# Matheuristics for the Design of a Multi-Step, Multi-Product Supply Chain with Multimodal Transport
## 7 Conclusions  
In this work, a problem for supply-chain network design is addressed. This problem was presented in the literature by [PERSON] et al. [23]. We selected this complex NP-Hard problem to prove the efficiency of novel matheuristic algorithms. Mathemuristics are algorithms that combine heuristic rules with exact optimization methods based on mathematical programming. There is a growing interest in these methods in the search for improving the quality of solutions with a fast computation. Solving hard combinatorial optimization problems, Although heuristics and metaheuristics demonstrated efficiency in obtaining feasible but poor-quality solutions in short computation times, exact methods implemented in commercial and open-source software deliver optimal solutions at the cost of long execution times. In this way, the research in matheuristics looks for the benefits of hybridization. The problem addressed has characteristics that increase the level of complexity compared to other problems of supply-chain network design solved with matheuristics. In addition to the classic decisions on the location of facilities and transportation, the problem involves determining capacities for the facilities, choosing between different transportation modes, and considering a hierachical product architecture as described by a bill of materials. Hence, this problem is a good challenge to demonstrate the efficiency of the methods proposed. The structure of the network and decisions presented can be used for the design of global manufacturing and distribution networks in high technology industries such as pharmaceutics, automotive, or electronics. The main contribution of this paper is the proposal of novel matheuristic algorithms to solve a challenging combinatorial optimization problem for supply chain network design. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Instance** & **MILP \({}^{*}\)** & \
\begin{tabular}{c} **Matheuristic \({}^{**}\)** \\ **Simple Heuristic** \\ \end{tabular} &
\begin{tabular}{c} **Matheuristic \({}^{**}\)** \\ **Heuristic with** \\ **Probability** \\ \end{tabular} & **No. of Clients** \\ \hline
2-175 & \$1,100,417 & \$970,647 & \$898,271 & 175 \\ \hline
3-175 & *** & \$930,802 & \$857,758 & 175 \\ \hline
4-175 & \$1,099,033 & \$994,140 & \$939,693 & 175 \\ \hline
5-175 & \$915,431 & \$865,468 & \$929,748 & 175 \\ \hline
6-175 & \$882,227 & \$925,900 & \$798,514 & 175 \\ \hline
7-175 & \$856,623 & \$912,108 & \$881,164 & 175 \\ \hline
8-175 & \$940,480 & \$955,951 & \$912,570 & 175 \\ \hline
9-175 & *** & \$970,484 & \$854,358 & 175 \\ \hline
10-175 & \$977,692 & \$965,596 & \$919,445 & 175 \\ \hline
1-200 & *** & \$1,153,589 & \$1,115,358 & 200 \\ \hline
2-200 & \$1,061,059 & \$1,019,363 & \$963,204 & 200 \\ \hline
3-200 & *** & \$1,138,016 & \$1,015,605 & 200 \\ \hline
4-200 & *** & \$1,129,623 & \$1,019,119 & 200 \\ \hline
5-200 & *** & \$1,130,032 & \$994,430 & 200 \\ \hline \hline \end{tabular} \({}^{*}\)Target function values of incumbent solution running the solver with a 4-h limit; \({}^{**}\) Target function values of incumbent solution running executable with a 2-h limit; \({}^{***}\) No integer solution found.  
\end{table} 
Table 4: _Cont._The algorithms proposed were efficient in obtaining solutions of high quality in reasonable computation times for large instances. To study larger instances or longer supply chains, the problem can be divided into more sub-problems with heuristic decisions in between. As we observed, it is possible to obtain better solutions with a well-thought heuristic in less time as data sets grow. Additionally, more complex decisions can be added to the model to integrate routing, inventory management, sustainability issues, and variability through stochastic modeling. The key to designing efficient matheuristics is the adequate decomposition of the problem to take advantage of the quality of exact methods using heuristics and randomness to boost the execution speed. Conceptualization, D.A.R.-S. and E.O.-B.