Pendekatan Model Reduced Gradient – Mixed Non Linier untuk Optimasi Pemeliharaan Kebun Kopi

Abstract

Optimizing the allocation of limited workers for the maintenance of smallholder coffee farms using a Mixed-Integer Nonlinear Programming (MINLP) model results in high computational complexity. Unlike other similar studies, the MINLP model in this research focuses on the number of human workers allocated according to optimal time windows based on daily productivity. This research proposes a hybrid method called the Approximation Reduced Gradient Hybrid Optimizer (ARGHO), which combines Outer Approximation (OA) for the master MILP and Generalized Reduced Gradient (GRG) for the sub-NLP with adaptive tolerance (gap-based decay) to efficiently explore the global solution space of MINLP models. OA decomposition breaks down the MINLP into a linear master problem and GRG integration refines the continuous variables in the non-linear subproblem sections when the discrete decision vector from the master problem is fixed, and returns gradient information to construct OA-cuts that strengthen the master problem in the next iteration until the gap between the Upper Bound (UB) and Lower Bound (LB) shrinks within an adaptive tolerance based on Gap-based Decay. The testing was conducted using field data consisting of 538 land blocks (up to 1,027,663 variables and 859,267 constraints) divided into 11 instances. Based on the numerical results obtained, the proposed method demonstrates the ability to improve efficiency and achieve a good balance between speed and solution quality, especially for small-to-medium sizes compared to other methods. Computational time decreased by 64% compared to the GBD and B&B methods, resulting in a tighter gap on a large scale (538 areas: GBD 0.22%; B&B 0.18%) with slightly higher time (≈6,922–6,935 seconds) and a greater number of nodes/iterations. These findings indicate that the adaptive tolerance mechanism is effective in accelerating convergence without sacrificing solution quality control.