## 3 Case Study
### Decision Making Using the TOPSIS Method  
_Step 2.1: Establishing an IAMS evaluation decision matrix_  
The decision matrix for evaluating the IAMSs was established as described by Equation (13), using the data in Table 3.  
_Step 2.2: Data Normalization_  
As with the CEGRA, large coefficients for MS, IPE, PR, TE, ES, SC, IS, ITC, QM, and ESP, represent good performance, while smaller coefficients represent good performance for DP, FBP, RD, HIP, PM, MR and LN. The former 10 metrics were categorized as benefit criteria in the TOPSIS framework, while the latter 7 were categorized as cost criteria. Hence, data were normalized using Equation (14) or (15), as appropriate, with results as shown in Table 5. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c c} \hline \hline  & **MS** & **IPE** & **PR** & **TE** & **ES** & **SC** & **IS** & **ITC** & **DP** & **FBP** & **QM** & **RD** & **HIP** & **PM** & **ESP** & **MR** & **LN** & \(\Gamma_{0i}\) & **Rank** \\ \hline Company A & 1.00 & 1.00 & 1.00 & 0.50 & 0.50 & 0.50 & 0.50 & 0.50 & 1.00 & 1.00 & 1.00 & 1.00 & 0.50 & 0.50 & 1.00 & 1.00 & 0.7941 & 1 \\ Company B & 0.50 & 0.52 & 0.51 & 0.92 & 0.84 & 1.00 & 1.00 & 0.58 & 0.68 & 0.73 & 0.61 & 0.50 & 0.58 & 1.00 & 0.61 & 0.50 & 0.75 & 0.6959 & 2 \\ Company C & 0.56 & 0.50 & 0.50 & 1.00 & 1.00 & 0.90 & 0.68 & 1.00 & 0.50 & 0.59 & 0.50 & 0.52 & 0.53 & 0.53 & 0.73 & 0.50 & 0.63 & 0.6571 & 3 \\ Company D & 0.62 & 0.93 & 0.62 & 0.61 & 0.53 & 0.84 & 0.60 & 0.63 & 0.73 & 0.50 & 0.70 & 0.55 & 0.50 & 0.72 & 1.00 & 0.58 & 0.50 & 0.6565 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Summaries of the resultant relational coefficients. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c} \hline \hline
**Company** & **MS** & **IPE** & **PRS** & **TE** & **ES** & **SC** & **IS** & **ITC** & **DP** & **FBP** & **QM** & **RD** & **HIP** & **PM** & **ESP** & **MR** & **LN** \\ \hline Company A & 2.98 & 2.71 & 3.35 & 2.30 & 2.33 & 2.21 & 1.72 & 2.09 & 2.04 & 2.25 & 1.91 & 2.17 & 2.16 & 2.64 & 2.08 & 2.04 & 1.96 \\ Company B & 2.19 & 2.32 & 2.08 & 2.62 & 2.62 & 2.64 & 3.39 & 2.36 & 2.51 & 2.43 & 2.65 & 2.65 & 2.54 & 2.31 & 2.39 & 2.72 & 2.34 \\ Company C & 2.35 & 2.29 & 2.04 & 2.65 & 2.69 & 2.59 & 2.61 & 3.06 & 3.04 & 2.59 & 3.05 & 2.62 & 2.62 & 2.60 & 2.61 & 2.71 & 2.62 \\ Company D & 2.49 & 2.68 & 2.53 & 2.43 & 2.37 & 2.56 & 2.28 & 2.49 & 2.41 & 2.74 & 2.39 & 2.56 & 2.68 & 2.44 & 2.91 & 2.53 & 3.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The decision-making matrix. _Step 2.3: Establishing a weighted normalization matrix_  
The weighted normalized decision matrix depicted in Table 6 was calculated as described by Equation (16). In this research, it was assumed that all evaluation criteria had the same weight. _Step 2.4: Calculating the separation measures_  
The separation of the benefit criteria and cost criteria of the four companies from the ideal solution were calculated using Equations (17) and (18). Results are as shown in Tables 7 and 8. Using the separation measures summarized above, the relative closeness coefficient, \(a_{t}^{*}\), for the four companies, was calculated based on Equation (21), as shown in the TOPSIS performance values column of Table 9. Each company was subsequently ranked in order of preference based on this metric. _Step 2.5: Calculating the relative closeness coefficient and rank the center preference order_  
Using the separation measures summarized above, the relative closeness coefficient, \(a_{t}^{*}\), for the four companies, was calculated based on Equation (21), as shown in the TOPSIS performance values column of Table 9. Each company was subsequently ranked in order of preference based on this metric. _Step 3: Comparing the CEGRA Method and the TOPSIS Method_  
The results of evaluation using the CEGRA and TOPSIS algorithms are given in the right-most column of Tables 4 and 9, respectively. A comparison of these columns shows that the first- and second-ranked results were the same for both methods.