Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 52. no. 3. P. 75 – 94. ISSN 2079-6641

MATHEMATICAL MODELING
https://doi.org/10.26117/2079-6641-2025-52-3-75-94
Research Article
Full text in Russian
MSC 90C26, 05C21, 05C85

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The Method of Rank Bush Optimization for the Large Branched Stream Networks with Concave Objective Function

M. B. Abazokov^{\ast}

Institute of Applied Mathematics and Automation of the Kabardino-Balkar Scientific Center of RAS, 360000, Nalchik, Shortanova st., 89A, Russia

Abstract. The design and optimization of large-scale branched stream networks for regional and interregional water supply is an urgent task due to the scarcity of water resources. Optimizing such networks is NP-complete “essentially-multi-extremality”problem. To solve this problem, the directly optimizing rankbased measures (P-optimization) was developed. The method is based on the principle of system optimality: “Every part of the optimal system is optimal (if the boundary conditions with the rest of the network are fixed)”. However, for large values of the rank P, this approach requires considerable computational time, since the computational time for solving the problem is exponential in rank P. In order to reduce the dimensionality of the problem, a method grounded on the bush optimization in the construction of the large-scale water distribution network with a high optimality rank has been developed and proposed. This technique restricts the solution search space and thus allows achieving the higher rankings. Thus, the designed approach associates each network vertex with a corresponding subgraph of the base graph (a bush). The subgraph is marked in the current spanning tree and has a certain dimensionality. The direct optimization applied to each bush provides a higher order of optimality than when it is applied to the entire network. At the same time, the total bush optimization time does not exceed the entire network optimization time. This paper proposes the method that provides to uniquely constructing a bush with a predetermined marginal number of vertexes. In addition, the developed method does not require specifying a range of the number of vertices; it only needs to specify a predetermined marginal number of bush vertices and the rank of optimization. The direct rank and bush optimization methods presented here are intended for use in the design and optimization of large-scale regional and interregional water distribution networks.

Key words: large water distribution networks, structure optimization, ranks of extrema, rank optimization method, bush optimization method, computer-aided design, regional and interregional water distribution network, dimensionality reduction.

Received: 11.09.2025; Revised: 04.10.2025; Accepted: 28.10.2025; First online: 10.11.2025

For citation. Abazokov M. B. The method of rank bush optimization for the large branched stream networks with concave objective function. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 52: 3, 75-94. EDN: QNIKKU. https://doi.org/10.26117/2079-6641-2025-52-3-75-94.

Funding. The scientific research was carried out without financial support from foundations.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. Author is solely responsible for providing the final version of the article in print.

^{\ast}Correspondence: E-mail: Abazokov.Mukhammed@yandex.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Abazokov M. B., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

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