Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Single objective applications <s> Abstract In this paper, the process of optimum design of structures is developed in which design parameters with a fuzzy nature are considered. The fuzzy optimum problem is transformed into an ordinary unconstrained optimum problem by expressing the objective function in the form of fuzzy expected cost. The proposed method is much simpler than the conventional fuzzy optimum method and appears to be able to deal with the uncertainty with the minimum of complexity. The proposed method can be shown to possess the following advantages: 1. (1) no predetermined mapping function is needed 2. (2) consistency between objective and constrained functions is obtained 3. (3) the computational process is simplified 4. (4) the optimum level design is obtained. Illustrative numerical examples have been presented and the above mentioned advantages have been proved. Moreover, a comparison between various cases for both system failure cost and fuzziness of design parameters is performed. <s> BIB001 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Single objective applications <s> Abstract Three fuzzy models are employed to combine with an improved imposed-on penalty approach for attacking a nonlinear multiobjective in the mixed-discrete optimization problem. The model can generate a unique compromise solution with the maximum design satisfaction. A systematic study of the proposed penalty method is presented, including the forms of penalty function and the values of each parameter. A default value of the parameter and an optimization algorithm are presented in the paper. One can easily program and reach the solution by any nonlinear programming method. Several structural design systems serve as the illustrative examples. The presented strategy is suggested as appropriate for solving a generalized mixed-discrete optimization problem. <s> BIB002 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Single objective applications <s> Abstract The single-objective optimization of structures, whose parameters are assigned as fuzzy numbers or fuzzy relations, is presented in this paper as a particular case of the random set theory and evidence theory approach to uncertainty. Some basic concepts concerning these theories are reviewed and the relationships among interval analysis, convex modeling, possibility theory and probability theory are pointed out. In this context a frequentistic view of fuzzy sets makes sense and it is possible to calculate bounds on the probability that the solution satisfies the constraints. Some special but useful cases illustrate in detail the meaning of the approach proposed and its links with a recent formulation conceived within the context of convex modeling. Some theorems allow a very efficient computational procedure to be set up in many real design situations. Two numerical examples illustrate the model presented. <s> BIB003 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Single objective applications <s> In the traditional optimization algorithms, constraints are satisfied within a tolerance defined by a crisp number. In actual engineering practice, constraint evaluation involves many sources of imprecision and approximation. When an optimization algorithm is forced to satisfy the design constraints exactly, it can miss the global optimum solution within the confine of commonly acceptable approximations. Extending the augmented Lagrangian genetic algorithm (GA) of Adeli and Cheng, a fuzzy augmented Lagrangian GA is presented for optimization of steel structures subjected to the constraints of the AISC allowable stress design specifications taking into account the fuzziness in the constraints. The membership function for the fuzzy domain is found by the intersection of the fuzzy membership function for the objective function and the constraints using the max-min procedure of Bellman and Zadeh. Nonlinear quadratic fuzzy membership functions are used for objective function and constraints. The algorithm is applied to two space axial-load structures including a large 37-story structure with 1,310 members. The features and advantages of the new fuzzy GA include acknowledging the imprecision and fuzziness in the code-based design constraints, increased likelihood of obtaining the global optimum solution, improved convergence, and reduced total computer processing time. <s> BIB004 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Single objective applications <s> A fuzzy logic integrated genetic programming (GP) based methodology is proposed to increase the performance of the GP based approach for structural optimization and design. Fuzzy set theory is employed to deal with the imprecise and vague information, especially the design constraints, during the structural design process. A fuzzy logic based decision-making system incorporating expert knowledge and experience is used to control the iteration process of genetic search. Illustrative examples have been used to demonstrate that, when comparing the proposed fuzzy logic controlled GP approach with the pure GP method, the proposed new approach has a higher search efficiency. <s> BIB005 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Single objective applications <s> Abstract The optimization problem of structural systems with imprecise properties on the basis of a possibilistic approach is considered. System imprecisions are defined by fuzzy numbers and characterized by membership functions. A methodology for the efficient solution of the optimization process is presented. A two-step method is used to include the imprecision within the optimization, where high quality approximations are used for the evaluation of structural responses. The approximations are constructed using concepts of intermediate response quantities and intermediate variables. The approach is basically an algebraic process which can be implemented very efficiently for the optimal design of general structural systems with imprecise parameters. The method provides more information to the designer than is available using conventional design tools. The effectiveness of the methodology and the interpretation of the results are illustrated by the solution of two example problems. <s> BIB006 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Single objective applications <s> Fuzzy membership functions are implemented in the multistage optimization technique to improve its performance for the minimum weight design of truss structures of fixed topology. First, a large truss is split into several smaller substructures that are optimized separately. However, to preserve the unity of the original truss, interaction-balance equality constraints are introduced to account for the compatibility of deformations, and the equality of the action and reaction of forces at the points of the separation of the substructures. A penalty function, containing the equilibrium equations and interaction constraints as well as other constraints, is defined for each substructure. Then, a fuzzy membership function, returning a value between 0 and 1, is introduced for each of the equality constraints to provide some fuzzy evaluation of the degree of satisfaction. This measure is then used in the determination of an appropriate penalty parameter. The design procedure is iterative and continues until conv... <s> BIB007 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Single objective applications <s> Two single level-cut approaches of the first and second kind for obtaining the unique compromise design in solving nonlinear optimum engineering design problems with fuzzy resources have been developed and presented in this paper. The conventional standard level-cuts method has been discussed for inspiring the proposed novel formulation consequently. The proposed strategies with the illustrative design examples indicate that the unique design as well as corresponding optimum level-cut value can be guaranteed obtained. Additionally, two wide-applicable linear or nonlinear membership functions of objective functions are presented depending on the practical situations of design tasks. The proposed level-cut approaches have been shown easy formulation and successfully employed to large-scaled structural design problems by sequential quadratic programming (SQP) technique combined with the finite element analysis. <s> BIB008 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Single objective applications <s> This paper presents the modified double-cuts approach for large-scale fuzzy optimization, typically in 25-bar and 72-bar truss design problems that the unique design can be guaranteed obtained. The modified approach also is reliable for general optimum structural design with fuzzy resources, as concluded the most recommended method for fuzzy engineering design. The proposed approach is better than the single-cut approach and easy programming for use to instead of multiple-cuts approach. <s> BIB009 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Single objective applications <s> In real world engineering design problems, decisions for design modifications are often based on engineering heuristics and knowledge. However, when solving an engineering design optimization problem using a numerical optimization algorithm, the engineering problem is basically viewed as purely mathematical. Design modifications in the iterative optimization process rely on numerical information. Engineering heuristics and knowledge are not utilized at all. In this article, the optimization process is analogous to a closed-loop control system, and a fuzzy proportional–derivative (PD) controller optimization engine is developed for engineering design optimization problems with monotonicity and implicit constraints. Monotonicity between design variables and the objective and constraint functions prevails in engineering design optimization problems. In this research, monotonicity of the design variables and activities of the constraints determined by the theory of monotonicity analysis are modelled in the fuzzy PD controller optimization engine using generic fuzzy rules. The designer only needs to define the initial values and move limits of the design variables to determine the parameters in the fuzzy PD controller optimization engine. In the optimization process using the fuzzy PD controller optimization engine, the function value of each constraint is evaluated once in each iteration. No sensitivity information is required. The fuzzy PD controller optimization engine appears to be robust in the various design examples tested. <s> BIB010 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Single objective applications <s> Abstract Evolutionary structural design has been the topic of much recent research; however, such designs are usually hampered by the time-consuming stage of prototype evaluations using standard finite element analysis (FEA). Replacing the time-consuming FEA by neural network approximations may be a computationally efficient alternative, but the error in such approximation may misguide the optimization procedure. In this paper, a multi-objective meta-level (MOML) soft computing-based evolutionary scheme is proposed that aims to strike a balance between accuracy vs. computational efficiency and exploration vs. exploitation. The neural network (NN) is used here as a pre-filter when fitness is estimated to be of lesser significance while the standard FEA is used for solutions that may be optimal in their current population. Furthermore, a fuzzy controller updates parameters of the genetic algorithm (GA) in order to balance exploitation vs. exploration in the search process, and the multi-objective GA optimizes parameters of the membership functions in the fuzzy controller. The algorithm is first optimized on two benchmark problems, i.e. a 2-D Truss frame and an airplane wing. General applicability of the resulting optimization algorithm is then tested on two other benchmark problems, i.e. a 3-layer composite beam and a piezoelectric bimorph beam. Performance of the proposed algorithm is compared with several other competing algorithms, i.e. a fuzzy-GA–NN, a GA–NN, as well as a simple GA that only uses only FEA, in terms of both computational efficiency and accuracy. Statistical analysis indicates the superiority as well as robustness of the above approach as compared with the other optimization algorithms. Specifically, the proposed approach finds better structural designs more consistently while being computationally more efficient. <s> BIB011
Direct Hybrid (Maglaraset.al., 1997; Fang et.al., 1998; Arakawa et.al., 1999; BIB006 Shih et.al., 2003; BIB008 Marler et.al., 2004; BIB009 BIB001 Mohandas et.al., 1990; Hsu et.al., 1995; BIB002 BIB003 BIB004 BIB005 BIB007    BIB010 BIB011 BIB001 Hsu et.al., 1995; BIB002 Arakawa et.al., 1999; Shih et.al., 2003; BIB008 BIB010 ) 4 bar: (Mohandas et.al., 1990; Fang et.al., 1998;   10 bar: (Mohandas et.al., 1990; BIB003 BIB005 BIB007 BIB008 ) 25 bar: BIB006 BIB008 Marler et.al., 2004; BIB009 Table 6 shows the objectives in literature. It is seen that minimizing weight is still the most common objective for single objective optimization studies. Minimizing deflection and natural frequency are also used.