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Fig. 5. Surface view of rule base in Table 1a for the difficulty (left) and rule base in Table 1b for the cost (right).

Figure 5 Surface view of rule base in Table 1a for the difficulty (left) and rule base in Table 1b for the cost (right).

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Please cite this article in press as: Hameed, I. A. Using Gaussian membership functions for improving the reliability and robustness of students’ evaluati systems. Expert Systems with Applications (2011), doi:10.1016/j.eswa.2010.12.048 where g; €[1, 100] denotes the assigned maximum score of ques- tion i satisfying The method proposed by Bai and Chen (2008b) has several empirical weights which are determined subjectively by the do- main expert. Quite different ranks can be obtained depending on these weight values. By using this method, the examiners could not easily verify how new ranks are acquired and could not per- suade sceptical students. Naturally, there is no method to deter- mine the optimum values of these weights. Also, the weighted arithmetic mean formula used to compute the outputs do not sat- isfy the concept of fuzzy set. To reduce the degree of subjectivity in this method and provide a method based on the theory of fuzzy set, Saleh and Kim (2009) proposed a system applying the most com- monly used Mamdani’s fuzzy inference mechanism (Mamdani,
Please cite this article in press as: Hameed, I. A. Using Gaussian membership functions for improving the reliability and robustness of students’ evaluati systems. Expert Systems with Applications (2011), doi:10.1016/j.eswa.2010.12.048 where g; €[1, 100] denotes the assigned maximum score of ques- tion i satisfying The method proposed by Bai and Chen (2008b) has several empirical weights which are determined subjectively by the do- main expert. Quite different ranks can be obtained depending on these weight values. By using this method, the examiners could not easily verify how new ranks are acquired and could not per- suade sceptical students. Naturally, there is no method to deter- mine the optimum values of these weights. Also, the weighted arithmetic mean formula used to compute the outputs do not sat- isfy the concept of fuzzy set. To reduce the degree of subjectivity in this method and provide a method based on the theory of fuzzy set, Saleh and Kim (2009) proposed a system applying the most com- monly used Mamdani’s fuzzy inference mechanism (Mamdani,
Fig. 2. Node representation as a fuzzy logic controller.
Fig. 2. Node representation as a fuzzy logic controller.
Here, G’ denotes the transpose of G. Total score for each individ- 1al student is then obtained by formula (1) as Example. Assume that 10 students laid to an exam of 5 questions and the accuracy rate matrix, the time rate matrix, and the grade vector are given as follows (Bai & Chen, 2008b; Saleh & Kim, 2009):
Here, G’ denotes the transpose of G. Total score for each individ- 1al student is then obtained by formula (1) as Example. Assume that 10 students laid to an exam of 5 questions and the accuracy rate matrix, the time rate matrix, and the grade vector are given as follows (Bai & Chen, 2008b; Saleh & Kim, 2009):
lease cite this article in press as: Hameed, I. A. Using Gaussian membership functions for improving the reliability and robustness of students’ evalu< ystems. Expert Systems with Applications (2011), doi:10.1016/j.eswa.2010.12.048 where pix € [0,1] denotes the degree of importance of question i belonging to the importance level k. Here, five levels of importance (l=5) are represented by five fuzzy sets; k=1 representing the linguistic term “low”, k =2 representing “more or less low”, k =3 representing “medium”, k=4 representing “more or less high” and k=5 representing “high”. The MFs are shown in Fig. 3. Once
lease cite this article in press as: Hameed, I. A. Using Gaussian membership functions for improving the reliability and robustness of students’ evalu< ystems. Expert Systems with Applications (2011), doi:10.1016/j.eswa.2010.12.048 where pix € [0,1] denotes the degree of importance of question i belonging to the importance level k. Here, five levels of importance (l=5) are represented by five fuzzy sets; k=1 representing the linguistic term “low”, k =2 representing “more or less low”, k =3 representing “medium”, k=4 representing “more or less high” and k=5 representing “high”. The MFs are shown in Fig. 3. Once
Based on the fuzzy MFs in Fig. 3 we obtain the fuzzy accuracy rate matrix and the fuzzy time rate matrix as: In the first node, both inputs are given by examination results, whereas in the later nodes, one input is the output of its previous node and the other is given by a domain expert. The output of each node can be in the form of a crisp value (defuzzified) or in the form of linguistic variables (MFs). Each node has two scale factors (SFs in Fig. 1). We can adjust the effects of inputs by varying the values of scale factors. Here, we let both scaling factors have the same value of unity assuming equal influence of each input on the output. Each fuzzy node proceeds in three steps:
Based on the fuzzy MFs in Fig. 3 we obtain the fuzzy accuracy rate matrix and the fuzzy time rate matrix as: In the first node, both inputs are given by examination results, whereas in the later nodes, one input is the output of its previous node and the other is given by a domain expert. The output of each node can be in the form of a crisp value (defuzzified) or in the form of linguistic variables (MFs). Each node has two scale factors (SFs in Fig. 1). We can adjust the effects of inputs by varying the values of scale factors. Here, we let both scaling factors have the same value of unity assuming equal influence of each input on the output. Each fuzzy node proceeds in three steps:
A fuzzy rule base to infer the difficulty, cost, and adjustment. 1: “Low”, 2: “more or less low”, 3: “medium”, 4: “more or less high”, and 5: “high”. Table 1
A fuzzy rule base to infer the difficulty, cost, and adjustment. 1: “Low”, 2: “more or less low”, 3: “medium”, 4: “more or less high”, and 5: “high”. Table 1
Fig. 4. Fuzzification, Mamdani’s max-min inference, and COG to compute the difficulty of question 1.
Fig. 4. Fuzzification, Mamdani’s max-min inference, and COG to compute the difficulty of question 1.
In Step 2, based on the rule base given in Table 1b, we obtain the inference output (i.e., the fire-strength) of the cost matrix as a function of the fuzzy difficulty and fuzzy complexity matrices by formula (7) as: At the next cost node, in Step 1, the crisp values of D obtained at the previous node are fuzzified to obtain the fuzzy difficulty matrix Pi “Sed
In Step 2, based on the rule base given in Table 1b, we obtain the inference output (i.e., the fire-strength) of the cost matrix as a function of the fuzzy difficulty and fuzzy complexity matrices by formula (7) as: At the next cost node, in Step 1, the crisp values of D obtained at the previous node are fuzzified to obtain the fuzzy difficulty matrix Pi “Sed
In Step 3, we use COG to compute the crisp value of the diffi- culty of question 1 by formula (8) as 0.576, as illustrated in Fig. 4. Likewise, we compute the crisp values of other questions as
In Step 3, we use COG to compute the crisp value of the diffi- culty of question 1 by formula (8) as 0.576, as illustrated in Fig. 4. Likewise, we compute the crisp values of other questions as
The surface view of the rule base in Table 1b is shown in Fig. 5. At the final adjustment node, in Step 1, the crisp values of C ob- tained at the previous node are fuzzified to obtain the fuzzy cost matrix as
The surface view of the rule base in Table 1b is shown in Fig. 5. At the final adjustment node, in Step 1, the crisp values of C ob- tained at the previous node are fuzzified to obtain the fuzzy cost matrix as
The surface view of the rule base in Table 1c for the adjustment is typically that of effort shown in Fig. 5. In Step 3, the crisp values of adjustment are obtained as: as a function of the fuzzy cost and fuzzy importance matrices by formula (7) as:
The surface view of the rule base in Table 1c for the adjustment is typically that of effort shown in Fig. 5. In Step 3, the crisp values of adjustment are obtained as: as a function of the fuzzy cost and fuzzy importance matrices by formula (7) as:
Fig. 7. Surface view of rule base in Table 1a (left) and in Table 1b (right) for Gaussian MFs of o = 0.1.
Fig. 7. Surface view of rule base in Table 1a (left) and in Table 1b (right) for Gaussian MFs of o = 0.1.
Fig. 6. Gaussian membership functions of the five levels for o = 0.1.
Fig. 6. Gaussian membership functions of the five levels for o = 0.1.
New total scores and new ranking orders of 10 students using the three fuzzy nodes system based on Gaussian MF’s. Table 2 ters and there is no way to acquire its optimum values. It was noted that when theses parameters are changed slightly, different ranking orders are obtained which could impair the system’s reli- ability. In order to avoid losing reliability and having a robust sys- tem, the system should be able to give the same ranking orders without changing students’ scores and for various values of these parameters. As an alternative approach, Gaussian MF’s are pro- posed. Gaussian MF’s are suitable for problems which require con- tinuously differentiable curves and therefore smooth transitions, whereas the triangular do not posses these abilities. Gaussian MP’s are defined by two parameters which is a one parameter less than that of the triangular MF’s. The tuning of a reduced number of parameters will result in a reduced Degree Of Freedom (DOF) and hence a more robust system. Gaussian MF’s is defined as
New total scores and new ranking orders of 10 students using the three fuzzy nodes system based on Gaussian MF’s. Table 2 ters and there is no way to acquire its optimum values. It was noted that when theses parameters are changed slightly, different ranking orders are obtained which could impair the system’s reli- ability. In order to avoid losing reliability and having a robust sys- tem, the system should be able to give the same ranking orders without changing students’ scores and for various values of these parameters. As an alternative approach, Gaussian MF’s are pro- posed. Gaussian MF’s are suitable for problems which require con- tinuously differentiable curves and therefore smooth transitions, whereas the triangular do not posses these abilities. Gaussian MP’s are defined by two parameters which is a one parameter less than that of the triangular MF’s. The tuning of a reduced number of parameters will result in a reduced Degree Of Freedom (DOF) and hence a more robust system. Gaussian MF’s is defined as
Fig. 8. Mean scores of the 10 students ranked using classical method (dotted line) and the three nodes fuzzy system based on Gaussian MF’s (solid line).
Fig. 8. Mean scores of the 10 students ranked using classical method (dotted line) and the three nodes fuzzy system based on Gaussian MF’s (solid line).
Ranking orders and new scores obtained by the three methods. (3): Applied to 3 students with equal score and (10): applied to 10 (i.e., all) students. Table 3
Ranking orders and new scores obtained by the three methods. (3): Applied to 3 students with equal score and (10): applied to 10 (i.e., all) students. Table 3
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Artificial IntelligenceEducational evaluationEnvironmental InformaticsMulti Agent Systems and Technologies
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