## 3 Two-Individual Based Master-Apprentice Evolutionary Algorithm
### Interval Scheduling-Based Decoding Scheme  
Section 2.3 outlines a method for deriving uncertain interval processing times through the establishment of confidence levels. By employing this approach, the processing time \(\widetilde{P}_{i}\) for operation \(i\) can be expressed as an interval, specifically \(\widetilde{P}_{i}=\left[\underline{P_{i}},\overline{P_{i}}\right]\). The boundary range of this interval can be adjusted by varying the confidence levels, which serves as an effective strategy for enhancing the robustness of the scheduling plan. When uncertain processing times are represented as intervals, this paper introduces interval scheduling theory to facilitate the decoding and representation of scheduling plans. Unlike the decoding processes associated with deterministic scheduling problems, interval scheduling redefines the fundamental operational rules for interval numbers to compute operation start and completion times, along with machine availability times. In the context of interval scheduling, the primary operations involved are the addition and comparison of interval numbers. Assuming that A and B are two interval numbers, \(A^{L}/B^{L}\) and \(A^{R}/B^{R}\) represent the left and right limits, respectively. This paper employs the operational rules proposed by [PERSON] et al. [27]. \[A+B=\left[A^{L}+B^{L},\ A^{R}+B^{R}\right] \tag{22}\]  
\[P(A\leq B)=\left\{ \begin{array}{cc}0&A^{L}\geq B^{R}\\ 0.5*\frac{B^{R}-A^{L}}{A^{R}-A^{L}}*\frac{B^{R}-A^{L}}{B^{R}-B^{L}}&B^{L}\leq A ^{L}\leq B^{R}\leq A^{R}\\ \frac{B^{L}-A^{L}}{A^{R}-A^{L}}+0.5*\frac{B^{R}-B^{L}}{A^{R}-A^{L}}&A^{L}\leq B ^{L}\leq B^{R}\leq A^{R}\\ \frac{B^{L}-A^{L}}{A^{R}-A^{L}}+\frac{A^{R}-B^{L}}{A^{R}-A^{L}}*\frac{B^{R}-B^ {L}}{B^{R}-B^{L}}+0.5*\frac{A^{R}-B^{L}}{A^{R}-A^{L}}*\frac{A^{R}-B^{L}}{B^{R}- B^{L}}&A^{L}\leq B^{L}\leq A^{R}\leq B^{R}\\ 1&B^{L}\leq A^{L}\leq A^{R}\leq B^{R}\\ \end{array} \right. \tag{23}\]  
Based on the rules of interval number operations, the decoding steps for interval scheduling aimed at minimizing \(C_{max}\) are outlined as follows:  
Step 1: Establish the rules for interval number operations and gather the interval information for the processing times of each operation. Step 2: Initialize the current availability time for each machine as \([0,\ 0]\) and set the release time for each operation to \([0,\ 0]\). Step 3: Extract the operation sequence numbers in the order of the OS segment encoding and retrieve the corresponding machine numbers using the MS segment encoding. Employ the interval number comparison method to compare the release times of operation \(i\) with the earliest availability times of the corresponding machines. The larger of the two values is designated as the operation start time \(O_{i\_S}\). Subsequently, apply interval addition to combine the processing time \(\widetilde{P_{i}}\) with the start time \(O_{i\_S}\) to obtain the completion time \(O_{i\_E}\). Step 4: Update the availability time of the machine processing the operation and the release time of the corresponding component based on the newly determined completion time \(O_{i\_E}\). Step 5: Repeat Steps 3 and 4 until all operations have been computed. Step 5: Output the maximum value of all operation completion times, denoted as \(C_{max}\). As evidenced by the aforementioned process, interval scheduling differs from deterministic scheduling schemes in that each operation is defined by two interval numbers: its start time and completion time. Therefore, the \(C_{max}\) obtained through interval scheduling is also represented as an interval number. Figure 6 illustrates the Gantt chart corresponding to the encoding scheme presented in Figure 4, where the interval processing times for each operation are as follows: \(P1:[4,\ 6]\), \(P2:[1,\ 3]\), \(P3:[9,\ 11]\), \(P4:[7,\ 9]\), \(P5:[8,\ 10]\), \(P6:[2,\ 4]\), and \(P7:[3,\ 5]\). In contrast to the conventional Gantt chart representation vertex indicating the latest completion time.