## 3 Decision Case Modeling and the Comparisons: Methods and the Results
### Alternative Ranking/Selection Using AHP-GTMA  
Although the decision case was solved in [4] using AHP-GTMA, this subsection re-establishes the model based on the precise datasets presented in Section 3.2, while the model is implemented as an integrated, one-stop program in R to maintain precision consistency during calculation (including the prior steps to obtain Dataset 1 and the precise CWV in Equation (4)). With the previous Excel-R solution of [4], the former stages of AHP-GTMA (i.e., determining the CWV using AHP and composing the ASAMs) were performed in spread sheets and then the latter stages (i.e., computing the permanent values of the alternatives using the ASAMs and ranking them) were done in R. In addition, unlike [4] that implemented GTMA based on dynamic programming (DP), in this study, a recursive algorithm is proposed. Other details of this algorithm are discussed in the next section, but its overall flow is presented here along with the solution process. Following the symbolic conventions and using the datasets sourced in Section 3.2, the algorithm involves the following phases:  
1. [label=(0)]
2. [label=(0)]
3. Compose an ASAM matrix for each alternative and archive it. 4. Load and compute the permanent value for each alternative. 5. Rank the alternatives according to the permanent values obtained in Phase (2). According to [PERSON] and [PERSON] [52], for Phase (1), for each alternative (i.e., \(A_{i}\), \(i=1\dots\) 8), an ASAM, \(C_{i}\), is obtained by replacing the diagonal axis of \(B\) with its NPS in \(D^{\prime}\) (i.e., the \(i\)-th row of \(D^{\prime}\)). Algorithm 1 shows the algorithm of this phase. Phase (2) is to obtain the \"index score\" for each alternative. The ASAM square matrix for every alternative \(i\) (\(C_{i}\), as obtained from Phase (1)) is loaded, and the permanent value of which, \(PV_{i}\), is determined. These are exactly the index scores used to compose the \"score vector\", as the basis for ranking. Algorithm 2 shows the algorithm for this phase. Phase (3) ranks the alternatives according to the index scores in the score vector. Fortunately, for the implementation in R, this can be performed by \"rbind(\(PV\), \(ROV_{\text{AHP-GTMA}}=\text{rank}(-PV)\))\" because the values in \(PV\) are TMTB in terms of GTMA. This single line command is not sufficient to form \"an algorithm\". Running these phases for the exemplar problem using the datasets established in Section 3.2 gives a solution to \(PV\) that contains index scores for the decision alternatives and a rank order vector for them (i.e., \(ROV_{\text{AHP-GTMA}}\)). In Table 3, the numerical part of the second column shows the value of \(PV\), while the numerical part of the third column shows the value of \(ROV_{\text{AHP-GTMA}}\). Such result gives the following preferential order over the alternatives:  
\[\text{A7}\succ\text{A8}\approx\text{A6}\succ\text{A5}\succ\text{A3}\approx \text{A2}\succ\text{A4}\succ\text{A1}. \tag{5}\]  
If they are strictly ordered without any indifference interval, it is:  
\[\text{A7}\succ\text{A8}\succ\text{A6}\succ\text{A5}\succ\text{A3}\succ\text{ A2}\succ\text{A4}\succ\text{A1}. \tag{6}\]It is seen that such a preferential order concurs with the order that was obtained by [PERSON] et al. [4]. The fact the two index score vectors assessed using the previous DP-based and the current recurrence-based algorithms are quite close to each other verifies the adequacy and functional correctness of this new implementation. However, since there should be no calculation error, in the final \"Reference\" column, the permanent values that were obtained previously are slightly different simply because of the imprecise calculation basis upon the imprecise data sources (see Section 3.2), and not because of the different implementation logic.