobile objects in water transport.