### Information Gap Decision Theory Code Implementation The information gap decision theory (IGDT) focuses on robustness against severe uncertainty. Below is an example of how one might implement IGDT in Python to evaluate the robustness of decisions under varying levels of uncertainty. ```python import numpy as np def performance_measure(x, u): """ Define a function that calculates the performance measure. Args: x : float or array_like The decision variable(s). u : float or array_like Uncertainty parameter(s). Returns: float: Performance value. """ return -(x - 2)**2 + u * x def find_robustness(horizon_of_uncertainty, nominal_value, required_performance_level): """ Determine the level of robustness given a horizon of uncertainty and desired performance threshold. Args: horizon_of_uncertainty : int Maximum allowed deviation from the nominal value. nominal_value : float Nominal estimate of uncertain parameters. required_performance_level : float Minimum acceptable performance level. Returns: tuple: Robustness level and corresponding worst-case scenario. """ for alpha in range(0, horizon_of_uncertainty + 1): lower_bound = max(nominal_value - alpha, 0) upper_bound = min(nominal_value + alpha, 1) worst_case_utility = -np.inf optimal_x = None # Search over feasible values within bounds defined by current radius α for candidate_x in np.linspace(lower_bound, upper_bound, num=1000): utility_at_candidate_x = min( [performance_measure(candidate_x, perturbed_u) for perturbed_u in [lower_bound, upper_bound]]) if utility_at_candidate_x >= required_performance_level \ and utility_at_candidate_x > worst_case_utility: worst_case_utility = utility_at_candidate_x optimal_x = candidate_x if not np.isinf(worst_case_utility): break return alpha if not np.isinf(worst_case_utility) else 0, optimal_x horizon_of_uncertainty = 5 nominal_value = 0.7 required_performance_level = 0.8 alpha_star, best_decision = find_robustness(horizon_of_uncertainty, nominal_value, required_performance_level) print(f"The system's robustness level is {alpha_star}. " f"Under this condition, the most robust choice would be x={best_decision:.3f}.") ``` This script defines two functions: `performance_measure` which computes some metric based on both control variables \( x \) and uncertainties \( u \); while `find_robustness` iteratively searches through different radii around the estimated point until finding where there exists at least one solution meeting our predefined criteria even when facing maximum possible disturbance specified by each tested radius size.