## 4 Solution for Declining Discrimination of Salvage Ship Supplier Selection
### Correction of Index Weights by Limited Diversity Factors  
Weights that were obtained through entropy could objectively reflect the difference degree of each index in the system, namely the influence degree of change of each index on the final evaluation results, providing an objective mathematical methods and theoretical basis for weight determination [26]. If the diversity factors of entropy are directly applied to the adjustment of weights that were obtained through the analytic hierarchy process, it may lead to over correction because of an excessive adjustment range. Hence, the LDFM was put forward in order to carry out the fine adjustment of index weights on the basis of remaining the relative importance of index. Steps of adjustment of weights obtained through analytic hierarchy process by diversity factors are: first, \(p(x_{ij})\), the weight of Index \(x_{ij}\) in Index \(j\), is figured out through Equation (2). \[p(x_{ij})=\frac{x_{ij}}{\sum\limits_{j=1}^{n}x_{ij}} \tag{2}\]  
For example, on the basis of eight evaluation indexes in criterion layers, six experts who had served as judges in projects of supplier selection of this ship model, and three of them have experience in operating salvage ships, while three of them are senior engineers in shipbuilding field, were invited to mark indexes of 25 shipyards, which has the ability to building salvage ships, according to the data, such as technical equipment data, historical building data, numbers of senior engineering, and the collecting of historical shipbuilding scheme, which were obtained from these shipyards. If the variance of set of scores, which is made by six experts, is more than 0.6, all of the scores should be re-evaluate by using Delphi method. Finally, for each index, take the average scores of six experts as the final score of the supplier. In order to study the difficulty of distinguishing indexes when the overall scores of different suppliers are similar in the bidding process, 10 shipyards with similar overall scores have been selected out of 25 shipyards, the table of scoring, as shown in Table 4.  
According to Equation (2),  
\[p(x_{ij})=\left[ \begin{array}{cccccccc}0.1013&0.0817&0.0961&0.1005&0.0999&0. 1091&0.1107&0.0834\\ 0.0937&0.1046&0.1061&0.0919&0.1048&0.1079&0.1069&0.1040\\ 0.1035&0.0986&0.1149&0.1054&0.0950&0.0878&0.1069&0.0834\\ 0.1035&0.1130&0.1074&0.1152&0.0950&0.0878&0.0906&0.0873\\ 0.0977&0.1130&0.1124&0.0821&0.1035&0.0890&0.1094&0.1207\\ 0.1035&0.0805&0.0936&0.1115&0.0950&0.1115&0.0906&0.1181\\ 0.0980&0.0865&0.0836&0.1164&0.1035&0.0996&0.0881&0.1130\\ 0.1013&0.1082&0.0874&0.0809&0.1060&0.1115&0.0818&0.1001\\ 0.0969&0.1142&0.0986&0.0833&0.1023&0.1127&0.1170&0.0847\\ 0.1035&0.0998&0.0999&0.1127&0.0950&0.0830&0.0981&0.1053\\ \end{array} \right]\) \begin{table}
\begin{tabular}{l c c c c c c c c} \hline  & \(\mathbf{X_{1}}\) & \(\mathbf{X_{2}}\) & \(\mathbf{X_{3}}\) & \(\mathbf{X_{4}}\) & \(\mathbf{X_{5}}\) & \(\mathbf{X_{6}}\) & \(\mathbf{X_{7}}\) & \(\mathbf{X_{8}}\) \\ \hline S\({}_{1}\) & 9.30 & 6.80 & 7.70 & 8.20 & 8.20 & 9.20 & 8.80 & 6.50 \\ S\({}_{2}\) & 8.60 & 8.70 & 8.50 & 7.50 & 8.60 & 9.10 & 8.50 & 8.10 \\ S\({}_{3}\) & 9.50 & 8.20 & 9.20 & 8.60 & 7.80 & 7.40 & 8.50 & 6.50 \\ S\({}_{4}\) & 9.50 & 9.40 & 8.60 & 9.40 & 7.80 & 7.40 & 7.20 & 6.80 \\ S\({}_{5}\) & 9.00 & 9.40 & 9.00 & 6.70 & 8.50 & 7.50 & 8.70 & 9.40 \\ S\({}_{6}\) & 9.50 & 6.70 & 7.50 & 9.10 & 7.80 & 9.40 & 7.20 & 9.20 \\ S\({}_{7}\) & 9.00 & 7.20 & 6.70 & 9.50 & 8.50 & 8.40 & 7.00 & 8.80 \\ S\({}_{8}\) & 9.30 & 9.00 & 7.00 & 6.60 & 8.70 & 9.40 & 6.50 & 7.80 \\ S\({}_{9}\) & 8.90 & 9.50 & 7.90 & 6.80 & 8.40 & 9.50 & 9.30 & 6.60 \\ S\({}_{10}\) & 9.50 & 8.30 & 8.00 & 9.20 & 7.80 & 7.00 & 7.80 & 8.20 \\ \hline \end{tabular}
\end{table} 
Table 4: Supplier’s Rating Scale. By working out \(e_{j}\), the entropy of Index \(x_{ij}\), Equation (3) was obtained:  
\[e_{j}=-k\cdot\sum\limits_{j=1}^{n}p(x_{ij})\cdot\ln p(x_{ij}) \tag{3}\]  
where \(k>0\) and \(e_{j}\geq 0\). If \(k=1/\ln(n)\), \(0\leq e_{j}\leq 1\). So \(k=1/\ln(n)\) is token in Equation (3).