Interval mathematical programming for robust engineering design under parameter and environmental uncertainty

Abstract

Engineering design optimization problems often exhibit uncertainty associated with material properties, loads, operating conditions, temperature, and environmental effects, requiring robustness beyond deterministic optimization to guarantee performance reliability. Probabilistic methods can be utilized to handle uncertainty; however, the applicability of these methods is subject to the availability of probability distributions, which might not exist if there are insufficient number of observations or the uncertainty is mainly of epistemic nature. This paper presents an Interval-Minimax Mathematical Programming Framework to robustly design engineering components in the presence of uncertainty in engineering parameters and environment. In the suggested methodology, uncertain engineering parameters are modeled by bounded intervals and a common uncertainty set consisting of parameter, environment, and operation uncertainty is defined. Using the minimax formulation, optimization is performed such that the worst possible objective value is minimized and robust feasibility requirements are guaranteed for all possible uncertainties in the uncertainty set. The nested optimization problem is then reformulated through a mathematical programming model based on epigraph approach and a procedure for robust design optimization under consideration of uncertainty is provided. The numerical analysis takes into account cases with larger uncertainty intervals and various environmental situations, whereas the comparative assessment is carried out relative to deterministic, nonlinear, interval, classical robust, and traditional minimax optimization methodologies. The findings show that the proposed methodology greatly enhances worst-case performance and constraints validity in spite of a well-balanced compromise between robustness and optimality. The developed interval-minimax approach guarantees zero violations of constraints and provides greater safety margin compared to the competing methodologies, though at the cost of additional computational complexity due to worst-case evaluation.