Robust DEA methodology via computer model for conceptual design under uncertainty

Abstract

This paper presents an integrated approach for an alternative exploration and selection of product development via computer aided engineering under uncertainty. For the proposed approach, a set of possible alternatives (decision making units, DMUs) are generated by designers during product development. The computer models are introduced to convert the design values of the controllable variables of DMUs into the multiple responses of interest; these are categorized into inputs and outputs. These inputs and outputs are randomized values under uncertain environments. Because of incompatible dimensions in terms of input and output values, they are further normalized prior to data envelopment analysis (DEA). Subsequently, the randomized and normalized inputs and outputs are used for DEA analysis. The first DMU ranking, chosen on the basis of the DEA analysis, is considered to be the best DMU of all available DMUs under the impact of uncertainty. Two examples: a bike frame design and an electronic circuit design are introduced to demonstrate the proposed approach. The computer models, where ANSY represents an example of the former and WEBENCH represents an example of the latter, are adopted as conversion processes during DEA analysis.

Appendix

Single controllable variable case

The expansion of design function, $Y = f(X)$, established at $U_{X}$ for controllable variable X by Taylor’s series up to the first three terms is:

$$\begin{aligned} Y= & {} f(X) = f(Ux)+(X-Ux)\left. \left( \frac{\partial f(X)}{\partial X} \right| _{Ux} \right) \nonumber \\&\left. +\,\frac{1}{2!}(X-Ux)^{2}\left( {\frac{\partial f^{2}(X)}{\partial X^{2}}} \right| _{Ux}\right) +\mathfrak {R}\end{aligned}$$

(4)

where $\mathfrak {R}$ is the residue and $U_{X }$ could be or could not be the mean value of X.

Taking the expectation of Eq. (4) to obtain the mean value of Y when X is at $U_{X}$, the outcome is:

$$\begin{aligned} U_{Y }= & {} E(Y) = f(U_{X})+\frac{1}{2}\left. {\frac{\partial ^{2}f(X)}{\partial X^{2}}} \right| _{U_X } V\left( X \right) + E(\mathfrak {R})\nonumber \\\approx & {} f(U_{X})+\frac{1}{2}\left. {\frac{\partial ^{2}f(X)}{\partial X^{2}}} \right| _{U_X } V(X)\nonumber \\= & {} f(U_{X})+\frac{1}{2}\left. {\frac{\partial ^{2}f(X)}{\partial X^{2}}} \right| _{U_X } \sigma _X ^{2} \end{aligned}$$

(5)

where $\sigma _X$ is the standard deviation of X.

To have an approximate value of V(Y), consider the Taylor’s series expansion up to the first two terms. That is:

$$\begin{aligned} \sigma _Y ^{2}= & {} V(Y) = \left( \left. {\frac{\partial f(X)}{\partial X}} \right| _{Ux}\right) ^{2 }V(X) \nonumber \\= & {} \left( \left. {\frac{\partial f(X)}{\partial X}} \right| _{Ux} \right) ^{2}\sigma _X ^{2}\nonumber \\ \end{aligned}$$

(6)

where $\sigma _Y$ is the standard deviation of response value Y.

Multiple controllable variable case

Extending the above derivation to the n controllable variables; assuming $X_{1}, X_{2},\ldots , X_{\mathrm{n}}$ are independent; such as $Y=f(X_{1}, X_{2}, \ldots , X_{n})$; that is:

$$\begin{aligned} Y= & {} f(X_{1}, X_{2}, \ldots ,X_{n})\nonumber \\= & {} f(U_{1}, U_{2}, \ldots , U_{n})\nonumber \\&+\sum _i^n \left[ (X_{i}-U_{i})\left( \left. {\frac{\partial f(X_1 ,X_2 ,X_3 , \ldots , X_n )}{\partial X_{i}}} \right| _{U_{1},U_{2},U_{3}, \ldots , {U_{n}}} \right) \right] \nonumber \\&+\,\frac{1}{2!} \sum _j^n \sum _i^n \left[ (X_{i}-U_{i})(X_{j}-U_{j})\right. \nonumber \\&\left. \left( \left. {\frac{\partial f^{2}(X_1 ,X_2 ,X_3 , \ldots , X_n )}{\partial X_{i}\partial X_{j}}} \right| _{U_{1},U_{2},U_{3}, \ldots , {U_{n}}} \right) \right] +\mathfrak {R}\end{aligned}$$

(7)

$$\begin{aligned} U_{Y }= & {} E(Y)= f(U_{1}, U_{2}, \ldots , U_{n})\nonumber \\&+\,\frac{1}{2}\sum _i^n \left[ \left. {\frac{\partial ^{2}f(X_1 ,X_2 ,X_3 , \ldots , X_n )}{\partial Xi^{2}}} \right| _{U_{1},U_2 ,U_3 , \ldots , U_n } V(X_{i})\right] + E(\mathfrak {R})\nonumber \\\approx & {} f(U_{1}, U_{2}, \ldots , U_{n})\nonumber \\&+\,\frac{1}{2}\sum _i^n \left[ \left. {\frac{\partial ^{2}f(X_1 ,X_2 ,X_3 , \ldots , X_n )}{\partial X_i ^{2}}} \right| _{U_1,U_2 ,U_3 , \ldots , U_n } V(X_{i})\right] \end{aligned}$$

(8)

To have an approximate value of V(Y), consider the Taylor’s series expansion up to the first two terms. That is:

$$\begin{aligned} \sigma _Y ^{2}\approx & {} V(Y) = V (f(X_{1}, X_{2}, \ldots ,X_{n}))\nonumber \\&+\,V\left( \sum _i^n \left[ (X_{i}-U_{i})\left( \left. {\frac{\partial f(X_1 ,X_2 ,X_3 , \ldots , X_n )}{\partial Xi}} \right| _{U_{1},U_{2},U_{3}, \ldots ,U_{n}} \right) \right] \right) \nonumber \\= & {} \sum _i^n \left[ \left( \left. {\frac{\partial f(X_1 ,X_2 ,X_3 , \ldots , X_n )}{\partial Xi}} \right| _{U_{1},U_{2},U_{3},\ldots ,U_{n}} \right) ^{2}\sigma _{Xi} ^{2}\right] \end{aligned}$$

(9)