Multi-model ensembles and multi-objective evolutionary algorithms provide a systematic approach to reconciling competing criteria in time-series forecasting. However, most existing methods are tailored to specific tasks and lack essential mathematical details. This study introduces a general multi-objective ensemble framework based on a Multi-Objective Enhanced Crisscross Optimization (MOECSO) algorithm, exemplified through Brent crude oil price forecasting. Initially, ensemble-weight selection is framed as a bi-objective optimization problem, where the two objectives penalize Mean Absolute Error (MAE) and the Sample Standard Deviation of the Validation Residuals (SSDVRs), both assessed on the original United States Dollar (USD) scale under a leakage-free rolling-origin protocol. Subsequently, a Variational Mode Decomposition (VMD) reconstruction operator is defined, which adaptively reconstructs the raw series by integrating intrinsic mode functions with weights derived from their entropy and center-frequency characteristics, while adhering to nonnegativity and normalization constraints. Furthermore, horizontal and vertical crossover operators, along with a hypervolume–ideal-distance archive rule, are introduced, collectively forming a comprehensive MOECSO scheme for bi-objective ensemble weighting. Utilizing a public Brent crude oil dataset, the proposed ensemble demonstrates superior performance compared to robust statistical, machine-learning, and deep-learning benchmarks in terms of MAE, Root Mean Square Error (RMSE), and Mean Absolute Percentage Error (MAPE), while also reducing error dispersion and enhancing robustness during crisis periods. Diebold–Mariano (DM) and superior predictive ability tests with multiple-comparison control validate that these improvements are statistically significant. In summary, this paper presents a mathematically transparent framework for constructing and analyzing multi-objective ensembles in univariate time-series forecasting.
Zhao et al. (Fri,) studied this question.