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Three-Stage Lexicographic Allocation on Directed Irrigation Trees with Conveyance Losses and Capacity Constraints: A Closed-Form Stage-1 Fairness Guarantee and Operator–Balance Equivalence

Adilbay KudaybergenovDepartment of Information Security, National University of Uzbekistan named after Mirzo Ulugbek, 4 University St., Tashkent 100174, UzbekistanAnvar KabulovDepartment of Information Security, National University of Uzbekistan named after Mirzo Ulugbek, 4 University St., Tashkent 100174, UzbekistanMukhabbad KazimbetovaDepartment of Information Systems and Technologies, Karakalpak State University named after Berdakh, 1 Ch. Abdirov St., Nukus 230112, UzbekistanGulsara AmetovaDepartment of Digital Technologies and Artificial Intelligence, “Tashkent Institute of Irrigation and Agricultural Mechanization Engineers” National Research University, 39 Kari Niyaziy St., Tashkent 100000, UzbekistanJadira IspanovaDepartment of Information Systems and Technologies, Karakalpak State University named after Berdakh, 1 Ch. Abdirov St., Nukus 230112, Uzbekistan

2026en

ABI

Annotatsiya

Water allocation in lossy canal trees must respect local capacities and a service guarantee. We develop a deterministic three-stage lexicographic model for a rooted, fixed-route tree: Stage 1 maximizes the minimum service ratio, Stage 2 maximizes weighted seasonal satisfaction subject to that guarantee, and Stage 3 minimizes temporal service-ratio variation while preserving both optima. The nonnegative packing structure yields the Stage-1 optimum in closed form from capacity-to-full-load ratios, and the loss-aware path operator equals node balance. Across 400 randomized instances, 140 with capacities not tied to their own loads, and 60 more with load-independent scales, the closed form agreed with linear programming (LP) to 2.2 × 10−16, confirming every bottleneck the formula named. On 500-user trees, Stage 1 took 12.1 ms against 93.1 ms for the LP. Stage 3 lowered the temporal variation only where multiple Stage-2 optima were detected. Of five allocation rules, only the hierarchy, equal proportional allocation, and full leximin always preserved the guarantee; of those three, the hierarchy delivered the most water in the median. In a controlled Gone Abat Jap scenario, ±10% demand perturbations moved the binding resource from source to reach in 61% of draws. The results verify a model-specific characterization, not field performance.

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Identifikatorlar

DOI: 10.3390/appliedmath6100178

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