# Matheuristics for the Design of a Multi-Step, Multi-Product Supply Chain with Multimodal Transport
## 6 Results  
The generated instances were resolved with the mixed-integer linear programming model programmed in OPL in the IBM ILOG CPLEX 12.8.0 IDE, and the case of the matheuristics, an application in \(C++\) Concert Technology was programmed with the use of CPLEX. Both were run on a Lenovo ThinkPad T580 computer with an Intel Core i7-8550U @ 1.80 GHz processor and 16 GB of RAM. The results obtained are shown in Table 4, indicating the total cost obtained for each instance with the different methods compared. The results show that as the instances increase in size, the computation time to obtain optimal Solutions increases exponentially. In all the cases, the MILP reached the time limit of 4 h. Only for instances with 100 clients, the matheuristic algorithms achieved times below 25 min. However, for instances larger than 100 clients, the matheuristic algorithms reached the time limit of 2 h. Hence, obtaining results through the use of mathematical modeling becomes impractical for large instances. It can be observed in Table 4 that when the size of the instance grows, i.e., it has a higher number of clients, the total cost increases also. Likewise, it can be seen that as the instance size increase, the difference between the quality of solutions provided by CPLEX and those provided by the matheuristics decreases. For instances of 100, 125 and 150 clients, the results of the matheuristic algorithm with the \"simple heuristic\" were just 14.20% on average above those obtained with CPLEX, and the results with the matheuristic algorithm \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & \(|C|\) & \begin{tabular}{c} **Number of** \\ **Continuous Variables** \\ \end{tabular} & \
\begin{tabular}{c} **Number of** \\ **Binary Variables** \\ \end{tabular} &
\begin{tabular}{c} **Number of** \\ **of Constraints** \\ \end{tabular} \\ \hline
*-100 & 100 & 18,000 & 8990 & 17,190 \\ \hline
*-125 & 125 & 37,050 & 15,196 & 28,103 \\ \hline
*-150 & 150 & 60,750 & 22,410 & 40,785 \\ \hline
*-175 & 175 & 100,170 & 32,736 & 57,893 \\ \hline
*-200 & 200 & 144,000 & 43,780 & 76,380 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistics of the instances and their values for MILP. with the \"heuristics with probability\" were only 6.06% on average above those obtained with CPLEX. In the instances of both 175 and 200 clients, at least one of the two implemented matheuristics obtained a better solution than that of CPLEX in most of the cases. For the instances with 175 customers, the average improvement was 1.41% comparing the result of CPLEX and the result of the matheuristic algorithm with the \"simple heuristic\". For the same instances, the average improvement was 5.96% comparing the result of CPLEX and the result of the matheuristic algorithm with the \"heuristics with probability\". For the instance 2-200, the improvement was 3.93% comparing the result of CPLEX and the result of the matheuristic algorithm with the \"simple heuristic\". For the same instance, the improvement was 9.22% comparing the result of CPLEX and the result of the matheuristic algorithm with the \"heuristics with probability\". In six cases out of 15, with 175 and 200 clients, the matheuristic algorithms were able to find a feasible solution while CPLEX could not. Comparing the two matheuristics, it can be observed that the heuristic method used to decide the opening of warehouses has a great impact on the objective functions for the type of instances used because the fixed cost of opening a warehouse is the cost with the greatest impact. Both the first and second heuristics diversify the results at each iteration so that in small instances, the algorithms may be used on more than one occasion to increase the quality of the solutions. Regarding the use of the probability distribution used for constructive heuristics with probability, although some randomness is applied, the selected warehouses will tend to be those with the lowest cost of opening divided by their capacity, so the quality of the solutions tends to be better.