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

Water allocation in capacity-constrained canal trees with conveyance losses requires local limits and guaranteed service to be considered simultaneously. We develop a deterministic three-stage lexicographic model for a rooted tree with fixed routes. 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 the preceding optima. The nonnegative packing structure yields the Stage-1 optimum in closed form from capacity-to-full-load ratios; the loss-aware path operator is equivalent to node balance and determines unique gross flows. In five exact benchmarks, the maximum discrepancy between the closed-form and linear-programming solutions was 1.11×10⁻¹⁶, with zero operator–balance residual. Scaling tests with up to 500 users and 1022 edges yielded a maximum discrepancy of 2.22×10⁻¹⁶. In the three-period test, the temporal variation ranged from 0.40 to 1.05 over the Stage-2 optimal face, and Stage 3 attained 0.40. The price of fairness was 3.29% in the controlled Gone Abat Jap scenario. The results provide a model-specific analytical characterization and a reproducible computational framework for loss-aware allocation on fixed-route irrigation trees.