# Production Planning Problem of a Two-Level Supply Chain with Production-Time-Dependent Products
## 1 Introduction  
This study considers a problem of production planning in a two-level supply chain consisting of multiple suppliers and a manufacturing plant. Each supplier that consists of multiple production lines can produce several types of semi-finished products and the manufacturing plant produces the finished products using the semi-finished products from the suppliers to meet dynamic demands. The planning manager of this supply chain selects suppliers to produce semi-finished products and determines the quantity of semi-finished products to be produced for each period to minimize total costs associated with the supply chain (the information of demand in each period is known in advance). We focus on a case in which semi-finished products are produced by almost the same general process, but those products are sorted according to their production times. That is, the type of semi-finished product depends on the production time. In this paper, we assume that different types of semi-finished products can be produced in the same batch, and products in the same batch can only be started simultaneously (at the same time) even if they complete at different times. Such production characteristics (called _production time-dependent products_) can be easily found in food industries such as fermented foods, wines, and poultry. In the case of poultry farming industry, it has only been studied toanalyze diseases and environmental issues [1; 2], or to determine the location and size of livestock [3]. However, production management of the poultry industry is also one of the most important issues, but there is little research on this issue. Therefore, we focus on the production planning with consideration of characteristics of poultry farming. In this industry, sizes (or weights) of the products are one of major attributes that distinguish the types of final products. In the suppliers, the size (weight) of the poultry may be the only distinguishing attribute if we do not consider exceptional cases like infection of diseases. In general, the size of poultry depends on the breeding time, and hence production plans for the semi-finished products at the suppliers are established considering the breeding time (or production time) of the semi-finished products. The production time-dependent products can also be found in other industries such as steel industry and chemical industry. In the poultry industry, many companies own and operate manufacturing plants and many poultry farms together. In the concept of supply chains, the farms occupy the role of suppliers. However, the farms are not independent but are subordinate to headquarters. Therefore, the production plan of such suppliers depends on the manufacturer's plan. Based on this situation, we consider a system in which one management unit controls all suppliers and a manufacturing plant. The management determines:  
1. whether each supplier should be opened and operated in each period;
2. the total production quantities of the semi-finished product among candidate levels;
3. production quantities of each type of semi-finished product at the operated suppliers in each period. When we determine suppliers to start production in each period, we also have to determine production quantity levels of the suppliers. For example, in real-world cases of poultry farms, there may be multiple numbers of production quantity levels depending on the number of production lines of a farm. That is, production quantity levels of farms are determined by the number of production lines to be operated to satisfy the quantity of products to be produced in each period. In other words, the farms may have to determine production quantity levels of semi-finished products in each period. We call this problem as the production planning problem with discrete production quantity levels for production-time-dependent products (FLPP-VC). As mentioned above, the FLPP-VC determines production quantity levels in each period as a normal production planning problem.