## 5 Optimization Process of the Structure Weight  
Optimization is a mathematical discipline that concerns the finding of minima and maxima of functions subjected to different constraints. Various optimization techniques, including \"linear programming\" and so on, are used to solve problems in different industries. Today, optimization comprises a wide variety of techniques from Operations Research (OR), Artificial Intelligence (AI), and Computer Science (CS). In the present optimization problem, the weight of the compartment is considered as an objective function that should be minimized. The geometric dimensions of the compartment, including length (a) and width (b), are considered as constraints in the optimization problem. In other words, in the optimization process, the external dimensions of the compartments are constant. Moreover, optimization variables for all cases (steel, Al + Depth 20, and Al + Depth 60) include sheet thickness, number of reinforcements in each side, and type of reinforcements. One of the constraints in solving this problem is that only one type of T-shaped beam can be used at a time. In order to determine the type of T-shaped beam to be used, the standard sizes of the beams according to Figure 1 have been entered into the software to obtain a response and the correspondence of the sizes with the compartment dimensions. In addition, at each stage of the optimization, the values of von Misses stress and deflection of the compartment must be less than the allowable value at the design stage. In summary, this problem consists of five variables, two dimensional constraints, three conditional constraints, and an objective function, as shown in Figure 6.  
This process was performed using FE software. The maximum and minimum values of the sheet thickness were 12 and 1 mm, respectively, with an interval of 1 mm. A range of 1-10 was also assumed for the number of reinforcement beams. The trial and error method for obtaining the minimum weight  
\begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Mode No.**} & \multicolumn{3}{c}{**BF**} \\ \cline{2-4}  & **Steel** & **AI + Depth 20** & **AI + Depth 60** \\ \hline \(1\) & 179.99 & 12.259 & 23.474 \\ \hline \end{tabular}
\end{table}
Table 6: Buckling factor for different cases of compartments.  
Figure 5: The first three mode shapes of buckling of compartment.  
of the structure was used. All possible states in different combinations of variables were analyzed, and by comparing the responses, the most optimal state was found. Table 7 presents the optimal design parameters for each compartment. The results indicated that the steel compartment had the highest reliability, while the aluminum one had the lowest weight at a depth of 20 m. However, the aluminum compartment at a depth of 60 m had the lowest thickness.  
Next, Figure 7 shows the stress contours of the optimal compartments. As seen in this figure, the maximum stress was created in the steel compartment, the aluminum compartment with a depth of 60 m, and the aluminum compartment with a depth of 20 m, respectively.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Compartment Type** & **Type of Reinforced Beam** & **Number of Beams** & **Sheet Thickness (mm)** & **Safety Factor** & **Weight (kg)** \\ \hline Steel & 50 T & 4, 4, 7 & 10 & 3 & 1199 \\ Al + Depth 20 & \(40\times 80\times 7\) T & \(3,3,4\) & 12 & 2.83 & 305 \\ Al + Depth 60 & \(80\times 80\times 9\) T & \(3,3,6\) & 8 & 3.51 & 446 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Details of optimum design for different cases of compartments.  
Figure 6: Details of the optimization process.  
Figure 7: The results of stress analysis of the optimized compartments made of different materials including (**a**) steel compartment, (**b**) aluminum compartment (depth of 20 m), and (**c**) aluminum compartment (depth of 60 m).