## 2 Integration of Ordinal Optimization with Ant Lion Optimizer
### Informed Ant Lion Optimization
essence, all queuing systems are broken down into the entities queuing for an activity, e.g., dispensing and counseling. The goal of queuing design optimization in a communication system is to look for the optimal routing percentages for minimizing the expected total cost. Consider a queueing network design situation consisting of a communication system, one should determine the routing percentages to route randomly arriving messages to a particular destination [40]. There are \(n\) arrived random messages that need to go to a particular destination and there are J networks available to process these messages. Figure 3 shows an example of queueing network design with J networks. The per message processing cost is \(c_{1},c_{2},\ldots,c_{l}\) depending on which network the message is routed through. It also takes time for a message to go through a network. This transit time is denoted by \(t_{j}\) for each network \(j\), where \(t_{j}\) follows a triangular distribution with a mean \(u_{j}\), lower limit \(u_{j}-0.5\) and upper limit \(u_{j}+0.5\). There is a cost for the length of time a message spends in a network measured by K per unit time. The decision variables are the routing percentages \(\mathrm{P}_{1},\mathrm{P}_{2},\ldots,\mathrm{P}_{l-1}\in[0,\,100]\) which are the probabilities that a message will go through a particular network. When a message is in front of network \(j\), there is a \(\mathrm{P}_{j}\%\) chance that it will be processed by network \(j\). If the message is not processed by that network, then it will go to network \(j+1\), and will be processed with probability \(\mathrm{P}_{j+1}\%\), and so on. All messages arrive at network \(1\) with an exponentially distributed interarrival time with a mean of \(1/\lambda\) time unit. The average transit time on the network \(j\) is \(\mathrm{E}[t_{j}]\) in steady state. The goal is to minimize the expected total cost, which is the sum of processing cost and average transit cost. ### Mathematical Formulation

The optimization of queueing network design can be formulated as a CESOP. \[\min_{\mathbf{x}\in\mathbf{\Theta}}\ \ f(\mathbf{x})=\sum_{j=1}^{1}n\times p_{j} \times(c_{j}+\mathrm{K}\times\mathrm{E}[t_{j}]) \tag{15}\]

Subject to

\[p_{1}=\mathrm{P}_{1}/100 \tag{16}\]

\[p_{j}=\left(\mathrm{P}_{j}/100\right)\times\left(1-p_{j-1}\right),\,j=2,\ldots,\mathrm{J}-1 \tag{17}\]

\[p_{N}=1-\sum_{j=1}^{l-1}p_{j} \tag{18}\]

\[w^{k+1}=w_{\min}+(w_{\max}-w_{\min})\times\left(1-\exp\!\left(-\frac{w_{\max }}{w_{\min}}\times\frac{k+1}{k_{\max}}\right)\right) \tag{12}\]0 Queueing network design of J networks. where \(\mathbf{x}=[P_{1},\ldots,P_{l-1}]^{T}\) denotes a J\(-1\) dimensional design vector, \(f(\mathbf{x})\) is the expected total cost, \(P_{j}\in[0,100]\) denotes the routing percentage, \(\Theta=[0,100]^{J-1}\) denotes the feasible region, \(n\) is the number of messages, \(c_{j}\) denotes per message processing cost of network \(j\), K denotes transit cost, and \(\mathrm{E}[t_{j}]\) denotes the average transit time on the network \(j\) in steady state. The goal of the CESOP is to find the optimal routing percentages, \(\mathbf{x}^{*}\), such that the expected total cost is minimum. Generally, multiple simulation replications are performed to obtain the objective value. Thus, the sample mean is utilized to approximate the objective value. \[w^{k+1}=w_{\min}+(w_{\max}-w_{\min})\times\left(1-\exp\!\left(-\frac{w_{\max }}{w_{\min}}\times\frac{k+1}{k_{\max}}\right)\right) \tag{12}\]1

\[w^{k+1}=w_{\min}+(w_{\max}-w_{\min})\times\left(1-\exp\!\left(-\frac{w_{\max }}{w_{\min}}\times\frac{k+1}{k_{\max}}\right)\right) \tag{12}\]2

where \(L\) denotes the amount of replications, \(f_{\ell}(\mathbf{x})\) represents the objective value of the \(\ell\)th replication, and \(t_{j}^{\ell}\) denotes the transit time on the network \(j\) of the \(\ell\)th replication. For simplicity, we let \(\overline{f}_{a}(\mathbf{x})\) indicate the sample mean for a given \(\mathbf{x}\) using precise estimation.