Mathematical Optimisation of Smart Irrigation Scheduling Using Climate and Soil-moisture Data: A Simulation Study for Potato Production

Background: Efficient irrigation scheduling requires simultaneous consideration of soil-water status, atmospheric demand and crop response. Sensor-based monitoring and optimisation methods can support this integration, but model outputs must be interpreted according to the quality and provenance of their inputs. Objective: This study develops a five-state compartmental framework for potato irrigation and evaluates threshold-based and optimisation-based irrigation scenarios using a 2025 rainfall forcing series associated with Elgeyo-Marakwet County, Kenya. Methods: The model represents soil water, root-zone water, plant tissue water, biomass and an environmental water compartment through ordinary differential equations. Soil-water retention follows the van Genuchten relation, reference evapotranspiration is represented with the Penman-Monteith formulation, and irrigation is formulated as a constrained quadratic tracking problem. Numerical integration was performed with MATLAB ode45 and Python solve_ivp; constrained optimisation was represented by an SLSQP-based implementation. The analysis is treated as a simulation study rather than as a field-validation study because the underlying station metadata, raw meteorological file and independent crop-calibration dataset were not available for verification. Results: For the reported loam-soil scenario, total available water was 120 mm m-1, rooting depth was 0.6 m and the adopted depletion fraction was 0.35, giving root-zone total available water of 72 mm and readily available water of 25.2 mm. The supplied simulations produced a rain-fed yield-equivalent output of 3.1 kg m-2 and a threshold-controlled output of 6.3 kg m-2, corresponding to a 103% increase within the model scenario. Conclusion: The simulations illustrate how climate and soil-moisture information can be incorporated into an irrigation-control framework, but the quantitative outputs should be interpreted as scenario results pending calibration and independent field validation.