First-Order Taylor’s Expansion

Neglecting higher-order terms, the Taylor’s series expansion for a performance response, $Y$ , is

\begin{matrix} (5) & Y (x) = y + \frac{d Y}{d x} Δ x \end{matrix}

The mean of this performance response is then calculated by setting the uncertain design parameters to their mean value, $μ_{x}$ :

\begin{matrix} (6) & μ_{Y} = Y (μ_{X}) \end{matrix}

and the standard deviation of $Y (x)$ is given by

\begin{matrix} (7) & σ_{Y} = {\sqrt{\sum_{i = 1}^{n} {(\frac{\partial Y}{\partial x_{i}})}^{2}}}^{} {(σ_{x_{i}})}^{2} \end{matrix}

where $σ_{x i}$ is the standard deviation of the ith parameter and n is the number of uncertain parameters (Phadke, 1989; Chen, 1996).

Because first-order derivatives of responses with respect to random variables are needed and the mean value point is needed in the previous equation, the first-order Taylor’s expansion estimates require $n + 1$ analyses for evaluation. Consequently, this approach is significantly more efficient than Monte Carlo simulation and often more efficient than DOE while including distribution properties. However, the approach loses accuracy when responses are not close to linear. This method is recommended when responses are known to be linear or close to linear and when computational cost is high and rough estimates are acceptable.