# Enhanced Hyper-Cube Framework Ant Colony Optimization for Combinatorial Optimization Problems
## 1 Introduction  
Combinatorial optimization is devoted to the mathematical process of searching for the optimal solution (maxima or minima) of an objective function with a discrete domain of decision variables. The possible number of solutions for a combinatorial optimization problem is equal to \(\left[D\right]^{n}\), where \(D\) is the discrete design domain vector and \(n\) represents the number of design variables [1]. Therefore, the optimization problem becomes more computationally difficult to be solved when the number of design variables increases. Accordingly, many combinatorial optimization problems are hard to solve within deterministic polynomial time (or NP-hard). A Travelling Salesman (or TSP) is a typical example of this type of optimization problem where the number of cities to be visited is given and the shortest path needs to be determined [2]. As the number of cities increases, the number of possible solutions increases too and this leads to the computational complexity of the problem, where it is not possible to enumerate all these solution possibilities with limited computation resources, such as memory size or processor speed. Furthermore, many structural design problems fall under this category of optimization problems because of the discrete nature of their design domain. It makes the objective function a multimodal (non-convex) function. As a consequence, the design space will have more than one optimal solution which then turns the structure design into an NP-hard design optimization problem. Trusses, beams, and composite structures are good examples ofreal-life combinatorial optimization problems [3]. In recent years, several studies have been published regarding the optimization of composite laminated structures [4]. This interest of the research community has been driven by the shared property of high-specific strength (or strength/weight ratio), thermal stability, corrosion resistance, and the high impact strength of the composite materials [5]. Hence, to solve such problems, many optimization techniques have been developed. The Ant Colony Optimization (ACO) algorithm demonstrated a significant performance improvement in solving NP-hard combinatorial optimization problems. The Traveling Salesman Problem (TSP) is a good example of such problems and it is solved using an early version of ACO [6]. The improvements in subsequent ACO algorithms focused on enhancing the algorithm variants to yield better searching and computational performance. As a result of the improved algorithm performance, many new applications of ACO appeared, such as the probabilistic TSP using estimation based ACO in [PERSON] and [PERSON]'s [7] work. [PERSON] and [PERSON] [8] used a hybrid ACO with a novel local search tool to solve the Sequential Ordering Problem (SOP). [PERSON] and [PERSON] [9] introduced a novel approach of ACO to optimize a water distribution system design. Furthermore, the optimization of the space truss design, using improved ACO, was demonstrated by [PERSON] and [PERSON] [10]. In the field of structural design problems, a well-known optimization problem of Stacking Sequence Design (SSD) of composite laminate has been solved. It is done by using ACO to determine the optimal stacking sequence design for maximum critical buckling load factor or natural frequency [11]. [PERSON] and [PERSON] [12] examined the ACO performance in solving the Stacking Sequence Design problem and he compared a modified standard ACO performance with Genetic Algorithm (GA) and Tabu Search (TS). He found that ACO performed much better than GA and TS in terms of solution cost and quality. [PERSON] [13] presented a hybrid ACO-TS algorithm to optimize the stacking sequence of a composite laminate subjected to bidirectional compression loading. He concluded that ACO is an effective optimization technique if combined with an appropriate local searching tool.