A nested goal programming model integrated with an improved genetic bee colony algorithm supported by machine learning methods

Abstract

Formulating engine oil additives is challenging because it requires simultaneously optimizing production efficiency, cost, and compliance with strict quality standards. This study presents an advanced optimization framework for 10W-40 API SL engine oil that combines a nested goal programming model with machine learning (ML) techniques to predict production rates and quality metrics that cannot be expressed in closed-form equations. To address the inability of conventional ML approaches to generate novel additive combinations, we propose an enhanced genetic bee colony algorithm incorporating arithmetic crossover, Makinen–Periaux–Toivanen mutation operators, and a Cauchy distribution-based local search. These modifications significantly improve the algorithm’s ability to explore and evaluate new formulations. The resulting framework achieves 98.76% of nominal production capacity—very close to the theoretical optimum—while reducing quality-related costs by an average of 20.44%. These results represent substantial improvements in production efficiency, cost savings, and overall formulation quality, providing a powerful and practical tool for the engine oil industry.