Abstract
In this article, we propose a ranking method based on a matrix, called D-matrix, with the special identical diagonal values. This ranking system has five properties: (1) it can provide both biased and bias-free ranking, and except for that, the working matrix can be built in two ways: results merging and results separating for both biased and bias-free matrices. (2) it can perform the webpage ranking with a sparse matrix to generate ratings for pages instead of constructing complicated, irreducible, and stochastic matrices as the Google PageRank matrix does, thereby accelerating the computation speed. (3) this D-matrix has a solution no matter how much data is selected. If there are no comparisons among items, then all the items end up with the same equal ratings. (4) the ranking system has the least effects on data variation. If one item changes, only those connecting to it get different ratings, those without connection retain the same ratings. (5) this D-matrix has a $\ddot {R}$ support matrix with a delicate diagonal value which may or may not appear crucial. These five features are illustrated with five different and comprehensive examples. Besides that, a 2017 game of the National Football League data is tested where the D-matrix generates a reasonable result. Furthermore, we introduce a new approximate non-dominated sorting method based on D-matrix and thereby put forth a new algorithm for solving the multi-objective optimization problems. Experimental results indicate that our algorithm can maintain a better spread of solutions on many standard test functions.
Highlights
To rank an object is to order objects based on their importance in a finite set of size n
D-matrix, which shows a low computation of page ranking compared to PageRank, and it shows the good stability of our ranking method. We show another perspective on verifying the correctness of the ranking method, in which we construct a variant version of fundamental-rank-differential matrix
Due to the preference of D-matrix which is about mutual relation among data not about the matchup score, D-matrix can be used to do the webpage ranking and get a good calculation speed
Summary
To rank an object is to order objects based on their importance in a finite set of size n. There are many ways proposed to handle such problems, for example, assuming that the irreducible teleportation matrix S , the stationary vector of Sexists and is unique, S = βS + (1 − β)/nE, where E is the matrix of all ones and n is the number of teams Another ranking method is called Offense-Defense rating [21], which first rates individual attributes of each object, and combines these strengths to produce a single number which reflects overall rating. An D-matrix is initially designed for rating individuals in the population of an evolutionary algorithm adopted for solving multi-objective optimization problems In this situation, each individual represents one candidate solution and will be assigned several fitness function values. We will explain how to set N value in the examples
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