Non-ADAPTIVE METHOD OF FUZZY SETS

Over the last 20 years the theory of fuzzy sets has been developing actively. The results of the theory are widely used in the testing. The paper proposes a modification of a known adaptive method of fuzzy sets to assess the level of preparedness of a student. Modification of the method consists in the prohibition of adaptation of fuzzy sets. Using the Rasch’s testing model and Monte Carlo’s method the author investigates the accuracy of the modification proposed. The results have shown that modification of the method of fuzzy sets has a higher accuracy than the classical method. The modification of the method of fuzzy sets is a non-adaptive algorithm. The test results in the form of primary points are not used for construction of membership functions of fuzzy sets. We describe the steps of the algorithm, which makes it possible to “tweak” the primary points. 1) At first numerical characteristics of the linguistic variable B (score, which is put to a student at the task) are defined: d - length tolerance intervals for the extreme elements of the term-set, m+1 - the number of elements of the term-set of the linguistic variable B, r=(1-2d)/m - the distance between vertices unimodal membership functions of the neighboring elements of the term-set. 2) Membership function is formed in accordance with the parameters set in the first step. 3) The numbers of E_k, k = 0, ..., m, are calculating for each membership function. These numbers are the result of diffusivities for fuzzy sets by the method of severity’s center. 4) Adjustment of the primary point B for the i-th task is made by the formula: where max is maximum score for the i-th task. 5) The sum of all “adjusted” points calculates: where M is the number of test tasks. 6) Test points ТБ_кор is calculated by using of scaling. The main results of the article 1) Non-adaptive modification of the method of fuzzy sets makes it possible to increase the accuracy of the fuzzy sets of about 0.2%. 2) Non-adaptive modification of the method of fuzzy sets is comparable in accuracy with adaptive modification, but its implementation in practice is simpler. 3) Non-adaptive modification of the method of fuzzy sets in accuracy is not inferior to the method of the Rasch’s logarithm. 4) The method of the Rasch’s logarithm does not depend on the choice of the scaling function and other parameters. Therefore, this method is the most convenient to use. The accuracy of this method is not worse than the accuracy of the methods based on the use of fuzzy sets.

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