# Matheuristics and Column Generation for a Basic Technician Routing Problem
## 6 Computational Results
### Results with MILP-VND Local Search  
Tables 5 and 6 provide the comparative results among local search approaches. For the MILP-VND matheuristics, it allows comparing the quality of the local minimum obtained by using different initialization heuristics. The comparative analyses on subsets of instances in Table 6 illustrate the impact of the difficulty of constraints for the efficiency of local search approaches.  
Using only complementary neighborhoods with pairs of technicians \(\mathcal{N}_{i,i^{\prime}}^{pair}\) and reoptimization of buckets of 10 jobs \(\mathcal{N}^{jobs}\) in different partitions allowed considering larger neighborhoods than the traditional ones, contains the other main neighborhood and allowed quick resolution times. As in [4], a very short number of iterations of the MILP-VND  
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & **Fail** & **BKS** & **Mean** & \(\boldsymbol{\sigma}\) & **min** & **Q1** & **Med** & **Q3** & **Max** \\ \hline LocalSolver 30 s & 2 & 0 & 13.68\% & 9.93\% & 2.43\% & 6.47\% & 10.78\% & 19.09\% & 35.86\% \\ LocalSolver 60 s & 2 & 0 & 11.24\% & 9.61\% & 2.31\% & 3.76\% & 6.99\% & 19.03\% & 32.18\% \\ LocalSolver 300 s & 2 & 0 & 5.36\% & 5.78\% & 1.30\% & 1.96\% & 3.35\% & 6.96\% & 24.52\% \\ LocalSolver 600 s & 2 & 0 & 3.89\% & 2.63\% & 0.70\% & 1.91\% & 3.35\% & 5.06\% & 8.76\% \\ LocalSolver 900 s & 1 & 1 & 2.99\% & 2.37\% & 0.00\% & 1.68\% & 2.49\% & 2.95\% & 8.26\% \\ \hline RMP\_INT\_HEUR & 0 & 6 & 1.09\% & 1.32\% & 0.00\% & 0.00\% & 0.50\% & 1.82\% & 3.66\% \\ LAG\_HEUR & 0 & 7 & 0.81\% & 1.10\% & 0.00\% & 0.00\% & 0.43\% & 1.07\% & 3.66\% \\ LAG\_RINS & 0 & 8 & 0.19\% & 0.29\% & 0.00\% & 0.00\% & 0.08\% & 0.29\% & 1.12\% \\ LAG\_RINS+VND & 0 & 15 & 0.08\% & 0.27\% & 0.00\% & 0.00\% & 0.00\% & 0.00\% & 1.12\% \\ \hline
5 job,nbTree+VND & 0 & 3 & 2.91\% & 2.84\% & 0.00\% & 0.49\% & 2.10\% & 4.71\% & 9.23\% \\
5 job,timJeb+VND & 0 & 2 & 3.60\% & 3.59\% & 0.00\% & 0.70\% & 2.66\% & 5.56\% & 14.32\% \\
10 job,nTree+VND & 0 & 1 & 3.34\% & 3.17\% & 0.00\% & 1.24\% & 2.84\% & 4.43\% & 13.53\% \\
10 job,timJeb+VND & 0 & 0 & 4.71\% & 3.57\% & 0.09\% & 2.33\% & 4.13\% & 7.03\% & 11.67\% \\ init0+VND & 0 & 2 & 5.91\% & 6.04\% & 0.00\% & 1.11\% & 4.44\% & 10.46\% & 20.91\% \\ \hline Multi-start VND & 0 & 4 & 1.71\% & 2.33\% & 0.00\% & 0.11\% & 0.66\% & 2.  
using iteratively all the considered neighborhoods were enough for converging to a local minimum; for most instances, 2/3 of such iterations were needed, and the worst cases used five such iterations. MILP-VND allows having quick convergence and scaled better for all the larger instances than exact methods. Contrary to [4], such large neighborhoods are not sufficient to converge to local minimums of excellent quality. The quality of the local minimum varies significantly following the different initialization. Considering the four matheuristic initialization and the null initialization, followed by the MILP-VND provided a multi-start matheuristic that is accurate for finding good solutions, the most difficult dataset \(\text{sc}_{max}\) provided the worst solutions. Dataset \(\text{sc}_{max}\) is generally the more difficult one for local search approaches, with less feasible neighbor moves because of the hard constraints. This was not the case of the CG-based heuristics. Using MILP-VND after the LAG_RINS heuristic of an excellent quality allowed polishing the results and increases the number of BKS found, illustrating the advantages, drawbacks and complementarity of local search approaches and CG heuristics depending on the structures of the instances.