Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 1350807, 8 pages

http://dx.doi.org/10.1155/2016/1350807

## Problems on Solving Matrix Aggregation in Group Decision-Making by Glowworm Swarm Optimization

^{1}School of Management, Hefei University of Technology, Hefei 230009, China^{2}Anhui Economic Management Institute, Hefei 230059, China

Received 19 May 2016; Revised 16 July 2016; Accepted 17 July 2016

Academic Editor: Muhammad N. Akram

Copyright © 2016 Yaping Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Judgment matrix aggregation, as an important part of group decision-making, has been widely and deeply studied due to the universality and importance of group decision-making in the management field. For the variety of judgment matrix in group decision-making, 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.

#### 1. Introduction

In social and economic life, group decision-making is widely applied in various management fields, providing support for solving complicated decisions. Therefore, aggregation of expert opinions in group decision-making is of long standing [1, 2]. At the same time, Zeng et al. gave an applicable example of group decision-making under Web 2.0 environment; it was shown that group decision problem has important significance in both the present and the future [3]. As the application of information technology becomes widely used, expert opinion forms for group decision-making are changing. Forms of the expert judgment matrix mainly include deterministic type and interval type; fuzzy judgment matrix containing deterministic value and interval value is derived. For judgment matrixes of different types, the matrix aggregation method mainly includes additive aggregation, multiplicative aggregation [4, 5], operators-based aggregation [6], graph theory-based aggregation [7], and evidence theory-based aggregation [8]. Lu and Guo gave a matrix aggregation scheme based on an undirected connected graph theory. In the scheme, an aggregation matrix with complete consistency is rebuilt by screening more consistent expert opinions according to opinion deviations, thereby obtaining the importance sequence [7]. In 2014, a matrix aggregation algorithm based on spanning tree aggregation operators was proposed by Huang et al. [8], which applies spanning tree aggregation operators to obtain a completely consistent aggregation matrix according to the relations between the judgment matrix and simple undirected graph spanning tree. Zhai and Zhang [9] proposed a matrix aggregation method based on the evidence theory in group judgment. The above methods mainly focus on the problem of aggregation in the deterministic matrix. In addition, the aggregation method for interval judgment matrix is also an important research direction [10, 11]. In 2015, L. Li and J. Li [12] gave an aggregation method for interval judgment matrix, the three-point interval number judgment matrix aggregation is transformed into optimal two-point judgment matrix, and optimal aggregation intervals are synthesized through plant growth simulation algorithm. For Pythagorean fuzzy multiple-criteria decision-making problems, Zeng et al. [13] developed a new method with aggregation operators and distance measures in 2016. Meanwhile, Zeng et al. [14] presented another aggregation operator that uses generalized means in a unified model between the probability and the OWA operator. Moreover, more attention has also been paid to differential weighting research for expert opinions [15].

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 decision-making in matrix aggregation. The glowworm swarm optimization (GSO) [16] proposed by Krishnanand and Ghose is a new type of swarm intelligence optimization algorithm; it has a higher efficiency than the traditional swarm intelligence algorithms in solving multimodal problems [17]. GSO algorithm is hereby introduced into the matrix aggregation; the expert matrix, as the known feasible solution and the parameter of the objective function, is optimized continuously by GSO algorithm for solving the feasible solution with closer Euclidean distance to the expert matrix as the aggregation result. Therefore it can play a positive role in improving the consistency of the judgment matrix after aggregation, reducing the subjectivity of the aggregation process and the information loss of the original judgment matrix. The researches herein mainly include three parts: Section 2 gives the optimization idea of matrix aggregation-oriented GSO algorithm, Section 3 shows a matrix aggregation method based on an improved GSO algorithm, and Section 4 presents an experimental analysis for deterministic judgment matrix aggregation and interval judgment matrix aggregation.

#### 2. Matrix Aggregation-Oriented GSO Algorithm

##### 2.1. Basic Idea of GSO Algorithm

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 high-brightness 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.

*Step 1 (initialize the algorithm). *This includes assigning relevant parameters such as the quantity of glowworms, initial position, initial fluorescein, decision-making radius, fluorescein volatilization rate, and domain change rate.

*Step 2 (calculate the fitness according to the objective function). *That is, calculate the fitness of each glowworm in place according to the objective function for the specific issues.

*Step 3 (determine the moving direction and step length of the glowworm). *Each glowworm searches for the glowworm with higher fluorescein value within its own decision-making radius and determines the moving direction and step length of the next step according to the fluorescein value and the distance.

*Step 4 (update the position of the glowworm). *To be specific, update the position of each glowworm according to the determined moving direction and step length.

*Step 5 (update the decision-making radius and the fluorescein of the glowworm). *Judge whether the algorithm is terminated, and decide whether to enter into the next iteration.

##### 2.2. Features of Judgment Matrix Aggregation

Expert judgment matrix aggregation is a key element for the effectiveness of group decision-making. 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 as the value range of the element in expert judgment matrix, matrix as the expert judgment matrix, and matrix as the synthetic judgment matrix corresponding to the glowworm. Here, the judgment matrix is described according to 1–9 scale method [18]:

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 ), the restrictions to are canceled during initialization of glowworm distribution:

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.

#### 3. Matrix Aggregation Method Based on Improved GSO Algorithm

Through the above analysis and in combination with the features of group decision-making matrix aggregation, the algorithm (M-GSO) solving the judgment matrix aggregation by using GSO algorithm can be described as follows.

*Step 1 (initialize). *Assume the judgment matrix is designed according to 1–9 scale method, and provide transformation processing for initial judgment matrixes to facilitate comparison for Euclidean distances. Later, refer to the construction method for the distance matrix proposed by Lu and Guo [7], and transform 17 scale values into 1–9 scales, such as correspondingly; that is,Thereby transform into .

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 , step length , initial fluorescein value , fluoresce in volatilization rate , domain change rate , decision domain initial value , domain threshold , and other related parameters in GSO algorithm, as well as the maximum number of iterations and the algorithm termination condition.

*Step 2 (set the objective function). *, in which the objective function is introduced into to express the consistency ratio of the th expert judgment matrix:where is the maximum eigenvalue of the judgment matrix:Different judgment matrixes have different consistencies, and the importance is different. The greater the , the worse the consistency, leading to the smaller weight of the corresponding judgment matrix.

Meanwhile, introduce corresponding average random consistency index RI and the consistency ratio CR [19]:If , it is considered that the judgment matrix has consistency, and the element weight result is acceptable. Otherwise, the judgment matrix must be subjected to necessary correction until .

*Step 3 (calculate the fitness). *To be specific, put the position value of each glowworm into the objective function to calculate the fitness of each glowworm:

*Step 4 (update the position of the glowworm). *Search for the glowworm with the highest fluorescein within the field range of the glowworm , in order to determine the direction of moving toward :

*Step 5 (update the fluorescein). * represents the fluorescein value of the th glowworm in the th iteration, represents the fluorescein volatilization rate, and represents the extract ratio of the fitness:

*Step 6 (update the decision domain). * represents the decision-making radius of the th glowworm in the th iteration, represents the domain change rate, represents the quantity threshold of glowworms in the domain, and represents the quantity of glowworms within the decision-making radius of the th glowworm in the th iteration:

*Step 7. *Judge the number of iterations and termination conditions and either terminate the algorithm or turn to Step .

#### 4. Experimental Analysis

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, M-GSO 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 M-GSO algorithm are selected as follows: , with the maximum number of iterations as 100.

##### 4.1. Aggregation for the Deterministic Judgment Matrix

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 positions in the upper triangle (or lower triangle) of the matrix based on the basic theory of the undirected connected graph during aggregation for judgment matrixes . obtained correspondingly in each position of positions is required to be kept relatively consistent, with deviation as small as possible (where the method for deviation calculation is the sum of the absolute values from subtraction between two elements in vectors). Then, calculate the arithmetic mean (additive) or geometric mean (multiplicative) for the elements in these positions. At last, on this basis, according to the principle of proportional rows, calculate the element values in other positions, thereby building judgment matrix with complete consistency.

Four expert judgment matrixes are selected as the deterministic judgment matrix for M-GSO 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 , and the importance sequence of corresponding elements is , in which the consistency ratio of is .

According to the above analysis, the original judgment matrix is first transformed into the distance matrix by M-GSO algorithm, as below:

By running the M-GSO algorithm five times with randomized initial points, five synthetic judgment matrixes with complete consistency can be obtained as below:where , , , , satisfy . The sequence of corresponding weights can be calculated according to the matrix , , , , , as shown in Table 1.