A Predictive–Prescriptive Mathematical Framework for Football Squad Composition: Mixed-Integer Optimization with Machine-Learned Performance Inputs and Robust Decision Support

We develop a predictive–prescriptive decision-support framework for football squad composition under budget and positional constraints, in which mixed-integer optimization is the central methodological contribution. The framework integrates four components: (i) construction of a Sports Performance Index (SPI) via principal component analysis (PCA) of standardized per-90 performance features; (ii) leakage-free, player-wise, five-fold cross-fitted prediction of seasonal goals plus assists using ordinary least squares (OLS) as the primary input model, regularized linear models as additional baselines, and gradient boosting (XGBoost) as a nonlinear benchmark and uncertainty-estimation component, with each player-season receiving an out-of-fold prediction; (iii) a mixed-integer linear program (MILP) that selects a player-season portfolio under budget, position composition, squad-size, and duplicate-player constraints, with a tunable weight balancing predicted attacking output against the performance index; and (iv) a robust counterpart that incorporates prediction uncertainty into the optimization. The framework is implemented on English Premier League data covering the 2023–2024 and 2024–2025 seasons (N = 218 forward and midfielder player-seasons with at least 1800 min of playing time). Cross-fitted OLS attains a five-fold mean R2 = 0.76 ± 0.03 with stable performance across folds. The exact MILP outperforms a benefit-to-cost greedy heuristic by up to 9.4% and the best of 1000 random feasible rosters by 41.0–53.0% in objective value, while guaranteeing feasibility of all composition and budget constraints. Sensitivity analyses across budget tightness, position bounds, squad size, and cost proxy demonstrate qualitative robustness. The robust counterpart with risk-aversion parameter κ ∈ {0, 0.5, 1.0} produces a Jaccard roster overlap of 0.75 relative to the nominal solution and reduces average selected-player prediction uncertainty from 2.65 to 2.49. The contribution is a mathematically rigorous, integer-programming-centered decision-support framework that integrates machine learning with optimization for sports business analytics and is directly applicable to club-level squad planning.