# Matheuristics for the Design of a Multi-Step, Multi-Product Supply Chain with Multimodal Transport
## 2 Literature Review  
This section will describe some of the literature, first related to the problem of supply-chain network design, and later, related to matheuristics, commenting about the gap covered by the research presented in this paper. [PERSON] and [PERSON] [2] created a model called PLANWAR, which focused on optimizing the supply chain through a heuristic to decide on the opening of plants and warehouses and the flow between them. However, the model does not handle different levels of capacity between facilities nor multimodal transportation. [PERSON] et al. [3] solved the supply-chain planning problem where the same product can be produced in multiple facilities, but their work focuses more on analyzing different algorithms and the complexity of each one of them. In their problem of supply-chain design, [PERSON] et al. [4] consider delivery times and transportation modes, but their work focuses mainly on outbound logistics and is not multi-tier. [PERSON] and [PERSON] [5] solved a multi-product supply-chain problem using a mixed-integer linear programming model in which the opening of facilities is decided, as well as the level of capacity and the mode of transport of flows between levels of the supply-chain. However, the work takes into account only finished products. [PERSON] et al. [6] focus on optimizing the transportation of a two-tier supply chain, calculating the flow and time between facilities using bi-objective optimization. The problem addressed in this work is for a single product. [PERSON] and [PERSON] [7] developed a mixed-integer linear programming model and a heuristic based on the evolutionary algorithm of the firefly, where transport and construction costs are taken into account, but their solution method focuses on problems in telecommunications or power distribution companies. [PERSON] et al. [8] developed min-max methods and a heuristic to solve the multi-tier inventory problem taking into account purchasing, manufacturing, and transportation costs, but they do not address the issue of opening of facilities. Additionally, they do not take into account the bill of materials. Many more recent models address the problem of supply-chain network design with different considerations, for example, ref. [9] describe a problem with environmental and financial considerations, or [10] analyze a model considering inventory decisions besides the classic decisions for location and transportation. However, the research in the literature is far from a general model for supply-chain network design that includes all the situations, variables, and decisions. Our purpose was to analyze a problem with a high level of complexity to be a challenge for new methods of solution. In particular, very few models consider the product architecture represented in the bill of materials. One of the reasons is that introducing this element creates an explosion in the number of variables and interactions, making it very hard to solve even small instances. As can be observed in the literature presented, the most used solution methods are based on metaheuristics because of the computational complexity of the models used. We are proposing new methods of solution into the field of matheuristics, which rarely has been used in SCND. The term matheuristic is relatively new; the term began to be used between 2008 and 2009. [PERSON] et al. [11] talk about the hybridization that can exist between mathematical programming and metaheuristics. [PERSON] and [PERSON] [12] explain that matheuristics exploit heuristics and metaheuristics to improve and facilitate the mixed-integer programming (MIP) model. They show three applications: optimization of layout, packaging, and routing. Moreover, there is a rising interest in these hybrid methods, demonstrated by a growing literature and specialized tutorials in operations research [13]. Matheuristics have been applied to different decision-making problems, for example in routing [14], production planning [15], lot sizing [16], and risk management [17], among others.