## 3 Principle of I-NSGA-III-VLC Method
### Variable Length Cegment Hybrid Coding Method  
Due to the need for considering the rotation between different units, the chromosome-coding method needs to be changed to distinguish different units. The rotation stage is placed before the task stage. This paper adopted the variable length and hybrid segmented-coding methods. Let us add a number \"prefix\" before each rotation, such as 1, 2, \(\ldots\), \(c*(c-1)\). Therefore, the shortest chromosome length at the rotation stage is \(c*(c-1)\), and the maximum quantity of equipment for each rotation is \(g\). Therefore, the longest chromosome length is \(c*(c-1)*(g+1)\). After the rotation phase, according to the task intensity sequence \(U\) and \(V\) of each unit, according to the unit quantity, let us add a number \"prefix\" before each unit task, such as \(c*(c-1)+1\),\(c*(c-1)+2\), \(\ldots\)\(c^{2}\). Therefore, the total length of chromosomes in the task phase is \(\sum\limits_{i=1}^{c}N_{i}^{\prime}*a+c\). Each coding segment is divided by a digital \"prefix\". With this coding method, we can not only distinguish each unit but also cross and mutate under the same \"prefix\". The range of crossover and mutation is reduced, and it is easy to control the chromosome length. The chromosome initialization process is shown below. Step 1: Arrange all equipment numbers \(S_{i}\) and remaining usable times \(T\) in each unit to form a sequence in \(S^{\prime}_{i}\) in the format of \"Number: Remaining Usable Times\". Like all equipment numbers, \(S^{\prime}_{1}\) of \(u_{1}\) is \(S^{\prime}_{1}=\) ['A': 4, 'C': 5, 'E': 5, 'F': 6, 'B': 6, 'D': 7], and like all equipment numbers, \(S^{\prime}_{2}\) of \(u_{2}\) is \(S^{\prime}_{2}=\) ['AA': 3, 'CC': 6, 'EE': 7, 'BB': 7, 'FF': 8, 'DD': 9]. Step 2: Adopt a segmental coding method where there are \(c*(c-1)\) coding segments in the rotation phase, and the shortest length of each coding segment is 1 and the longest is \(g+1\). Assume \(c=2\), \(g=3\). Randomly select \(l_{1}\) equipment from \(u_{1}\) and randomly select \(l_{2}\) equipment from \(u_{2}\),\(0\leq l_{1},l_{2}\leq g\), where the equipment number and \"prefix\" together constitute the code segment, i.e., [1, 'A', 'B', 2, 'CC', 'FF']. This indicates that the first rotation is to rotate the equipment numbered 'A' and 'B' in \(u_{1}\) to \(u_{2}\), and the second rotation is to rotate the equipment numbered 'CC' and 'FF' in \(u_{2}\) to \(u_{1}\). Add 'A' and 'B' to \(u_{1}\) according to the remaining usable times, delete them in \(u_{2}\), add 'CC' and 'FF' to \(u_{2}\) according to the remaining usable times, delete them in \(u_{1}\), update \(S^{\prime}_{1}\) and \(S^{\prime}_{2}\), and obtain \(S^{\prime\prime}_{1}=\) ['C': 5, 'E': 5, 'F': 6, 'CC': 6, 'D': 7, 'FF': 8] and \(S^{\prime\prime}_{2}=\) ['AA': 3, 'A': 4, 'B': 6, 'EE': 7, 'BB': 7, 'DD': 9]. Step 3: In the task phase, the order of executing the task of the equipment is not considered in the same coding segment, but it cannot be reused. Gene \(j\) can be reused in different coding segments, but the number of repetitions cannot exceed the corresponding remaining usable times. The total length of code in a task stage is \(U_{i}*v\). Assume \(U_{1}=2\), \(U_{2}=1\), \(v=4\), initialize the chromosome, select \(v\) equipment from \(u_{1}\) as a group, repeat \(U_{1}\) times and select \(v\) equipment from \(u_{2}\) as a group, repeat \(U_{2}\) times, such as [3, ['CC', 'C', 'E', 'F'], ['C', 'E', 'FF', 'D'], 4, ['AA', 'B', 'EE', 'DD']]. This coding method can be directly interpreted as selecting the equipment to complete the task. The segmented real-number coding conforms to the practical significance, and it is easy to control the upper limit of usage times. Step 4: Combine the chromosomes of the two stages to acquire \(C_{1}=\) [1, 'A', 'B', 2, 'CC', 'FF', 3, ['CC', 'C', 'E', 'F'], ['C', 'E', 'FF', 'D'], 4, ['AA', 'B', 'EE', 'DD']]. By the same token, \(C_{2}=\) [1, 'C', 2, 'DD', 'AA', ['AA', 'A', 'E', 'F'], ['B', 'E', 'DD', 'D'], 4, ['C', 'FF', 'EE', 'BB']].