# Truss Structure Optimization with Subset Simulation and Augmented Lagrangian Multiplier Method
## 4 Test Problems and Optimization Results
### Planar 10-Bar Truss Structure  
The first test problem regards the well-known 10-bar truss structure (see Figure 2), which was previously used as a test problem in many studies. The modulus of elasticity is 10 Msi and the mass density is 0.1 lb/inch\({}^{3}\). The truss structure is designed to resist a single loading condition with 100 kips acting on nodes 2 and 4. Allowable stresses are tensile stress \(\sigma^{U}=+25\) kip and compressive stress \(\sigma^{L}=-25\) kip. Displacement constraints are also imposed with an allowable displacement of 2 inch for all nodes. There is a total of 22 inequality constraints. The cross-sectional areas \(A_{1},\ldots,A_{10}\) are defined as the design variables with minimum size limit of 0.1 inch\({}^{2}\) and maximum size limit of 35.0 inch\({}^{2}\). The optimization results of ALSSO are compared with those available in the literature in Table 1, including optimal design variables and minimal weight. It should be pointed out that the maximum displacements evaluated from GA, HA, HPSACO, HPSSO, SAHS, and PSO at the optimum designs exceed the displacement limits of 2.0 inch. Furthermore, the GA solution also violates the stress limits. The proposed method and ALPSO converged to fully feasible designs. The present results are in good agreement with those obtained using PSO and DALMM (i.e., ALPSO) by [PERSON] and [PERSON] (2015). The optimized design obtained by ALSSO is competitive with the feasible designs from ABC-AP, TLBO,  
Figure 2: Schematic of the planar 10-bar truss structure. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \hline
**Design Variables** & **G.A.[12]** & **HPSO [21]** & **HPS [28]** & **HPSACO [17]** & **PSO [23]** & **ALPSO [24]** & **HPSACO [36]** & **ALSC-AP [36]** & **SAIS [31]** & **TELBO [34]** & **MSPSO [27]** & **HPSSO [41]** & **WDO [38]** & **ASSSO** \\ \hline \(A_{1}\) & 30.440 & 30.704 & 30.150 & 30.493 & 35.500 & 30.551 & 30.307 & 30.585 & 30.394 & 30.248 & 30.5257 & 30.5888 & 30.5758 & 30.4379 \\ \(A_{2}\) & 0.100 & 3.200 & 3.100 & 0.100 & 0.100 & 0.100 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1001 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 \\ \(A_{3}\) & 21.790 & 23.167 & 22.710 & 22.320 & 22.286 & 23.220 & 22.544 & 23.18 & 22.038 & 22.286 & 23.225 & 23.1550 & 23.3386 & 21.599 \\ \(A_{4}\) & 14.260 & 35.135 & 15.270 & 15.346 & 14.477 & 15.198 & 15.505 & 15.218 & 15.481 & 13.5077 & 15.4114 & 13.2596 & 13.397 & 15.2446 \\ \(A_{5}\) & 0.100 & 3.100 & 3.100 & 0.300 & 0.100 & 0.100 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 \\ \(A_{6}\) & 0.451 & 0.551 & 0.544 & 0.538 & 0.530 & 0.534 & 0.531 & 0.539 & 0.5751 & 0.5833 & 0.58897 & 0.5756 & 0.5458 \\ \(A_{7}\) & 21.600 & 20.978 & 25.780 & 20.990 & 20.992 & 21.071 & 21.079 & 20.983 & 21.189 & 20.9645 & 20.9177 & 21.08447 & 20.9992 & 21.1212 \\ \(A_{8}\) & 7.658 & 27.404 & 7.561 & 7.561 & 7.564 & 7.455 & 7.463 & 7.463 & 7.464 & 7.4995 & 27.46832 & 27.4684 & 7.4640 \\ \(A_{9}\) & 0.100 & 3.200 & 3.200 & 3.200 & 0.100 & 0.100 & 0.100 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 \\ \(A_{10}\) & 21.360 & 21.508 & 21.498 & 21.648 & 20.467 & 21.584 & 21.229 & 22.191 & 22.342 & 21.5398 & 21.5098 & 21.5095 & 21.526 & 21.5391 \\ Weight(b) & 4897.00 & 5006.92 & 5007.88 & 5068.45 & 5068.45 & 5068.56 & 5068.88 & 5061.42 & 5060.96 & 5061.00 & 5062.96 & 5063.99 & 5060.888 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of optimized designs for the 10-bar truss problem. The statistical performance of the proposed algorithm is evaluated in Table 2. The reported results were obtained from 30 independent optimization runs each including 100, 200, and 500 samples at each simulation level. The \"SD\" column refers to the standard deviation on optimized weights, while the \"NSA\" column refers to the number of structural analyses. The number in parentheses is the NSA required by the proposed method for the best design run. It appears that the proposed approach is very robust for this test problem by checking the SD column.