A Decoupled Method for Credibility-Based Design Optimization with Fuzzy Variables

Abstract

Fuzzy uncertainty (FU) exists widely in engineering applications, but there lack design optimization methods under FU, thus a credibility-based design optimization (CBDO) is focused to obtain the safety design under FU in this paper. Firstly, the concepts of credibility index and most credible point (MCP) are presented to measure the safety degree under FU, where the credibility index and the MCP, respectively, show similar properties as the reliability index and the most probable point under random uncertainty. Secondly, the inverse MCP (IMCP) is defined with respect to the required credibility, and the detailed method is established for searching IMCP, on which the performance measure approach (PMA) can be combined to solve the CBDO. Since the PMA combined with the IMCP includes a time-consuming double-loop strategy, the sequential optimization and credibility assessment (SOCA) is proposed to decouple the double-loop strategy thirdly. In the SOCA, a shifting vector constructed by the IMCP is used to transform the credibility constraint into an equivalent deterministic one, on which the double-loop strategy can be avoided to reduce the computational cost for solving the CBDO. One numerical example and two engineering examples fully illustrate the efficiency and accuracy of the SOCA.

Appendices

Appendices

1.1 Appendix 1

See Tables 8 and 9.

Table 8 Regular fuzzy credibility distributions and their characteristics

Table 9 Standard regular fuzzy credibility distributions

1.2 Appendix 2

1.2.1 A Brief Introduction to Fuzzy Advanced First-Order Second-Moment (FAFOSM) Method

Consider a non-linear performance function $Y = g\left( {\mathbf{X}} \right)$, and the MCP $P^{*} \left( {x_{1}^{*} ,x_{2}^{*} , \cdots ,x_{n}^{*} } \right)$ is confined in the limit state equation as $g(P^{*} ) = 0$. $\tilde{Y}$ is expanded as the first-order Taylor series at $P^{*}$:

$$ \tilde{Y} = g\left( {x_{1}^{ *} ,x_{2}^{ *} , \ldots ,x_{n}^{ *} } \right) + \mathop \sum \nolimits_{i = 1}^{n} \left( {{{\partial g} \mathord{\left/ {\vphantom {{\partial g} {\partial x_{i} }}} \right. \kern-0pt} {\partial x_{i} }}} \right)_{{P^{ *} }} \left( {x_{i} - x_{i}^{ *} } \right) . $$

(B.1)

Thus, the linearized limit state equation is $\mathop \sum \nolimits_{i = 1}^{n} \left( {{{\partial g} \mathord{\left/ {\vphantom {{\partial g} {\partial x_{i} }}} \right. \kern-0pt} {\partial x_{i} }}} \right)_{{P^{ *} }} \left( {x_{i} - x_{i}^{ *} } \right) = 0$. According to the linear addition law of fuzzy expectation and variance [39], the approximate fuzzy first- and second-order moments are $\tilde{E}\left( {\tilde{Y}_{{P^{ *} }} } \right) = \mathop \sum \nolimits_{i = 1}^{n} \left( {{{\partial g} \mathord{\left/ {\vphantom {{\partial g} {\partial x_{i} }}} \right. \kern-0pt} {\partial x_{i} }}} \right)_{{P^{ *} }} \left( {\mu_{i} - x_{i}^{ *} } \right)$ and $\sqrt {\tilde{D}\tilde{Y}_{{P^{ *} }} } = \mathop \sum \nolimits_{i = 1}^{n} \left| {\left( {{{\partial g} \mathord{\left/ {\vphantom {{\partial g} {\partial x_{i} }}} \right. \kern-0pt} {\partial x_{i} }}} \right)_{{P^{ *} }} } \right|\sigma_{i}$. Then the credibility index can be expressed as

$$ \tilde{\beta } = \frac{{\tilde{E}\tilde{Y}_{{P^{ *} }} }}{{\sqrt {\tilde{D}\tilde{Y}_{{P^{ *} }} } }} = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {{{\partial g} \mathord{\left/ {\vphantom {{\partial g} {\partial x_{i} }}} \right. \kern-0pt} {\partial x_{i} }}} \right)_{{P^{ *} }} \left( {\mu_{i} - x_{i}^{ *} } \right)}}{{\mathop \sum \nolimits_{i = 1}^{n} \left| {\left( {{{\partial g} \mathord{\left/ {\vphantom {{\partial g} {\partial x_{i} }}} \right. \kern-0pt} {\partial x_{i} }}} \right)_{{P^{ *} }} } \right|\sigma_{i} }} . $$

(B.2)

After transforming all input variables and limit state equation into standardized space, the linearized limit state equation becomes $\mathop \sum \nolimits_{i = 1}^{n} \left( {{{\partial g} \mathord{\left/ {\vphantom {{\partial g} {\partial x_{i} }}} \right. \kern-0pt} {\partial x_{i} }}} \right)_{{P^{*} }} \sigma_{i} (u_{i} - u_{i}^{*} ) = 0$. Since the contour topology of the JMF of U is a series of hypercubes centered at the origin, coordinates of the MCP are solved by the following equations:

$$ \left\{ {\begin{array}{ll} {\mathop \sum \nolimits_{i = 1}^{n} \left( {{{\partial g} \mathord{\left/ {\vphantom {{\partial g} {\partial x_{i} }}} \right. \kern-0pt} {\partial x_{i} }}} \right)_{{P^{*} }} \sigma_{i} (u_{i} - u_{i}^{*} ) = 0} \\ {\left\| {\mathbf{u}} \right\|_{\infty } = \mathop {\hbox{max} }\nolimits_{1 \le i \le n} \left\{ {\left| {u_{i} } \right|} \right\} = t} \\ \end{array} } \right. . $$

(B.3)

In most cases, the MCP is on the vertex of the joint membership contour, i.e., $\left| {u_{1} } \right| = \left| {u_{2} } \right| = \cdots = \left| {u_{n} } \right| = t$. After basic mathematical operations, it is not hard to obtain $t = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {{{\partial g} \mathord{\left/ {\vphantom {{\partial g} {\partial x_{i} }}} \right. \kern-0pt} {\partial x_{i} }}} \right)_{{P^{ *} }} \left( {\mu_{i} - x_{i}^{ *} } \right)}}{{\mathop \sum \nolimits_{i = 1}^{n} \left| {\left( {{{\partial g} \mathord{\left/ {\vphantom {{\partial g} {\partial x_{i} }}} \right. \kern-0pt} {\partial x_{i} }}} \right)_{{P^{ *} }} } \right|\sigma_{i} }}$, which is also the definition of $\tilde{\beta }$ exactly as shown in Eq. (B.2). As a result, compared with AFOSM in reliability, the MCP (or design point) can be considered as the point located at the limit state equation has the shortest Chebyshev distance to the origin.