# Production Planning Problem of a Two-Level Supply Chain with Production-Time-Dependent Products
## 1 Introduction
However, in each period, not every supplier starts its setup operation for processing a production batch. Therefore, we have to determine not only the production quantity, but also suppliers to be operated. Focusing on the determination in each period, the considered problem becomes the special case of a typical facility location problem (FLP). Both problems deal with selecting locations of the facilities to be operated among possible (candidate) facilities to satisfy the demand. Additionally, in general, these decisions are handled by the central manager of department, taking into account the entire systems. However, in a typical FLP, the locations of facilities are decided at the beginning of the planning horizon and the locations are fixed throughout the entire period, whereas the decisions on suppliers' operations in the FLPP-VC can be changed during the planning horizon. The studies of FLPP-VC are very rare, while FLPs have been studied by many researchers. Thus, we categorized and reviewed the variants of FLPs into four parts: uncapacitated FLP, capacitated FLP, dynamic FLP, and FLP determining capacities. For an uncapacitated facility location problem (UFLP), a branch-and-bound algorithm (B&B) was proposed in [4]. Additionally, for capacitated facility location problems (CFLPs), various algorithms were devised, including B&B [5; 6], genetic algorithms [7; 8; 9; 10], and Lagrangian relaxation based heuristic algorithms [11; 12; 13; 14]. The dynamic facility location problem (DFLP), which is an FLP with multi-period dynamic demands, is very similar to our problem (FLPP-VC). However, like a typical FLP, the locations of the facilities in the DFLP are determined once at the beginning and not changed, even if it can worsen the overall efficiency of the supply chain for the entire planning period. Additionally, the semi-finished products inventory is not allowed in FLPP-VC, while DFLP can have positive inventory. For the DFLP, dynamic programming algorithms were developed in [15; 16; 17], while a simulated annealing was developed in [18] and an MIP based heuristic algorithm in [19]. In addition, an optimal solution algorithm was suggested in [20], considering a dual problem of the original DFLP. Additionally, ref. [21] studied an extended version of the problem considering setup, transportation, production costs and taxes (between two countries), and proposed a branch-and-bound algorithm, while [22] studied the problem considering relocation of facilities for the planning horizon and propose heuristics based on [PERSON]'s decomposition and Lagrangian relaxation method. In their model, the opening and closing of the facilities are determined in each period but opening decisions for each facility can be done only once in the planning horizon. Ref. [23] considered a multi-period facility location problem with two different classes of customers, depending on the tolerance of delayed delivery. Ref. [24] proposed heuristic algorithms for dynamic facility location problems consisting of determining locations and sizes of facilities. Several extensions of the DFLP are also found, such as considering multi-commodities [25; 26] and time restrictions for deliveries [27]. Furthermore, many heuristics and exact methods have been studied to solve the discrete multi-period facilities location problem, for example in [28; 29; 30; 31; 32]. In the considered problem, capacities of facilities can be changed. There have been very few previous studies, if any, in which changeable capacity levels were considered. However, the FLPs with determining capacities of suppliers have been considered in several studies. Among them, refs. [33; 34; 35; 36; 37] consider models in which at most one capacity level can be selected among a set of possibilities. Additionally, there are some studies on FLPs in which expansion of capacities is considered [38; 39; 40; 41; 42].