## 3 Modeling  
### Problem Description  
The production model of medical infusion device manufacturers is generally make-to-order (MTO) [26]. Due to the complexity of product varieties and the limited production resources, traditional order management methods cannot effectively address issues such as order diversity, complex resource constraints, and dynamic changes in order demand [27]. Therefore, this paper proposes a multi-dimensional and multi-scale bi-level collaborative optimization model for order splitting and reorganization, aiming to develop rational strategies for order splitting and reorganization that improve production efficiency, reduce production costs, and meet order delivery deadlines.  
As shown in Figure 1, the model is divided into two levels. The upper level mainly focuses on the order splitting process, which is responsible for decomposing the order demand plan into multiple orders and product specifications, ultimately breaking down orders into several sub-orders (\(O_{i}\) denotes the i-th original order, \(S_{ij}\) denotes the \(j\)-th specification within order \(O_{i}\), \(T_{ijk}\) refers to the \(k\)-th sub-order derived from specification \(S_{ij}\)). The objective of this splitting process is to achieve an optimal granularity in order splitting. The lower level primarily focuses on the order reorganization process; it takes the set of sub-orders generated by the upper level as input and then reorganizes them into a new set of orders according to the relevant constraints. Its objective is to minimize production costs as well as order delivery delays. Throughout the entire process of order splitting and reorganization, multi-dimensional criteria (i.e., materials, processes, equipment, and labor efficiency) and multi-scale criteria (i.e., finished products, components, subassemblies, and parts) are adhered to, thereby considering various limitations encountered in actual production processes. Moreover, the upper-level order splitting provides input for the lower-level order reorganization, and the reorganization results are fed back to the upper level for adjustments. Through repeated iterative collaborative optimization, the final optimal production orders are achieved.  
In constructing the order splitting and reorganization model, and in order to realistically reflect the production processes of the enterprise while simplifying the complexities of real-world situations to render the model feasible and solvable, the following assumptions are made to satisfy the basic conditions of this bi-level collaborative optimization model:  
1. All order demand quantities are assumed to be deterministic throughout the order splitting and reorganization optimization process.
2. The material list required for each order is assumed to be fixed, with no substitute materials available.
3. The operations for each order follow a clear sequence, disallowing cross-process manufacturing or the merging of individual processes.
4. Sterilization equipment can be used only once per day and, once assigned to an order task, cannot be interrupted or preempted by another task until the current task is completed.
5. The granularity of order splitting must satisfy a minimum production batch size requirement, and orders cannot be split indefinitely, in order to ensure processing economy.  
Figure 1: Bi-level collaborative optimization framework for order splitting and reorganization.