Gray target decision-making is an important method for solving multiattribute decision-making problems from an objective perspective, and it can effectively reduce the loss of original information in the decision-making process (Zhou et al., 2014). Its basic idea is that an optimal data sequence is found from an existing set of sequences to construct a standard model. Then, the standard model is applied as the target, and the gray target model is built by comparing other models with the standard; the degree of likeness between the models is then evaluated to assess the target before calculating the target distance to determine the evaluation grade. Considering the errors in accuracy based on single-weight gray target models, the combination weighting method based on game theory was adopted in this work; then, the combined weights of the critic and entropy methods were optimized to obtain the optimal weight.

Assuming that there are m samples to be evaluated A i i = 1 , 2 , . . . , m and n evaluation indexes A j j = 1 , 2 , . . . , m , the sample matrix is given by A = a i j i = 1 , 2 , . . . , n .

Suppose that c j is the mean value of evaluation index, then c j = 1 m ∑ i = 1 n a i j (1)

where i = 1 , 2 , 3 . . . , m ; j = 1 , 2 , 3 . . . , n .

Let x i j be the standardized processing result for the economic indicators, which is expressed as (Zhengzheng et al., 2024) x i j = a i j − c j max max a i j − c j , c j − ⁡ min a i j (2)

For the cost-type indicator, this formula becomes x i j = c j − a i j max max a i j − c j , c j − ⁡ min a i j (3)

From Eqs 1–3, the decisive matrix is expressed as X = x 11 x 12 . . . x 1 n x 21 x 22 . . . x 2 n . . . . . . . . . . . . x m 1 x m 2 . . . x m n

For the decisive matrix X , if x j 0 + = ⁡ max x i j 1 < i < m , then x 0 + = x 1 0 + , x 2 0 + , . . . , x m 0 + is the positive center of the gray target, and x 0 − = x 1 0 − , x 2 0 − , . . . , x m 0 − is the negative target center. The distance between x 0 + and x 0 − is regarded as the interval d 0 between the positive and negative target centers and is given by d 0 = x 0 + − x 0 − = ∑ j = 1 n ω j x j 0 + − x j 0 − 2 1 / 2 (4)

where ω j is the optimal weight of the j t h index obtained using game theory.

The positive target center distance d i + is the distance between x i and x 0 + given by d i + = x i − x 0 + = ∑ j = 1 n ω j x i j − x j 0 + 2 1 / 2 (5)

The negative target center distance d i − is the distance between x i and x 0 − given by d i − = x i − x 0 − = ∑ j = 1 n ω j x i j − x j 0 − 2 1 / 2 (6)

The distance from any sample point x i to the positive target center is d i + < d 0 , such that x i is located on a sphere with x 0 + as the center and d 0 as the radius; the distance from any sample point x i to the negative target center is d i − < d 0 , such that x i is located on a sphere with x 0 − as the center and d 0 as the radius. Thus, the sample point x i , positive target center d i + , and negative target center d i − are three points in space that are collinear or triangular. Therefore, the degree of danger of a sample can be measured from the projection of the positive target center distance on the line between the positive and negative target centers. By assuming that the angle between the positive target center distance and line between the positive and negative target centers is θ , the target center distances can be obtained according to the law of cosines (Figure 1): d i * = d i + ⁡ cos ⁡ θ = d i + 2 + d 0 2 + d i − 2 2 d 0 (7)

From the definitions of the target center distances, it is found that the comprehensive target distance quantitatively reflects the quality grade of a sample. Assuming that the samples to be evaluated have t quality grades, let D = d 1 , d 2 , . . . , d m be the set of comprehensive target center distances S of the samples to be evaluated, and B = B 1 B 2 . . . B t be the ordered set of the t quality grades; let f be a positive integer such that 1 ≤ f < t , and the thresholds of the f t h set are g f = ⁡ max B and c f = ⁡ min B f , then h f = α c f + 1 + 1 − α g f for α ∈ 0 , 1 , h 0 = 0 , h q = + ∞ , d = d 1 d 2 . . . d i are the critical comprehensive target center distances of the different quality grades. The interval distribution set of comprehensive target distances for the t quality grades can then be obtained as follows: D i j = d h 0 > d 1 > h 1 , . . . , h t − 1 > d t > h t (8)

(1) Critic method

The critic method is a kind of weight-assignment method (Zhou et al., 2008) that uses the variability and conflict between different evaluation indexes for weighting to comprehensively measure the evaluation index. Its procedure is as follows:

① Standardization processing of the data

Each evaluation index is quantified to be dimensionless to eliminate the influences of different variables. If the evaluation index is of the benefit type, then the calculation formula is y = x i j − ⁡ min x i j max x i j − ⁡ min x i j (9)

If the evaluation index is of the cost type, then the calculation formula is y = max x i j − x i j max x i j − ⁡ min x i j (10)

Here, y is the normalized processing value; max x i j and min x i j are the maximum and minimum values from a set of evaluation indexes.

σ j = ∑ i = 1 n x i j − x j 2 n (11)

where x j is the mean of the j t h evaluation index, and n is the total number of j t h evaluation indexes.

③ The correlation coefficient of the evaluation index is calculated as

The entropy weighting method is used to determine the entropy value of an index according to the variance of the evaluation index, and its specific calculation method is based on a previously reported approach (Gu et al., 2019).

(3) Game theory combination weighting method

To avoid information loss caused by using a single weighting method and improve the accuracies of the weights, the combination weighting method based on game theory (Gu et al., 2021c) is applied to optimize the weights obtained from several other weighting methods to obtain the optimal weight as follows.

① The weight sets ω 1 and ω 2 are obtained from the entropy weighting and critic methods, and a 1 and a 2 are assumed to be their respective linear combination coefficients. Then, the weight sets ω 1 and ω 2 are linearized as

The selection of indicators should consider the ease of access of the site, convenience, and practices. Accordingly, eight influential factors were selected as the assessment indexes: uniaxial saturated compressive strength of the rock block R c , rock quality index R Q D , rock softening coefficient k R , integrity coefficient of the rock mass k v , depth H , unit weight of the rock mass γ , coefficient of quantification of the angle between the principal structural plane and shaft axis k 2 , and weight of the groundwater k 1 . The assessment results were divided into five grades as excellent (I), good (II), common (III), inferior (IV), and bad (V). Twelve groups of monitoring data and assessment results from the mine were adopted as the learning samples, and these specific data are shown in Table 1.

The flowchart depicting the determination of the quality levels of the surrounding rocks in the mine shaft is shown in Figure 3, and the specific procedures are as follows.

(1) The sample matrix of the origin matrix is constructed;

Four of the indicators listed in Table 1 are of the benefit type. Using Eqs 1–3 and the information in Table 1, the decisive matrix values are obtained as shown in Table 2.

From Table 2, the corresponding set of weight coefficients is obtained as ω 1 = 0.6122 0.0042 0.0379 0.0008 0.0447 0.2149 0.0445 0.0409

From Eqs 9–11 and the information in Table 1, the correlation coefficients are calculated as r = 0.0797 1 0.0768 0.282 0.7898 0.8075 0.4875 0.1675 0.2811 0.0768 1 0.2661 0.3117 0.0724 0.1288 0.7461 0.0121 0.282 0.2661 1 0.0351 0.0098 0.036 0.6242 0.0797 0.7898 0.3117 0.0351 1 0.7068 0.5018 0.1596 0.2879 0.8075 0.0724 0.0098 0.7068 1 0.6207 0.04 0.5645 0.4875 0.1288 0.036 0.5018 0.6207 1 0.2129 0.025 0.1675 0.7461 0.6242 0.1596 0.04 0.2129 1 0.1814 0.2811 0.0121 0.0797 0.2879 0.5645 0.025 0.1814 1

Using Eq. 12, the standard deviations of the individual columns are obtained as C = 4.1079 5.386 5.6672 4.2073 4.1783 4.9874 4.8683 5.5684

Similarly, using Eq. 13 the set of weights of the evaluation indexes is obtained as ω 2 = 0.095 0.1504 0.1504 0.1136 0.0901 0.121 0.159 0.1204

3) Determining the combined weights

From Eqs. 14–18, the combined set of weights ω is obtained as ω = 0.5131 0.0322 0.0595 0.0224 0.0534 0.1969 0.0664 0.0561

The individual and combined weights are graphically depicted in Figure 4.

The positive and negative target centers are respectively obtained as x 0 + = 1 1 0.78 0.46 0.51 1 1 0.83 and x 0 − = − 0.59 − 0.9 − 0.99 − 1 − 1 − 0.78 − 0.88 − 1 . According to Eq. 4, the interval between the positive and negative target centers is d 0 = 1.678 . Then, from Eqs 5, 6, the positive target center distances d + and negative target center distances d − of the different samples are calculated, as shown in Table 3.

The comprehensive target distance of each sample was arranged according to the quality grade, and the critical value of the target distance for each quality grade was obtained, as shown in Table 5. Using Eq. 8, the target distance for each quality grade can be derived, as plotted in Figure 5.

To test the rationality and accuracy of the gray target evaluation model established in this study, a mine shaft project was selected as the example; the monitoring data for 13-17# samples are shown in Table 6. Different parameters are obtained and substituted into the game gray target evaluation model, and the corresponding comprehensive target distances d i * are obtained. For the different d i * ranges shown in Figure 5, the quality grades of the surrounding rocks are evaluated, whose results are shown in Table 7. The results obtained from the suggested model were compared with those from two other methods, as shown in Figure 6.

The game gray target model was used to evaluate the quality levels of the surrounding rocks, and the assessment results are depicted in Table 7; it is seen from this table that the quality grades of the surrounding rocks for the 13–17# samples differ greatly. The quality levels for the 13# sample is IV, 16# sample is V, and the remaining samples are II; this means that the 13# sample is of inferior quality, 16# sample is of bad quality, and the rest are of good quality, for a combined quality rate of 60%. For the 13# and 16# samples, corresponding consolidation measures should be implemented; for example, shotcrete and anchor supports should be adopted to enhance the stabilities of the surrounding rocks, and no measures need to be adopted for the other samples.

The comparative results of the assessment models in Figure 6 indicate that the proposed method is consistent with the actual investigations for the five different samples and that its accuracy is 100%, which is greater than the accuracy of the RBF (Radial Basis Function) method (80%) (Zhou et al., 2012). Therefore, estimating the quality levels of surrounding rocks is feasible using the proposed game gray target model. The proposed approach also provides additional details for assessing the quality levels of the surrounding rocks. For example, the quality of the 15# sample is more likely to be level II as its synthetic target distance (0.8241) is greater than that for the 17# sample (0.7019) that could likely be closer to level III. Thus, the results of the proposed model can accurately demonstrate the quality grades of the surrounding rocks and further determine the risk grade rankings for different samples at the same levels.