## 2 Integration of Ordinal Optimization with Ant Lion Optimizer
### Informed Ant Lion Optimization
Figure 4 shows the relationship between inputs and output in queueing network design, where \(\mathbf{x}\) expresses the design vector, \(\lambda\) denotes the arrival rate of messages, \(L\) indicates the number of replications, and \(\overline{f}(\mathbf{x})\) is the sample mean. ### Apply the OALO Algorithm

#### 3.3.1 Establish the Approximation Model

Various basis functions in different intervals of the input variables space are used to build the MARS approximation model. The output of MARS \(F(\mathbf{x})\) is predicted by a linear function, which is composed of an intercept constant and weighted basis functions. To obtain the optimal MARS approximation model, we need to determine the intercept constant and expansion coefficients of basis functions. There are four steps to establish the MARS approximation model for roughly estimating a design vector. (i) Randomly select \(M\)\(\mathbf{x}\)'s from solution space and calculate \(\overline{f}_{a}(\mathbf{x})\) by precise estimation. (ii) Express the \(M\) design vectors and the corresponding sample means as \(\mathbf{x}^{(i)}\) and \(\overline{f}_{a}(\mathbf{x}^{(i)})\), respectively. (iii) The optimal MARS approximation model is constructed using a two-stage process. (iv) The ordinary least squares method is utilized to calculate the coefficient of each term. The following hinge function is adopted as the basis function in the MARS approximation model. \[w^{k+1}=w_{\min}+(w_{\max}-w_{\min})\times\left(1-\exp\!\left(-\frac{w_{\max }}{w_{\min}}\times\frac{k+1}{k_{\max}}\right)\right) \tag{12}\]3

where \(k_{m}\) denotes the amount of splits corresponding to the \(m\)th basis function, \(s_{k,m}=\pm 1\), \(v(k,m)\) is the \(v\)-th variable, \(1\leq v\leq n\), \(n\) indicates the dimension of \(\mathbf{x}\), \(x_{v(k,m)}\) is the variable split, and \(l_{k,m}\) denotes a knot. The '+' subscript indicates that a truncated function is used in the associated bracketed term. \[\Bigl{[}s_{k,m}\times\Bigl{(}x_{v(k,m)}-l_{k,m}\Bigr{)}\Bigr{]}_{+}=\left\{ \begin{array}{cc}s_{k,m}\times\Bigl{(}x_{v(k,m)}-l_{k,m}\Bigr{)},if&s_{k,m} \times\Bigl{(}x_{v(k,m)}-l_{k,m}\Bigr{)}>0\\ 0,\,otherwise\end{array} \right. \tag{22}\]

Figure 4: Input output relationship in queueing network design. MARS finds possible univariate candidate knots and across interactions among all variables. The selected input variables and knot locations are obtained using an exhaustive search algorithm. A \"loss of fit\" (LOF) criterion is utilized to optimize predictors, knots and interactions simultaneously. MARS chooses the LOF which most improves the approximation model at each search process. #### 3.3.2 Apply the RALO

With the assistance of MARS approximation model, \(N\) exceptional designs can be quickly chosen from solution space using the RALO. At first, \(\Psi\) ants and antlions are arbitrarily created as the initial population. Since the decision variables are the routing percentages with a range of 0-100%, the positions of ants and antlions in initial population are randomly generated by \(U[0,100]\), where \(U[0,100]\) is a uniformly distributed random variable that ranges between 0 and 100. The fitness of each ant and antlion is obtained by the MARS approximation model. After the RALO executed \(k_{\max}\) iterations, the final \(\Psi\) antlions are sorted based on their fitness. When a real value of the optimal design variable is obtained, we can round this real value to the nearest integer using the rounding function \(z_{j,k}^{i}=\left\lfloor x_{j,k}^{i}\right\rfloor\), where \(x_{j,k}^{i}\in\Re\) and \(z_{j,k}^{i}\in Z\). Finally, we choose the former \(N\) antlions to construct the candidate subset. The value of \(N\) may not be large to save computational effort; however, some critical designs may be lost when the value of \(N\) is small. #### 3.3.3 Obtain the Quasi-Optimal Design

Finally, the R&S technique is utilized to find a quasi-optimal design from the \(N\) exceptional designs.