## 5 IICA-QL for EEMsMIAJSP
### Revolution Based on Hyper-Heuristic Variable Neighborhood Search
\begin{array}{l}0.1,\textrm{ new solution is not dominated by old solution}\\ -0.1,\textrm{ otherwise}\end{array} \right. \tag{31}\]

After completing the revolution operations, the cost of each country within the empire is calculated, and the country with the highest cost is selected as the new imperialist, while the original imperialist becomes a colony. \(\mathbf{P^{G}}(\mathbf{x}\mathbf{y})\) is updated according to Equation (31). ### Joint Imperialist Invasion

Dividing the population into multiple subpopulations is helpful for individuals to extensively search different areas and balances the exploration and exploitation capabilities of the algorithm through co-evolution [33]. To address the issues of poor diversity and premature convergence in the basic imperialist competitive algorithm, a joint imperialist competition mechanism is designed to replace the standard imperialist competition operations. Each empire is regarded as a population that collaboratively searches different regions of the solution space. This mechanism ensures that the collaborative evolutionary search is maintained within each empire and that information sharing and interaction are enhanced between empires, thereby improving diversity and the search efficiency of the algorithm. The specific operations of the joint imperialist invasion are as follows: The total cost of each imperial group is calculated and ranked according to Equation (32), where \(Emp_{k}\) represents the set of colonies owned by \(Imp_{k}\), and \(\xi\) is the proportional coefficient of the total cost of the empire accounted for by the colonies, which is set to 0.1. The imperialists of the top \(N_{im}-1\) empires perform POX crossover [18] with random solutions, and the corresponding numbers of the worst colonies in the worst-ranked empire are replaced by the newly generated solutions. Through joint imperialist competition, the collaborative evolution within each empire can be maintained, while information sharing and interaction between empires can be enhanced, thus improving diversity and increasing the search efficiency of the algorithm. The pseudocode is shown in Algorithm 4. \[TC_{k}^{g}=c_{i}^{g}+\xi\sum_{\sigma\in Em_{k}}c_{\sigma}^{g}/NC_{k} \tag{32}\]
```
Input: empires \(Emp\), number of imperialists \(N_{im}\) Output: updated \(Emp\) and new population \(NevPop\) For\(k=1\) to \(N_{im}\)do  Calculate the total cost of each empire and rank them in non-ascending order End For\(i=1\) to \(N_{im}-1\)do  Perform POX between the imperialist and a random solution; the newly generated solution replaces the worst colony of the worst empire End For  Merge all imperialist countries into \(NewPop\)
```

**Algorithm 4** Joint imperialist invasion

### Algorithm Description

A flowchart of the proposed IICA-QL is shown in Figure 6. The process of IICA-QL is described as follows. Step 1: Generate the initial population using heuristic rules and random generation methods. Select \(N_{im}\) imperialist countries to form the initial empires. Step 2: Construct the external archive set and update it based on non-dominated solutions. Step 3: Select the UX operator parameter according to the Q-table, then perform assimilation in each empire and update Q-table. Meanwhile, retain colonies based on the elite retention strategy. Step 4: Execute high-dimensional strategies constructed by the two-dimensional probability model to select low-level neighborhood search operation sequences. Update the probability model based on feedback information. Step 5: Perform joint imperialist competition to achieve the collaborative evolution of empires. Step 6: If the termination condition is met, stop the search; otherwise, return to Step 3. Figure 6: Framework of IICA-QL. ## 6 Experimental Results and Discussion

The proposed IICA-QL is implemented in MATLAB R2021a. The experimental configuration is an Intel Core i7-13700K and CPU @3.40 GHz PC with 32 GB memory (Intel, Santa Clara, CA, USA). Since no prior research has been conducted on the EEMsMIAJSP, the test data are generated randomly.