## 4 Further Analysis and Discussion
### Code Optimization: The Recursive Implementation of the Permanent Value Determination Algorithm
For this decision case, it only requires about 2.022 s to complete the permanent value calculation step for GTMA, subject to the given problem size. Finally, in terms of efficiency, the recursive implementation to facilitate the solution process for GTMA in Section 3.3 should be another contribution of this study, and the implementations for the algorithms in Section 4.2 can be generalized to compute the permanent value for a square matrix of any dimension. In terms of decision support system (DSS) design (and thus not in terms of decision problem modeling), this implementation method is shown to be efficient by benchmarking. Note that another function that has just been supported by the BonsonSample package in September 2017 is similar to this function (permanent value computation in R), with a lower number of decimal places as a default. However, for this study, the relevant implementation work has started since May 2017. During the experimental stage of this study, in 2017, neither R nor its packages seemed to support the required permanent value computation. Relevant functions for this study were implemented on the R platform in parallel and developed for the MADM research purpose. ## 5 Conclusions

This work started with the doubt placed on the effectiveness of AHP-GTMA on aiding the industrial selection decisions. A systematic comparative research flow was proposed. According to this flow, the effectiveness of using the relatively new AHP-GTMA approach was examined by reference to the results from taking another \"widely used\" approach, AHP-TOPSIS. Based on such encouraging final outcome, the possible contributions of this study are thus reviewed and summarized. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**For A\({}_{i}\)** & **Start Time (s-ms)** & **Stop Time (s-ms)** & **PermTimeVector** \\ \hline
**PermRecur(\(C_{1}\))** & 43.974 & 44.230 & 0.2552431 \\
**PermRecur(\(C_{2}\))** & 44.230 & 44.482 & 0.2526710 \\
**PermRecur(\(C_{3}\))** & 44.482 & 44.735 & 0.2526729 \\
**PermRecur(\(C_{4}\))** & 44.735 & 44.994 & 0.2587240 \\
**PermRecur(\(C_{5}\))** & 44.994 & 45.251 & 0.2566819 \\
**PermRecur(\(C_{6}\))** & 45.251 & 45.505 & 0.2541721 \\
**PermRecur(\(C_{7}\))** & 45.506 & 45.760 & 0.2547059 \\
**PermRecur(\(C_{8}\))** & 45.760 & 46.012 & 0.2521710 \\ \hline
**Total Elapsed** & 2.022281 & **Average** & 0.2527851 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The Fundamental Building Block of Permanent Value Determination. The first contribution relates to the methodological flow that was proposed. It determines whether there is similarity between the final solutions that are obtained using two MADM methods. As only the final rank information is used (rather than using other information during the solution process) and only the ROVs in these final solutions (which is available from almost all MADM models) are used, the proposed flow should be a convenient tool which can be easily applied in operational practice. Except for the ROVs from using AHP-GTMA and AHP-TOPSIS (as in this study), it can be applied to determine the similarity between any two ROVs that are produced using any other two MADM methods, where appropriate. This is the methodological reason for naming this flow as a \"rank-based\" one. In addition, the flow includes several heterogeneous measures to determine the similarity between the ROVs from two methods using several measures. It suggests to establish both tangible measure(s) and the intangible measure(s), while the results from using these measures can be cross-validated or confirmed. The tangible measures can be arithmetic (i.e., the difference vector, absolute distance in rank for an alternative and the aggregated absolute distance) and/or geometrical (i.e., the Euclidean distance), while other concepts in the data analytics field can also be integrated (e.g., number of outliers, percent of alternatives ranked very similarly and maximum deviation in rank). Another niche is the establishment of the intangible measure from the perspective of statistics.