New Applications and Developments of Fuzzy Systems

CONTENTS

LIST OF FIGURES

ACRONYMS AND ABBREVIATION

LIST OF NOTATIONS

ACKNOWLEDGMENT

DECLARATION

PUBLICATIONS

ABSTRACT

1. INTRODUCTION
1.1 B ackground
1.2 Methods
1.3 Results and contributions
1.4 Discussion and conclusion
1.5 Organization of thesis

2. A FUZZY SYSTEM FOR EVALUATING STUDENTS’ LEARNING ACHIEVEMNET
2.1 Introduction
2.2 A review of evaluation methods using membership functions and fuzzy rules
2.3 Three node fuzzy evaluation system
2.4 Method validation
2.5 Conclusions

3. A SIMPLIFIED ARCHITECTURE OF TYPE-2 FUZZY CONTROLLER USING FOUR EMBEDDED TYPE-1 FUZZY CONTROLLERS AND ITS APPLICATION TO A GREENHOUSE CLIMATE CONTROL SYSTEM
3.1 Introduction
3.2 A simplified architecture of type-2 FLS
3.3 Genetic algorithm (GA)
3.4 Greenhouse climate control problem
3.5 Simulation experiments
3.6 Conclusions

4. USING THE EXTENDED KALMAN FILTER TO IMPROVE THE EFFICIENCY OF GREENHOUSE CLIMATE CONTROL
4.1 Introduction
4.2 Greenhouse climate control problem
4.3 Inaccurate measurements and energy consumption in greenhouses
4.4 The continuous-time EKF
4.5 State estimation using EKF
4.6 Simulation results
4.7 Conclusions

5. CONCLUSIONS
5.1 Conclusions
5.2 Future work

REFERENCES

LIST OF TABLES

Table 2-1: A fuzzy rule base to infer the difficulty, cost and adjustment

Table 2-2: Ranking order of the three methods

Table 3-1: A fuzzy rule base

Table 3-2: Parameters oftype-1 (T1) and type-2 (T2) fuzzy logic controllers obtained by GA where measurement uncertainty is introduced in experiment 1 and modeling uncertainty is introduced in experiment 2

Table 3-3: Mean squared error (MSE) and signal-to-noise ratio of temperature (SNRT) and humidity ratio (SNRH) of different types of controllers when measurement uncertainty is introduced in experiment 1 and modeling uncertainty is introduced in experiment

LIST OF FIGURES

Fig. 2-1. Membership functions of the fuzzy sets “low”, “more or less low”, “medium”, “more or less high” and “high”

Fig. 2-2. Block diagram of the three nodes fuzzy evaluation system

Fig. 2-3. Node representation

Fig. 2-4. Fuzzification, Mamdani’s max-min inference, and COG to obtain the difficulty of question 1

Fig. 2-5. Surface view of rule base of Table 1 (a) (i.e., difficulty)

Fig. 2-6. Surface view of rule base of Table 2-1 (b) (i.e., cost and adjustment)

Fig. 3-1. Gaussian type-2 FS: (a) blurring the width of type-1 FS where oL and ou are the minimum and maximum resultant widths respectively, (b) blurring the center of type-1 FS where c1 and c2 are minimum and maximum resultant centers respectively

Fig. 3-2. Illustration of decomposing T2MFs into 4 T1MFs

Fig. 3-3. (a) Membership functions of left intersection points. (b) Membership functions of upper intersection points. (c) Membership functions of lower intersection points, and (d) Membership functions of right intersection points

Fig. 3-4. Simplified type-2 fuzzy logic system: controller output is the average of the four outputs of the embedded upper, left, right, and lower type-1 fuzzy logic systems, x1 and x2 are the controller inputs and y is the controller output

Fig. 3-5. Genetic Algorithm’s chromosome: (a) Set of parameters of type-2 fuzzy logic controller (b) Set of parameters oftype-1 fuzzy logic controller

Fig. 3-6. A Schematic diagram of the greenhouse climate control process

Fig. 3-7. Greenhouse outputs: indoor air temperature (upper) and indoor air humidity ratio (bottom)

Fig. 3-8. Control outputs: ventilation rate (upper), humification rate (middle) and heating rate (bottom)

Fig. 3-9. Climate variables: outdoor air temperature (upper), outdoor humidity ratio (middle) and outdoor solar radiation (bottom)

Fig. 3-10. MF’s: type-1 fuzzy logic controller of temperature loop (left fuzzy logic controller of humidity ratio loop (right upper), type controller of temperature loop (left bottom) and type-2 fuzzy logic controller of humidity ratio loop (right bottom)

Fig. 3-11. Control surface: type-1 fuzzy logic controller of temperature loop (left upper), type-1 fuzzy logic controller of humidity ratio loop (right upper), type-2 fuzzy logic controller of temperature loop (left bottom) and type-2 fuzzy logic controller of hu humidity ratio loop (right bottom)

Fig. 3-12. Greenhouse outputs: indoor air temperature (upper) and indoor air humidity ratio (bottom)

Fig. 3-13. Control outputs: ventilation rate (upper), humification rate (middle) and heating rate (bottom)

Fig. 3-14. MF’s: type-1 fuzzy logic controller of temperature loop (left upper), type-1 fuzzy logic controller of humidity ratio loop (right upper), type-2 fuzzy logic controller of temperature loop (left bottom) and type-2 fuzzy logic controller of humidity ratio loop (right bottom)

Fig. 3-15. Control surface: type-1 fuzzy logic controller of temperature loop (left upper), type-1 fuzzy logic controller of humidity ratio loop (right upper), type-2 fuzzy logic controller of temperature loop (left bottom) and type-2 fuzzy logic controller of humidity ratio loop (right bottom)

Fig. 4-1. A schematic diagram of the greenhouse climate control (GCC) process incorporated with extended Kalman filter (EKF) for state feedback control

Fig. 4-2. State observer block diagram using continuous-time extended Kalman filter

Fig. 4-3. Climate variables: outdoor air temperature (upper), outdoor humidity ratio (middle), and outdoor solar radiation (bottom)

Fig. 4-4. Greenhouse outputs for step changes in both humidity and temperature: indoor air temperature (upper) and indoor air humidity ratio (bottom)

Fig. 4-5. Greenhouse outputs in the first 30 minutes (i.e., transient response)

Fig. 4-6. Error in temperature (upper) and humidity (bottom)

Fig. 4-7. Controller outputs: ventilation rate (upper), humidification rate (middle), and heating rate (bottom)

Fig. 4-8. Daily mean square error (MSE) (for filtered case and for unfiltered case) for temperature (upper left) and humidity ratio (upper right), Signal-to-Noise Ratio (SNR) for temperature (bottom left) and humidity ratio (bottom right)

Fig. 4-9. Daily operating cost efficiency (i.e., filtered operating cost/unfiltered operating

ACRONYMS AND ABBREVIATION

Abbildung in dieser Leseprobe nicht enthalten

ACKNOWLEDGMENT

This thesis comes out after very tough years of research that has been done after joining artificial intelligence and applied statistics lab at Korea University. By that time, I have worked and met with a great number of people who always inspired me to be good at what I am doing. It is a pleasure to convey my gratitude to them all in my humble acknowledgment.

First and foremost I offer my sincerest gratitude to my supervisor, Professor Seong-in Kim, PhD, who has supported me throughout my thesis with his patience and knowledge whilst allowing me the room to work in my own way. I attribute the level of my work to his supervision, advice, guidance, encouragement and effort and without him this thesis would not have been completed or written. Many thanks go to him for giving me the opportunity to meet very great people from many different areas and the opportunity to benefit from visiting factories, companies and sharing in conferences in Korea and abroad.

It is a pleasure to express my gratitude wholeheartedly, again, to my supervisor Professor Seong-in Kim and his wife for their kind hospitality and their interest to provide me and my family a comfortable life in Korea.

Furthermore, I gratefully thank and acknowledge the very exceptional and experienced faculty members of the department of industrial engineering and the department of control engineering, who gave me unique opportunities to enhance my knowledge and learn new things every day during the lectures of my PhD course work and projects.

I convey special acknowledgement to Mr. Shin Young-Ho, Mrs Kim Eun-ko and Mrs. Eun Hye Kang for their indispensable help dealing with administration and bureaucratic matters during my stay in Korea and my commute between Egypt and Korea.

The financial support and scholarships provided by Professor Seong-in Kim, Korea University, YoongPong Co. and BK21 are gratefully acknowledged.

I was fortunate in having very sincere and kind friends who helped a lot to make my life in Korea wanner. Thanks to Min-Kyu Kim, MSc, and Young-min Kim for the friendship at the office and many other places. Thanks to Seok Yoo, internship at Nuri Solution Ltd. Thanks to Ji-Ho kil for taking care of submitting my graduation staff.

My Father deserves special mention for his inseparable support and prayers. I feel sorry that I cannot share this great moment with my late mothers. I keep praying for both of you that God may bless you in his grace and merci.

Words fail me to express my appreciation to my wife Abeer Badawy, MBBCh, whose dedication, love and persistent confidence in me, has taken the load off my shoulder. I would also thank my parents in law for taking care of my young family during my stay abroad. A big thanks to my sons Mohamed and Omar, thanks for always being there for me

Finally, I would like to thank everybody who was important to the successful realization of my thesis, as well as expressing my apology that I could not mention personally one by one.

Ibrahim Abdel Fattah Abdel Hameed Ibrahim Seoul, Republic of Korea, February, 2010

DECLARATION

This is to certify that:

(i) the thesis comprises only my original work towards the PhD except where indicated,
(ii) author names “Ibrahim Saleh” and “Ibrahim A. Hameed” are my names, the former one had been mis-used and unintentionally during a paper submission.
(iii) due acknowledgement has been made in the text to all other material used,

Ibrahim Abdel Fattah Abdel Hameed Ibrahim

PUBLICATIONS

During the course of this project, a number of public presentations and articles have been made which are based on the work presented in this thesis. They are listed here for reference.

Saleh, I. A., (Hameed, I. A.,) & Kim, S.-i. (2009). A fuzzy system for evaluating students’
learning achievement. Expert Systems with Applications, 36(3), 6236-6243.

Hameed, I. A. (2009). Simplified architecture of a type-2 fuzzy controller using four embedded type-1 fuzzy controllers and its application to a greenhouse climate control system. Proc. IMechE Part I: Journal of Systems and Control Engineering, 223(5), 619-631.

Hameed, I. A. (2010). Using the extended Kalman filter to improve the efficiency of greenhouse climate control. International Journal of Innovative Computing, Information and Control (IJICIC), 6(6), 2671-2680.

ABSTRACT

Fuzzy Logic (FL) is a particular area of interest in the study of Artificial intelligence (AI) based on the idea that in fuzzy sets each element in the set can assume a value from 0 to 1, not just 0 or 1, as in classic or crisp set theory. The gradation in the extent to which an element is belonging to the relevant sets is called the degree of membership. This degree of membership is a measure of the element’s belonging to the set, and thus of the precision with which it explains the phenomenon being evaluated. A linguistic expression is given to each fuzzy set. The information contents of the fuzzy rules are then used to infer the output using a suitable inference engine. The key contribution of fuzzy logic in computation of information described in natural language made it applicable to a variety of applications and problem domains; from simple control systems to human decision support systems. Yet, despite its long-standing origins, it is a relatively new field, and as such leaves much room for development.

The thesis presents two novel applications of fuzzy systems; a human decision support system to help teachers to fairly evaluate students and two hybrid intelligent fuzzy systems; a type-2 fuzzy logic system and a combined type-1 fuzzy logic system and extended Kalamn filter for controlling systems operating under high levels of uncertainties due to various sources of measurement and modeling errors.

The combination of fuzzy logic and the classical student evaluation approach produces easy to understand transparent decision model that can be easily understood by students and teachers alike. The developed architecture overcomes the problem of ranking students with the same score. It also incorporated different dimensions of evaluation by considering subjective factors such as difficulty, complexity and importance of the questions. Although we discuss this approach with an example from the area of student evaluation, this method evidently has wide applications in other areas of decision making including student’s project evaluation, learning management systems evaluation, as well as, other assessment applications.

Uncertainty is an attribute of information. For systems being controlled using the mentioned above type-1 fuzzy logic systems, such uncertainty leads to fuzzy rules whose antecedents or consequents are uncertain, which translates into uncertain antec consequent membership functions. Type-1 fuzzy systems, whose membershjp functions are type-1 fuzzy sets, are unable to directly handle such uncertainties. Type-2 fuzzy systems in which the antecedent or consequent membership functions are type-2 fuzzy sets. Such sets are fuzzy sets whose membership grades themselves are type-1 fuzzy sets, are very useful in circumstances where it is difficult to determine an exact membership function for a fuzzy set.

By combining type-2 fuzzy logic with traditional soft computing techniques such as genetic algorithms, we build a powerful hybrid intelligent control system that can use the advantages that each technique offers. Due to the complexity of implementing type-2 systems, a simplified approach for building type-2 fuzzy system using the well known type-1 is developed. A genetic algorithm is used to give adaptability to the fuzzy system to adapt to changing situations. In addition, it provides the system with an aid to show how much uncertainty is incorporated in the system. The system is applied to a nonlinear multi-input multi-output system equipped with almost all types of uncertainty and it shows very stable response even under very high levels of uncertainties.

A novel approach for controlling systems equipped with high levels and different sources of uncertainties due to measurement and modeling errors is developed by combining a type-1 fuzzy system with the well known extended Kalman filter (EKF). The addition of an EKF in the feedback loop improved the system response by blocking possible effects of measurement error through the use of the estimated states instead of the measured states. The developed type-1 fuzzy-Kalamn filter scheme is applied to a complex, nonlinear multi-input multi-output system exposed to high levels of noise. Surprisingly, the new scheme decreased the power consumption while keeping system states very close to the desired states. In addition, the output response becomes very smooth which could help to increase the life time of the system actuators. The filtering effect is also expected to result in less number of false alarms when fault detection and isolation system is applied and hence increase system robustness and reliability.

1. INTRODUCTION

1.1 Background

Since the development of the theory of fuzzy sets, started with the 1965 paper “Fuzzy Sets” (Zadeh, 1965), and the introduction of the concept of a linguistic variable, that is, a variable whose values are words rather than numbers (Zadeh, 1973), the concept of a linguistic variable has played and is continuing to play a pivotal role in the development of fuzzy logic and its applications (Zadeh, 1999). Fuzzy logic is a precise logic of imprecision and approximate reasoning and it may be viewed as an attempt at formalization/mechanization of two remarkable human capabilities. First, the capability to converse, reason and make rational decisions in an environment of imprecision, uncertainty, incompleteness of information, conflicting information, partiality of truth and partiality of possibility - in short, in an environment of imperfect information. And second, the capability to perform a wide variety of physical and mental tasks without any measurements and any computations (Zadeh, 2008).

In this thesis, fuzzy logic has been applied into two main fields of artificial intelligence; human decision support system and control of complex and ill-defined systems working under imprecise conditions. Students’ evaluation as a crucial problem for students, teachers and institutes as well, has been selected to illustrate how much development could be made to this process when fully mechanized using fuzzy logic system to benefit from representing knowledge in words and reasoning. Over The last decades, a considerable number of applications of fuzzy set theory in educational evaluation had been introduced. A fuzzy logic system for translating traditional scores into letter-grades was proposed by Echauz and Vachtsevanos (1995). Law (1996) proposed a fuzzy grading system to aggregate different test scores to produce a single score for each individual student. Wilson, Karr and Freeman (1998) presented an automatic grading system based on fuzzy rules and genetic algorithms. Chen and Lee (1999) presented two methods for applying fuzzy sets to overcome the problem of rewarding two different fuzzy marks the same total score. A fuzzy set approach to assess the outcomes of student-centered learning using the evaluation of their peers and lec was proposed by Ma and Zhou (2000). Weon and Kim (2001) presented an evaluation strategy based on fuzzy MFs and subjective factors such as difficulty, importance, and complexity of questions as a necessity for fair evaluation. Wang and Chen (2008) presented a method for evaluating students’ answer scripts using fuzzy numbers associated with degrees of confidence provided by the evaluator. Bai and Chen proposed a method to automatically construct the grade MFs of fuzzy rules and applying it for the evaluation of students’ learning achievement (Bai and Chen, 2008a; 2008b). Previous studies succeeded to get benefit from using fuzzy numbers, fuzzy sets and fuzzy rules for knowledge representation in evaluation, yet, they did not use the framework of fuzzy reasoning to deal with knowledge represented in natural language. As a result, they lake the transparent and understand necessary to interpret results. In addition, these methods relay on a number of variables such as weight factors, etc which requires tuning and adjusting not based on clear theoretical bases. When changing these variables, different results could be obtained for different combinations of these variables. The presented method in this thesis is fully automatic and does not need any adjusting or tuning. In addition, it has the enough transparences and understanding necessary for interpreting the results.

In this thesis, the problems of controlling systems operating under sever conditions of noise and uncertainty is also studied. Researchers have shown that type-1 fuzzy logic systems have difficulty in modeling and minimizing the effect of such uncertainties (Mendel, 2001). Type-2 fuzzy systems, characterized by membership grades that are themselves fuzzy, were first introduced by Zadeh in 1975 to account for this problem (Zadeh, 1975a). The footprint of uncertainty (FOU) of type-2 fuzzy sets which represent the uncertainties in the shape and position of normal type-1 fuzzy sets provides an extra mathematical dimension to better handle the amount of uncertainty in a system (Wu & Tan, 2004). Therefore, type-2 fuzzy systems in which the antecedent or consequent membership functions are type-2 fuzzy set have the potential to outperform type-1 fuzzy system (Karnik & Mendel, 2001; Sepulveda et al., 2007a).

Despite the advantages offered by type-2 fuzzy system in handling uncertainties, one major problem that may hinder its use broadly in real-time applications is its high computational cost. Type-reduction, which is an additional processing convert type-2 fuzzy sets into type-1 fuzzy sets to be processed by the defi a crisp output, is very computationally intensive, especially when there are many membership functions and the rule base is large (Karnik & Mendel, 1999). To reduce the computational burden while preserving the advantages of type-2 fuzzy logic system, a novel paradigm of type-1 fuzzy logic and genetic algorithms is used to implement type-2 fuzzy logic system. A genetic algorithm is used to adapt the controller parameters and at the same time provides a way to measure the amount of uncertainty in the system and representing it in terms of the width of the type-2 membership functions. The developed hybrid intelligent controller is easy to implement and does not require irregular computational requirements.

The above type-2 fuzzy logic system proved superiority in dealing with systems exposed to several types of uncertainties. However, combining type-1 fuzzy systems with extended Kalamn filter in the feedback loop, surprisingly, improved the system response and reduced the effect of measurement and modeling uncertainties and as a result reduced the power consumed in controlling the system. This approach is applied to a multi-input multi-output system exposed to different sources of measurement and modeling errors.

1.2 Methods

Type-1 fuzzy logic has been used in developing a student evaluation system. ft consists of several steps; fuzzification which converts a variable from its physical domain into a degree of membership, fuzzy rules which represent the relation between inputs and outputs, a fuzzy inference engine which uses fuzzy operators to produce the fuzzy output, here max-min inference is used, and finally a center-of-gravity (COG) defuzzification is used to convert the output from its fuzzy domain into the problem physical domain. fn the thesis, type-2 fuzzy logic system is implemented using the basic knowledge of type-1 fuzzy logic using a novel paradigm of four type-1 fuzzy logic systems and genetic algorithms. Extended Kalman filter is used in the feedback loop with a type-1 fuzzy logic system to control complex systems exposed to several sources of measurement and modeling uncertainties.

1.3 Results and contributions

In the thesis, three methods are developed; (1) a fuzzy system for student’s evaluation, (2) a hybrid intelligent type-2 fuzzy controller and (3) a novel paradigm of type-1 fuzzy logic system with an EKF in the feedback loop for controlling ill-defined systems suffering from different types of uncertainties. The developed student’s evaluation system overcomes the problem of students with the same total score. The transparency and easy understanding of the developed system made it easy for teachers to explain adjusted scores. The inclusion of subjective factors such as difficulty, complexity and importance of question gave new dimensions to the evaluation process by producing more reflective and fair scores. With the developed system, the evaluation process becomes fully automatic and robust that it does not require any human intervention to tune or adjust extra parameters.

The developed two approaches for controlling systems suffering from high levels and different types of uncertainties have proven superiority and reliability. The easy implementation of type-2 fuzzy logic system using a combination of type-1 fuzzy logic systems and genetic algorithms is expected to attract researcher to apply it in many different applications. The combination of type-1 fuzzy logic systems and Extended Kalamn filter in the feedback loop of a system has also proven reliability through the blocking of the uncertainty effect by using the estimated system states instead of the corrupted measured states. This scheme provides smooth response which could increase the life time of system’s actuators and reduce the power consumed in controlling such systems.

1.4 Discussion and conclusion

The thesis presented a novel approach of using fuzzy logic in student’s evaluation. The developed method overcomes the drawbacks of previous studies and enables teachers to explain why scores are adjusted because of the transparency feature of fuzzy systems. The method incorporated subjective factors such as difficulty, complexity and importance of questions for a more fair evaluation. The method is applied to an example from the field of students’ evaluation, however, it is not limited to student also applicable to different fields where a fair assessment is crucial.

The thesis presented also a simple approach for implementing type-2 fuzzy logic system using a hybrid intelligent combination of type-1 fuzzy logic and genetic algorithms. The developed paradigm is applied to a complex nonlinear multi-input multi­output system subjected to numerous types of uncertainties. The response of the system controlled using the new paradigm superimposes that of the type-1 fuzzy logic controller. Genetic algorithms improved the system response by adapting system parameters for various conditions. It also provided a way to detect uncertainty in the system and expressing it in terms of the width of the type-2 membership functions.

Combining type-1 fuzzy logic system and extended Kalaman filter in the feedback loop provided a way to block the harmful effects of uncertainties. This approach is applied to the above system and simulation results at different levels of errors in measurements illustrated the system capability to reduce power consumption and provide smooth response. A smooth response could increase the life time of actuators and reduce number of false alarms received from fault detection and isolation systems. The developed scheme is applicable to different complex systems operating under different sever conditions.

1.5 Organization of thesis

The present thesis aimed to study the promising potential of applying fuzzy logic as an artificial intelligence tool into two main complex fields. First, in human decision support system through the mechanization of the process of student’s evaluation which is very crucial for students, teachers, and educational institutes as well. Second, in controlling ill-defined systems suffering from various sources of uncertainties such as measurement and modeling errors.

The development of a fuzzy system for evaluating student’s achievement is introduced in Chapter 2 of the thesis. A novel approach for implementing type-2 fuzzy logic system using a hybrid scheme of four type-1 fuzzy logic systems and a genetic algorithm is introduced in Chapter 3. The developed approach is successfully applied to a complex nonlinear multi-input multi-output system exposed to high levels and different sources of uncertainties. In Chapter 4, a novel scheme is developed by combining 1 fuzzy logic system with an Extended Kalman filter in the feedback loop is

The developed system showed promising results including reducing the power consumption for marinating controlled variables very close to its desired or set values. Finally, main conclusions and future prospects are stated in Chapter 5.

2. A FUZZY SYSTEM FOR EVALUATING STUDENTS’ LEARNING ACHIEVEMNET

2.1 Introduction

Evaluation of students’ learning achievement is the process of determining the performance levels of individual students in relationship to educational objectives. A high quality evaluation system certifies, supports, and improves individual achievement and ensures that all students receive fair treatment in order not to limit students’ present and future opportunities. Thus, the system should regularly be reviewed and improved to ensure that it is proper, fair, and beneficial to all students. It is also desirable that the system has transparency and automation in the evaluation.

Since its introduction in 1965 by Lotfi Zadeh (1965) the fuzzy set theory has been widely used in solving problems in various fields, and recently in education evaluation. Biswas (1995) presented two methods for students’ answerscripts evaluation using fuzzy sets and a matching function: a fuzzy evaluation method and a generalized fuzzy evaluation method. Chen and Lee (1999) presented two methods for applying fuzzy sets to overcome the problem of rewarding two different fuzzy marks the same total score which could arise from Biswas’ method. Echauz and Vachtsevanos (1995) proposed a fuzzy logic system for translating traditional scores into letter-grades. Law (1996) built a fuzzy structure model for education grading system with its algorithm to aggregate different test scores in order to produce a single score for individual student. He also proposed a method to build the membership functions (MFs) of several linguistic values with different weights. Wilson, Karr and Freeman (1998) presented an automatic grading system based on fuzzy rules and genetic algorithms. Ma and Zhou (2000) proposed a fuzzy set approach to assess the outcomes of student-centered learning using the evaluation of their peers and lecturer. Wang and Chen (2008) presented a method for evaluating students’ answerscripts using fuzzy numbers associated with degrees of confidence of the evaluator. From the previous studies, it can be found that fuzzy numbers, fuzzy sets, fuzzy rules, and fuzzy logic systems are used for grading systems.

Weon and Kim (2001) presented an evaluation strategy based on fuzzy MFs. They pointed out that the system for students’ achievement evaluation should consider the three important factors of the questions which the students answer: the difficulty, the importance, and the complexity. Weon and Kim used singleton functions to describe the factors of each question reflecting the effect of the three factors individually, but not collectively. Bai and Chen (2008b) pointed out that the difficulty factor is a very subjective parameter and may cause an argument about fairness in evaluation.

Bai and Chen (2008a) proposed a method to automatically construct the grade MFs of fuzzy rules for evaluating student’s learning achievement. Bai and Chen (2008b) proposed a method for applying fuzzy MFs and fuzzy rules for the same purpose. To solve the subjectivity of the difficulty factor of Weon and Kim’s method (2001), they obtained the difficulty as a function of accuracy of the student’s answer script and time consumed to answer. However, their method still has the subjectivity problem, since the results in scores and ranks are heavily depend on the values of several weights which are determined by the subjective knowledge of domain experts.

Here, we propose an evaluation method considering the importance, difficulty, and complexity of questions based on Mamdani’s fuzzy inference (Mamdani, 1974) and center of gravity (COG) defuzzification which is an alternative to Bai and Chen’s method (2008b). The transparency and objective nature of the fuzzy system makes it easy to understand and explain the result of evaluation, and thus to persuade the students.

2.2 A review of evaluation methods using membership functions and fuzzy rules

In this paper, we consider the same situation as in Bai and Chen’s (2008b). Assume that there are n students to answer m questions. Accuracy rates of students’ answerscripts (student’s scores in each question divided by the maximum score assigned to this question) are the basis for evaluation. We get an accuracy rate matrix of dimension mxn

Abbildung in dieser Leseprobe nicht enthalten

where [Abbildung in dieser Leseprobe nicht enthalten]denotes the accuracy rate of student j on question i. Time rates of students (the time consumed by a student to solve a question divided by the maximum time allowed to solve this question) is another basis to be considered in evaluation. We get a time rate matrix of dimension m x n,

Abbildung in dieser Leseprobe nicht enthalten

where [Abbildung in dieser Leseprobe nicht enthalten] denotes the time rate of student j on question i. We are given a grade vector

G = [gi], m x 1,

where [Abbildung in dieser Leseprobe nicht enthalten] denotes the assigned maximum score of question i satisfying

Abbildung in dieser Leseprobe nicht enthalten

Based on the accuracy rate matrix A and the grade vector G, we obtain the tota[1] score vector of dimension nxl,

Abbildung in dieser Leseprobe nicht enthalten

where [Abbildung in dieser Leseprobe nicht enthalten] is the total score of student j which is obtained by

Abbildung in dieser Leseprobe nicht enthalten

The classical rank of students is then obtained by sorting values of S in a descending order.

Example. 1

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):

Abbildung in dieser Leseprobe nicht enthalten

Importance of the questions is an important factor to be considered. We have l levels of importance to describe the degree of importance of each question in the fuzzy domain. The domain expert determine the importance matrix of dimension mxl[Abbildung in dieser Leseprobe nicht enthalten]

where [Abbildung in dieser Leseprobe nicht enthalten]denotes the degree of membership of question i belonging to the

importance level k. In this paper, five levels (fuzzy sets) of importance (l = 5) are used; k = 1 for linguistic term “low”, k = 2 for “more or less low”, k = 3 “medium”, k = 4 “more or less high”, and k = 5 for “high”. Their MFs are shown in Fig. 2-1. We note that the same five fuzzy sets are applied to the accuracy, the time rate, the difficulty, the complexity, and the adjustment of questions. The values of pik’s are obtained by the fuzzification once crisp values are given for the importance of questions by domain expert.

Complexity of the questions which indicates the ability of students to give correct answers is also an important factor to be considered. The domain expert determine the fuzzy complexity matrix of dimension mxl,

Abbildung in dieser Leseprobe nicht enthalten

[Abbildung in dieser Leseprobe nicht enthalten]denotes the degree of membership of question i belonging to the complexity level k.

Abbildung in dieser Leseprobe nicht enthalten

Example.2

For the above example we get the following by domain expert:

Abbildung in dieser Leseprobe nicht enthalten

Total score is then obtained as

ST =[67.60 54.05 38.40 49.70 49.70 48.80 46.10 52.30 85.95 49.70], and thus the classical rank of students is then becomes:

Abbildung in dieser Leseprobe nicht enthalten

Bai and Chen’s method (2008b) uses 3 steps to evaluate students’ answerscripts. In the first step, using the average accuracy rate vector of dimension mxl,

Abbildung in dieser Leseprobe nicht enthalten

where ait denotes the average accuracy rate of question i which is obtained by[Abbildung in dieser Leseprobe nicht enthalten](2.2)

and the average time rate vector of the same dimension,

Abbildung in dieser Leseprobe nicht enthalten

where [Abbildung in dieser Leseprobe nicht enthalten]denotes the average time rate of question i which is obtained by

Abbildung in dieser Leseprobe nicht enthalten

we obtain the fuzzy accuracy rate matrix of dimension mxl,

Abbildung in dieser Leseprobe nicht enthalten

where fak e \0, l] denotes the membership value of the average accuracy rate of question i belonging to level k, and the fuzzy time rate matrix of dimension mxl,

Abbildung in dieser Leseprobe nicht enthalten

where[Abbildung in dieser Leseprobe nicht enthalten] denotes the membership value of the average time rate of question i belonging to level k, respectively.

Example.3

In the above example, we get AT [Abbildung in dieser Leseprobe nicht enthalten]

[Abbildung in dieser Leseprobe nicht enthalten]

Based on the fuzzy MFs in Fig. 2-1, we obtain the fuzzy accuracy rate matrix and the fuzzy time rate matrix:

Abbildung in dieser Leseprobe nicht enthalten

In the second step, based on the fuzzy accuracy rate matrix, [Abbildung in dieser Leseprobe nicht enthalten] fuzzy time rate matrix, [Abbildung in dieser Leseprobe nicht enthalten] and the fuzzy rules,[Abbildung in dieser Leseprobe nicht enthalten] given in the form of IF-THEN rules, we obtain the fuzzy difficulty matrix of dimension mxl,

Abbildung in dieser Leseprobe nicht enthalten

where [Abbildung in dieser Leseprobe nicht enthalten] denotes the membership of difficulty of question i belonging to level k. When the level of accuracy, Ia, and the level of time,[Abbildung in dieser Leseprobe nicht enthalten] are given, the level of difficulty, [Abbildung in dieser Leseprobe nicht enthalten] is determined by the relation given by fuzzy rules,

Abbildung in dieser Leseprobe nicht enthalten

Denoting the weights of the accuracy rate and time rate, which are determined by domain expert, by [Abbildung in dieser Leseprobe nicht enthalten] and [Abbildung in dieser Leseprobe nicht enthalten] respectively. The value of dik is obtained by

Abbildung in dieser Leseprobe nicht enthalten

Next, based on the fuzzy difficulty matrix, D, fuzzy complexity matrix, C, their weights, Wd and [Abbildung in dieser Leseprobe nicht enthalten] respectively, and the fuzzy rules, [Abbildung in dieser Leseprobe nicht enthalten] we obtain the cost matrix of dimension mxl,in the same manner [Abbildung in dieser Leseprobe nicht enthalten]

where [Abbildung in dieser Leseprobe nicht enthalten] denotes the degree of membership of the cost of question i belonging to level k, which is a measure of cost for students to answer question i.

Based on the fuzzy cost matrix, AC, fuzzy importance matrix, P, their weighs,[Abbildung in dieser Leseprobe nicht enthalten] and [Abbildung in dieser Leseprobe nicht enthalten], respectively, and the fuzzy rules,[Abbildung in dieser Leseprobe nicht enthalten] we obtain the adjustment matrix of dimension mxl [Abbildung in dieser Leseprobe nicht enthalten]

where [Abbildung in dieser Leseprobe nicht enthalten] denotes the degree of membership of adjustment required by question i belonging to level k.

Then we use the following formula to obtain the adjustment vector,

Abbildung in dieser Leseprobe nicht enthalten

where[Abbildung in dieser Leseprobe nicht enthalten] denotes the final adjustment value required by question i obtained by

Abbildung in dieser Leseprobe nicht enthalten

where [Abbildung in dieser Leseprobe nicht enthalten] are the centers of the fuzzy MFs shown in Fig. 2-1.

Example.4

Assume that we are given the rule base for [Abbildung in dieser Leseprobe nicht enthalten] in Table 2-1 [Abbildung in dieser Leseprobe nicht enthalten] and [Abbildung in dieser Leseprobe nicht enthalten] in Table 2-1 (b), respectively. The difficulty level of 1 (//> = 1) for question 1, for example, is obtained from [Abbildung in dieser Leseprobe nicht enthalten] and[Abbildung in dieser Leseprobe nicht enthalten] By setting [Abbildung in dieser Leseprobe nicht enthalten]and [Abbildung in dieser Leseprobe nicht enthalten]

[...]