The final section is conclusion. Here are three basic examples of t -norms and t -conorms which are often used for reasoning in fuzzy logic systems:. A fuzzy matrix is fuzzy similarity matrix when it is reflexive and symmetric. A fuzzy matrix is fuzzy equivalence matrix when it is reflexive, symmetric, and max-min transitive.

Fuzzy matrix B is called the max-min transitive closure of fuzzy matrix A if fuzzy matrix B includes fuzzy matrix A and fuzzy matrix B satisfies the following properties: 1 fuzzy matrix B is max-min transitive; 2 fuzzy matrix B is included by any fuzzy matrix which includes fuzzy matrix A and satisfies max-min transitivity.

## Multi-valued and Fuzzy Logic Realization using TaOx Memristive Devices

Therefore, known by the above-mentioned Theorem 2 , then after the finite times of compositions, we have. Known by the above theoretical result, we can construct the following algorithm for fuzzy clustering analysis. The match points of 12 football teams in 's Chinese National Football League are given.

The match point table contains a great amount of data, and some data are missing. The details are listed in Table 1.

It is difficult to rank the 12 football teams directly according to the original data in Table 1 since the numbers of fields are in huge difference between some teams. Aimed at the amount of missing data, some researcher proposed some methods to deal with these problems. For example, Dadelo et al. Jalao et al. Keener [ 21 ] applied Perron-Frobenius theorem in ranking of football teams. In this paper, we will apply fuzzy cluster analysis to investigate such ranking problem.

And firstly we approximately recognize the competency of every team through the numbers of wins, loses, and draws of fields that they attended. The data are listed in Table 2. The result is listed in Table 3. But the gap among them is not very big. In order to give a more reasonable and reliable ranking result, we propose a more efficient algorithm that can make full use of given data. Considering that 1 there are some teams that did not play a match with each other at all and the match points are unknown and 2 the match times between some football teams have a large difference, the number may be 0, 1, 2, and even 4.

Therefore, we have to establish a rule and define a group of characteristic data for each team. Then we can compute the degrees of similarity of competencies of the 12 teams i. Each match, every goal, and every goal against play an equally important role in ranking. We only use the difference of the numbers of goals and goals against to decide the characteristic data for each team. Therefore, weighting factors shall be included when we compute characteristic data. The degree of fuzzy similarity between team T i and T j is computed by.

The ranking principle is that the earlier the teams cluster, the closer they are in ranking. Then we can calculate the degrees of fuzzy similarity r ij between football teams T i and T j and obtain the fuzzy similarity matrix R as follows:. And known by hypothesis 6 , the first team clustered with T 4 is T 5 ; thus, T 5 is ranked as number The rest process is just similar. Team T 7 is ranked as number 1 since it is the last team falling into the cluster of T 4. The whole ranking result is listed in Table 4. There are four parameters in our model and they are predefined which seems very subjective.

To make sure the ranking result is reasonable and reliable, we shall analyze whether the result will change when these parameters vary in a certain range. Some calculating results are listed in Table 5. That shows our ranking result is stable when the parameters vary slightly. Therefore, the result is reasonable and reliable.

### Bibliographic Information

Our study shows that our ranking result is reliable and stable when the parameters change in a certain range. And our algorithm can be easily generalized to the case in which the number of teams is an arbitrary positive integer N. By the way, it needs to be pointed out that there are some disadvantages in our algorithm. The authors declare that there is no conflict of interests regarding the publication of this paper.

National Center for Biotechnology Information , U.

Journal List ScientificWorldJournal v. Published online Jun Author information Article notes Copyright and License information Disclaimer. Received Apr 7; Accepted May Zeng and J.

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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 Fuzzy set theory and fuzzy logic are a highly suitable and applicable basis for developing knowledge-based systems in physical education for tasks such as the selection for athletes, the evaluation for different training approaches, the team ranking, and the real-time monitoring of sports data. Introduction In recent years, computational intelligence has been used to solve many complex problems by developing intelligent systems.

For clarity, some definitions and notations are listed as follows.

Definition 1. Definition 2. Definition 3. Definition 4. Definition 5. Definition 6. Definition 7. Theorem 1 see [ 6 ].

## Get Fuzzy 2: Fuzzy Logic

Theorem 2 see [ 6 ]. Step 1. Step 2. Step 3. Table 1 Match scores of 12 football teams. Open in a separate window. Table 2 Numbers of wins, loses, and draws. Table 3 Numbers of goals, goals against, and their difference. Model Hypotheses For conveniences, we give the following hypotheses. Table 4 Ranking results of 12 football teams.

Parameter Sensitivity Analysis There are four parameters in our model and they are predefined which seems very subjective. Table 5 Some ranking results with varied parameters. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References 1.

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## Introduction to Type‐2 Fuzzy Logic Control | Wiley Online Books

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