## 4 Proposed Method
### Performing the Track-Touring Process  
The track selects one out of a certain number of neighborhood strategies with the probability function shown in Equations (15) and (17), proposed by [PERSON] et al. [39].  
\[G_{bt}=\frac{S_{bt}}{\sum_{c=1}^{C}S_{ct}} \tag{15}\]  
\[S_{bt}=FN_{bt-1}+\ (1-F)A_{bt-1}+KI_{bt-1} \tag{16}\]  
\[P_{bt}=\left\{\begin{array}{ll}G^{Max}\ \ if\ \ G_{bt}>G^{Max}\\ G^{min}\ \ if\ G_{bt}<G^{min}\\ G_{bt}\ \ otherwise\end{array}\right. \tag{17}\]  
where \(G_{bt}\) is the probability of black box \(b\) selecting in iteration \(t\) before it is adjusted by the edge boundary. \(C\) is equal to the total number of black box \(c\), and \(b\) is the index of black box \(b\) or \(c\). \(S_{bt}\) is the weight to select black box \(b\) in iteration \(t\). \(N_{bt-1}\) is the number of tracks that select black box \(b\) in the previous iteration. \(A_{bt-1}\) is the average objective function of all tracks that select black box \(b\) in the previous iteration. \(I_{bt-1}\) is a binary decision  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Job** & 7 & 3 & 10 & 8 & 4 & 1 & 9 & 2 & 5 & 6 \\ \hline
**Machine** & B & C & A & C & B & A & B & C & A & B \\ \hline
**Energy used (THB)** & 30 & 19 & 29 & 18 & 18 & 26 & 28 & 29 & 27 & 23 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Results of the proposed decoding method in Step 3.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Job** & **10** & **1** & **5** & & & \\ \hline Machine A & S\({}_{10,A}\) & C\({}_{10,A}\) & S\({}_{1,A}\) & C\({}_{1,A}\) & S\({}_{5,A}\) & C\({}_{5,A}\) & & \\ S\({}_{\text{Im}}\)/C\({}_{\text{jm}}\) & 0 & 19 & 19 & 41 & 41 & 70 & & \\ D\({}_{\text{j}}\) & & 200 & & 30 & & 120 & & \\ L\({}_{\text{j}}\) & & 0 & & 11 & & 0 & & \\ T\({}_{\text{j}}\) & & 0 & & 1 & & 0 & & \\ \hline
**Job** & **7** & & **4** & & **9** & & **6** & \\ \hline Machine B & S\({}_{7,\text{B}}\) & C\({}_{7,\text{B}}\) & S\({}_{4,\text{B}}\) & C\({}_{74\text{B}}\) & S\({}_{9,\text{B}}\) & C\({}_{9,\text{B}}\) & S\({}_{6,\text{B}}\) & C\({}_{6,\text{B}}\) & \\ S\({}_{\text{jm}}\)/C\({}_{\text{jm}}\) & 0 & 29 & 29 & 58 & 58 & 76 & 76 & 98 \\ D\({}_{\text{j}}\) & & 150 & & 85 & & 50 & & 45 \\ L\({}_{\text{j}}\) & & 0 & & 0 & & 26 & & 53 \\ T\({}_{\text{j}}\) & & 0 & & 0 & & 1 & 1 & 1 \\ \hline
**Job** & **3** & & **8** & & **2** & & & \\ \hline Machine C & S\({}_{3,C}\) & C\({}_{3,C}\) & S\({}_{8,C}\) & C\({}_{8,C}\) & S\({}_{2,C}\) & C\({}_{2,C}\) & & \\ S\({}_{\text{jm}}\)/C\({}_{\text{jm}}\) & 0 & 18 & 18 & 35 & 35 & 51 & & \\ D\({}_{\text{j}}\) & & 80 & & 30 & & 45 & & \\ L\({}_{\text{j}}\) & & 0 & & 5 & & 6 & & \\ T\({}_{\text{j}}\) & & 0 & & 1 & & 1 & & \\ \hline \hline \end{tabular}
\end{table}
Table 5: Sequence and scheduling from Table 4.  
variable. It is equal to 1 if the box contains the iterative best solution of the last iteration; otherwise, it is equal to 0. \(F\) is a predefined random variable that lies between 0 and 1. \(K\) is predefined factors that are located between 1 and 5. \(P_{bt}\) is the probability to select black box \(b\) in iteration \(t\) after the edge boundary. \(G^{Max}\) and \(G^{min}\) are the maximal and minimal probabilities that are allowed to select a black box, respectively.  
The black boxes (neighborhood strategies) applied in this research were: (1) solution decompose and repair (SDR); (2) track-transition method (TTM); and (3) multiplier factor (MF).