# Matheuristics for the Design of a Multi-Step, Multi-Product Supply Chain with Multimodal Transport
## 3 Problem Description  
This paper addresses the problem of designing a four-tier multi-product supply-chain network (SCND): suppliers, factories, warehouses, and customers (Figure 1). The number and locations of plants and warehouses must be chosen from a set of potential plants and warehouses, respectively, and the capacity level of each factory and each warehouse must be chosen from a set of predetermined capacity levels for each location. Each location with each of its capacity levels has a fixed opening cost. Each of the materials can be supplied by a set of suppliers that have this material. Each material has a different purchase price for each supplier. Materials are shipped to factories, where they are converted into finished products. Each finished product has its bill of materials (BOM), which states what materials are needed for each finished product and how many units, so all the materials needed to produce a finished product will have to be taken to the factory where it is produced. Figure 2, shows the representation of the BOM; it exemplifies that to produce product \(p_{1}\), 4 units of material \(r_{1}\) and 3 units of material \(r_{2}\) are needed. The generic description is for any product \(p\) the necessary units of material \(r\) are given by the parameter \(A^{rp}\). Each product can be manufactured in a pre-established set of factories. Each of the products has a manufacturing cost in each plant. Once the products are manufactured, they can be taken to a set of pre-established warehouses. Finished products must be supplied \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \begin{tabular}{c} **Facilities** \\ **Opening** \\ \end{tabular} & \begin{tabular}{c} **Transportation** \\ **Cost** \\ \end{tabular} & \begin{tabular}{c} **Multi-Mode** \\ **Transportation** \\ \end{tabular} & \
\begin{tabular}{c} **Multi-Product** \\ **Multi-Product** \\ \end{tabular} & **BOM** & **4-Levels** &
\begin{tabular}{c} **Facility** \\ **Capacity Levels** \\ \end{tabular} & **Matheuristics** \\ \hline \hline
[2] & \(\chi\) & \(\chi\) & & & & & & & \\ \hline
[3] & \(\chi\) & \(\chi\) & & & & & & \\ \hline
[4] & & \(\chi\) & \(\chi\) & & & & & \\ \hline
[5] & \(\chi\) & \(\chi\) & \(\chi\) & & & & \(\chi\) & \\ \hline
[6] & \(\chi\) & \(\chi\) & \(\chi\) & & & & & \\ \hline
[7] & \(\chi\) & \(\chi\) & \(\chi\) & & & \(\chi\) & & \\ \hline
[18] & \(\chi\) & \(\chi\) & & & & & & \(\chi\) \\ \hline
[19] & & \(\chi\) & & & & & & & \(\chi\) \\ \hline
[20] & \(\chi\) & \(\chi\) & \(\chi\) & & & & & \(\chi\) & \(\chi\) \\ \hline
[21] & \(\chi\) & \(\chi\) & \(\chi\) & & & & \(\chi\) & \(\chi\) & \(\chi\) \\ \hline
[22] & \(\chi\) & \(\chi\) & & & & & & & \(\chi\) \\ \hline Our proposal, & & & & & & & & & \(\chi\) \\ based on [23] & \(\chi\) & \(\chi\) & \(\chi\) & \(\chi\) & \(\chi\) & \(\chi\) & \(\chi\) & \(\chi\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description the features for the literature presented. Figure 1: Representation of the supply chain posed in the problem. to each customer in a single delivery; that is, products cannot be shipped to the customer from two different warehouses. All materials and finished products can be transported by different transport methods for each destination-origin pair, which can be the following: supplier-factory, factory-warehouse, warehouse-customer. Each product or material has a transport cost for each means of transport for each possible pair. Each means of transportation for each origin-destination pair has a minimum quantity to be transported and a maximum transport capacity. Each supplier has a maximum capacity to supply each material. Each product occupies a different manufacturing or storage capacity. Each factory has a maximum production capacity for each product, a total maximum production capacity, and a minimum production capacity so that the factory can be opened for each of its capacity levels. Each warehouse has a maximum storage capacity and a minimum of storage to be opened.