## 4 Numerical Experiments and Applications
### Application in Chemical Spill Emergency Management  
At the end of this section, we will verify the effectiveness of Algorithm 2 by solving a practical problem of chemical spill emergency decision-making. All of the relevant data are from the \"Community Contact\" emergency exercise organized by the Brandon Emergency Support Team (BSET), held on Wednesday, 21 June 2006, in Brandon, Manitoba (also see [25]). In this exercise, there are four key experts, including the Brandon Police Service (\(DM_{1}\)), the Brandon Fire Division (\(DM_{2}\)), the Western Manitoba Hazardous Materials Technical Team (\(DM_{3}\)) and the Brandon School Division (\(DM_{4}\)), as a GDM framework. They are required to evaluate six emergency response alternatives \(A_{i}\) (\(i=1,2,\cdots,6\)) under three criteria \(C_{j}\) (\(j=1,2,3\)). \(C_{1}\) represents physiological discomfort; \(C_{2}\) represents emergency cost; and \(C_{3}\) represents the safety criterion (in terms of the expected number of lives saved). During the release of hazardous airborne material, the \"shelter-in-place alternative\" (\(A_{1}\)) is the practice of staying inside (or going indoors as quickly as possible) and moving to an area of maximum safety. On the other hand, \"evacuation\" involves transporting the victims to a nearby destination (\(A_{2}\)) or the more distant Brandon Keystone Center (\(A_{3}\)). \(A_{1}\), followed by \(A_{2}\), gives rise to the fourth alternative of sheltering in place followed by an evacuation to a nearby location, which is referred to as \(A_{4}\). If \(A_{1}\) is followed by \(A_{3}\), an alternative of sheltering-in-place followed by an evacuation to the Keystone Center, it is named \(A_{5}\). Finally, \(A_{6}\) is the alternative of \"do-nothing\". Table 10 shows the initial score matrices. Different from ordinary score matrices, the information of the score matrices in Table 10 is incomplete. Since Model (3) does not depend on the completeness of the initial information, we can obtain the overall score matrix of the four experts by Steps 1 and 2 of Algorithm 2 (see Table 11) as follows. \begin{table}
\begin{tabular}{c|c c c|c c c|c c c|c c c} \hline \hline \multirow{2}{*}{**Alternatives**} & \multicolumn{3}{c|}{**DM\({}_{\mathbf{1}}\)**} & \multicolumn{3}{c|}{**DM\({}_{\mathbf{2}}\)**} & \multicolumn{3}{c}{**DM\({}_{\mathbf{3}}\)**} & \multicolumn{3}{c}{**DM\({}_{\mathbf{4}}\)**} \\ \cline{2-13}  & \(c_{1}\) & \(c_{2}\)  
For the defective score matrix of the alternatives from the experts, we modify Model (4) by:  
\[ \begin{array}{ll}\min\limits_{w}&\max\limits_{1\leq k\leq p}\left\{\sum \limits_{i=1}^{n}b_{ik}\left(a_{ik}-\sum\limits_{k=1}^{p}\frac{w_{k}a_{ik}}{ \sum\limits_{k=1}^{p}w_{k}b_{ik}}\right)^{2}\right\}\\ \text{s.t.}&\sum\limits_{k=1}^{p}w_{k}=1,\\ &0\leq w_{k}\leq 1,\;\;k=1,2,\cdots,p,\end{array}  \tag{16}\]  
where:  
\[b_{ik}=\left\{ \begin{array}{ll}0,&\text{if }\;a_{ik}\text{ is defective};\\ 1,&\text{if }\;a_{ik}\text{ is given}.\end{array} \right.\]  
In Step 3 of Algorithm 2, we solve Model (16) to obtain the weights of experts. Then, the final rank of the alternatives is obtained as in Table 12.  
In Table 12, we denote GSNP the group analytic network process approach in [25]. The results in Table 12 shows that there exists a ranking difference of the alternatives by the different methods. From the final alternative scores in Table 12, we can compute the standard deviations of these scores, which reflect the data dispersion. The standard deviations are 0.1161, 0.0343 and 0.0962, respectively. The result indicates that our method has the highest degree of discrimination for the alternatives. Although we have reported that for the given same initial score matrix, the different weights of experts, obtained by Model (4) in this paper and Model (6) in [8], respectively, seem not to seriously