# Reliability-Based Design of an Aircraft Wing Using a Fuzzy-Based Metaheuristic
## 1 Introduction  
Composite materials are increasingly used as aircraft structures, but complexity in design analysis causes uncertainties due to material non-homogenous and thickness variation. It has been shown that a classical deterministic optimization technique is not enough to handle the present requirements due to the presence of the uncertainties that can be found throughout the whole part of the aircraft [1]. The uncertainties can suppress the actual performance of aircraft far from the optimum design and lead to impracticality. The traditional Monte Carlo simulation (MCS) technique has been used to identify the effects of uncertainty on optimum aircraft structure design. However, it utilizes expansive computation for analysis. For this reason, there other techniques are needed to collect data on the uncertainties in aircraft wing structure design. To enhance performance in the aircraft design process, reliability-based design optimization (RBDO) has been proposed. The reliability method is used to analyze reliability or failure probability in an optimization problem with uncertainties. The failure probability can be obtained by using probabilistic and non-probabilistic techniques. At present, a method in first group is the MCS method, which is used as a base line to quantify uncertainty. Due to the drawbacks mentioned in the previous cause, several techniques in this group are developed, i.e., the first-order and second moment (FOSM), the first-order reliability method (FORM) and the second-order reliability method (SORM) [2]. The various forms are performed to calculate the reliability index of the design space at the most probable point (MPP). The drawback of this technique is that it needs more precise calculation, which makes its inefficient computation. To increase the performance in generating a probability distributed function (PDF) rather than using MCS, the optimum Latin hypercube sampling (OLHS)with and without infill sampling has been used [3]. The second group, on the other hand, is proposed to solve the reliability-based design without requiring precise distribution of random variables. The most popular approaches are convex set [4], an interval method [5] and fuzzy set theory [6]. The general reliability-based design optimization problem is a double-looped nested problem due to the calculation of probability failure and optimization solving. In cases of a non-probabilistic approach, triple-looped nested problem is needed due to the possibility safety index (PSI) calculation. This technique is called possibility safety index-based design optimization (PSIBDO). The target-performance-based design approach (TPBDA) is proposed to solve the triple-looped problem by performing only the double-loop nested problem [7]. Then the interval perturbation method (IMP) can reduce the double-loop nested problem to a single one by estimating the constraints of the optimization problem [8]. Most techniques are accomplished with a based gradient, which is needed to perform differentiations. Recently, alternative techniques for solving the original triple-loop nest problem can be performed by means multi-objective optimization with evolutionary algorithms (EAs) or metaheuristics (MHs) [1; 9; 10]. This technique is solved with the single-looped problem. Reliability-based design optimization of aircraft aeroelasticity is a computational burden problem due to complex of aircraft structures and the double-loop nested problem for the probabilistic technique. The problem changes to the triple-looped problem for the non-probabilistic approach. Further, so far, there have been only a few techniques introducing the non-probabilistic reliability index into the aeroelastic optimization design of aircraft wings [11]. A very recent work applied the worse-case scenario to handle uncertainty into classical aircraft wing aeroelastic design [1]. Aeroelastic aircraft wing design results usually deviate by various uncertainties in prototyping, manufacturing, and other stages in the design process.