A 4 step-cone pulley [39] 

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... to a specific problem. ” [17]. Design and implementation of such optimization methods has been at the origin of a multitude of contributions to the literature in the last 50 years. Genetic algorithms (GA) [18], simulated annealing (SA) [19], ant colony optimization (ACO) [20], particle swarm optimization (PSO) [21], harmony search algorithm (HS) [22], big bang-big crunch optimization (BB-BC) [23], imperialist competitive algorithm (ICA) [24], firefly algorithm (FA) [25], cuckoo search (CS) [26], charged system search algorithm (CSS) [27], magnetic charged system search algorithm (MCSS) [28], and ray optimization (RO) [29], are some familiar examples of meta-heuristics. Generally, a meta-heuristic algorithm uses two basic strategies while searching for the global optima; exploration and exploitation. The exploration enables the algorithm to reach at the best local solutions within the search space, and the exploitation provides the ability to reach at the global optimum solution which may exist around the local solutions obtained. In exploitation, the promising regions are explored more comprehensively, while in exploration the non-explored regions are visited to make sure that all the regions of the search space are fairly explored. Due to common properties between chaos and meta-heuristic optimization algorithms, simultaneous use of these concepts seems to improve the performance and to overcome the limitations of meta-heuristics. The previous research can be categorized into two types. In the first type, chaotic system is inserted into the meta-heuristics instead of a random number generator for updating the value of parameters; and in the second type, chaotic search is incorporated into the procedures of the meta-heuristics in order to enrich the searching behavior and to avoid being trapped in local optimums using traditional chaos optimization algorithms (COA). For simulating complex phenomena, sampling, numerical analysis, decision making and in particular in meta-heuristic optimization, random sequences are needed with a long period and reasonable uniformity. On the other hand as mentioned before chaos is a deterministic, random-like process found in nonlinear dynamical system which is non-period, non- converging and bounded. The nature of chaos looks to be random and unpredictable, possessing an element of regularity. Mathematically, chaos is randomness of a simple deterministic dynamical system, and chaotic system may be considered as sources of randomness [30-32]. However, meta-heuristics are non-typical; hence, the critical issue in implementing meta- heuristic methods is the determination of “proper” par ameters which must be established before running these algorithms. The efficient determination of these parameters leads to a reasonable solution. That is why; these parameters may be selected chaotically by using chaotic maps. In this case, sequences generated from chaotic systems substitute random numbers for the parameters where it is necessary to make a random-based choice. By this way, it is intended to improve the global convergence and to prevent to stick on a local solution. Alatas et al. [33] proposed different chaotic maps to update the parameters of PSO algorithm. This has been done by using of chaotic number generators each time a random number is needed by the classical PSO algorithm. Twelve chaos-embedded PSO methods have been proposed and eight chaotic maps have been analyzed in the unconstrained benchmark functions. The simulation results show that the application of deterministic chaotic signals may be a possible strategy to improve the performances of PSO algorithms. Also Alatas [32] presented another interesting application. He has integrated chaos search with HS for improved performance. Seven new chaotic HS algorithms have been developed using different chaotic maps. A similar utilizing of chaotic sequences for artificial bee colony (ABC) [34], BB-BC [35], ICA [1], and CSS [36] have been performed by researchers. Based on the results obtained from literature it is not easy to say which chaotic map performs the best. However, we can say that chaotic maps have a considerable positive impact on the performance of meta-heuristics. In these studies generally unconstraint problems were considered. On the other hand, most of the real life problems including design optimization problems require several types of variables, objective and constraint functions simultaneously in their formulation. In engineering design as the first attempts to analyze the performance of meta-heuristics in which chaotic maps are used for parameters updating process, Talatahari et al. [37] combined the benefits of chaotic maps and the ICA to determine optimum design of truss structures. These different chaotic maps were investigated by solving two benchmark truss examples involving 25- and 56-bar trusses to recognize the most suitable one for this algorithm. As an example taken from the original paper a 56-bar dome truss structure is shown in Figure 2. Members of the dome are into 7 groups. Table 1 shows the statistical results and the optimum weight for the 56-bar dome truss using the ICA algorithms where cm is a chaotic map based on the Sinusoidal map for CICA-1, Logistic map for CICA-2, Zaslavskii map for CICA-3 and Tent map for CICA-4 [37]. The results show that the use of Sinusoidal map (CICA-1) results in a better performance for the chaotic ICA than others. Two other larger examples were also considered by Talatahari et al. [37] to obtain more clear details about the performance of the new algorithm. These were 200- and 244-bar trusses with 29 and 32 design variables, respectively. Almost for all examples, the performance of the new algorithm is far better than the original ICA; especially when the standard deviations of the results are compared. The standard deviation of the new algorithm is much better than the original ICA and this illustrates the high ability of the new algorithm. As another attempt in optimization problems related to the engineering design a new improved CSS using chaotic maps was presented for engineering optimization by Talatahari et al. [38]. They defined five different variants of the new methodology by adding the chaos to the enhanced CSS. Then, different chaotic systems were utilized instead of different parameters available in the algorithm. To evaluate the performance of the new algorithm two sets of examples were considered: In the first set four well-known benchmark examples including design of a piston lever, design of a welded beam, design of a four-storey, two-bay frame, and design of a car side impact were selected from literature to compare the variants of the new method. In the second set two mechanical examples consisting of a 4 step-cone pulley design and speed reducer design problems were utilized in order to compare the variants of the new method with other meta-heuristics. As an example taken from the original paper, in design of a 4 step-cone pulley the objective is to design a pulley with minimum weight using 5 design variables, as shown in Figure 3. Four design variables are associated with the diameters of each step, and the fifth corresponds to the width of the pulley. In this example, it is assumed that the widths of the cone pulley and belt are identical. There are 11 constraints, out of which 3 are equality constraints and the remaining are inequality constraints. The constraints are imposed to assure the same belt length for all the steps, tension ratios, and power transmitted by the belt. The 4 step pulley is designed to transmit at least 0.75 hp (0.75 · 745.6998W), with an input speed of 350 rpm and output speeds of 750, 450, 250, and 150 rpm. This problem is considered to compare the chaotic CSS (CCSS) method with other meta-heuristic algorithms which was solved by using Teaching – learning-based optimization (TLBO) and ABC, previously [39]. It is observed from Table 2 that CCSS gives better results than the other methods for the best, mean, and standard deviation [39]. Due to the simplicity and potency of these methods, it seems that they can easily be utilized for many engineering problems to find the optimum designs. The basic idea of chaos optimization algorithm (COA) generally includes two major stages. Firstly, based on the selected chaotic map (cm) define a chaotic number generator for generating sequences of points then map them to a design space. Afterwards, evaluate the objective functions with respect to these points, and choose the point with the minimum objective function as the current optimum. Secondly, the current optimum is assumed to be close to the global optimum after certain iterations, and it is viewed as the center with a little chaotic perturbation, and the global optimum is obtained through fine search. Repeat the above two steps until some specified convergence criterion is satisfied, and then the global optimum is obtained [40]. The pseudo-code of COA is summarized as ...