Cleaner production assessment of group company based on improved AHP and grey relational analysis

3.1 Evaluation weight set

In this study, the importance of the indicators is related to experts to determine the weight value of every indicator factor in building a provision referring to 1∼9 Thomas Saaty [9, 10] scale method to determine the indicators values. If the parameter on the horizontal axis for each line is less important than the one on the vertical axis, it holds a value between 1 and 9. On the contrary, it holds the value between the reciprocals of the 1∼9. [11]. Emrouznejad, A., & Marra, M. [12] stated the literature review with a social network analysis of AHP. Aghdaie MH, Alimardani M. [13] applied AHP to a target market selection based on market segment evaluation. the traditional method of AHP explained the rationality of expert scoring of the decision matrix [14], while Wei. C.P.& Zhang. Z.M. put forward a method to keep the matrix consistency [15], and Shuang Chen et al. [16] used the improved method of AHP for the indicators sorting, Jiangdong Bao et al. [17] intergrated the improved AHP method with 2-tuple linguistic information in a mining industry to get more accurate expert scoring. This paper applies the same improved AHP method which is optimized as shown in Table 2.

Calculating the consistency indicators through the formula: CI = λ max - n n - 1 ;

In the formulas, n stands for the order and λ stands for the maximal eigenvalue, RI stands for the random consistency indicator [18]. And the value of RI is listed in Table 3. If 0≤CR≤10%, the consistency of AHP model receives satisfactory answers, If CR > 10%, it receives an opposite one. According to the above procedures, the weight of the indicators can be calculated, and the consistency test should also be carried out. More importantly, the experts should be well trained for the scoring knowledge.

4.1 Determining the number of analysis sequence

Select reference series and let X = {x₀, x₁, ⋯, x _m } be grey relational factor set, where x₀ is a reference sequence, x _i is a comparison sequence, x₀ (k) is the K point number of x₀, and x _i  (k) is the K point number of x _i as shown below.

x 0 = ( x 0 ( 1 ) , x 2 ( 2 ) , ⋯ , x 0 ( n ) ) x 1 = ( x 1 ( 1 ) , x 1 ( 2 ) , ⋯ , x 1 ( n ) ) x 2 = ( x 2 ( 1 ) , x 2 ( 2 ) , ⋯ , x 2 ( n ) ) ⋯ ⋯ x m = ( x m ( 1 ) , x m ( 2 ) , ⋯ , x m ( n ) )(1)

Various factors in the data in the column may be different due to the dimension, so it is not easy to get the correct conclusion when in comparison. The data is generally performed by non dimensional treatment when the grey relational analysis is carried out.

x i k = ( 1 , x ( 2 ) x ( 1 ) , x ( 3 ) x ( 1 ) , ⋯ , x ( k ) x ( 1 ) ) , where k = 1 , 2 , . . . n ; i = 1 , 2 , . . . m(2)

The relational coefficient of x₀ (k) and x _i  (k) is shown below [21]. ζ i ( k ) = min ⁡ i min ⁡ k | X 0 ( k ) − X i ( k ) | + ρ max ⁡ i max ⁡ k | X 0 ( k ) − X i ( k ) | | X 0 ( k ) − X i ( k ) | + ρ max ⁡ i max ⁡ k | X 0 ( k ) − X i ( k ) |(3)r ( x 0 ( k ) , x i ( k ) ) = Δ min + ρ Δ max Δ 0 i ( k ) + ρ Δ max(4) In the Formula, Δ_0i (k) = |x₀ (k) - x _i  (k) | is the absolute difference, Δ min = min i min k Δ 0 i ( k ) is the minimum difference between two poles, Δ max = max i max k Δ 0 i ( k ) is the maximum difference between two poles, ρ is the resolution ratio, ρ ∈ (0, 1) (remarks: ρ value normally equals 0.5 in actual calculation.), and ω _k is the weight of K point number which satisfies 0 ⩽ ω k ⩽ 1 , ∑ k = 1 n ω k = 1 .

Because the relational coefficient is the degree value at all times (that is, each point of the curve) between comparison sequence and reference sequence, and there is more than one, the information is too scattered for the overall comparison. Average value is treated as the degree value between comparison sequence and reference sequence, and r _i formula of the relational coefficient is as follows. r i = w k ∑ k = 1 n ζ i ( k )(5)

Normally, if r (x₀, x _i ) > r (x₀, x _j ), the relationa of x _i and x₀ is higher than that of x _j and x₀. That is to say, the influence degree of x _i on x₀ is higher than that of x _j on x₀.

The procedures above show how the grey relationa can be calculated and compared. In formula 8, the weight values can be calculated by improved AHP model instead of the average values of the indicators numbers.

5.1 The indicator weight and consistency testing

In this study, thirty experts with at least five working experience in the enterprises including two internal auditors were trained to give the score of the importance of the indicators with the improved AHP model. And according to the model calculation procedure of improved AHP, CR  =  CI/RI  =   (λ - n)/(n - 1)/1.32  =  0.00062 <  0.1, Hence, the value has passed the consistency testing. And the calculating results of the indicators are shown in Table 4.

In order to obtain comparable data sequences, the method of averaging is used to make the original data non dimensional, and the values after averaging of the three enterprises c₁, c₂ and c₃ are shown in Table 5.

The reference sequence is determined according to the relative optimization principle as shown below. U₀=(0.429,0.965,0.868,0.750,0.666,0.706,0.840)

According to Formula(3), Δ min = 0,  Δ max = 1.547.

According to the formula (4), let ρ = 0.5, then the following values can be obtained as shown in Table 6:

Grey relational analysis does not require too many samples, according to the calculation formula of relationa r i = w k ∑ k = 1 n ζ i ( k ) , the weight is usually measured by the average value of the indicators numbers without considering the reality of the difference in the weight of the indicators. While the improved AHP method is just an effective solution to such problems. The grey relationa r _i of the comparative indicator u _i and reference indicator u₀ can be obtained as the following: r c 1 = w U 1 − U 7 ∑ k = 1 7 ξ c 1 ( k ) = 0.943 ; r c 2 = w U 1 − U 7 ∑ k = 1 7 ξ c 2 ( k ) = 0.492 ; r c 3 = w U 1 − U 7 ∑ k = 1 7 ξ c 3 ( k ) = 0.523 ;

Obviously, r_c1 > r _{c 3}  > r _{c 2} .

According to the calculation result, in the cleaner production assessment of the three enterprises of the group, c₁ has the highest level of cleaner production, followed by c₃ and lastly by c₂, which keeps in line with the cleaner production report from a third party. The result indicates that the group should pay more attention to c₂ and c₃ for further improvement of cleaner production.