## 3 Decision Case Modeling and the Comparisons: Methods and the Results
### Description of the Decision Problem and Data Sourcing  
In the used case in [4], the size of problem \(P\) is defined by: (\(n\), \(m\)) = (7, 8), for which there are 8 shredder alternatives and 7 relevant criteria (attributes) in the criteria set, which is: {c1, c2, c3, c4, c5, c6, c7} = {Security level, Width of output strip or segment, Paper can size, Nestness of the shredder (in terms of the total volume of the shredder set), Noise level, Number of supported material typesand Price). Of the seven relevant criteria, c1, c3 and c6 are the-more-the-better (TMTB) criteria, and the other four are the-less-the-better (TLTB). This briefly describes the decision problem, but the required datasets, \(D^{\prime}\), \(B\) and \(CWV\), must be collected before going on. Based on the source decision matrix (\(D\)) in [4], a normalized decision matrix (i.e., \(D^{\prime}\)) is recalculated and obtained. It contains the normalized performance score (NPS) vector for each alternative as a row. This is shown in Table 1. As can be seen, compared with Table 1, the results that calculated in [4] had some rounding inconsistencies, i.e., sometimes rounded to the nearest whole number and sometimes not (e.g., rounded down unconditionally). That is, in [4], the other version of \(D^{\prime}\) did not give the precision that is required by subsequent permanent value calculations for the alternative selection attribute matrices (ASAMs) (because, to compose the ASAM for an alternative, the diagonal of the AHP pairwise comparison matrix is substituted by the NPS vector of that alternative, which affects the permanent function values of the ASAMs.). Table 1 shows the correction. For source dataset \(B\), which is the pairwise comparison matrix to be used by AHP-GTMA as the basis to compose the ASAM for each alternative before permanent value determination and to be used by AHP-TOPSIS as the attribute weights (i.e., the CWV) for the latter stage of TOPSIS, the previous dataset is followed directly. This is because, although it might suffer the problem for not passing the consistency check if the threshold for CR (consistency ratio) is set to 0.1, it can pass the check if the threshold is 0.2, a standard that is acceptable by many industrial studies. Besides, the CR of this matrix is just a little greater than 0.1 (CR = 0.1117), and of most importance is that it is a real dataset that was investigated from a DM using the formal yet rigorous style of AHP expert questionnaire survey [4]. Given such practical values, in this study, this dataset is sourced and used, as shown in Table 2.  
For the criteria weight vector, \(CWV\), it is also re-assessed. Again, this is to mitigate the risk of precision problem during modeling. In the previous study, the imprecise version of this vector was only used to have a rough knowledge about the priority over all considered criteria and draw practical implications based on it (e.g., the bulk purchase concerns for or design issues of paper shredders). It was totally unused by the later calculations in [4]. In this study, however, since the elements of this \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Alts. \(\times\) Attrib.** & **c1** & **c2** & **c3** & **c4** & **c5** & **c6** & **c7** \\ \hline A1 & 1/3 & 3/4 & 0.714286 & 0.634245 & 0.714286 & 1 & 0.425018 \\ A2 & 2/3 & 3/4 & 1 & 0.450433 & 0.769231 & 1 & 0.433333 \\ A3 & 2/3 & 3/4 & 0.973392 & 0.445553 & 0.769231 & 1 & 0.470483 \\ A4 & 2/3 & 3/4 & 0.648928 & 0.649547 & 0.588235 & 1 & 0.548614 \\ A5 & 2/3 & 1 & 0.428571 & 1 & 0.769231 & 2/3 & 0.770619 \\ A6 & 2/3 & 3/4 & 0.571429 & 0.860711 & 1 & 2/3 & 0.784777 \\ A7 & 1 & 1 & 0.865237 & 0.442971 & 0.714286 & 1 & 0.987549 \\ A8 & 2/3 & 3/4 & 0.428571 & 0.904556 & 0.684932 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Problem Dataset 1 (Matrix \(D^{\prime}\)): recalculated normalized decision matrix. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Attr. \(\times\) Attr.