## 8 Application FVIM to Solve Structural Engineering Design Problems
### Experimental Setup
#### 8.1.4 Three-Bar Truss Design Problem  
The Three-Bar Truss Design Problem is a classic optimization challenge in structural engineering, focusing on optimizing the design of a truss system composed of three bars [66]. The goal is to minimize the weight or maximize the strength of the truss while satisfying certain constraints. Typically, the design variables include dimensions such as the lengths or cross-sectional areas of the bars, which influence the truss's behavior under load. The constraints often involve limits on stresses, deflections, or geometric properties to ensure the truss meets performance requirements and remains structurally stable. This problem serves as a fundamental example in structural optimization, demonstrating the trade-offs between different design objectives and constraints in engineering design. It has applications in various fields such as civil engineering, aerospace engineering, and mechanical engineering, where lightweight and efficient structural designs are crucial for performance and safety. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Statistical Measurements**} & \multicolumn{3}{c}{**OX-X0 for Minimum Value**} \\
**Sheet Name** & **Mean** & **Std** & **Max** & **Best** & **X1** & **X2** & **X3** & **X4** & **X5** \\ \hline FVIMDE & 1.340E+00 & 1.765E-07 & 1.340E+00 & 1.340E+00 & 6.017E+00 & 5.306E+00 & 4.498E+00 & 3.501E+0  
The objective of the Three-Bar Truss Design Problem is to design a truss with minimum weight without violating any constraints (see Figure 7). The mathematical formulation of the problem is as follows:  
Let \(LB=[lb_{1},lb_{2},\dots,lb_{n}]\) and \(UB=[ub_{1},ub_{2},\dots,ub_{n}]\) denote the lower and upper bounds of the design variables, respectively. Here, \(lb_{i}\) and \(ub_{i}\) represent the lower and upper bounds of the \(i\)th design variable. _Vio_ denotes any violation conditions, and \(Obj\) represents the objective function to be minimized. The global minimum achieved is denoted by \(GloMin\). The objective function, denoted as \(z\), is defined as:  
\[z=100\cdot(2\sqrt{2}x_{1}+x_{2})\]  
where \(x_{1}\) and \(x_{2}\) are the design variables representing certain characteristics of the three-bar truss. Constraints \(g\) are defined as follows:  
\[g_{1}(x) =\frac{\sqrt{2}x_{1}+x_{2}}{\sqrt{2}x_{1}^{2}+2x_{1}x_{2}}\cdot 2-2\geq 0\] \[g_{2}(x) =\frac{x_{2}}{\sqrt{2}x_{1}^{2}+2x_{1}x_{2}}\cdot 2-2\geq 0\] \[g_{3}(x) =\frac{1}{\sqrt{2}x_{2}+x_{1}}\cdot 2-2\geq 0\]  
where \(g_{i}(x)\) represents the \(i\)th constraint function, and \(x_{1}\) and \(x_{2}\) are the design variables. The lower and upper bounds for the design variables are defined by \(LB\) and \(UB\), respectively, and any other conditions such as violation conditions are defined by _Vio_. Table 21 shows that the statistical measurements yield a mean objective function value of 252.3, accompanied by a standard deviation of 12.72, signifying the average and variability of performance across multiple runs. The maximum value achieved by FVIMDE is 265.6, with the best-recorded value being lower at 232.0, reflecting the optimizer's capacity to identify highly effective solutions, albeit with a notable range of outcomes. The design variables X1 and X2, corresponding to the optimized parameters of the truss design, attained values of 0.69 and 0.3688, respectively, at the best solution. These values are critical as they directly influence the structural integrity and weight of the truss, which are pivotal considerations in engineering design. When comparing with the results of other optimizers, FVIMDE's performance manifests as competitive, specifically in terms of achieving the lowest 'best' value, which is indicative of the potential to reach superior design solutions. Notwithstanding the broad standard deviation, which suggests a greater variability in solution quality, the ability of  
Figure 7: Three-bar truss design problem.