# Production Planning Problem of a Two-Level Supply Chain with Production-Time-Dependent Products
## 1 Introduction
On the other hand, the production-time-dependent products dealt with in this paper have just been introduced in [43], and [44] studied a production planning problem of a two-level supply chain with production-time-dependent products considering outsourcing. These two studies consider that each supplier has only one production line, whereas this study considers multiple production lines in each supplier. Note that, in practice, a facility with multiple production lines opens the appropriate number of lines, not all lines. Finally, we reviewed several papers related to centralized supply chains that are controlled by a central managerial department (or manager): ref. [45] developed a model to design a centralized supply chain network in markets under deterministic price-depended demands and with a rival chain present; ref. [46] analyzed and compared centralized and decentralized model predictive control strategies to control inventory positions and to reduce the bullwhip effect in a benchmark four-echelon supply chain; ref. [47] investigated the supply chain scenarios for their relevant performance at the conceptual design stage and showed that the centralized supply chain scenario demonstrates superiority in cost performance; ref. [48] presented a mathematical model to plan logistics activities in a forest products supply chain; ref. [49] analyzed control policies for a centralized two-stage supply chain with subcontractors at each stage a number of numerical results show the impact of variations in certain system parameters on file optimal control parameters and the corresponding profit; and ref. [50] explored the impact of misplaced inventory on a centralized two-level supply chain, which consists of a risk-neutral supplier and a risk-averse retailer. The main contributions of this paper are as follows. To the best of our knowledge, this study is the first attempt to solve a production planning problem that needs to determine the location of facilities (suppliers) with multiple production lines in each period and the quantities of production-time-dependent products to be produced in each supplier. For the problem, we propose a mixed-integer linear programming (MILP) and a heuristic algorithm, considering variable capacity levels for production-time-dependent products with dynamic demands in a two-level supply chain. The heuristic algorithm consists of two steps: determining locations of facilities and then determining capacities of the facilities, i.e., determining which suppliers should be operated in each period, and the quantity produced for each type of semi-finished product from the supplier (selected for operating in each period). This paper is organized as follows. In Section 2, we present the MILP and explain the problem in detail, and then we introduce a heuristic algorithm in Section 3. In Section 4, to evaluate the performance of our proposed algorithm, computational experiments are performed, and the results are reported. Finally, we conclude this paper with a short summary and discuss possible extensions of our study in Section 5.