## 4 The Proposed NSGA-II-MOEA/D-CIS Algorithm
### Design of the Collaborative Iterative Strategy  
The core of the collaborative iterative strategy (CIS) is to connect the upper and lower optimization processes through a dynamic feedback mechanism. The optimization results from the lower-level MOEA/D are evaluated for satisfaction. If they do not meet the upper-level requirements, the upper-level NSGA-II parameters are adjusted to regenerate solutions, which are then fed back to the lower level for further optimization. This process continues iteratively until a mutually satisfactory solution is obtained, forming a closed-loop optimization. The specific implementation steps of this collaborative iterative strategy are as follows:  
Step 1: Objective normalization. To eliminate the impact of dimensional differences between different objectives on the evaluation results, we first normalize the lower-level objectives--production cost and delayed delivery time--into the same range. After normalization, both optimization objectives will fall within the [0,1] interval. The normalization formula used is as follows:  
\[g_{1}=\frac{C_{produce}-C_{produce,min}}{C_{produce,max}-C_{produce,min}} \tag{48}\]  
\[g_{2}=\frac{T_{delay}-T_{delay,min}}{T_{delay,max}-T_{delay,min}} \tag{49}\]  
Step 2: Constructing the satisfaction function. Based on the normalized objectives, a comprehensive satisfaction function S is constructed to reflect the overall quality of the  
Figure 7: Example of uniform crossover and swap mutation operation (The black arrow indicates the transition from the original individual to the mutated individual. The two ends of the red arrow indicate the positions of the two genes involved in the exchange mutation).  
lower-level solutions. A commonly used approach is the weighted sum model, which is mathematically expressed as follows:  
\[S=\alpha\cdot g_{1}+\beta\cdot g_{2}, \tag{50}\]  
where \(\alpha\) and \(\beta\) are weight functions that reflect the importance of production cost and delayed delivery time, respectively, with their sum equal to 1.  
Step 3: Satisfaction evaluation. For each solution obtained from the lower-level MOEA/D optimization, this paper calculates its satisfaction value S and compares it with a predefined satisfaction threshold. If S is lower than the threshold, the solution is considered unsatisfactory. The satisfaction threshold can be determined based on historical production report data. Specifically, this paper extracts the production cost and delayed delivery time data corresponding to each historical order within a given period. After normalizing and weighting these two factors, the average across all orders is taken to obtain the satisfaction threshold.  
Step 4: Feedback and parameter adjustment. If unsatisfactory solutions exist in Step 3, feedback is provided to the upper level, and the parameters are adjusted using the following three methods:  
1. Dynamic adjustment of objective weights. If the lower-level delayed delivery time is too high, increase the weight of the splitting granularity objective in the upper level to enforce finer-grained splitting solutions (finer granularity may reduce single-batch load, thus decreasing delayed delivery time). Conversely, if the lower-level production cost is too high, increase the weight of the splitting cost objective in the upper level to generate coarser-grained splitting solutions (coarser granularity may reduce production batches, thereby lowering equipment switching costs).
2. Population injection and elite retention. Select upper-level solutions that lead to better lower-level performance and retain them as elite individuals.
3. Algorithm parameter adjustment. If unsatisfactory solutions persist over multiple iterations, adjust the crossover rate and mutation rate of the upper-level NSGA-II algorithm to prevent prolonged stagnation in local optima.  
Step 5: Iterative optimization. Continue executing Steps 1 to 4 until a sufficient number of satisfactory solutions are generated or the maximum number of iterations is reached.