# Enhanced Hyper-Cube Framework Ant Colony Optimization for Combinatorial Optimization Problems
## 3 Enhanced Hyper-Cube ACO Algorithms
The buckling load factor lb for simply supported rectangular laminated plate subjected to bi-axial loading is determined as follows:

\[\lambda_{b}(p,q) =\pi^{2}\frac{\left[D_{11}\big{(}\frac{p}{a}\big{)}^{4}{+}2(D_{12}{ +}2D_{66})\big{(}\frac{p}{a}\big{)}^{2}{+}D_{22}\big{(}\frac{q}{b}\big{)}^{4} \right]}{\big{(}\frac{p}{a}\big{)}^{2}N_{x}+\big{(}\frac{q}{b}\big{)}^{2}N_{y}} \tag{15}\]

where \(D_{ij}\) denotes the bending stiffness, \(N_{x}\) is the axial loading in x-direction, \(N_{y}\) is the axial loading in y-direction, \(p\) and \(q\) are the number of half waves in \(x,y\) directions. The critical buckling load factor \(\lambda_{cb}\) is defined as the minimum obtained value of \(\lambda_{b}\) (\(p,q\)). The critical values of \(p\) and \(q\) are linked to different factors such as laminate material, the number of plies, loading conditions, and the plate aspect ratio. In uniaxial loading and a simply supported plate, the critical buckling load occurs when \(p=1\) whereas in biaxial the critical buckling loads, it needs to be determined as the minimum value of \(\lambda_{b}(p,q)\)[1,13]. Finally, the constraints in stacking sequence optimization with constant laminate thickness t could be imposed as follow:

* Symmetry constraint is enforced by optimizing half of the laminate. * Balancing constraint is enforced by selecting \(\theta_{2}\) for the standard fiber orientation set of \(0\,,\pm 45\) and \(90\). * Only \(N/4\) ply orientations are needed to describe laminate as a result of balancing constraints. * Contiguity constraint is handled by using the penalty parameter (\(\beta\)). * The critical buckling load factor objective function \(f_{obj}\) could be formulated as: \[f_{obj}=(1-\beta){\cdot}\text{max}(\lambda_{cb}(p,q))\] (16)

To compare the performance of the proposed algorithm alongside the other ACO algorithms, we implemented all the algorithms presented here using MATLAB R2019b software. The benchmarking problem from the literature of Stacking Sequence Design optimization is accredited to [PERSON] and has been used by previous studies [1, 23]. The original problem describes a simply supported plate subjected to an in-plane biaxial loading as shown in Figure 2.

The thickness of each ply \(t_{i}\) is assumed constant, and the ply orientations are limited to \(0\,,\pm 45\) and \(90\) sets of angles. The number of plies \(N_{L}\) is constant. The required properties, dimensions, and loading conditions are listed in Tables 1 and 2. The objective function is set to maximize the critical buckling load. The constraints are integrated into the solution (e.g., balanced laminate, symmetrical, etc.). The implemented ACO algorithms were executed on the same computer for the same number of experiments; \(N_{exp}=200\). This number was

Figure 2: Simply supported plate subjected to biaxial loading. used to overcome the stochastic behaviour of meta-heuristic algorithms [1]. Furthermore, this number of experiments was conducted over ten different random generating seeds of 301, 2, 50, 75, 111, 200, 167, 225, 11, and 25. Then, the average of the performance measures values was used in the comparison of different ACO algorithms. Lastly, all ACO algorithms were examined at two different levels of convergence rate, slow and fast. The slow rate forces the algorithm searching loop to stop after 56 iteration without improvement, while the fast rate needs just 10 iterations to be terminated [1]. ### ACO Parameters Setting

To ensure a fair assessment of the ACO algorithms performance, the following standard ACO parameters were assumed for all implemented algorithms: number of Ants \(n_{Ants}=25\), the maximum number of iterations \(Iter_{max}=1000\), evaporation rate \(r=0.1\), the parameter of the pheromone trail relative importance \(\alpha=1\), and initial one trail \(\tau_{0}=0.004\) (except for MMACO and BWACO algorithms were \(\tau_{0}=1\)).