Multi-objective genetic algorithm driven formulation of a plant-based probiotic premix powder: Nutrient-cost modeling, microstructural characterization and sensory validation

This study presents a multi-objective genetic algorithm (NSGA-II) driven framework for the rational formulation of a plant-based probiotic premix powder under simultaneous nutritional and economic constraints. Protein content, energy value, and protein density were maximized while formulation cost was minimized within predefined macronutrient ranges. Pareto-optimal solutions were further refined into integer-feasible formulations and ranked using the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), from which the most balanced candidate was selected and experimentally validated. The predicted and measured nutritional values showed strong agreement, with relative errors below 6% for macronutrients, and approximately 6% for energy, confirming the robustness and practical reliability of the optimization strategy. During 28 days of storage, probiotic viability remained above 7 log₁₀ CFU/g under refrigerated and ambient conditions, while elevated temperature accelerated viability loss and color changes.. Microstructural and physicochemical analyses revealed near-spherical particles and a predominantly amorphous composite matrix. Fourier transform infrared spectroscopy (FTIR) spectra confirmed the preservation of major carbohydrate–protein functional groups, while High-Resolution X-Ray Diffraction (XRD) patterns exhibited a broad diffraction halo without sharp crystalline peaks. Differential scanning calorimetry (DSC) thermograms displayed a single endothermic transition within the scanned temperature range, indicating good thermal stability of the composite matrix. Sensory evaluation further demonstrated acceptable flavor, viscosity, and overall preference of the optimized formulation. The novelty of this study lies in integrating multi-objective optimization, integer-feasible formulation refinement, and experimental validation into a single reproducible and cost-aware framework, offering a scalable strategy for the development of nutritionally optimized plant-based probiotic powders.