# Reliability-Based Design of an Aircraft Wing Using a Fuzzy-Based Metaheuristic
## 5 Design Results and Discussions  
The uncertainties of material property and allowable transverse displacement on the wing are [PERSON]'s modulus (\(E\)) and \(u\), respectively. The uncertainties are modelled with normal membership functions in such a way that \(E=\exp\biggl{\{}\ -1/2\left(\frac{r-E_{\text{m}}}{E\text{m}\times 0.1}\right)^{2} \biggr{\}}\) and \(u=\exp\biggl{\{}\ -1/2\left(\frac{r-U_{\text{m}}}{U_{\text{m}}U_{\text{m}} \times 0.1}\right)^{2}\biggr{\}}\), where \(E_{\text{m}}=70\) GPa, and \(U_{\text{m}}=0.1\) m are the core of the functions, respectively. The figure of the normal membership function of \(E\) and \(u\) are presented in Figure 3. By using the PSIBDO technique, Equation (9) can be changed to Equation (10) using a triple-loop nested to solve the problem using a computer with the following specification: AMD Ryzen 7 3700X with Radeon Graphics 3.00 GHz, 32.00 GB, 64-bit Window 10 operating system. The optimizer performs an initial setting of the algorithm with a population size of 50 and termination iteration of 200. Using the suggested PSIBDO, the results of the optimum solutions for the aircraft wing design are given in Table 2. The deterministic results for an optimum design variables are prescribed as PSI \(\pi_{H}^{max}=1\). The aeroelastic results corresponds to PSIs as less allowable PSIs cause safer aeroelastic characteristics in both static and dynamic as shown in the last column of Table 2.  
Figure 3: Normal membership functions of (**a**) Young’s modulus (\(E\)) and (**b**) allowable transverse displacement (\(u\)). As exposed by Table 2, the total mass of the aircraft wing structure has an increasing trend with the allowable PSI \(\pi_{\beta}{}^{max}\) being reduced from the value of 0.1 to 0.001. Meanwhile, the critical speed increases with the allowable PSI \(\pi_{\beta}{}^{max}\) being reduced, and the lift effectiveness decreases with the allowable PSI \(\pi_{\beta}{}^{max}\) reducing. Figure 4 demonstrates the comparison of search history for \(\pi_{\beta}{}^{max}\) = 0.1, 0.01 and 0.001, respectivelyand presents the upper and lower skin ply orientations. When running TLBO for solving the problem, the iterative procedure stops when the number of iterations reaches 100. It equals termination iteration 200 due to the nature of TLBO, which has embedded teaching and learning phases. The results in Table 3 show the total mass of the aircraft wing structure have an increasing trend with the allowable PSI \(\pi_{\beta}{}^{max}\) reducing. Moreover, the skin thicknesses in Layers 2 and 5 increase with the allowable PSI \(\pi_{\beta}{}^{max}\) reducing, while the skin thicknesses in Layers 1, 3, 4, and 6--rib thicknesses and spar thicknesses approach to their lower limits. The values of the skin ply orientations between layer increases with the allowable PSI \(\pi_{\beta}{}^{max}\) reducing. Furthermore, the results in Table 3 and the search history in Figure 4 reveal that the search history in Figure 4 are already constant before iteration 40 and that the remaining iterations do not yield significant benefits. The chosen ply thickness with increasing steps of 0.5 mm to 5 mm is not enough to bring the mass reduction as shown. The optimum thickness varies by two different thicknesses (0.5 and 1.0 mm). For future study, the step change should be reduced to 0.1 mm to provide an optimized search for a more optimum mass. The optimum thickness varies in range from 0.5-1.0 mm, which eliminates the practical thickness. It should be more than 3 mm to maintain resistance damage [23], otherwise, theply orientation might still change.