## 5 Computational Results  
The proposed two-phase stochastic variable neighborhood search algorithm approach is coded in C++ and is run on Intel(r) Core(tm) 2 Duo CPU E7500 2.93 GHz under the Windows 7 OS. The algorithm parameters' values used are the ones presented in Table 1, Section 3.1. In order to demonstrate its efficiency and very good performance, the proposed algorithm is compared with six very effective algorithms for solving the nurse rostering problem issued in the literature [32; 33; 34; 35; 36; 37] in solving the same seven input instances. In Table 2, the performance and efficiency of the proposed two-phase stochastic variable neighborhood search algorithm is shown by comparing the best timetables constructed by it with the best timetables created by the other six algorithms for each different input instance. Since in [32; 33; 34; 35; 36; 37] the best rosters created are presented, we also decided in the current contribution to present and compare the best rosters constructed by the proposed approach in order to have a fair comparison between the algorithms. Note that, for each different input instance, in order to compare the roster constructed by the proposed variable neighborhood search algorithm and the roster constructed by the respective published nurse rostering algorithm, we used the same fitness function, the same hard and soft constraints and the same constraint weights as the ones used by the respective algorithms. Table 2 demonstrates that the proposed algorithm outperforms other published approaches in 6/7 cases (85.7%), considering the best roster per instance, while it achieves the same result in 1/7 cases (14.3%). From experimental results presented in Table 2, one can easily come to the conclusion that the proposed algorithm is very efficient and achieves better results compared to the other six techniques issued in the literature that have been applied to the same instances of the nurse rostering problem. Moreover, in order to demonstrate the efficiency and very satisfactory performance of the proposed two-phase variable neighborhood search algorithm, the best rosters found by it are compared with the best-known timetables ever reported for the same seven different input nurse rostering instances [38]. Table 3 demonstrates that the proposed two-phase algorithm manages to reach the best known fitness ever reported in the literature in 6/7 cases (85.7%), while it manages to beat the best known fitness ever reported in the literature in 1/7 cases (14.3%). From experimental results presented in Table 3, one can easily come to the conclusion that the proposed algorithm is very efficient and achieves results equal to the best known ever reported for the majority of these quite different nurse rostering instances, while it manages to beat the best known ever reported result in one case. Except for that, the proposed algorithm has demonstrated experimentally that the best result ever reported for these six instances is not unique, since the best nurse rosters that the proposed algorithm created are in all six cases different from the best ones ever reported [38]. This means that in these six cases, there are at least two different best ever reported rosters and that maybe the global optimum is yet to be found. The executables implementing the proposed stochastic variable neighborhood search algorithm, as well as the best rosters achieved for each input instance, can be accessed in [47].