## 5 Yield Optimization
### Method  
Yield optimization is a method for reducing the risk/probability of system failure by aiming to center a tolerance box in the design space such that the probability mass of the random variables inside the feasible region is at a maximum [20]. When used, it adds a stochastic consideration to a design process. Here, system failure is defined as the maximum DGR temperature exceeding 100 \({}^{\circ}\)C.  
Yield optimization comprises three steps: feasible region approximation, joint cumulative distribution approximation, and yield maximization [21]. Their details are as follows, provided here for readers' better understanding. 1. For a feasible/acceptability region of \(m\) design variables defined by \(k\) constraints: \[R_{A}=\{\mathbf{x}\in\mathbb{R}^{m}|h_{i}\geq 0,\;\;\;i=1,\ldots,k\}\] (7) a polyhedral approximation is performed to obtain the constraints in the form [21]: \[h_{i}(\mathbf{x})\approx h_{i}(\mathbf{x}^{*})+\mathbf{g}_{i}(\mathbf{x}^{*})^{\mathsf{T}}( \mathbf{x}-\mathbf{x}^{*})\] (8) where \(\mathbf{g}_{i}(\mathbf{x}^{*})\) is the gradient vector of constraint \(h_{i}(\mathbf{x}^{*})\) and \(\mathbf{x}^{*}\) is an expansion point that lies on the surface of \(h_{i}(\mathbf{x}^{*})=0\) and is closest to the center of the design's tolerance box. This leads to a polytope that approximates the feasible region [21]: \[R_{P}=\left\{\mathbf{x}|\mathbf{A}\mathbf{x}\geq\mathbf{c},\mathbf{x}^{L}\leq\mathbf{x}\leq\mathbf{x}^{L}\right\}\] (9) where vector \(\left(\mathbf{g}_{i}^{*}\right)^{\mathsf{T}}=\left[\frac{\partial h_{i}}{ \partial x_{1}},\ldots,\frac{\partial h_{i}}{\partial x_{m}}\right]_{\mathbf{x}^ {*}_{i}}\) and scalar \(\left(\mathbf{g}_{i}^{*}\right)^{\mathsf{T}}\mathbf{x}_{i}^{*}\) make up the \(i\)th row of \(A\) and \(\mathbf{c}\), respectively. Superscripts \(L\) and \(U\) denote lower and upper bounds on the design variables, respectively. 2. If unknown, the true distributions of the random variables (design variables) are approximated by arbitrary distributions. A closed-form cumulative distribution function (CDF) is ideal for the next step. Thus, for algebraic simplicity, the authors of [21] used the Kumaraswamy distribution, which has the following CDF [16]: \[F(x;a,b)=1-\left(1-x^{a}\right)^{\hat{b}},\;\;\;x\in(0,1)\] (10) where parameters \(\hat{a}\) and \(\hat{b}\) are those in the [PERSON] PDF described earlier in Equation (4). Additionally, recall that the Kumaraswamy PDF is useful in that it can represent many distributions (Figure 5) depending on its two shape parameters. Here, we use \(\hat{a}\) and \(\hat{b}\) to denote the shape parameters for the distribution of random variables, while the previous \(a\) and \(b\) continue to denote the shape parameters that define the UFC age arrangement. 3. Yield maximization proceeds by determining the location of a maximum yield box that is strictly located within the approximated feasible region, \(R_{P}\). This containment requirement is [21]: \[A^{+}x^{u}-A^{-}x^{l}\leq c\] (11) where \(A^{+}_{ij}\!=\max\!\left\{0,A_{ij}\right\}\) and \(A^{-}_{ij}\!=\max\!\left\{0,-A_{ij}\right\}\). Superscripts \(l\) and \(u\) denote lower and upper bounds on the optimum maximum yield box, respectively. The maximum yield box is contained inside the design's tolerance box, which has dimensions specified by \(t\). The yield maximization problem that must be solved is therefore [21]: \[ \begin{array}{ll}\max_{\begin{array}{c}x^{r},x^{l},x^{u}\\ \text{subject to}\end{array}}&\text{Yield}(x^{r},x^{l},x^{u})\\ A^{+}x^{u}-A^{-}x^{l}\leq c\\ x^{u}\leq x^{r}+t\\ x^{r}\leq x^{l}\leq x^{u}\end{array} \] (12) where \(x^{r}\) is a reference point corresponding to the lower bounds of the tolerance box.