# Production Planning Problem of a Two-Level Supply Chain with Production-Time-Dependent Products
## 2 Problem Description  
This study considers a problem of production planning for the production-time-dependent products in a two-level supply chain consisting of multiple suppliers and a manufacturing plant. Each supplier has an upper limited capacity for production quantity which can be produced simultaneously at the same batch. However, it is assumed that the manufacturing plant is uncapacitated, that is, the plant can satisfy all demands of external customers. In a poultry industry, the production process of a manufacturer is simple and automated, such as packaging or cutting. Additionally, in general, the capacity of manufacturer is relatively larger than those of the suppliers. Thus, it is sufficient to consider only the capacities of suppliers for solving the real problem. To establish a production plan for the considered supply chain, a planning manager determines which suppliers start to produce semi-finished products in each period, capacities of the selected suppliers, and production quantities of each semi-finished product type at each supplier. The results of the planning are quantities of raw materials to be input into the suppliers for production of the semi-finished products.  
As mentioned in the previous section, the considered problem is motivated from a poultry farming industry in Korea. Thus, this study makes the following assumptions based on the situation of the poultry farming industry.  
1. The supply chain consists of multiple suppliers and a manufacturing plant;
2. Locations of the manufacturing plant and the (candidate) suppliers are pre-determined and given;
3. The suppliers produce several types of semi-finished products, and different semi-finished products may be produced at the same production line of a supplier. Note that we need to decide suppliers, product lines, and product types to produce, respectively;
4. If two or more types of semi-finished products are to be produced at the same supplier, they should be started simultaneously;
5. In a supplier, a new batch cannot be started while a batch is being produced;
6. A setup operation is needed (for cleaning and maintenance of a production line) at each supplier before each production run of a batch;
7. The capacity levels of production lines of each supplier and setup cost for each level are known and fixed;
8. Capacity of each supplier is equal to the sum of capacity levels of production lines in the supplier;
9. The demand for each product type in each period may vary by period and is known in advance;
10. There is no shortage of raw material for the semi-finished products;
11. There is no defective in the production process at the suppliers. Hence, the input and the output quantities at a supplier are the same;
12. All suppliers are ready to start production at the beginning (starting) of the planning horizon. In addition, there is no need for a setup operation for the first batch of the planning horizon;
13. The transportation time is negligibly short (compared to a period);
14. Only one production line can be selected and operated for a supplier at each operating period;
15. Production quantity of operated production line in a supplier to be operated in each period is equal to the capacity of the operated production line.  
For a clearer description of the problem, we develop a mixed integer programming formulation. We use this formulation to find an optimal solution for this problem using a commercial integer programming solver in the computational experiments.  
The following notation is used throughout the paper.