## 8 Application FVIM to Solve Structural Engineering Design Problems
### Experimental Setup
#### 8.1.5 Tabular Column Design Problem  
The Tabular Column Design Problem is a crucial engineering optimization challenge aiming to determine the optimal dimensions of a tubular column while considering various constraints and objectives [67]. In this problem, the goal is to minimize the weight of the column while ensuring it meets specific strength and stability requirements. Key parameters include the outer diameter of the column (\(x_{1}\)) and the thickness of its wall (\(x_{2}\)). The objective function seeks to minimize the weight of the column while adhering to constraints such as maintaining a minimum aspect ratio, limiting the maximum stress the column can endure, and ensuring geometric feasibility. By optimizing these parameters, engineers can design tubular columns that are both structurally sound and efficient in terms of material usage. The objective of the Tabular Column Design Problem is to minimize the cost of designing a uniform column of the tabular section, incorporating material and construction costs. The column, made of material with a length (\(L\)), yield stress (\(S\)), modulus of elasticity (\(E\)), and density (\(D\)), is designed to carry a compressive load (\(P\)). This problem is characterized by two optimized variables: the mean diameter of the column (\(x_{1}\)) and the tube thickness (\(x_{2}\)). It includes six constraints: ensuring the stress in the column is less than the buckling stress (\(g_{1}(\mathbf{X})\)) and the yield stress (\(g_{2}(\mathbf{X})\)); restricting the mean diameter between 2 and 14 cm (\(g_{3}(\mathbf{X})\) and \(g_{4}(\mathbf{X})\)); and limiting the column thickness to the commercially available range of 0.2-0.8 cm (\(g_{5}(\mathbf{X})\) and \(g_{6}(\mathbf{X})\)). The Tubular Column Design Problem involves optimizing the design of a tubular column subjected to specific constraints and objectives. Let \(x_{1}\) represent the outer diameter of the column and \(x_{2}\) represent the thickness of the column's wall. The objective function \(z\) aims to minimize the weight of the column while meeting strength requirements and is expressed as:  
\[z=9.8x_{1}x_{2}+2x_{1}\] \begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & \multicolumn{2}{c}{**Statistical Measurements**} & & & \multicolumn{2}{c}{**(X1–X2) for Best Value**} \\
**Optimizer** & **Mean** & **Std** & **Max** & **Best** & **X1** & **X2** \\ \hline FVIDE & 2.523E+02 & 1.272E+01 & 2.656E+02 & 2.320E+02 & 6.900E-01 & 3.688E-01 \\ GWO & 2.639E+02 & 1.921E+07 & 2.639E+02 & 2.639E+02 & 7.887E-01 & 4.083E-01 \\ HHO & 2.598E+02 & 5.064E+00 & 2.640E+02 & 2.498E+02 & 7.681E-01 & 3.257E-01 \\ MVO & 2.636E+02 & 3.371E-01 & 2.639E+02 & 2.628E+02 & 7.831E-01 & 4.135E-01 \\ CPO & 2.598E+02 & 4.458E+00 & 2.643E+02 & 2.515E+02 & 7.236E-01 & 4.688E-01 \\ FOX & 2.636E+02 & 8.884E-01 & 2.651E+02 & 2.620E+02 & 7.819E-01 & 4.086E-01 \\ ZOA & 2.639E+02 & 6.782E-08 & 2.639E+02 & 2.639E+02 & 7.887E-01 & 4.082E-01 \\ FOX & 2.636E+02 & 7.599E-01 & 2.640E+02 & 2.615E+02 & 7.833E-01 & 3.996E-01 \\ AVOA & 2.632E+02 & 5.715E-01 & 2.639E+02 & 2.620E+02 & 7.768E-01 & 4.227E-01 \\ SCA & 2.633E+02 & 1.660E+00 & 2.650E+02 & 2.601E+02 & 8.002E-01 & 3.376E-01 \\ MFO & 2.586E+02 & 4.265E+00 & 2.641E+02 & 2.524E+02 & 7.962E-01 & 2.722E-01 \\ WOA & 2.639E+02 & 1.376E-04 & 2.639E+02 & 2.639E+02 & 7.885E-01 & 4.088E-01 \\ MFO & 2.635E+02 & 3.574E-01 & 2.639E+02 & 2.628E+02 & 7.898E-01 & 3.940E-01 \\ \hline \hline \end{tabular}
\end{table} 
Table 21: Results of the three-bar truss design problem, run = 30, agents = 50, FES = 1000.