The basic idea of the fuzzy comprehensive evaluation method is to follow the step of “qualitative-quantitative-qualitative”, which is start with qualitative research, and quantitative processing, and then obtains qualitative evaluation result [1] . The concrete model factors including the followings [4] :

1) Establish rating object set B = { B 1 , B 2 , ⋯ , B s } , factor set U = { U 1 , U 2 , ⋯ , U m } , and comment set V = { V 1 , V 2 , ⋯ , V n } .

2) Evaluate fuzzy evaluation matrix R = ( r i j ) m × n , here, r i j represents the membership degree of factor U i to comment V j .

3) Establish the weight set W = { w 1 , w 2 , ⋯ , w m } of each factor in U .

4) Build the fuzzy comprehensive evaluation model C = W ⋅ R , and get the fuzzy evaluation result of all factors in U set.

It is a very important step to determine the weight of evaluation index in fuzzy comprehensive evaluation model, and there are many methods to do it, such as

AHP, entropy method, factor analysis method etc. In this paper, we will use the AHP to determine the weight of each index. Specific steps are as follows:

1) Structure the pair-wise comparison matrix. The matrix A = ( a i j ) n × n is established using the relative importance of a criterion to other criteria established by Saaty 1 - 9 scale. Here, a i j = w i / w j , w i is the weight of criterion i, and w j is the weight of criterion j.

2) Compute the weight vector. Using MATLAB to calculate the maximum eigenvalue λ of matrix A and its corresponding characteristic vector (which is normalized), the vector is weight vector which represents the weight of each factor C 1 , C 2 , ⋯ , C n on O.

3) Check the consistency of the pair-wise comparison matrixes through calculating the consistency index CI and consistency ratio CR. The formula is

C I = λ − n n − 1 ,       C R = C I R I

where n is the order of matrix A, and RI is the average random consistency index for matrices of order n, calculated as follows:

When CR is greater than 0.1, λ is not satisfied, and inconsistent judgments must be readjusted in order to improve the consistency.

4) Compute the combination weight vector. If the sequencing weight vector of the (k − 1)th layer factor towards the total goal is

W ( k − 1 ) = ( W 1 ( k − 1 ) , W 2 ( k − 1 ) , ⋯ , W n ( k − 1 ) ) T

The entire factors of the kth layer towards the synthetic sequence vector W ( k ) of the total goal are given by the following equation:

W ( k ) = ( P 1 ( k ) , P 2 ( k ) , ⋯ , P n ( k ) ) W ( k − 1 ) = P ( k ) W ( k − 1 )

where P ( k ) is the sequencing weight vector of the kth layer factor towards the (k − 1) the layer factor.

According to the red tourism resources evaluation index system, the main content of the research questionnaire is the satisfaction degree of the visitors to the evaluation index, and the grade option is set to be very satisfied, satisfied, general, dissatisfied, very dissatisfied which is according to the Li Kete’s five scale. A total of 200 questionnaires are distributed, 187 copies are collected, and 175 valid questionnaires are collected, get data as shown in Table 2.

According to the expert investigation method, the pair-wise comparison matrixes are established as follows:

A = [ 1 4 2 1 / 4 1 1 / 2 1 / 2 2 1 ] ; B 1 = [ 1 2 5 3 1 / 2 1 3 2 1 / 5 1 / 3 1 1 / 2 1 / 3 1 / 2 2 1 ] ;

B 2 = [ 1 1 / 2 2 1 ] ; B 3 = [ 1 2 4 1 / 2 1 2 1 / 4 1 / 2 1 ] .

The weight of each index as shown in Table 3.

The evaluation objects are B 1 , B 2 , B 3 , evaluation Indexes are U 1 , U 2 , ⋯ , U 9 , and comment set V contents very satisfied, satisfied, general, dissatisfied, and very dissatisfied. Let the membership degree r i j be the ratio of the number of factor U i to comment V j to the total number of people participating in the survey, then the fuzzy evaluation matrixes can be structured as follows:

R 1 = [ 0.26 0.40 0.20 0.10 0.04 0.11 0.36 0.33 0.12 0.09 0.15 0.37 0.31 0.11 0.06 0.15 0.39 0.30 0.10 0.06 ] ,

R 2 = [ 0.10 0.34 0.35 0.12 0.09 0.14 0.38 0.36 0.09 0.03 ] ,

R 3 = [ 0.09 0.26 0.42 0.11 0.12 0.13 0.34 0.35 0.13 0.05 0.05 0.25 0.36 0.21 0.14 ] .

The weight of U 1 , U 2 , U 3 , U 4 relative to B 1 :

W 1 = ( w 11 , w 12 , w 13 , w 14 ) = ( 0.4824 , 0.2718 , 0.0883 , 0.1575 ) ;

The weight of U 5 , U 6 relative to B 2 :

W 2 = ( w 21 , w 22 ) = ( 0.3333 , 0.6667 ) ;

The weight of U 7 , U 8 , U 9 relative to B 3 :

W 3 = ( w 31 , w 32 , w 33 ) = ( 0.5717 , 0.2857 , 0.1429 ) ;

The weight of U 1 , ⋯ , U 9 relative to A :

W = ( 0.2757 , 0.1553 , 0.0504 , 0.0900 , 0.476 , 0.0952 , 0.1633.0.0816 , 0.0408 ) .

The fuzzy evaluation result contains two parts: the target hierarchy and the rule hierarchy. The fuzzy sets of three rule hierarchy are calculated as follows:

C i = W i R i ,       i = 1 , 2 , 3

And it can be calculated:

C 1 = W 1 R 1 = ( 0.1910 , 0.3843 , 0.2605 , 0.1071 , 0.0571 ) ;

C 2 = W 2 R 2 = ( 0.1295 , 0.3676 , 0.3562 , 0.09851 , 0.0495 ) ;

C 3 = W 3 R 3 = ( 0.0955 , 0.2816 , 0.3910 , 0.1290 , 0.1029 ) ,

Let R * = ( R 1 R 2 R 3 ) , the fuzzy sets of target hierarchy can be calculated as:

C = W R * = ( 0.1549 , 0.3526 , 0.3114 , 0.1119 , 0.0691 ) .

Finally, according to the Li Kete’s five scales, the five comments are evaluated as Table 4:

Then the evaluation results are calculated:

E 1 = 5 × 0.1910 + 4 × 0.3843 + 3 × 0.2605 + 2 × 0.1071 + 0.0571 = 3.5449 E 2 = 5 × 0.1295 + 4 × 0.3676 + 3 × 0.3562 + 2 × 0.0971 + 0.0495 = 3.4305 E 3 = 5 × 0.0955 + 4 × 0.2816 + 3 × 0.3910 + 2 × 0.1290 + 0.1029 = 3.1380 E = 5 × 0.1549 + 4 × 0.3526 + 3 × 0.3114 + 2 × 0.1119 + 0.0691 = 3.4123

It is shown in Table 5.