## 6 Conclusions  
In this paper, we studied an application of the Nelder-Mead algorithm to the constrained '\(\lambda_{max}\)-optimal completion' and provided numerical simulations to study its performance. Our simulation results indicate that the proposed algorithm is capable of estimating the missing values in the incomplete PCMs, and it is simple, adaptable and efficient. Furthermore, the obtained solution is unique if and only if the undirected graph underlying the incomplete PCM is connected (by Theorem 1). It should be noted that the associated graph is necessarily connected if \(k\leq n-2\) and possibly connected if there are at most \(k=(n-1)(n-2)/2\) missing comparisons. If \(k>(n-1)(n-2)/2\), the graph is not connected because the number of known entries in the incomplete matrix is less than \(n-1\) (see, e.g., (63) (p. 7)). Most importantly, the average \(CR\) in Tables 4 and 5 are calculated on the basis of connected undirected graphs. Our proposal has its roots in the most widely used inconsistency index, as the \(CR\) proposed by [PERSON]. If, on the one hand, the \(CR\) was considered the standard for the \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{\(n=8\)} & \multicolumn{2}{c}{\(n=9\)} & \multicolumn{2}{c}{\(n=10\)} & \\ \hline \(k\) & **Average CR** & \(k\) & **Average CR** & \(k\) & **Average CR** & \(k\) & **Average CR** \\ \hline
13 & 0.3935 & 14 & 0.5186 & 20 & 0.4634 & 31 & 0.1553 \\
15 & 0.3011 & 16 & 0.4479 & 25 & 0.3236 & 32 & 0.1308 \\
18 & 0.1621 & 17 & 0.4112 & 28 & 0.2361 & 33 & 0.1032 \\
19 & 0.1206 & 22 & 0.2312 & 29 & 0.2091 & 35 & 0.0568 \\
21 & 0.0436 & 23 & 0.1953 & 30 & 0.1819 & 36 & 0.0350 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Average CR of 10,000 random matrices from the Saaty scale for an arbitrarily large number of missing comparisons \(k\) with respect to matrix size \(n\) that continued from Table 4. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{\(n=8\)} & \multicolumn{2}{c}{\(n=9\)} & \multicolumn{2}{c}{\(n=10\)} & \\ \hline \(k\) & **Average CR** & \(k\) & **Average CR** & \(k\) & **Average CR** & \(k\) & **Average CR** \\ \hline
13 & 0.0444 & 14 & 0.0592 & 20 & 0.0523 & 31 & 0.0169 \\
15 & 0.0335 & 16 & 0.0506 & 25 & 0.0364 & 33 & 0.0136 \\
18 & 0.0171 & 17 & 0.0461 & 28 & 0.0263 & 33 & 0.0106 \\
19 & 0.0125 & 22 & 0.0252 & 29 & 0.0232 & 35 & 0.0049 \\
21 & 0.0024 & 23 & 0.0212 & 30 & 0.0199 & 36 & 0.0021 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Average CR of 10,000 modified consistent matrices for an arbitrarily large number of missing comparisons \(k\) with respect to matrix size \(n\) that continued from Table 5.  
Figure 5: Computational time for the average \(CR\) of 10,000 matrices vs. number of missing comparisons \(k\) with respect to matrix size \(n\) corresponding to Table 4: (**a**) \(n=4\); (**b**) \(n=5\); (**c**) \(n=6\); and (**d**) \(n=7,8,9,10\). quantification of inconsistency, on the other hand, its role has been limited to this simple task. Its use for other purposes, as for instance, the optimal completion of pairwise comparisons matrices, has been impaired by its perception as a function difficult to treat. One of the purposes of this paper is also to help demystify this view. The future research could be a comparative analysis of the algorithm with other optimal completion methods (see for instance, refs. [13; 15; 18; 20; 64]). Conceptualization, H.A.T., M.F. and M.B. ; methodology, H.A.T., M.F. and M.B. ; software, H.A.T. ; validation, H.A.T., M.F. and M.B. ; writing--original draft preparation, H.A.T. ; writing--review and editing, H.A.T., M.F. and M.B. ; supervision, M.F. and M.B. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. We are grateful to [PERSON] for some suggestions on an early stage of this research. We also appreciate the editor, and all of our reviewers' insightful comments and suggestions, which helped us to improve the manuscript's quality.