## Embedded Ordinal Optimization into Tree-Seed Algorithm  
### Mathematical Formulation  
The mathematical formulation of a general PCSOP is described as follows [1].  
\[\min\ E[f(\mathbf{x})] \tag{1}\]  
\[\text{subject to }P[g(\mathbf{x})\geq 0]\geq\theta \tag{2}\]  
\[\mathbf{V}\leq\mathbf{x}\leq\mathbf{U} \tag{3}\]  
where \(\mathbf{x}=\left[x_{1},\ldots,x_{n}\right]^{\mathrm{T}}\) is an \(n\)-dimensional decision vector, \(\mathbf{V}=\left[V_{1},\ldots,V_{n}\right]^{\mathrm{T}}\) denotes the vector of lower bound, \(\mathbf{U}=\left[U_{1},\ldots,U_{n}\right]^{\mathrm{T}}\) represents the vector of upper bound, \(f(\mathbf{x})\) is the objective function, \(E[f(\mathbf{x})]\) represents the expected objective value, \(P[g(\mathbf{x})\geq 0]\) denotes the probability of the satisfaction of the constraint \(g(\mathbf{x})\geq 0\), and \(\theta\) is a fixed probability level lying in a range [0,1].  
Multiple simulation runs are carried out to obtain an accurate estimate of the expected objective value and probability. In practice, it is impossible to run simulations infinitely long. An alternative way of approximating the expected objective value and probability is to use the sample mean as follows.  
\[\overline{f}(\mathbf{x})=\frac{1}{L}\underset{j=1}{\overset{L}{\underset{j=1} {\sum}}}f_{j}(x) \tag{4}\]  
\[p(\mathbf{x})=\frac{1}{L}\underset{j=1}{\overset{L}{\underset{j=1}{\sum}}}y_{j} \tag{5}\]  
where \(L\) represents the amount of simulation runs, \(f_{j}(\mathbf{x})\) denotes the objective value of the \(j\)th simulation run, \(y_{j}=0\) indicates that the constraint \(g(\mathbf{x})\geq 0\) is not satisfied, and \(y_{j}=1\) denotes that a satisfaction of the constraint. The sample mean of \(\overline{f}(\mathbf{x})\) achieves a better approximation of \(E[f(\mathbf{x})]\) when \(L\) increases.  
Because the probabilistic constraint is a soft constraint, a quadratic penalty function [18] is used to convert the constrained optimization problem into an unconstrained optimization problem.  
\[\min\ F(\mathbf{x})=\lambda\times\overline{f}(\mathbf{x})+(1-\lambda)\times PF (\mathbf{x}) \tag{6}\]  
where \(\lambda\in(0,1)\) is a penalty weight, \(F(\mathbf{x})\) is a weighted objective value, and \(PF(\mathbf{x})\) indicates a quadratic penalty function as follows.  
\[PF(\mathbf{x})=\left\{\begin{array}{cc}0,&\quad if\ p(\mathbf{x})\geq\theta,\\ 10^{4}\times\left(\theta-p(\mathbf{x})\right)^{2},\mathit{else}.\end{array}\right. \tag{7}\]  
There is a sharp jump in the penalty function to guarantee that the probability does not exceed the fixed probability level. Let \(L_{s}\) denote the large enough of simulation runs, and the precise evaluation of Equation (6) is defined by using \(L=L_{s}\). For simplicity, we define \(F_{s}(\mathbf{x})\) as the objective value of Equation (6) obtained by precise evaluation for \(\mathbf{x}\).  
As pointed out by the OO theory [9], order of a solution is probably preserved even evaluating by a surrogate model. To select \(N\) outstanding solutions in an exceptionally short period of time, we need to establish a surrogate model which can evaluate a solution very fast and utilize an optimization technique assisted by the surrogate model. The RMTS [17] is adopted as a surrogate model, and the ITSA is the optimization technique.