Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> PLANAR (2D) <s> Abstract In order to find a satisfactory solution for the multi-objective fuzzy optimization problem of structures, the membership functions are constructed for the subsets of the fuzzy objectives subject to the fuzzy constraints, and three approaches are presented by using the different types of generalized fuzzy decision-making. Two illustrative numerical examples given show that in creating the user-oriented design environment in computer-aided design systems the application of generalized fuzzy decision-making to multi-objective fuzzy optimization of structures looks promising. <s> BIB001 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> PLANAR (2D) <s> Abstract A vast amount of imprecise information exists in both the goals and constraints of the optimum design problems. It is very difficult and sometimes confusing to describe and to deal with the several criteria which contain fuzzy degrees of relative importance. This paper proposes two weighting strategies in the multiobjective optimization of fuzzy structural system. The results of the objective weighting strategy shows that its distribution line along the Pareto solution has a better degree of closeness towards the ideal solution than that of the membership weighting technique. A set of Pareto fuzzy optimal solutions can be obtained by a min-max variant fuzzy weighting method. The computational approach is illustrated with several structural design problems. <s> BIB002 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> PLANAR (2D) <s> Abstract A max-min variant of the fuzzy λ-formulation strategy, called the alternative global criterion method, the purpose of which is to obtain solutions for multicriteria crisp or fuzzy structural design problems, is presented. This approach goes towards a Pareto optimal set that is equivalent to the min-max variant of the global criterion method. The presented approach is not only capable of acquiring the non-dominated solution, but also capable of achieving the highest design degree of satisfaction. The result shows that a solution with maximum overall satisfaction is not identical to the one which is closest to the ideal solution. The presented methodology is then applied to structural optimization problems involving a vector of conflicting objective functions with crisp or fuzzy constraints. <s> BIB003 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> PLANAR (2D) <s> The fuzzy /spl lambda/-formulation is used to solve multiobjective optimization problems in which the design variables are random, the constraints as a whole has a fuzzy probability or the constraints are mixed of deterministic, stochastic and fuzzy constraints. The fuzzy optimization strategy helps one to get the highest degree of satisfaction in reaching an optimum solution. The idea of expected value and the chance constrained programming technique allows one to convert a stochastic optimization problem into its equal deterministic form. The design of a three-bar truss illustrates the proposed design optimization in detail and the design of a machine-tool spindle express the comprehensive work of this technique. Results show that the proposed formulation and method can generate a natural, well-behaved, and reliable design. > <s> BIB004 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> PLANAR (2D) <s> A mixed fuzzy-probabilistic fuzzy-probabilistic programming approach is presented to solve a multiobjective optimization problem with random variables, fuzzy probability in constraints, or a mix of deterministic, stochastic and fuzzy parameters. This paper also examines a problem that has a known or unknown probability density function (pdf). To construct the membership function with a known pdf, one can apply the possibility-probability consistency principle. To deal with the probabilistic constraints with fuzziness, one can use the fuzzy and stochastic formulation. The fuzzy optimization method helps one to get the highest degree of satisfaction in reaching an optimum solution. The characteristic of each constraint by optimizing a three-bar truss is investigated and compared with one another. The optimal design of a structural spindle is also used in order to illustrate this integrated strategy. Results show that the proposed approach provides a natural representation of a design problem and gives a reliable solution. <s> BIB005 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> PLANAR (2D) <s> This paper presents a constrained multiobjective (multicriterion, vector) optimization methodology by integrating a Pareto genetic algorithm (GA) and a fuzzy penalty function method. A Pareto GA generates a Pareto optimal subset from which a robust and compromise design can be selected. This Pareto GA consists of five basic operators: reproduction, crossover, mutation, niche, and the Pareto-set filter. The niche and the Pareto-set filter are defined, and fitness for a multiobjective optimization problem is constructed. A fuzzy-logic penalty function method is developed with a combination of deterministic, probabilistic, and vague environments that are consistent with GA operation theory based on randomness and probability. Using this penalty function method, a constrained multiobjective optimization problem is transformed into an unconstrained one. The functions of a point (string, individual) thus transformed contain information on a point's status (feasible or infeasible), position in a search space, and distance from a Pareto optimal set. Sample cases investigated include a multiobjective integrated structural and control design of a truss, a 72-bar space truss with two criteria, and a four-bar truss with three criteria. Numerical experimental results demonstrate that the proposed method is highly efficient and robust. <s> BIB006 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> PLANAR (2D) <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> BIB007 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> PLANAR (2D) <s> Only a small fraction of the hundreds of papers published on optimization of steel structures deal with cost optimization; the great majority deal only with minimization of the weight of the structure. Those few that are concerned with cost optimization deal with small two-dimensional or academic examples. In this article, the writers present a fuzzy discrete multicriteria cost optimization model for design of space steel structures subjected to the actual constraints of commonly-used design codes such as the AISC ASD code by considering three design criteria: (1) minimum material cost; (2) minimum weight; and (3) minimum number of different section types. The computational model starts with a continuous-variable minimum weight solution with a preemptive constraint violation strategy as the lower bound followed by a fuzzy discrete multicriteria optimization. It is concluded that solving the structural design problem as a cost optimization problem can result in substantial cost savings compared with the tr... <s> BIB008 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> PLANAR (2D) <s> Abstract.Structural optimization problems have been traditionally formulated in terms of crisply defined objective and constraint functions. With a shift in application focus towards more practical problems, there is a need to incorporate fuzzy or noncrisp information into an optimization problem statement. Such practical design problems often deal with the allocation of resources to satisfy multiple, and frequently conflicting design objectives. The present paper deals with a genetic algorithm based optimization procedure for solving multicriterion design problems where the objective or constraint functions may not be crisply defined. The approach uses a genetic algorithm based simulation of the biological immune system to solve the multicriterion design problem; fuzzy set theory is adopted to incorporate imprecisely defined information into the problem statement. A notable strength of the proposed approach is its ability to generate a Pareto-Edgeworth front of compromise solutions in a single execution of the GA. <s> BIB009 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> PLANAR (2D) <s> Although much attention has been focused on the development and applications of fuzzy optimization, multiobjective programming, and mixed-discrete optimization methods separately, fuzzy multiobjective optimization problems in mixed-discrete design space have not been addressed in the literature. It is mainly because of the lack of mature and robust theories of mixed-discrete and multiobjective optimization. In most practical applications, designers often encounter problems involving imprecise or fuzzy information, multiple objectives, and mixed-discrete design variables. A new method is presented in which the fuzzy λ formulation and game theory techniques are combined with a mixed-discrete hybrid genetic algorithm for solving mixed-dixcrete fuzzy multiobjective programming problems. Three example problems, dealing with the optimal designs of a two-bar truss, a conical convective spine, and a 25-bar truss, demonstrate that the method can be flexibly and effectively applied to various kinds of engineering design problems to obtain more realistic and satisfactory results in an imprecise environment. Nomenclature D = fuzzy feasible solution domain d =f avorable search direction Fiti(X) = fitness function fi(X) = ith objective function f max i = upper bound of ith objective function f min i =l ower bound of ith objective function G j = allowable interval of the constraint function g j g j(X) = jth inequality constraint function k = number of objective functions M = size of the population m = number of inequality constraints n = total number of design variables nd = number of discrete design variables nq = number of discrete design variables with equal spacing X = design vector X c = centroid of the composite polygon Xw =v ertex with the lowest fitness value xi = ith design variable µD(X) = membership function of the design vector µfi(X) = membership function of the ith objective µgj(X) = membership function of the jth constraint <s> BIB010 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> PLANAR (2D) <s> The principle of solving multiobjective optimization problems with fuzzy sets theory is studied. Membership function is the key to introduce the fuzzy sets theory to multiobjective optimization. However, it is difficult to determine membership functions in engineering applications. On the basis of rapid quadratic optimization in the learning of weights, simplification in hardware as well as in computational procedures of functional-link net, discrete membership functions are used as sample training data. When the network converges, the continuous membership functions implemented with the network. Membership functions based on functional-link net have been used in multiobjective optimization. An example is given to illustrate the method. <s> BIB011 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> PLANAR (2D) <s> This paper presents a general algorithm for nonlinear space truss system optimization with fuzzy constraints and fuzzy parameters. The analysis of the space truss system is performed with the ANSYS program. The algorithm of multiobjective fuzzy optimization techniques was formed with ANSYS parametric dimensional language. In the formulation of the design problem, weight and minimum displacement are considered the objective functions. Three design examples are presented to demonstrate the application of the algorithm. <s> BIB012
SPACE (3D) 2 bar:  BIB010  3 bar:   Rao et.al., 1992; BIB001 BIB002 Forouraghi et.al., 1994; BIB003 BIB004 BIB005 Shih et.al., 1997;  BIB009 BIB011 4 bar: (Kiyota et.al., 2001; Kiyota et.al., 2003 ) BIB006 Kelesoglu & Ülker, 2005b) 9 bar: BIB012 10 bar: BIB007 11 bar: BIB009 25 bar:  Rao et.al., 1992;  BIB010  Kelesoglu & Ülker, 2005b) 56 bar:  72 bar: BIB006 120 bar: BIB012 244 bar: BIB012 1310 bar: BIB008  