## 4 An Optimal Design
### A Design Example  
Suppose that a measurement system consists of two components, whose accuracies are gradually declining. The accuracy loss values of the whole system and those of the two components (tested once a week) are given over the course of one hundred weeks, as shown in Figure 1.  
The accuracy loss weight for each component can be calculated using the multiple linear regression method; that is, \(p_{1}\) = 1.998 for component one, and \(p_{2}\) = 0.475 for component two. Their accuracy loss functions \(\delta_{1}(t)\), \(\delta_{2}(t)\), and \(\delta_{s}(t)\) (\(t>0\)) can also be gained by using the least-squares fitting method:  
\[\left\{ \begin{array}{l}\delta_{1}(t)=0.0031t^{2}+0.043t+0.3052\\ \delta_{2}(t)=0.0011t^{2}+0.0925t+0.3232\\ \delta_{s}(t)=0.0066t^{2}+0.0526t+0.7633\end{array} \right..\]  
Meanwhile, the allowable values of the accuracy loss of the system and its components are supposed as \(\delta_{1}\) = 20 \(\upmu\)m, \(\delta_{2}\) = 30 \(\upmu\)m and \(\delta_{s}\) = 60 \(\upmu\)m, which should be determined according to the different measurement accuracy requirements. Then, the values of the accuracy lifetimes and the uniformity of the system can be calculated as follows:  
\[T_{1}\approx 73.07\text{ weeks},\,T_{2}\approx 127.50\text{ weeks},\,\,T_{s}\approx 90.84\text{ weeks},\,\,\rho\approx 0.75.\]  
Assuming that the uniformity of the system is not big enough and cannot meet the uniformity requirement, the system should be redesigned. Firstly, vector \(X\) consisting of the design variables should be determined. Here, the average accuracy loss velocities of component one and component two, \(\Delta\overline{v}_{1}\) and \(\Delta\overline{v}_{2}\), respectively, as well as the accuracy lifetimes \(T_{1}\), \(T_{2}\) and \(T_{s}\)', are selected as the design variables. Therefore, the designed vector is  
\[X=(\Delta\overline{v}_{1},\Delta\overline{v}_{2},T_{1}^{\prime},T_{2}^{\prime },T_{s}^{\prime}).\]  
Secondly, the optimal design model, including an objective function and constraint conditions, should be established. When the designer focuses on the uniformity of the system, the biggest uniformity can be taken as an objective function:  
\[\max\rho\prime.\]  
Figure 1: Accuracy loss data of a whole system and its components. For the objective functions, there are two aspects of constraints. One aspect is the variation ranges of the average accuracy loss velocities of component one and component two, \(\Delta\overline{v}_{1}\) and \(\Delta\overline{v}_{2}\), respectively. Here, the ranges of the average accuracy loss velocities for component one and component two are both assumed to be between 0.1 \(\upmu\)m per week and 0.5 \(\upmu\)m per week. In practical applications, the range of the average accuracy loss velocities should be determined by the possible ranges of the loss velocities of different components under the current technical levels and conditions. The other aspect is the boundary constraints of the design parameters, which are determined by the relationships between the accuracy reserves, average accuracy loss velocities and accuracy lifetimes. The total constraint equation is as follows:  
\[s.t.\left\{ \begin{array}{l}\Delta\overline{v}_{1}T_{1}^{\prime}=20\\ \Delta\overline{v}_{2}T_{2}^{\prime}=30\\ (1.998\Delta\overline{v}_{1}+0.475\Delta\overline{v}_{2})T_{s}^{\prime}=60\\ 0.1\leq\Delta\overline{v}_{1}\leq 0.5\\ 0.1\leq\Delta\overline{v}_{2}\leq 0.5\end{array} \right..\]  
Finally, the optimization model is solved by employing the optimization toolbox in MATLAB. The initial values of 1000 groups uniformly distributed over the range of constraints are generated using uniform sampling. Then, the SQP algorithm is adopted to find the optimum solution in term of the corresponding initial value. Figure 2 shows the optimal solutions of uniformity \(\rho^{\prime}\) for each group of initial values.