Abstract
In this chapter two new fuzzy assessment models are developed, the trapezoidal (TRFAM) and the Triangular (TFAM), which are proved to be equivalent to each other. These models are new original variations of the Centre of Gravity (COG) defuzzification technique, which has been properly adapted and used in earlier papers as an assessment method. The central idea of TRFAM is the replacement of the rectangles appearing in the graph of the membership function of the COG technique by isosceles trapezoids sharing common parts. In this way one treats better the ambiguous cases being at the boundaries between two successive linguistic assessment characterizations (grades). The corresponding idea of TFAM is the replacement of the rectangles by isosceles triangles. Two applications are also presented (students' and Bridge players' assessment) illustrating our results. In these applications the TRFAM and TFAM are validated through their comparison with other, already established, assessment methods.
TopIntroduction
The assessment of a system’s effectiveness (i.e. of the degree of attainment of its targets) with respect to an action performed within the system (e.g. problem-solving, decision making, learning performance, etc) is a very important task that enables the correction of the system’s weaknesses resulting to the improvement of its general performance.
The several methods in use for assessing a system’s performance focus on different targets: Some of them measure the mean system’s performance (e.g. the calculation of the mean of the scores obtained by a group of individuals), while others focus to its quality performance by assigning greater coefficients (weights) to the higher scores (e.g. the widely used in the USA Grade Point Average Index). Therefore, one who wants to obtain a comprehensive view of a system’s performance must make use of more than one assessment methods for this purpose.
The assessment methods that are commonly used in practice are based on the principles of the classical, bivalent logic (yes-no). However, there are cases where a crisp characterization is not probably the proper one for the assessment. For example, a teacher is frequently not sure about a particular numerical grade characterizing a student’s performance. Fuzzy logic, due to its nature of characterizing a case with multiple values, offers wider and richer resources covering such kind of cases.
In this chapter we develop two new original fuzzy assessment models, which are equivalent to each other: The Triangular Fuzzy Assessment Model (TFAM) and the Trapezoidal Fuzzy Assessment Model (TRFAM). These models are variations of the very popular in fuzzy mathematics Centre of Gravity (COG) defuzzification technique, which we have properly adapted in earlier papers and used it as a general assessment method of a system’s performance. TFAM and TRFAM are treating better than COG the ambiguous cases being at the boundaries between two successive assessment grades. Two real life applications (students’ and Bridge players’ assessment) are also presented illustrating our results in practice.
For general facts on fuzzy sets we refer to the book of Klir and Folger (1988)
Background
There used to be a tradition in science and engineering of turning to probability theory when one is faced with a problem in which uncertainty plays a significant role. This transition was justified when there were no alternative tools for dealing with the uncertainty. Today this is no longer the case. Fuzzy logic, which is based on fuzzy sets theory introduced by Zadeh in 1965, provides a rich and meaningful addition to standard logic. The applications which may be generated from or adapted to fuzzy logic are wide-ranging and provide the opportunity for modelling under conditions which are inherently imprecisely defined, despite the concerns of classical logicians.
Fuzzy Logic, due to its property of characterizing the ambiguous cases of a phenomenon by multiple values, has been widely used recently to solve problems in the evaluation tasks (Subbotin et al. 2004, Liu and Lee 2009, Voskoglou 2011. 2012, Liu et al. 2013, Voskoglou and Subbotin 2013, etc). In earlier works we have utilized the corresponding system’s total uncertainty as a measurement of its performance (Voskoglou 2011, 2012, etc). In fact, as it is well known from the classical Information Theory (Shannon 1948), the reduction of a system’s uncertainty as a result of an action performed within the system is connected to the information obtained by this action. Consequently, the lower is the system’s uncertainty after the action, the greater is the amount of information obtained by the action. In other words, the system’s effectiveness with respect to this action can be measured by the amount of its total uncertainty. This assessment method is connected to the system’s mean performance. On the contrary, the COG defuzzification technique has been adapted and used in earlier papers (Subbotin et al. 2004, Voskoglou 2012, Voskoglou and Subbotin 2013, etc) as a general assessment method of a system’s quality performance. Below we shall sketch the above two fuzzy assessment methods, because we are going to use them in our applications, together with the new TRFAM and TFAM models.
Key Terms in this Chapter
Grade Point Average (GPA): A weighted average assigning greater coefficients (weights) to the higher scores and therefore focusing on the system’s quality performance.
Uncertainty and Information: An action performed within a system obtains an amount of information that reduces the existing uncertainty. Therefore, the measurement of the system’s uncertainty following the action can be used as an assessment method of the system’s effectiveness with respect to this action. The uncertainty measures the system’s mean performance.
Fuzzy Logic: A logic that, in contrast to the classical bivalent logic (yes – no), characterizes a case with multiple values. It is based on the concept of fuzzy set.
Trapezoidal Fuzzy Assessment Model (TRFAM): A variation of the COG assessment method in which the rectangles appearing in the COG’s graph are replaced by isosceles trapezoids sharing common parts. In this way one treats better the ambiguous assessment cases being at the boundaries between two successive assessment grades.
Triangular Fuzzy Assessment Model (TFAM): An assessment model equivalent to the TRFAM, in which the trapezoids are replaced by isosceles triangles.
Defuzzification: The process of representing a system’s fuzzy data by a crisp number.
Centre of Gravity (COG) Defuzzification Technique: A defuzzification technique in which the fuzzy data are represented by the coordinates of the centre of gravity of the level’s section contained between the graph of the membership function involved and the OX axis. Under a proper manipulation it can be used as an assessment method of the system’s quality performance.
Fuzzy Set: A generalization of the concept of a crisp set introduced by Zadeh in 1965. It is characterized by a membership function defined on the universal set U and taking values in the interval [0, 1], thus assigning a membership degree to each element of U with respect to the fuzzy set. It covers the real situations where certain definitions have no clear boundaries (e.g. the high mountains of a country).