## 3 The Proposed C-MDSDO
### Reliability Assessment with the Formulated Sub-Optimization Problem
#### 3.2.2 Formulated Sub-Optimization Problem and Problem-Solving Process  
To avoid neglecting the influence of the objective function performance when optimizing the reliability index of the solution, the objective function performance is introduced into the shifting vector search process to form the suggested sub-optimization problem, as in Formula (10). \[ \begin{array}{ll}\underset{\mathbf{v}^{k}}{min}&\phi(\mathbf{d}_{k}-\mathbf{ v}_{k},\mathbf{x}_{dv}(\mathbf{t}),\mathbf{u}_{k}(\mathbf{t}),\mathbf{t})\cdot C_{rate}( \mathbf{R}_{k})\\ subject\ to&\dot{x}-f(\mathbf{R}_{k}-\mathbf{v}_{k},\mathbf{x}_{Rv}(\mathbf{t}), \mathbf{u}_{k}(\mathbf{t}),\mathbf{t})=0\\ &\mathbf{d}_{min}\leq\mathbf{d}_{k}-\mathbf{v}_{k}\leq\mathbf{d}_{max}\end{array}  \tag{10}\]  
where \(k\) represents the iteration step of the MDSDO stage, \(\mathbf{v}_{k}\) represents the shifting vector, \(\mathbf{d}_{k}\) represents the design variable solution from the MDSDO stage, \(\mathbf{u}_{k}(\mathbf{t})\) represents the control strategy from the MDSDO stage, and \(\mathbf{x}_{dv}(\mathbf{t})\) and \(\mathbf{x}_{Rv}(\mathbf{t})\) are the deduction trajectories obtained based on \((\mathbf{d}_{k}-\mathbf{v}_{k})\) and \((\mathbf{R}_{k}-\mathbf{v}_{k})\), respectively. The deduction method is described in the following. \(C_{rate}\) is the index of the constraint violation and is denoted the \"crosstate\". \(C_{rate}\) counts the number of constraint violations of the trajectory in the observation point, which is deduced based on the accompanying samples \(\mathbf{R}_{k}\). Therefore, to illustrate the crossrate calculation method, the deduction process needs to be described. In the traditional approach, the solution of the design variable \(\mathbf{d}_{k}\) from the MDSDO stage is directly used in the reliability assessment stage. The solution of the state and control variable is processed under an ideal situation. However, in a practical situation, the accompanying sample \(\mathbf{R}_{k}\) represents the product with the limit reliability index, so that it is on the verge of failure. Without additional calibration, the product \(\mathbf{R}_{k}\) will apply the same control strategy \(\mathbf{u}_{k}(\mathbf{t})\) as the ideal product. Therefore, in this paper, the control strategy \(\mathbf{u}_{k}(\mathbf{t})\) that was designed on the basis of the deterministic situation is employed to deduce the state trajectory. Since the control strategy from the previous process and the equation of state are known, the fourth-order Runge-Kutta method can be adopted for the deduction process of the accompanying samples. This method applies Formula (11) to obtain the scatter of the state trajectory within the given time interval based on the known equation of state, the initial state, and the given step size. \[\left\{ \begin{array}{l}\mathbf{x}(\mathbf{m}+1)=\mathbf{x}(\mathbf{m})+\frac{ h}{6}(\mathbf{q}_{1}+2\mathbf{q}_{2}+2\mathbf{q}_{3}+\mathbf{q}_{4})\\ \mathbf{x}(0)=\mathbf{x}_{0}\end{array} \right. \tag{11}\]  
where \(\mathbf{x}_{0}\) is the initial state, \(\mathbf{x}(\mathbf{m})\) are the discrete points at the time node \(m\), and \(h\) is the given step size. Other parameters are calculated as shown in Formula (12). \[\left\{ \begin{array}{l}\mathbf{q}_{1}=f(\mathbf{t}(\mathbf{m}),\mathbf{x}( \mathbf{m}))\\ \mathbf{q}_{2}=f\Big{(}\mathbf{t}(\mathbf{m})+\frac{h}{2},\mathbf{x}(\mathbf{ m})+\frac{h}{2}\mathbf{q}_{1}\Big{)}\\ \mathbf{q}_{3}=f\Big{(}\mathbf{t}(\mathbf{m})+\frac{h}{2},\mathbf{x}(\mathbf{ m})+\frac{h}{2}\mathbf{q}_{2}\Big{)}\\ \mathbf{q}_{4}=f(\mathbf{t}(\mathbf{m})+h,\mathbf{x}(\mathbf{m})+h\mathbf{q}_{ 3})\end{array} \right. \tag{12}\]  
where \(f(\cdot)\) represents the equation of state of the system and \(\mathbf{t}(\cdot)\) is the given time node. At the beginning of trajectory deduction, each accompanying sample is substituted into the original equation of state to obtain the corresponding initial state value.