# Enhanced Hyper-Cube Framework Ant Colony Optimization for Combinatorial Optimization Problems
## 3 Enhanced Hyper-Cube ACO Algorithms
The last is the normalized price \(nPS\), which is defined as the ratio of price and practical reliability [1; 25; 26]. Finally, the performance rate measure, Prate, is also considered to link the computation effort with the number of function evaluations [24]. \[P_{rate}=\frac{number\ of\ successful\ runs}{\left( \begin{array}{c}number\ of\\ function\\ evaluations\end{array} \right)\cdot\left( \begin{array}{c}total\ number\\ \text{of}\ runs\end{array} \right)} \tag{12}\]

### Solution Quality

The solution quality of an algorithm can be measured by determining the absolute error between the current solution and the best-known global solution [1; 25; 26]. \[Q=\left|\frac{S^{*}-S_{\text{opt}}}{S_{\text{opt}}}\right|\cdot 100 \tag{13}\]

### Fitness Landscape Analysis

The design space of a combinatorial optimization problem can significantly affect the search performance of an algorithm. The notion of fitness landscape appeared in literature as an answer to the question of \"what does the design space look like?\". The fitness landscape is defined by the feasible solutions set, the objective function (fitness), and the structure of the solution neighborhood. To find the connection between the fitness landscape and the problem hardness, [PERSON] and [PERSON] [27] introduced a Fitness Landscape-Distance Correlation (FDC) to determine the hardness of optimization problems to be solved using Genetic Algorithm (GA). The distance mentioned here is defined as the number of movements that should be imposed on a Solution \(S_{i}\) to eliminate dissimilarity with the optimal solution \(S_{opt}\). The proposed correlation by [PERSON] is computed using the correlation factor, \(r\):

\[r(F,D)=\frac{CFD}{\sigma_{F}\cdot\sigma_{D}} \tag{14}\]

where \(CFD\) indicates the \(Cov(F,\ D)\) and \(\sigma_{F},\sigma_{D}\) are the standard deviation of \(F\) and \(D\), respectively. The values of the correlation coefficient \(\mathbf{r}\) are limited to interval \([-1,1]\) where negative values are desirable for maximization and indicate better searching performance. Finally, using the scattering of fitness versus the distance to the global optima can reveal valuable information about \(FDC\) of the optimization problem solved by an algorithm [18; 26]. ## 5 Numerical Experiments

To demonstrate the performance of the new approach, we selected a well-known NP-hard combinatorial optimization problem in the field of composite laminated structures. The optimization objective is maximizing the critical buckling load of composite laminated plate exposed to bidirectional compression loading. The decision variables are the fiber orientation of each composite layer (lamina) which form the optimal stacking sequence of the laminate (a group of layers). To employ ACO as an optimization algorithm for a SSD optimization problem, there is a need to understand specific problem characteristics such as solution representation, constraints, and objective function formulation. In meta-heuristic algorithms, the solution (stacking sequence) takes the form of a bit string that consists of a combination of plies with the available angle fiber orientations (e.g., \(0^{\circ},\pm 45^{\circ}\) and \(90^{\circ}\)). The different solutions have integer coding with 1, 2, and 3 numbers, which represent the three possible fiber orientations, respectively. For instance, the laminate with \(\left[2132231\right]_{s}\) stacking sequence describes the laminate of \(\left[\pm 45,\ 0_{2},90_{2},45,\pm 45,90_{2},0_{2}\right]s\) fiber orientations. The simplicity of using an integer representation along with significant performance gains made it the most widely used method in meta-heuristic optimization algorithms for composite laminated design.