Keywords: fuzzy optimization, structural optimization, engineering design, single cut approach, double-cuts approach, level-cuts method.

In the real-world structural engineering design problems, the allowable design stresses or input parameters are often fuzzy/imprecise with nonlinear characteristics that necessitated the developments of fuzzy optimum structural design method. From the published literatures for fuzzy nonlinear programming problems, it can be summarized that level-cuts method is accepted as the common solution approach for the problems with fuzzy constraints. The preferred final design is obtained on the basis of predetermining value of cutting level; however, the value of cutting level is difficult to obtain. Consequently, the designer hardly achieves a definite and unique optimum result.

Wang et al [1] first applied level-cuts (alpha-cuts) method to structural designs and the unique design level may be obtained by minimizing an additional difficultly obtained composing cost function. This alpha-cuts strategy has been recognized as the standard method for solving general optimum structural design problems with fuzzy constraints. Rao [2] applied the same level-cuts method to design a four-bar mechanism for function generating problem. The difficulty of defining an alpha for original fuzzy problems is a nature reflection of the imprecision on the design problem; however, no formulation presented in his paper to achieve the unique value of . Rao further proposed -formulation for achieving the unique solution that is workable for fuzzy multi-objective optimization problems [3,4]. Yeh and Hsu [5] followed the framework of Wang et al obtaining the optimum design level while the total cost is based on the failure possibility instead of the membership value of satisfaction. Xu [6] proposed a 2nd phase optimization in bound search method to obtain the particular alpha level by maximizing an established nonlinear fuzzy goal membership function. Three alternative level-cuts methods were proposed by Shih et al [7] for obtaining the unique design, however, they lack for the experiments and verifications on large-scale problems. Moreover, the multiple-cuts approach may be the best of dealing with the problems; however, it is somewhat complicated to handle the programming.

In this paper, the single-cut approach of the second kind has been verified, with illustrative examples, to be the same as the presenting single-cut approach of the first kind. A double-cuts approach is then presented with two formulations for linear or nonlinear membership function. From the algorithm and formulation, one can see the mathematical formulation of both single-cut and double-cuts methods can avoid requiring additional factors [2] or functions [6] to achieve the unique optimum design. For verifying the practical usability, nonlinear large-scale optimum structural design problems containing 17 design variables and 169 nonlinear fuzzy constraints are solved, by sequential quadratic programming combined with the finite element analysis, and compared with discussion. The unique design as well as corresponding optimum level-cut value can be guaranteed obtained by proposed strategies. Additionally, two wide-ranging used linear and nonlinear membership functions are adopted for the practical design situations. Some comparison and discussion are given in the paper; as a result, modified double-cuts approach is the most recommended approach for its easy programming, stable computation and producing the major design improvement. Therefore, it is recommended for use to instead of multiple-cuts approach introduced in previous research [7].

1

Wang, G..Y., Wang, W.Q., "Fuzzy Optimum Design of Structures", Engineering Optimization, 8, 291-300, 1985. doi:10.1080/03052158508902494

2

Rao, S.S., "Description and Optimum Design of Fuzzy Mechanical Systems", J. of Mechanisms, Transmissions, and Automation in Design, 109, 126-132, 1987.

3

Rao, S.S., "Multiobjective Optimization of Fuzzy Structural Systems", Int. J. for Numerical Methods in Engineering, 24, 1157-1171, 1987. doi:10.1002/nme.1620240608

4

Rao, S.S., Sundararaju K., Prakash, B.G., Balakrishna, C., "Multiobjective Fuzzy Optimization Techniques for Engineering Design", Computers & Structures, 42(1), 37-44, 1992. doi:10.1016/0045-7949(92)90534-7

5

Yeh, Y.C., Hsu, D.S., "Structural Optimization with Fuzzy Parameters", Computers & Structures, 37(6), 917-924, 1990. doi:10.1016/0045-7949(90)90005-M

6

Xu, C., "Fuzzy Optimization of Structures by the Two-phase Method", Computers & Structures, 31(4), 575-580, 1989. doi:10.1016/0045-7949(89)90334-9

7

Shih, C.J., Chi C.C., Hsiao, J.H., "Alternative a-level-cuts Methods for Optimum Structural Design with Fuzzy Resources", Computers & Structures, 81, 2579-2587, 2003. doi:10.1016/S0045-7949(03)00331-6

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