The sustainability evaluation of modern photovoltaic agriculture is complicated, the evaluation process is affected by a variety of complex factors, with a certain degree of uncertainty. And the method of interval type-2 fuzzy numbers is effective to deal with many uncertain factors and obtain robust results. Interval type-2 fuzzy numbers have been widely used in many fields [26].

AHP is a multi-criteria analysis method which can deal with qualitative problems quantitatively. AHP is one of the most commonly used multi criteria decision-making methods because of its scientific hierarchy construction and clear logical structure [27]. By combining the fuzzy set theory with AHP analysis method, we can better deal with the uncertainty involved in the problem and form pairwise comparison, and ensure the consistency of the ranking made by decision makers. Therefore, this section applies the interval type-2 fuzzy AHP model to evaluate the sustainability of modern photovoltaic agriculture to calculate the weight of the indexes.

The calculation process of interval type-2 fuzzy AHP model is as follows: (1)

Combined with interval type-2 fuzzy numbers theory, pairwise comparison matrix is established for indexes of different levels and types.

Decision makers usually use semantic form to evaluate the objects in the measurement analysis of research projects. But it is difficult for decision-makers to define interval type-2 fuzzy numbers directly. Therefore, according to the semantic expression given by the decision-maker, it needs to be transformed into interval type-2 fuzzy numbers. (2)

Test the consistency of comparison matrix

The interval type-2 fuzzy numbers are calculated with Eq. (1), and then the consistency of the matrix can be tested. (1)Defuzzified(A~~i)=12{14[(ai4U−ai1U)+(H1(A~iU)×ai2U−ai1U)+(H2(A~iU)×ai3U−ai1U)]+ai1U+14[(ai4L−ai1L)+(H1(A~iL)×ai2L−ai1L)+(H2(A~iL)×ai3L−ai1L)]+ai1L}

The consistency index (CI) of the weight matrix of the object to be measured is tested, and the calculation process is shown in Eqs. (2) and (3). (2)CI=(λmax())(m−1) (3)Aw=λmax where λmax is the maximum eigenvalue of the weight matrix of the object to be measured, w is the maximum eigenvector corresponding to λmax. The consistency ratio (CR) of comparison matrix can be calculated with Eq. (4). (4)CR=CIRI where RI is the average random consistency index of the judgment matrix. When CR value is less than 0.1, the consistency of weight comparison matrix is determined. Otherwise, the relevant elements in the matrix need to be adjusted to make it consistent. (3)

Integrate the interval type-2 fuzzy number comparison matrix by geometric average method

The interval type-2 fuzzy number comparison matrix of the object to be measured can be integrated according to Eq. (5). (5)A~~ij=[A~~1⊗A~~2⊗⋯⊗A~~n]1n (4) Calculate the fuzzy weight of each index

The fuzzy weight of each index of the object to be measured can be calculated by Eq. (6). (6)w~~i=r~~i⊗(r~~1⊕r~~2⊕⋯⊕r~~m)−1 where r~~i is the geometric average of each row of the comparison matrix after integration.

TOPSIS is one of the traditional multi criteria decision-making models. It is a method to rank based on the distance between the evaluation objects and the ideal solutions, so as to determine the quality of the evaluation objects [28]. For interval type-2 fuzzy TOPSIS model, the elements in the decision matrix will be calculated with interval type-2 fuzzy method.

Assuming that there are m items to be evaluated, N indexes and T experts in related fields to score, the main steps of interval type-2 fuzzy ideal point method are as follows: (1)

Transform semantic variables into interval type-2 fuzzy numbers

There are both quantitative and qualitative indicators in the index system. It is difficult to realize the horizontal comparison of semantic expression of different types of indexes, so it is necessary to standardize them and transform semantic variables into interval type-2 fuzzy numbers. Then the interval type 2 fuzzy decision matrix of expert k can be expressed by Eq. (7). (7)Dk=(d~~ijk)=[d~~11kd~~12k⋯d~~11kd~~11kd~~11k⋯d~~11k⋮⋮⋱⋮d~~11kd~~11k⋯d~~11k] (2)

Integrate interval type-2 fuzzy matrix

Collect the measurement score data of all objects to be measured for input, and then the integrated interval type-2 fuzzy decision matrix can be determined as Eqs. (8) and (9). (8)D=⁡(d~~ij)m×n=[d~~11d~~12⋯d~~1nd~~21d~~22⋯d~~2n⋮⋮⋱⋮d~~m1d~~m2⋯d~~mn] (9)d~~ij=(d~~ij1⊕d~~ij2⊕⋯⊕d~~ijt)t where d~~ij denotes interval type 2 fuzzy number, 1≤i≤m, 1≤j≤n, 1≤k≤t. (3)

Construct weighted normalized fuzzy decision matrix

The weighted normalized fuzzy decision matrix of the object to be measured can be calculated by Eqs. (10) and (11). (10)Y¯=(v~~ij)m×n=[v~~11kv~~12k⋯v~~1nkv~~21kv~~22k⋯v~~2nk⋮⋮⋱⋮v~~m1kv~~m2k⋯v~~mnk] (11)v~~ij=w~~i⊗d~~ij where, 1≤i≤m, 1≤j≤n. (4)

Determine fuzzy positive and negative ideal solutions

The positive ideal solution and the negative ideal solution are expressed as x+=(v1+,v2+,⋯,vm+) and x−=(v1−,v2−,⋯,vm−), respectively. vi+ and vi− can be calculated by Eqs. (12) and (13). (12)vi+={max1≤j≤n⁡{Def(v~~ij)},fi∈F1min1≤j≤n⁡{Def(v~~ij)},fi∈F2 (13)vi−={min1≤j≤n⁡{Def(v~~ij)},fi∈F1max1≤j≤n⁡{Def(v~~ij)},fi∈F2 where F1 is the set of profit indexes, and F2 is the set of cost indexes, 1≤i≤m. (5)

Calculate the distance between each alternative and the fuzzy positive and negative ideal solutions

The calculation formulas of the distance between the measurement score of each index of the object to be measured and the positive and negative ideal solutions are shown in Eqs. (14) and (15), respectively. (14)d+(xi)=∑i=1m⁡(Def(v~~ij)−vi+)2 (15)d−(xi)=∑i=1m⁡(Def(v~~ij)−vi−)2 (6)

Calculate and sort the similarity coefficients of each alternative

The similarity coefficients of each alternative can be calculated with Eq. (16). (16)C(xi)=d−(xj)d+(xj)+d−(xj) where 1≤j≤n, and the larger the index value C(xi) is, the better the measurement result is.

Least squares support vector machine (LSSVM) is an improvement of standard support vector machine (SVM). LSSVM uses equality constraints instead of inequality constraints of SVM, and uses kernel function to transform the prediction problem into solving equations, which can greatly improve the accuracy and speed of evaluation [29].

It is supposed that the given sample set is T={(xi,yi)}i=1N, and Nis the total number of samples. The regression model of the sample is as follows: (17)y(x)=wT∙ϕ(x)+b where ϕ(x) is a high dimensional space projected by training samples, w is the weighted vector, b is the bias.

For LSSVM, the optimization problem changes into the following: (18)min12wTw+12γ∑i=1N⁡ξi2 (19)s⋅tyi=wTϕ(xi)+b+ξi,i=1,2,3,⋯,N where γ is the penalty coefficient, which is used to balance the complexity and accuracy of the model, ξi is the estimation error.

In order to solve the above problems, Lagrange function is established as below: (20)L(w,b,ξi,αi)=12wTw+12γ∑i=1N⁡ξi2−∑i=1N⁡αi[wTϕ(xi)+b+ξi−yi] where αi is Lagrange multiplier.

Take a derivation of each variable of the function and make them have a value of 0, as shown in Eq. (21). (21){∂L∂w=0→w=∑i=1N⁡αiϕ(xi)∂L∂b=0→∑i=1N⁡αi=0∂L∂ξ=0→αi=γξi∂L∂α=0→wT+b+ξi−yi=0

Eliminate w and ξi, and turn the problem into Eq. (22): (22)[0enTenΩ+γ−1×I]×[ba]=[0y] where, (23)Ω=fT(xi)f(xi) (24)en=[1,1,…,1]T (25)α=[α1,α2,…,αn] (26)y=[y1,y2,…,yn]T

By solving the above linear equations, the following results are obtained: (27)y(x)=∑i = 1N⁡αiK(xi,x)+b where K(xi,x) is the kernel function satisfying Mercer condition.

Considering that the RBF kernel function has a wide range of convergence and application, this paper chooses it as the kernel of LSSVM. The expression is as follows: (28)K(xi,x)=exp{−||x−xi||2/2σ2} where σ2 represents the width of the kernel, it reflects the characteristics of the training data set and has an impact on the generalization ability of the system.

By analyzing the relevant theories of LSSVM, we can see that the difficulty of establishing LSSVM model is determining kernel function parameter σ2 and the penalty parameter γ. Selecting appropriate parameters is essential to improve the learning and generalization ability of the model [30]. Considering that FWA has excellent local search capability and the self-adjusting mechanism of global search capability [31], this paper selects FWA to optimize these two parameters.

FWA is a simulation of the whole process of fireworks explosion. When fireworks explode, a large number of sparks will be generated, which can continue to explode and produce new sparks, thus it will make beautiful and colorful patterns. In the FWA, each firework can be regarded as a feasible solution in the solution space of the optimization problem, so the process of fireworks explosion can be regarded as the process of searching for the optimal solution. In the specific optimization problem, FWA involves the number of sparks produced by fireworks explosion, the radius of explosion, and how to select a group of optimal fireworks and sparks to carry out the next explosion [32].

In the FWA, the number of sparks and explosion radius of fireworks are different. The fireworks with poor fitness have larger explosion radius, which makes them have greater exploration ability, while the fireworks with good fitness have smaller explosion radius, which makes them have greater excavation ability. In addition, the introduction of Gauss mutation sparks can further increase the diversity of the population [33].

Therefore, we can know that the three most important parts of FWA are explosion parameters, mutation parameters and selection strategy [34]. (1)

Explosion parameters

In the explosion parameters, the number of sparks and explosion radius of fireworks are calculated according to the fitness value of fireworks. For firework xi(i=1,2,⋯,N), the formulas to calculate the number of sparks Si and the explosion radius Ri are as follows: (29)Si=M×ymax−f(xi)+ε∑i=1N⁡(ymax−f(xi))+ε (30)Ri=R^×f(xi)−ymin+ε∑i=1N⁡(f(xi)−ymin)+ε where ymax is the maximum fitness value and ymin is the minimum fitness value in the current population; f(xi) is the fitness value of firework xi; M is a constant which is used to adjust the number of explosive sparks; R^ is a constant which is used to adjust the explosion radius of fireworks; ε is the minimum quantity of the machine, which is used to avoid no operation. (2)

Mutation parameters

The setting of mutation parameters is to increase the diversity of explosion spark population. The mutation sparks in FWA generate Gaussian mutation sparks by Gaussian mutation. Suppose that firework xi is selected for Gaussian variation, the Gaussian variation operation with K dimensions is as follows: x^ik=xik×e, where x^ik is the firework with K dimensions, e is a Gaussian distribution following N(1,1).

In FWA, the explosion sparks and mutation sparks generated by explosion parameters and mutation parameters may exceed the boundary range of feasible region, so they must be mapped to a new location by mapping rules by Eq. (31). (31)x^ik=xLB,k+|x^ik|%(xUB,k−xLB,k) where xUB,k and xLB,k are the upper and lower boundary of the solution space with K dimensions. (3)

Selection strategy

In order to transmit the information of excellent individuals to the next generation, it is necessary to select a certain number of individuals as the fireworks of the next generation from the explosion sparks and variation sparks.

Suppose that there are K candidates and the population number is N. The individuals with the best fitness value in the candidate set will be identified as the next generation fireworks. And the remaining N−1 fireworks are selected according to probability method. For firework xi, the probability formulas of being selected are as follows: (32)p(xi)=R(xi)∑xj∈K⁡xj (33)R(xi)=∑xj∈K⁡d(xi−xj)=∑xj∈K⁡||xi−xj|| where, R(x) is the sum of distances between all individuals in the current candidate set. In the candidate set, if the individual density is high, that is, there are other candidates around the individual, the probability of the individual being selected will be reduced.

Based on the previous description, the specific steps of FWA are as follows [35,36]:

Step 1. Randomly select N fireworks in the solution space and initialize their coordinates.

Step 2. Calculate the fitness value f(xi), the blast radius Ri and the spark number Si. The coordinate with the K dimensions are randomly selected to update, and the update formula is as follows: (34)x^ik=xik+Ri×U(−1,1)

Step 3. Generate M Gauss mutation sparks, randomly select spark xi, and M Gauss mutation sparks x^ik are calculated according to Gauss mutation formula, then save them in Gauss mutation spark population.

Step 4. From the fireworks, explosion sparks and Gauss mutation spark population, N individuals are selected as the fireworks for next generation iterative calculation by using probability selection Eqs. (32) and (33).

Step 5. Judge the stop condition. If the stop condition is satisfied, jump out of the program and output the optimal result; if not, return to Step 2 to continue the loop.

In this section, on the basis of the evaluation model of modern photovoltaic agriculture sustainability based on interval type-2 Fuzzy AHP-TOPSIS, the intelligent evaluation model of modern photovoltaic agriculture sustainability based on FWA-LSSVM is proposed, that is, the index value of modern photovoltaic agriculture sustainability evaluation index system is taken as the input, and the LSSVM is optimized by FWA, so as to obtain the optimal value of LSSVM, finally, the sustainability evaluation results of modern photovoltaic agriculture can be obtained and analyzed. The flow chart of the proposed model is shown in Fig. 1.

The specific steps are as follows:

Step 1. Construct a sustainable evaluation index system for modern photovoltaic agriculture.

Step 2. Score the evaluation indexes according to the classification of indexes and assign weights to semantic variables.

Step 3. Transform semantic variables into interval type-2 fuzzy numbers, integrate interval type-2 fuzzy matrix and construct weighted normalized fuzzy decision matrix.

Step 4. Determine fuzzy positive and negative ideal solutions, calculate the distance between each alternative and the fuzzy positive and negative ideal solutions, then calculate and sort the similarity coefficients of each alternative.

Step 5. Initialize the parameters of LSSVM and FWA, optimize the parameters of LSSVM model with FWA, then output and analysis the intelligent evaluation results.

In order to achieve the hierarchical measurement of different levels of indexes, it is necessary to classify different indexes on the basis of the level differences, that is, to define the index set. Here, according to the evaluation index system, we divide the indexes into different levels, including the first level indexes and the second level indexes. (1)

Set of the first level indexes

The set of the first level indexes can be expressed as Eq. (35). (35)U={U1,U2,U3} where U1 stands for technical characteristic index; U2 stands for economic characteristic index; U3 stands for social characteristic index. (2)

Set of the second level indexes

The set of the first level indexes can be expressed as follows: a.

Technical characteristic index (36)U1={u11,u12,u13,u14,u15}

Due to the differences in various aspects of the projects to be measured, the measurement results also have the characteristics of diversity. Through the integration of semantic variables, the corresponding set of comments V is established, as shown in Eq. (39). (39)V={v1,v2,v3,v4}

All possible states of the objects to be measured should be fully included in the set of comments. The reasonable construction of the comments set will have an important impact on the accuracy and scientificity of the measurement results. Therefore, we should define the comments set on the basis of full investigation and expert advice. (1)

Comments set of the interval type-2 fuzzy AHP model

There are five semantic variables in the comments set of the interval type-2 fuzzy AHP model. They are absolutely strong (AS), very strong (VS), fairly strong (FS), slightly strong (SS) and equal (E). The expression is shown in Eq. (40). (40)V1={AS,VS,FS,SS,E}

The correspondence between semantic variables of interval type-2 fuzzy AHP model and interval type-2 fuzzy numbers can be shown in Table 2. Table 2

Semantic variables	Interval type-2 fuzzy numbers	The reciprocal of interval type-2 fuzzy numbers
Absolutely strong (AS)	((7, 8, 9, 9;1, 1),(7.2, 8.2, 8.8, 9;0.8, 0.8))	((0.11, 0.11, 0.12, 0.14;1, 1),(0.11, 0.11, 0.12, 0.14;0.8, 0.8))
Very strong (VS)	((5, 6, 8, 9;1, 1),(5.2, 6.2, 7.8, 8.8;0.8, 0.8))	((0.11, 0.12, 0.17, 0.2;1, 1),(0.11, 0.13, 0.16, 0.19;0.8, 0.8))
Fairly strong (FS)	((3, 4, 6, 7;1, 1),(3.2, 4.2, 5.8, 6.8;0.8, 0.8))	((0.14, 0.17, 0.25, 0.33;1, 1),(0.15, 0.17, 0.24, 0.31;0.8, 0.8))
Slightly strong (SS)	((1, 2, 4, 5;1, 1),(1.2, 2.2, 3.8, 4.8;0.8, 0.8))	((0.2, 0.25, 0.5, 1;1, 1),(0.21, 0.26, 0.45, 0.83;0.8, 0.8))
Equal (E)	((1, 1, 1, 1;1, 1),(1, 1, 1, 1;1, 1))	((1, 1, 1, 1;1, 1),(1, 1, 1, 1;1, 1))

(2)

Comments set of the interval type-2 fuzzy TOPSIS model

There are seven semantic variables in the comments set of the interval type-2 fuzzy TOPSIS model. They are very low (VL), low (L), medium low (ML), medium (M), medium high (MH), high (H), very high (VH). The expression is shown in Eq. (41). (41)V1={VL,L,ML,M,MH,H,VH}

The correspondence between semantic variables of interval type-2 fuzzy TOPSIS model and trapezoidal interval type-2 fuzzy numbers can be shown in Table 3.

In this paper, the sustainability evaluation model of modern photovoltaic agriculture based on interval type-2 Fuzzy AHP-TOPSIS method is used to get the objective and accurate results of 5 samples of modern photovoltaic agriculture sustainability evaluation. However, according to the above calculation, it can be found that the calculation of the model is more complex, the efficiency is low, and the workload is large. When facing the massive data of modern photovoltaic agriculture projects, this method is bound to be difficult to quickly and effectively calculate the sustainability evaluation results of modern photovoltaic agriculture. Therefore, this paper will further use the FWA-LSSVM model constructed above for research. The other 15 samples are selected as the training set, and the existing 5 samples are used to complete the sustainability evaluation of the modern photovoltaic agriculture. After that, the intelligent evaluation results are compared with the evaluation results in the previous section to verify the effectiveness of the FWA-LSSVM model.

The parameters of FWA are set as follows: the maximum number of iterations Maxgen is 500, the population number PopNum is 30, the determination constant of spark numbers M is 100, the determination constant of explosion radius R^ is 150. Matlab software is used for model calculation.

In order to verify the effectiveness of the intelligent evaluation model proposed in this paper, this part continues to use the above 5 photovoltaic agricultural projects for empirical analysis, taking these five photovoltaic agricultural projects as test samples, and selecting another 15 photovoltaic agricultural projects as training samples. FWA-LSSVM, PSO-LSSVM, LSSVM and SVM models are applied to the sustainability evaluation of modern photovoltaic agriculture projects for comparative experiments. The calculation results are shown in Table 15 and Fig. 3.

Fig. 3 shows the maximum and minimum relative distances of the intelligent evaluation results. In SVM model, the maximum and minimum relative distances are 0.1198 and 0.0193, respectively; while in LSSVM model, the maximum and minimum relative distances are 0.1100 and 0.0188 respectively. The two distance values of LSSVM are less than that of SVM, which indicates that the evaluation accuracy of LSSVM is higher than that of SVM. In the PSO-LSSVM model, the maximum and minimum relative distances are 0.0772 and 0.0103, respectively. The deviation between the maximum and minimum relative distance is 0.0669, which is smaller than that of LSSVM and SVM, showing that the stability of PSO-LSSVM model is higher than that of LSSVM and SVM. In FWA-LSSVM model, the maximum relative distance is only 0.0357, the minimum relative distance is 0.0036, and the deviation between the maximum and minimum relative distance is only 0.0321. Compared with the other three models, the three values of FWA-LSSVM model are smaller, which shows that the proposed FWA-LSSVM model has higher accuracy and higher stability.

Mean absolute percentage error (MAPE), root mean square error (RMSE) and average absolute error (AAE) are used to compare the evaluation errors of each intelligent evaluation model. Fig. 4 shows the MAPE, RMSE, and AAE for each model.

It can be seen from Fig. 4 that the sustainability evaluation results of modern photovoltaic agriculture calculated by FWA-LSSVM have the highest accuracy, and the MAPE, RMSE and AAE are 4.80%, 4.97% and 4.89%, which are the optimal values. The MAPE, RMSE and AAE of PSO-LSSVM are 9.80%, 9.87% and 9.76%, respectively. The MAPE, RMSE and AAE of standard LSSVM are 14.39%, 14.61% and 13.86%, respectively. The MAPE, RMSE and AAE of standard SVM model are 18.40%, 18.45% and 18.15%, respectively. It shows that the error of the evaluation results of the proposed FWA-LSSVM model is the smallest, and the overall accuracy of the evaluation is the highest. Compared with PSO-LSSVM model, FWA has better optimization performance for LSSVM model. Compared with the original LSSVM model, the generalization ability and classification accuracy can be improved by optimizing the parameters of LSSVM. Compared with SVM model, LSSVM model can use kernel function to transform the prediction problem into the solution of equations and greatly improving the accuracy of evaluation. Overall, the evaluation performance of FWA-LSSVM model is the best.

Therefore, the results of the example analysis show that the proposed FWA-LSSVM model can evaluate the sustainability of modern photovoltaic agriculture scientifically and effectively. On the basis of interval type-2 fuzzy AHP-TOPSIS evaluation method, the intelligent algorithm is introduced, and expert knowledge is obtained by intelligence learning to generalize the expert scoring process in comprehensive evaluation. Then, the sustainable evaluation results of modern photovoltaic agriculture are obtained through FWA-LSSVM model, finally achieve the purpose of fast calculation and supporting relevant decision-making.