Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Fuzzy design optimization applications <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> BIB001 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Fuzzy design optimization applications <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> BIB002 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Fuzzy design optimization 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> BIB003 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Fuzzy design optimization applications <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> BIB004 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Fuzzy design optimization applications <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> BIB005
This section will classify the applications using previously mentioned methods. As mentioned earlier, optimization can be classified according to how many objectives problem have. Investigated literature studies are shown in Table 1 from objective perspective. Shih et.al. (2003) developed and proposed three alternative α-level-cuts approaches: singlecut, double-cuts, and multiple-cuts, for solving nonlinear programming design problems of structuring engineering with fuzzy resources. The approaches have performed better than that of conventional α-level-cuts method. Hsu et.al. (1995) considered the optimization process as a closed-loop control system. Traditional "controllers", the numerical optimization algorithms, are usually "crisply" designed for well defined mathematical models. However, when applied to engineering design optimization problems in which function evaluations can be expensive and imprecise, very often the crisp algorithms will become impractical or will not converge. They presented how the heuristics of this human supervision can be modeled into the optimization algorithms using fuzzy control concept. BIB003 employed three fuzzy models to combine with an improved imposed-on penalty approach for attacking a nonlinear multiobjective in the mixed-discrete optimization problem. He presented a penalty method, including the forms of penalty function and the values of each parameter. The presented strategy is suggested as appropriate for solving a generalized mixed-discrete optimization problem. Arakawa et.al. (1999) showed the effectiveness of the use of fuzzy members as design variables, by comparing with the other robust design methods. They proposed a way to raise certainties in estimating robustness by using approximation concepts in operation of fuzzy function. Fang et.al. (1998) considered an approach to the optimum design of structures, in which uncertainties with a fuzzy nature in the magnitude of the loads. The optimization process under fuzzy loads is transformed into a fuzzy optimization problem based on the notion of Werners' maximizing set by defining membership functions of the objective function and constraints. An example of a ten-bar truss is used to illustrate the present optimization process. The results are compared with those yielded by other optimization methods. Mohandas et.al. (1990) has combined Zadeh's approach in Eq. (20) with goal programming. They implemented this approach to single objective optimization problems. As example problems, optimization of four bar and ten bar truss are selected. No comparison is made in this work. Yang and Soh (2000) proposed a fuzzy logic integrated genetic programming (GP) based methodology 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. Joghataie and Ghasemi (2001) implemented fuzzy membership functions in the multistage optimization technique to improve its performance for the minimum weight design of truss structures of fixed topology. It has been found that this technique has significantly improved the convergence speed at the expense of increasing the minimum weight by a negligible amount. Maglaras et.al. (1997) compared probabilistic and fuzzy set based approaches in designing a damped truss structure.  developed a fuzzy discrete multicriteria cost optimization model by considering three criteria 1) minimum cost 2) minimum weight and 3) minimum number of section types. In the design, the uncertainty of fuzziness of the AISC code based design constraints is considered.  Adeli (2000a, 2000b) presented a fuzzy augmented Lagrangian GA 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 algorithm is applied to two space axial-load structures including a large 37-story structure with 1310 members.  presented a new method in which the fuzzy lambda-formulation and game theory techniques are combined with a mixed-discrete hybrid genetic algorithm for solving mixed-discrete fuzzy multiobjective programming problems. They dealt with three example problems: the optimal designs of a two-bar truss, a conical convective spine and a twenty-five bar truss. BIB005 studied the principle of solving multiobjective optimization problems with fuzzy sets theory. Membership functions based on functional-link net have been used in multiobjective optimization. BIB004 have dealt with a genetic algorithm based optimization procedure for solving multicriterion design problems where the objective or constraint functions may not be crisply defined. Forouraghi et.al. (1994) introduced a new methodology in which multiobjective optimization is formulated as unsupervised learning through induction of multivariate regression trees. In particular, they showed that learning of Pareto-optimal solutions can be eficiently accomplished by using a number of fuzzy tree-partitioning criteria. The widely used problem of design of a three-bar truss is presented. Shih et.al. (1997) introduced a design method using fuzzy logic to find the best stochastic design by maximizing Hasofer-Lind's (H-L's) reliability and simultaneously optimizing design goals. The objective weighting strategy in multiobjective fuzzy formulation is adopted to represent the importance among the design goals.  has used Werner's approach in section IV.3. This approach is presented to solve multiobjective optimization problems. Sample problems are three bar and 25 bar truss optimization problems. No comparison is made in this work. BIB002 has combined Werner's approach in section IV.3 with Global Criterion method and implemented on multiobjective optimization problems. As sample cases, three bar truss and 11 bar truss are solved and results (objective function values) are compared.  proposed a general fuzzy programming with wide generality in order to consider the overall fuzzy factors and fuzzy information in optimum design of engineering structures. BIB001 Kiyota et.al. (2001 Kiyota et.al. ( , 2003 described a fuzzy satisficing method for multiobjective optimization problems using Genetic Algorithm (GA). A multiobjective design problem with constraints is expressed as a satisficing problem of constraints by introducing an aspiration level for each objective. Kelesoglu & Ulker (2005a) optimized space truss systems by using fuzzy sets. The algorithm of multi-objective fuzzy optimization was formed using the macros of Ms-Excel. Cheng and Li (1997) presented a constrained multiobjective optimization methodology by integrating Pareto Genetic algorithm with fuzzy penalty function method. A 72-bar space truss with two criteria and a 4-bar truss with three criteria were investigated.  proposed a genetic algorithm to solve fuzzy multiobjective optimization of space truss. This method enables a flexible method for optimal system design by applying fuzzy objectives and fuzzy constraints. An algorithm was developed by using MATLAB programming. The algorithm is illustrated on 56-bar space truss system design problem. At following pages, these studies will be classified according to used methods and application area. Table 2 shows the objectives in literature. It is seen that minimizing weight is the most common objective for single objective optimization studies. Minimizing failure possibility and natural frequency are also used even though found rarely. Table 3 shows the studies in literature according to used method type. Table 4 shows the hybrid methods.