## 6 Conclusions  
This paper presents a variable block insertion heuristic (VBIH) algorithm for solving the permutation flow shop scheduling problem (PFSP) with makespan criterion. In addition, we introduce mixed integer programming (MIP) and constraint programming (CP) models to solve the small benchmark set and to verify the results of our proposed heuristic algorithm. By employing the time limited CP model, we can find optimal solutions for some of small VRF instances for the first time in the literature. Furthermore, all algorithms can generate better solution values than upper those currently exist in the literature. We adapted a well-known speed-up method of Taillard and applied all the necessary parts while coding the heuristic algorithms. The parameters of the proposed VBIH algorithm is tuned through a design of experiments on randomly generated benchmark instances. Extensive computational results on two new VRF benchmark suites show that the VBIH algorithm is superior to the best performing algorithms from the literature. CP model found and verify optimal solutions for 108 out of 240 small VRF instances, whereas 236 out of 240 large VRF benchmark instances are further improved by the VBIH and IG\({}_{\rm ALL}\) algorithms for the first time in this paper with remaining solutions being equal, which are also given in Appendix B  
Figure 10: Interval plot at the 95% confidence level for Taillard’s instances. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **IG\({}_{\mathbf{RS}}\)** & **IG\({}_{\mathbf{RS}}\)\({}^{*}\)** & **IG\({}_{\mathbf{ALL}}\)** & **IG\({}_{\mathbf{ALL}}\)\({}^{*}\)** & **VBIH** & **VBIH \({}^{*}\)** \\  & **Avg** & **Avg** & **Avg** & **Avg** & **Avg** & **Avg** \\ \hline \(20\times 5\) & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \(20\times 10\) & 0.01 & 0.00 & 0.00 & 0.00 & 0.00 & 0.01 \\ \(20\times 20\) & 0.01 & 0.01 & 0.00 & 0.00 & 0.00 & 0.01 \\ \(50\times 5\) & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \(50\times 10\) & 0.34 & 0.43 & 0.40 & 0.43 & 0.26 & 0.31 \\ \(50\times 20\) & 0.57 & 0.79 & 0.53 & 0.71 & 0.33 & 0.53 \\ \(100\times 5\) & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \(100\times 10\) & 0.10 & 0.19 & 0.04 & 0.11 & 0.02 & 0.09 \\ \(100\times 20\) & 0.82 & 1.33 & 0.89 & 1.23 & 0.54 & 0.94 \\ \(200\times 10\) & 0.05 & 0.14 & 0.03 & 0.05 & 0.03 & 0.05 \\ \(200\times 20\) & 1.04 & 1.46 & 0.82 & 1.29 & 0.55 & 1.02 \\ \(500\times 20\) & 0.47 & 0.92 & 0.35 & 0.75 & 0.26 & 0.64 \\ Overall Avg. & 0.28 & 0.44 & 0.26 & 0.38 & **0.17** & **0.30** \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Computational results of [PERSON]’s instances with \(T_{max}=45\times n\times m\) milliseconds (The bolds show better results). (Table A5). Furthermore, three instances of [PERSON]'s benchmark suite are also further improved for the first time in this paper since 1993. As future research, VBIH algorithm can be easily extended to other variants of the PFSPs such as no-idle, blocking and no-wait PFSP. In addition, other performance criteria can be considered such as total flow time and total tardiness. Furthermore, different meta-heuristic algorithms or matheuristics can be proposed to solve the PFSP. The following are available online at [[http://www.mdpi.com/1999-4893/12/5/100/s1](http://www.mdpi.com/1999-4893/12/5/100/s1)]([http://www.mdpi.com/1999-4893/12/5/100/s1](http://www.mdpi.com/1999-4893/12/5/100/s1)). Conceptualization, M.F.T. and D.K. ; methodology, Q.-K.P. ; software, D.K., M.F.T. and Q.-K.P. ; validation, D.K., and M.F.T. and L.G. ; writing--original draft preparation, D.K. ; writing--review and editing, L.G. and M.F.T. ; supervision, M.F.T., Q.-K.P. ; project administration, L.G; funding acquisition, L.G. This research is partially supported by the National Natural Science Foundation of China (Grant No. 51435009). **Acknowledgments:** [PERSON], [PERSON] and [PERSON] acknowledge the HUST Project in Wuhan in China. We also thank [PERSON] for providing the permutations for large VRF benchmark suite so that we were able to check our code.