## 5 Conclusions  
The computational effort for solving the 1D-CSP is largely affected by the number of the cutting patterns. In order to cope with large instances of the problem, we proposed a strategy to restrict the number of patterns to be generated while applying a pattern generating procedure. Using benchmark instances, computational experiments were performed for ILP models, an implementation of the CG technique, and an application of the G&S framework. We begin the conclusions on our experimental analysis, stating that the exact approach was only able to cope well with small and medium-sized instances. In addition, none of the CG and G&S algorithms stood out from the other for all instances. In fact, the effectiveness of these methods varied according to the characteristics of the instances, such as the number of item types, the size of the items in relation to the size of the object, and the demand multiplicity. Nonetheless, G&S performed well for all classes, obtaining quasi-optimal solutions for the majority of the instances. In addition, the strategy of artificially reducing the number of cutting patterns proved to be very effective. It is important to remark, however, that G&S does not present any sort of guarantee of convergence and the optimality gap cannot be estimated by the G&S method alone. On the other hand, CG was able to produce strong lower bounds fast. Therefore, G&S could be integrated to CG to speed up the convergence of branch-and price algorithms, especially when CG fails to produce good-quality integer feasible solutions. Future research is also \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**CG**} & \multicolumn{3}{c}{**G\&S**} \\ \hline
**Instance** & **db** & **pb** & **Time (s)** & **Best** & **avg** & **ttb (s)** \\ \hline HARD0 & 55 & 63 & 600.0 & **56** & 57.6 & 340.6 \\ HARD1 & 56 & 62 & 600.0 & **57** & 57.0 & 21.2 \\ HARD2 & 56 & 63 & 600.0 & **57** & 58.6 & 175.3 \\ HARD3 & 55 & 61 & 600.0 & **56** & 57.7 & 140.8 \\ HARD4 & 56 & inf & 600.0 & **57** & 58.8 & 173.7 \\ HARD5 & 55 & 61 & 600.0 & **56** & 58.6 & 163.6 \\ HARD6 & 56 & 64 & 600.0 & **57** & 58.6 & 112.0 \\ HARD7 & 54 & 61 & 600.0 & **56** & 56.0 & 13.2 \\ HARD8 & 56 & 63 & 600.0 & **57** & 57.9 & 178.8 \\ HARD9 & 55 & 62 & 600.0 & **56** & 56.7 & 71.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results for the set of instances of class _hard10_. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**CG**} & \multicolumn{3}{c}{**G\&S**} \\ \hline
**Instance** & **db** & **pb** & **Time (s)** & **Best** & **avg** & **ttb (s)** \\ \hline WAE\_GAU1\_TEST0005 & 28 & inf & 33.8 & **29** & 29.0 & 8.4 \\ WAE\_GAU1\_TEST0014 & 23 & **23*** & 18.9 & 24 & 24.0 & 4.6 \\ WAE\_GAU1\_TEST0022 & 14 & **14*** & 6.6 & 15 & 15.0 & 2.4 \\ WAE\_GAU1\_TEST0030 & 27 & **27*** & 28.9 & 28 & 28.0 & 4.9 \\ WAE\_GAU1\_TEST0044 & 14 & **14*** & 13.6 & 15 & 15.3 & 133.9 \\ WAE\_GAU1\_TEST0049 & 11 & inf & 6.7 & **12** & 12.0 & 185.7 \\ WAE\_GAU1\_TEST0054 & 14 & inf & 12.6 & **15** & 15.0 & 167.4 \\ WAE\_GAU1\_TEST0055 & 15 & inf & 11.9 & **16** & 16.3 & 133.8 \\ WAE\_GAU1\_TEST0052 & 20 & inf & 12.1 & **22** & 22.1 & 112.9 \\ WAE\_GAU1\_TEST0058 & 20 & inf & 15.5 & **20*** & 20.7 & 89.3 \\ WAE\_GAU1\_TEST0065 & 15 & 19 & 6.8 & **16** & 16.0 & 1.4 \\ WAE\_GAU1\_TEST0068 & 12 & inf & 7.2 & **13** & 13.0 & 74.5 \\ WAE\_GAU1\_TEST0075 & 13 & inf & 7.0 & **14** & 14.1 & 48.8 \\ WAE\_GAU1\_TEST0082 & 24 & inf & 10.2 & **24*** & 24.9 & 11.3 \\ WAE\_GAU1\_TEST0084 & 16 & inf & 17.7 & **16*** & 16.5 & 120.8 \\ WAE\_GAU1\_TEST0095 & 16 & inf & 15.3 & **17** & 17.3 & 162.3 \\ WAE\_GAU1\_TEST0097 & 12 & inf & 10.7 & **13** & 13.0 & 15.5 \\ \hline \hline \end{tabular} \({}^{\star}\)Proven optimal solutions.  
\end{table} 
Table 5: Results for the set of instances of class _wave_gau1_. envisaged to adapt the proposed framework using arc-flow formulations, which have shown to be appealing in branch-and-price algorithms. Conceptualization, N.N. ; methodology, N.N. ; software, J.V.S.S.