层次分析法

In order to find optimal design of tollbooth, we need to make a choice among time cost, lane cost, accident prevention and throughput. We use Analytic Hierarchy Process to determine the proportion of the four factors above. First, we establish the hierarchical structure of the hierarchical relationship: Target layer: the optimal design of tollbooth A
Criteria layer: lane cost B1; accident prevention B2; throughput B3; time cost B4.
Second, we build pairwise comparison matrix of the first level indicators:
When comparing the importance of the ith and jth element with an element from the upper layer, the digit, relative importance B ij is used to describe it. The comparison matrix is obtained by B ij :
1531111152512133
1
1515ij B ⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦
Third, we check consistency .
Analysis from the theoretical results: if the matrix is completely consistent with the pairwise comparison matrix, there should be B ij B ik =B ik (1<=i, j, k<=4), from the analysis, for paired consistent comparison matrix, the maximum eigenvalue is equal to the dimension of the matrix. For the consistency of pairwise comparison matrix, the absolute value of its eigenvalue and the
dimension of the matrix are not quite different.
The procedure for checking the pairwise comparison matrix B consistency is as follows:
(1)Compute the maximum eigenvalue of the matrix λmax
(2) Consistency index CI =1max --n n
λ (3) Random consistency ratio CR=RI CI
Random consistency index RI: for fixed N, and the random structure comparison matrix B, we pick sufficiently large sample to get the value of the average maximum eigenvalue of B, the corresponding values are listed in the following table:
Fourth, by means of consistency check:
CI= 0.0014 RI=0.90
CR== 0.0015 <0.1, is completely consistent. Finally, calculate the weight vector V=[ 0.6806 0.1302 0.2377 0.6806 ]
However, Lane cost B1, accident prevention B2 and time cost B4 have opponent effects on the performance. We set them negative indexes. Therefore, we get a new vector V=[ -0.6806 -0.1302 0.2377 -0.6806 ]. RI
CI。