## 4 Consistency Enhancement Strategy of Ahp in Group Decision-Making
### A Consistency Correction Strategy Based on Improved Eigenvector Method  
According to the foregoing description, the key solution of consistency enhancement of the judgment matrix is finding error elements and correcting them based on the existing information. The correction strategy proposed in [31] is only a single correction that could not achieve iterative correction, so the correction effect is limited. Therefore, based on the feature vector method and the theorem of geometric mean induced bias matrix (GMIBM), this paper proposes a consistency correction strategy with better initial value and faster decline. The new strategy relies on the following theoretical foundations.  
Perron-Frobenius theorem describes the eigenvalues and eigenvector properties of positive real number matrices. As a special kind of positive real matrix, the positive and negative reciprocal matrix is inherited. As the analytic hierarchy process is an evaluation method constructed based on the positive and negative matrix eigenvectors, the judgment matrix of AHP should also satisfy the PF theorem and related inferences. This theorem is also the theoretical basis for the establishment of the eigenvector method. The PF theorem is described as follows.  
**Theorem 1**.: _For a given positive matrix \(A\), the only positive vector \(x\) and only positive constant \(c\) that satisfy \(Ax=cx\) is a vector \(x\) that is a positive multiple of the Perron vector (principal eigenvector) of \(A\), and the only such \(c\) is the Perron value (principal eigenvalue) of \(A\)._  
The authors of [21] give three properties and their demonstrations of a perfectly consistent positive reciprocal matrix. These properties provide a theoretical basis for the correction of the error elements of the judgment matrix, and give a new error matrix description method-GMIBM. Based on that, this paper introduces the valuation of GMIBM as the initial value of the revised strategy.  
Based on the above theory, this paper proposes a consistency correction strategy based on eigenvector method with better initial elements. Compared with the previous confidence-based strategy in [31], there are two major advantages: (1) faster descent speed--the speed of the modified matrix's eigenvalues approaching perfect consistency eigenvalues is faster; (2) stronger adaptability  
Figure 5: The flow chart of this enhancement strategy.  
for the judgment matrix with missing elements during rapid assessment. The core idea is to obtain the maximum deviation element through the search of the error matrix. The missing element is automatically determined as the maximum deviation/error element. Then the maximum error element is given a better initial value based on the geometric mean. Finally, the feature vector method iteration loop is enabled until the consistency ratio is met. The steps are shown in the Algorithm 1.  
```
1:Initialize:input pairwise comparison matrix, \(A_{0}=\{a_{ij}\}_{1\leq i,j\leq n}\) if \(\exists a_{ij}=0\), then \(a_{ij}=1\)
2:Calculate the bias matrix, \(B_{0}=\{\sqrt[n]{\prod\limits_{k=1}^{n}a_{ik}a_{kj}}\cdot a_{ji}-1\}_{1\leq i,j\leq n}\)
3:Find the location of \(\max|b_{ij}|\), \(then\ a_{ij}=\sqrt[n]{\prod\limits_{k=1}^{n}a_{ik}a_{kj}}\), it \(\leftarrow\) 0
4:Calculate the consistency index of \(A_{0}\), \(CI_{0}=\frac{\lambda_{\max}-n}{n-1}\)
5:While\(CI_{0}>0.1*RI(n)\)  Calculate the principal eigenvector of\(A_{0}\)Do operation 2  Find the location of \(\max|b_{ij}|\), then \(a_{ij}=\frac{w_{i}}{w_{j}}\)Do operation 4 endwhile
6:Output modified matrix \(A_{0}\)
```  
**Algorithm 1**The algorithm pseudocode of improved method.