## 4 Encoding and Decoding Design
### Design of Energy-Efficient Strategy Decoding Trigger Mechanism  
To address the issue of substantial idle energy consumption caused by the hierarchical coupling characteristics of the EEMsMIAJSP, an energy-efficient strategy based on block-moving can reduce machine idle time by making dispersed operations more compact without changing the maximum completion time, thus reducing the idle energy consumption generated by machines.  
However, as the problem scale increases, executing the energy-efficient strategy usually takes a long time and consumes significant computational resources. In fact, not all solutions benefit from this strategy in terms of improving the quality of non-dominated solutions. Therefore, we propose an adaptive trigger mechanism for the energy-efficient strategy decoding, which ensures potential high-quality solutions are not lost while improving the search efficiency. The specific operations are as follows:  
Step 1: Apply the decoding rules and the active decoding method [30] to the individual to obtain the scheduling results \(C_{\max}(\mathbf{\pi})\) and \(TEC(\mathbf{\pi})\), and record the idle time of all machines.  
Step 2: According to Equations (15) and (16), obtain the upper bound of energy savings \(ES_{UB}\) and the lower bound of energy consumption \(E_{LB}\), respectively, for the scheduling scheme after applying the energy-efficient strategy.  
Step 3: Determine the relationship between \((C_{\max}(\mathbf{\pi}),E_{LB})\) and the current non-dominated solutions. If they are not dominated by the current non-dominated solutions or if they are mutually non-dominating, proceed to Step 5; otherwise, proceed to Step 4.  
Step 4: Use the objective values \(C_{\max}(\mathbf{\pi})\) and \(TEC(\mathbf{\pi})\) obtained from the decoding rules as the decoding results of the scheduling scheme.  
Step 5: Following the product assembly sequence of the scheduling scheme, check each product in reverse order from the last to the first according to the process sequence, and determine whether the block-moving condition is met according to Equation (17). If the condition is met, update the completion time of the operation according to Equation (18). Otherwise, check the preceding operation of this operation.  
Step 6: If all operations have been checked, use Equation (19) to determine whether the idle intervals meet the shutdown conditions for profit and loss time, and output the results \(C_{\max}(\mathbf{\pi})\) and \(TEC(\mathbf{\pi})\) of the scheduling scheme after executing the energy-efficient strategy. Then, update the non-dominated solution set. Otherwise, return to Step 5.  
\[ES_{UB}=\sum_{a=2}^{N_{A}^{k}}IP(\pi_{a-1}^{k},\pi_{a}^{k})\times PI_{k} \tag{15}\]  
\[E_{LB}=TEC(\mathbf{\pi})-ES_{UB} \tag{16}\]  
\[P(\pi_{a}^{k})\leq IP(\pi_{I}^{k},\pi_{I+1}^{k}) \tag{17}\]\[CT(\pi_{a}^{k})=\left\{\begin{array}{l}\max(CTI_{i}^{k}),\;\forall i,\;\pi_{a}^{k} \text{ is the final assembly operation of product}\\ \min\Bigl{\{}C(\pi_{last}^{k}),\max(CTI_{i}^{k})\Bigr{\}},\text{ otherwise}\end{array}\right. \tag{18}\]  
\[T_{off-on}\leq EC_{off-on}/\widetilde{EC} \tag{19}\]  
where \(IP(\pi_{l}^{k},\pi_{l+1}^{k})\) is the idle time after the operation \(\pi_{a}^{k}\) on machine \(m,l=a,a+1,\cdots,N_{A}^{k}-1\). \(CTI_{i}^{k}\) denotes the end time of the insertable idle interval for \(\pi_{a}^{k}\) on machine \(k\), and \(\pi_{last}^{k}\) denotes the immediate subsequent operation of \(\pi_{a}^{k}\). \(T_{off-on}\) is the time required for the machine to turn off and on, \(EC_{off-on}\) is the energy consumption for the machine to turn off and on, and \(\widetilde{EC}\) is the energy consumption per unit time of the machine. The flow of the energy-efficient strategy decoding trigger mechanism is shown in Figure 3.