A Smooth Penalty Function Approach for Constrained Crop Planning Optimization

Abstract. Agricultural planning requires the efficient allocation of limited resources such as land, water, and capital among competing crops. Such decision-making problems are naturally constrained and may become nonlinear when yield response, production cost, and environmental effects are included. This paper presents a smooth penalty function approach for a constrained crop planning optimization model. The aim is to maximize agricultural profit while satisfying land availability, water requirement, budget, and non-negativity restrictions. The constrained problem is transformed into a sequence of smooth bound-constrained optimization problems by replacing the positive part of each constraint violation with a continuously differentiable approximation. The resulting formulation is suitable for gradient-based and quasi-Newton algorithms and avoids the nonsmooth behavior associated with classical exact penalty functions. A practical algorithm based on gradually increasing penalty parameters is described. A numerical crop planning example involving wheat, rice, and maize is used to illustrate the method. The corrected computational results show that the optimal allocation is feasible with respect to land, water, and budget restrictions, and that the water and land constraints become active at the optimum. The study indicates that smooth penalty methods provide a flexible and numerically stable framework for resource allocation problems in agriculture.

Keywords: Smooth penalty function, constrained optimization, crop planning, nonlinear programming, resource allocation, agricultural optimization.