Judgment matrix aggregation, as an important part of group decisionmaking, has been widely and deeply studied due to the universality and importance of group decisionmaking in the management field. For the variety of judgment matrix in group decisionmaking, the matrix aggregation result can be obtained by using the mode of glowworm swarm optimization. First, this paper introduces the basic principle of the glowworm swarm optimization (GSO) algorithm and gives the improved GSO algorithm to solve the matrix aggregation problems. In this approach, the consistency ratio is introduced to the objective function of the glowworm swarm optimization, thus reducing the subjectivity and information loss in the aggregation process. Then, the improved GSO algorithm is applied to the solution of the deterministic matrix and the fuzzy matrix. The method optimization can provide an effective and relatively uniform aggregation method for matrix aggregation. Finally, through comparative analysis, it is shown that the method of this paper has certain advantages in terms of adaptability, accuracy, and stability to solving the matrix aggregation problems.
In social and economic life, group decisionmaking is widely applied in various management fields, providing support for solving complicated decisions. Therefore, aggregation of expert opinions in group decisionmaking is of long standing [
Traditional expert judgment matrix aggregation methods are mainly used for direct arithmetic and logic operations based on consistency regulation for the existing matrix. It has higher dependence on the matrix type and quality; thus, reduction of consistency regulation, subjectivity, and information loss in the aggregation process are still key problems for group decisionmaking in matrix aggregation. The glowworm swarm optimization (GSO) [
GSO algorithm is to solve the problem mainly through each glowworm representing a feasible solution to the objective problem in space. Glowworms will gather toward highbrightness glowworms by mutual attraction and movement, thus finding multiple extreme points in the solution space for the objective problems. The main idea is described as below.
This includes assigning relevant parameters such as the quantity of glowworms, initial position, initial fluorescein, decisionmaking radius, fluorescein volatilization rate, and domain change rate.
That is, calculate the fitness of each glowworm in place according to the objective function for the specific issues.
Each glowworm searches for the glowworm with higher fluorescein value within its own decisionmaking radius and determines the moving direction and step length of the next step according to the fluorescein value and the distance.
To be specific, update the position of each glowworm according to the determined moving direction and step length.
Judge whether the algorithm is terminated, and decide whether to enter into the next iteration.
Expert judgment matrix aggregation is a key element for the effectiveness of group decisionmaking. The judgment matrix aggregation aims at reasonably and effectively integrating expert opinions, removing opinion deviations to the greatest extent, and obtaining a synthetic judgment matrix with the highest consistency. Therefore, each expert matrix herein can be considered as a suboptimal feasible solution, and the initial distribution of glowworms is optimized. Random distribution of glowworms in the solution space is replaced with probability distribution by taking the initial suboptimal feasible solution as the boundary point, so as to improve the optimizing efficiency for solving the judgment matrix aggregation. The specific optimizing ideas are as follows.
First, narrow the initial distribution field of glowworms; each initial glowworm represents a feasible synthetic judgment matrix. The initial value of the glowworm is assumed to be within the field defined by the known expert matrix.
Assume
Second, in view of the elements in the synthetic judgment matrix of matrix aggregation not completely restricted by 1–9 scales (i.e., the value range is not
Third, add probability distribution factors. The aggregation of expert matrix is probably closer to its mathematical expectation. Therefore, the probability for obtaining the optimal solution is greater if the Euclidean distance to the space point represented by the mathematical expectation is closer. Based on random glowworm distribution, probability distribution is introduced to improve the algorithm convergence speed and optimizing effect.
Through the above analysis and in combination with the features of group decisionmaking matrix aggregation, the algorithm (MGSO) solving the judgment matrix aggregation by using GSO algorithm can be described as follows.
Assume the judgment matrix is designed according to 1–9 scale method, and provide transformation processing for initial judgment matrixes
On this basis, the value range of the synthetic judgment matrix element corresponding to each glowworm can be obtained as follows:
In the meantime, set the quantity of glowworms
Different judgment matrixes have different consistencies, and the importance is different. The greater the
Meanwhile, introduce corresponding average random consistency index RI and the consistency ratio CR [
If
To be specific, put the position value of each glowworm
Search for the glowworm
Judge the number of iterations and termination conditions and either terminate the algorithm or turn to Step
Deterministic judgment matrix aggregation and interval judgment matrix aggregation are discussed herein. The deterministic judgment matrix is mainly represented as the matrix elements, which are deterministic values, while the element of the interval judgment matrix is a numerical interval. Herein, MGSO algorithm is compiled by using MATLAB. Referring to the parameter design for GSO algorithm in relevant reference documents and combining actual conditions of the matrix aggregation, relevant parameters for MGSO algorithm are selected as follows:
The aggregation method of graph theory, proposed by Lu and Guo, focuses on aggregation for the deterministic judgment matrix. The basic idea of the matrix aggregation scheme based on undirected connected graph theory is to first select
Four expert judgment matrixes are selected as the deterministic judgment matrix for MGSO algorithm experiment (shown below):
For the above expert matrixes, some judgment matrixes with complete consistency constructed by the aggregation method of graph theory can be obtained, one of which is as below:
The weight result obtained through the aggregation method of graph theory is
According to the above analysis, the original judgment matrix is first transformed into the distance matrix by MGSO algorithm, as below:
By running the MGSO algorithm five times with randomized initial points, five synthetic judgment matrixes with complete consistency can be obtained as below:
Aggregation for the deterministic judgment matrix.
Synthetic judgment matrixes  Optimal weight  Importance sequence 


0.4455, 0.1099, 0.1875, 0.2570 


0.4680, 0.1085, 0.1804, 0.2431 


0.4620, 0.1092, 0.1785, 0.2502 


0.4444, 0.1105, 0.1891, 0.2559 


0.4449, 0.1105, 0.1900, 0.2545 

Arithmetic mean  0.4530, 0.1098, 0.1851, 0.2521 

As shown in Table
However, this method based on undirected connected graph theory does not get the correct results every time. In this paper, we change the elements
For these matrixes, we can get some conflicting results through the aggregation method of graph theory. The conflicting importance sequence of corresponding elements is
For the indeterminacy and inconsistency in the judgment matrix, the relations between the basic consistency and consensus of group decisionmaking are discussed. A scheme for solving the importance sequence based on integrated arithmetic mean is given by Zhai and Zhang [
According to the aggregation method based on integrated arithmetic mean, the combined judgment matrix
The weight result obtained from aggregation in the above interval judgment matrix by the method based on integrated arithmetic mean is
Now, the algorithm accuracy (
The difference between the aggregation for interval judgment matrix and the above deterministic judgment matrix is not just in the transformation of matrix elements from the numerical values into intervals; at the same time, the value range is not according to 17 grades corresponding to 1–9 scales anymore but expanded to the continuous interval of 0–9. The consistency index for the deterministic matrix cannot be used. Thus, the upper and lower limits of the modal interval judgment matrix are split into two deterministic judgment matrixes by MGSO algorithm; that is,
The judgment matrix is used as the suboptimal feasible solution for optimization by MGSO algorithm, and the consistency of the judgment matrix is introduced into the objective function as the parameter, so the influence of the consistency of the initial judgment matrix on the aggregation result is relatively small. Moreover, more optimal feasible solutions can also be obtained by running MGSO algorithm for the deterministic judgment matrix after the transformation of interval judgment matrix. Also, by running MGSO algorithm for
Aggregation for interval judgment matrix.
Synthetic judgment matrixes  Optimal weight of the first group of judgment matrixes  Optimal weight of the second group of judgment matrixes 


0.54004, 0.14557, 0.31439  0.67608, 0.16878, 0.15514 

0.60355, 0.16131, 0.23514  0.67654, 0.16903, 0.15443 

0.58768, 0.15270, 0.25962  0.68154, 0.16772, 0.15074 

0.59078, 0.15350, 0.25572  0.67602, 0.16907, 0.15491 

0.63984, 0.14281, 0.21734  0.67606, 0.16903, 0.15491 
Arithmetic mean  0.59238, 0.15118, 0.25644  0.67725, 0.16872, 0.15402 
The two sets of weight results obtained from the operation results of MGSO algorithm and the arithmetic mean are considered as feasible solutions of weights for the elements
According to the definition of algorithm accuracy, the accuracy
The following conclusions can be obtained:
Compared with
The above experimental results show that MGSO algorithm provides a relative uniform solution for the deterministic judgment matrix aggregation and interval judgment matrix aggregation. The traditional aggregation methods often focus on special judgment matrix type; thus in practical application, consistent judging methods cannot be provided for different expert judgment ways of the same problem. A uniform solution can be provided by MGSO algorithm because the expert judgment matrix is considered as the feasible solution in MGSO algorithm, and more optimal feasible weight solutions can be obtained through optimization, while traditional matrix aggregation methods are often to operate the judgment matrix; hence, there are more requirements on the expert matrix.
The expert judgment matrix is subjected to differentiation by MGSO algorithm. Different expert matrixes have different subjective recognition, and the actual qualities are different; for example, expert judgment matrixes with poor quality have negative impact on the final weight result. Thus, MGSO algorithm is to differentiate expert matrixes by adding the consistency index of the judgment matrix to the objective function, consequently to weaken the negative impact of the expert matrix with poor consistency on the final result, and meanwhile to further reduce the dependency of matrix aggregation scheme on the consistency of the initial matrix.
Compared with the method of graph theory, the operation results of MGSO algorithm are more stabilized. According to the specific case analysis for the method of graph theory, the weight sequence results are instable when multiple element values are equal in the distance deviation matrix, but the weight sequence results are obtained through constant optimizing in MGSO algorithm, so those results are more stable, which is the same as proved above.
By defining the algorithm accuracy, the advantages and disadvantages of the weight range can be compared more intuitively. Due to
The result comparison of different decision alternatives or decision factors is usually given in the form of a judgment matrix in group decisionmaking. The aggregation of the matrix is equivalent to the synthesis of various expert opinions. In this paper, the weight or importance sequence obtained by the MGSO algorithm can be understood as the importance of decision scheme or decision factors in the specific application. This is the application value of the MGSO algorithm in group decisionmaking.
This paper presents the MGSO algorithm to solve the matrix aggregation problems. Unlike the traditional matrix aggregation methods, this method not only can be used in deterministic matrix, but also can be used for the interval matrix. The aggregation mode of MGSO algorithm is no longer to conduct operation for the judgment matrix itself but to take the judgment matrix as the restrictions for optimization. In the meantime, consideration is given to the quality difference of judgment matrixes in MGSO algorithm. It is unnecessary to manually regulate the judgment matrixes with relatively poor consistency; thus this optimization is more adaptive. Through comparative analysis, the MGSO algorithm has good robustness, stability, and computational accuracy.
In addition, the effects of random distribution of glowworm position can also be noticed. The MGSO algorithm should be improved in convergence and accuracy, so that it can be better applied to solve group decisionmaking problems.
The author declares no competing interests.
The work was supported by the fund of the Provincial Excellent Young Talents of Colleges and Universities of Anhui Province (no. 2013SQRW115ZD) and the fund of the Natural Science of Colleges and Universities of Anhui Province (no. KJ2016A162).