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TITLE Variations of the Rectangular Fuzzy Assessment Model and Applications to Human Activities 

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ABSTRACT The Rectangular Fuzzy Assessment Model (RFAM) is based on the popular in fuzzy mathematics COG defuzzification technique. In the paper at hands the RFAM and its variations GRFAF, TFAM and TpFAM, developed for treating better the ambiguous assessment situations being at the boundaries between two successive assessment grades, are presented and their outcomes are compared to each other through the corresponding outcomes of a traditional assessment method of the bivalued logic, the Grade Point Average (GPA) index. Examples are also presented illustrating our results. 

KEYWORDS Grade Point Average (GPA) index, Centre of Gravity (COG) defuzzification technique, Rectangular Fuzzy Assessment Model (RFAM), Generalized RFAM (GRFAM), Triangular FAM (TFAM), Trapezoidal FAM (TpFAM) 

1. Introduction In 1999 Voskoglou [1] developed a fuzzy model for the description of the process of learning a subject matter in classroom with the help of the possibilities of student profiles and later he assessed the student learning skills by calculating the corresponding system’s total possibilistic uncertainty [2]. Meanwhile, Subbotin et al. [3], based on Voskoglou’s model [1], adapted properly the frequently used in fuzzy mathematicsCenter of Gravity (COG) defuzzification technique and used it as an alternative assessment method of student learning skills, later named as the Rectangular Fuzzy Assessment Model (RFAM). Since then, Voskoglou and Subbotin, working either jointly or independently, applied the RFAM and three variations of it, the GRFAM, the TFAM and the TpFAM, for assessing several human or machine (Decision – Making and CaseBased Reasoning with the help of computers, etc.) activities, e.g. see [4  15], , etc. In the present work the outcomes of the RFAM and of its variations are compared to each other through the corresponding outcomes of a traditional assessment method of the bivalued logic, the Grade Point Average (GPA) index. The rest of the paper is formulated as follows: In Section II we present the GPA method. In Section II we sketch the RFAM, while in Section IV we briefly describe the three variations of the TRFAM, developed for treating better the ambiguous assessment situations being at the boundaries between two successive assessment grades. In Section V the outcomes of the RFAM and its variations are compared to each other through the outcomes of the GPA index and examples are presented illustrating our results. The last Section VI is devoted to our conclusion and to some hints for future research on the subject.
2. The GPA Index The calculation of the mean value of the individual scores of a group of objects (e.g. students, players, machines, etc.) characterizing their performance with respect to an action is the traditional method for assessing the group’s mean performance.
On the other hand, the GPA index is a very popular in the USA and other Western countries assessment method calculating the group’s quality performance. The GPA index, a weighted average in which greater coefficients (weights) are assigned to the higher scores, is calculated by the formula GPA = (1) , where n is the total number of the group’s members andn_{A}, n_{B}, n_{C}, n_{D} and n_{F} denote the numbers of the group’s members that demonstrated excellent (A), very good (B), good (C), fair (D) and unsatisfactory (F) performance respectively [16]. Formula (1) can be also written in the form GPA = y_{2} + 2y_{3} +3y_{4} + 4y_{5 } (2), where y_{1 }= , y_{2 }= , y_{3 }= , y_{4 }= and y_{5 }= denote the frequencies of the group’s members which demonstrated unsatisfactory, fair, good, very good and excellent performance respectively. In case of the worst performance ( n_{F} = n) formula (1) gives that GPA = 0, while in case of the ideal performance ( n_{A} = n) it gives GPA = 4. Therefore we have in general that 0 GPA 4 . Consequently, values of GPA greater than 2 could be considered as indicating a more than satisfactory performance.
3 . The Rectangular Fuzzy Assessment Model (RFAM) A commonly used in fuzzy mathematics technique is the defuzzification of a given fuzzy set (FS) by calculating the coordinates of the COG of the level’s section contained between the graph of the FS’s membership function and the X  axis [17]. In [15] we have described in detail the use of the COG technique as an assessment method and we have applied it for evaluating the student understanding of the polar coordinates in the plane. The graph of the membership function in this case is that presented in Figure 1. It is recalled here that the design of this graph was achieved by replacing the elements of the universal set U = {A, B, C, D, F} of the linguistic characterizations introduced in Section II by the real intervals [4, 5], [3, 4), [2, 3), [1, 2) and [0, 1) respectively, corresponding to a scale of numerical scores assigned to each linguistic characterization.. Figure 1: The graph of the COG technique In Figure 1 the area of the level’s section contained between the graph and the X  axis is equal to the sum of the areas of the rectangles S _{i}, i=1, 2, 3, 4, 5. Due to the shape of this graph we have named the above method as theRectangular Fuzzy Assessment Model (RFAM). Note that the membership function y = m(x) can be defined, according to the user’s personal goals, in any compatible to the common sense way. However, in order to obtain assessment results compatible to the corresponding results of the GPA index, we have defined y = m(x) in terms of the frequencies introduced in Section II. Then, using the well known from Mechanics formulas for calculating the coordinates of the COG it is straightforward to check [15] that the coordinates of the COG in this case are given by the formulas x_{c} = (y_{1}+3y_{2}+5y_{3} +7y_{4}+9y_{5}), y_{c} = (y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}+ y_{5}^{2}) (3) , with x_{1 }= F, x_{2 }= D,x_{3 }= C, x_{4 }= B, x_{5}_{ }=A and y_{i }= m(x_{i}) = , i = 1, 2, 3, 4, 5, where obviously = 1. Further, using elementary algebraic inequalities and performing elementary geometric observations [15] one obtains the following assessment criterion:
As it becomes evident from the above criterion, a group’s performance depends mainly on the value of the xcoordinate of the COG of the corresponding level’s area, which is calculated by the first of formulas (3). In this formula, greater coefficients (weights) are assigned to the higher grades. Therefore, the COG method focuses, similarly to the GPA index, on the group’s quality performance. In case of the ideal performance ( y_{5} =1 and y_{i} = 0 for i 5) the first of formulas (3) gives that x_{c} = . Therefore, values of x_{c} greater than = 2.25 could be considered as demonstrating a more than satisfactory performance.
4. The Variations GRFAM, TFAM and TpFAM of the RFAM A group’s performance is frequently represented by numerical scores in a climax from 0100. These scores can be assigned to the linguistic labels of U as follows: A (85100), B(7584), C (6074), D(5059) and F (049) [1] . Nevertheless, ambiguous cases appear frequently in practice, being at the boundaries between two successive assessment grades; e.g. something like 8485%, being at the boundaries between A and B. In an effort to treat better such kind of cases, Subbotin [8] “moved” the rectangles of Figure 1 to the left, so that to share common parts (see Figure 2). In this way, the ambiguous cases, being at the common rectangle parts, belong to both of the successive grades, which means that these parts must be considered twice in the corresponding calculations. The graph of the resulting fuzzy set is now the bold line of Figure 2. However, the method mentioned in Section II for calculating the coordinates of the COG of the area contained between the graph and the Xaxis is not the proper one here, because in this way the common rectangle parts are calculated only once. The right method for calculating the coordinates of the COG in this case was fully developed by Subbotin & Voskoglou [9] and the resulting framework was called the Generalized Rectangular Fuzzy Assessment Model (GRFAM). The development of GRFAM involves the following steps: 1. Let y_{1}, y_{2}_{ },y_{3, }y_{4 }, y_{3} _{ }be the frequencies of a group’s members who obtained the grades F, D, C, B, A respectively. Then = 1 (100%). 2. We take the heights of the rectangles in Figure 2 to have lengths equal to the values of corresponding frequencies. Also, without loss of generality we allow the sides of the adjacent rectangles lying on the OX axis to share common parts with length equal to the 30% of their lengths, i.e. 0.3 units. [2] Figure 2: Graphical representation of the GRFAM 3. We calculate the coordinates ( ) of the COG, say F_{i }, of each rectangle, i = 1, 2, 3, 4, 5 as follows: Since the COG of a rectangle is the point of the intersection of its diagonals, we have that Also, since the xcoordinate of each COG F_{i }is equal to the x coordinate of the middle of the side of the corresponding rectangle lying on the OX axis, from Figure 2 it is easy to observe that _{ } = 0.7i – 0.2. 4. We calculate the coordinates (X_{c}, Y_{c}) of the COG F of the whole area considered in Figure 2 as the resultant of the system of the GOCs F_{i } of the five rectangles from the following well known [20] formulas X_{c } = , Y_{c }= (4) . In the above formulas Si, i= 1, 2, 3, 4, 5 denote the areas of the corresponding rectangles, which are equal to y_{i }. Therefore S = = = 1 and formulas (4) give that X_{c } = , Y_{c }= or X_{c } = , Y_{c }= (5) . 5. We determine the area in which the COG F lies as follows: For i, j = 1, 2, 3, 4, 5, we have that 0 (y_{i } y_{j})^{2 }= y_{i}^{2 }+ y_{j}^{2 } 2y_{i}y _{j} ,_{ }therefore y_{i}^{2 }+ y_{j}^{2} 2y_{i}y_{j} , with the equality holding if, and only if, y_{i }= y_{j}. Therefore 1=( )^{2}= + 2 + 2 = 5 or ^{ } (6) , with the equality holding if, and only if, y_{1 }= y_{2 }= y_{3 }= y_{4 }= y _{5 }= . In case of the equality the first of formulas (5) gives that X_{c }= 0.7( + + + + ) – 2 = 1.9. Further, combining the inequality (6) with the second of formulas (5), one finds that Y_{c} . Therefore the unique minimum for Y_{c} corresponds to the COG F_{m }(1.9, 0.1). The ideal case is when y_{1 }= y_{2 }= y_{3 }= y_{4 }= 0 and y_{5}=1. Then formulas (5) give that X_{c }= 3.3 and Y_{c }= . Therefore the COG in this case is the point F_{i } (3.3, 0.5). On the other hand, the worst case is when y_{1 }= 1 and y_{2 }= y_{3} = y_{4 }= y_{5 }= 0 . Then from formulas (5) one finds that the COG is the point F_{w }(0.5, 0.5). Therefore, the area in which the COG F_{ }lies is the area of _{ }the triangle F_{w }F_{m }F_{i } (Figure 3). Figure 3: The triangle where the COG lies 6. From elementary geometric observations on Figure 3 one obtains the following assessment criterion:
From the first of formulas (5) it becomes evident that the GRFAM measures the quality group’s performance. Also, since the ideal performance corresponds to the value X_{c }= 3.3, values of X_{c} greater than = 1.65 could be considered as indicating a more than satisfactory performance. At this point one could raise the following question: Does the shape of the membership function’s graph of the assessment model affect the assessment’s conclusions? For example, what will happen if the rectangles of the GRFAM will be replaced by isosceles triangles? The effort to answer this question led to the development of the Triangular Fuzzy Assessment Model (TFAM), created by Subbotin & Bilotskii [4] and fully developed by Subbotin & Voskoglou [6]. Figure 4: Graphical Representation of the TFAM The graphical representation of the TFAM is shown in Figure 4 and the steps followed for its development are the same with the corresponding steps of GRFAM presented above. The only difference is that one works with isosceles triangles instead of rectangles. The final formulas calculating the coordinates of the COG of TFAM are: X_{c } = , Y_{c}= (7) and the corresponding assessment criterion is the same with the criterion obtained for GRFAM. An alternative to the TFAM approach is to consider isosceles trapezoids instead of triangles [6, 7]. In this case we called the resulting framework Trapezoidal Fuzzy Assessment Model (TpFAM). The corresponding scheme is that shown in Figure 5. In this case the y  coordinate of the COG _{ }F_{i}, i=1, 2, 3, 4, 5, of each trapezoid is calculated in terms of the fact that the COG of a trapezoid lies on the line segment joining the midpoints of its parallel sides a and b at a distance d from the longer side b given by d = , where h is its height [18]. Also, since the x coordinate of the COG of each trapezoid is equal to the xcoordinate of the midpoint of its base, it is easy to observe from Figure 5 that x = 0.7i  0.2.
Figure 5: The TpFAM’s scheme One finally obtains from formulas (4) that X_{c } = , Y_{c }= (8) and the assessment criterion remains the same again.
5. Comparison of the Assessment Models One can write formulas (5), (6) and (7) in the unified form: X_{c } = , Y_{c }= (9) , where a = for the GRFAM, a = for the TFAM and a = for the TpFAM. Combining formulas (9) with the common assessment criterion stated in Section 4 one obtains the following result: Theorem 1: The three variations of the COG technique, i.e. the GRFAM, the TFAM and the TpFAM are equivalent to each other assessment models. Further, the first of formulas (9) can be written as X_{c } = 0.7(y_{1} + 2 y_{2} + 3y_{3} + 4y_{4} + 5y_{5} ) – 0.2 = 0.7 [( y_{2} + 2y_{3} + 3y_{4} + 4y_{5} ) + ] – 0.2. Therefore, by formula (2) one finally gets that X_{c } = 0.7(GPA + 1) – 0.2 = 0.7GPA + 0.5 (10). In the same way, the first of formulas (3) for RFAM can be written as x_{c} = (y_{1 }+ 3y_{2 }+ 5y_{3 }+7y_{4 }+ 9 y_{5}) = (2GPA + 1), or x_{c} = GPA + 0.5 (11). We are ready now to prove: Theorem 2: If the values of the GPA index are different for two groups, then the GPA index, the RFAM and its variations (GRFAM, TFAM and TpFAM) provide the same assessment outcomes on comparing the performance of these groups. Proof: Let G and G΄ be the values of the GPA index for the two groups and let x_{c}, x_{c}΄ be the corresponding values of the xcoordinate of the COG for the RFAM. Assume without loss of generality that G>G΄, i.e. that the first group performs better according to the GPA index. Then, equation (11) gives that x_{c} > x_{c}΄, which, according to the first case of the assessment criterion of Section III, shows that the first group performs also better according to the RFAM. In the same way, from equation (10) and the first case of the assessment criterion of Section IV, one finds that the first group performs better too according to the equivalent assessment models GRFAM, TFAM and TpFAM. The following result shows that Theorem 2 remains true even in case of the same GPA index. Theorem 3: If the values of the GPA index are the same for two groups, then the RFAM and its variations GRFAM, TFAM and TpFAM provide the same assessment outcomes on comparing the performance of these groups . Proof: Since the two groups have the same value of the GPA index, equations (10) and (11) show that the values of X_{c} and x_{c }are also the same. Therefore, one of the last two cases of the assessment criteria of Sections III and IV could happen. The possible values of x in these criteria lie in the intervals [0, ] and [0, 3.3] respectively, while the critical points correspond to the values x_{c} = 2.5 and X_{c} = 1.9 respectively. Obviously, if both values of x are in [0, 1.9), or in [2.5, ], then the two criteria provide the same assessment outcomes on comparing the performance of the two groups. Assume therefore that 1.9 < X_{c} and x_{c} < 2.5. Then, due to equation (10), 1.9 < X_{c } _{ } 1.9< 0.7GPA + 0.5 _{ } 1.4 <0.7GPA _{ } GPA > 2. Also, due to equation (11), x_{c} < 2.5 _{ } GPA + 0.5 < 2.5 _{ } GPA < 2. Therefore, the inequalities 1.9 < X_{c} and x_{c} < 2.5 cannot hold simultaneously and the result follows._{} Combining Theorems 2 and 3 one obtains the following corollary: Corollary 4: The RFAM and its variations GRFAM, TFAM and TpFAM provide always the same assessment results on comparing the performance of two groups. The following example ([9], Section 4, paragraph vii) shows that in case of the same GPA values the application of the GPA indexcould not lead to logically based conclusions . Therefore, in such situations, our criteria of Sections 3 and 4 become useful due to their logical nature. Example 5: The student grades of two Classes with 60 students in each Class are presented in Table 1 Table 1: Student Grades
The GPA index for the two classes is equal to , which means that the two Classes demonstrate the same performance in terms of the GPA index. Therefore equation (10) gives that
X_{c} = 0.7*3.67 + 0.5 , while equation (11) gives that x_{c} for both Classes. But = for the first and = = for the second Class. Therefore, according to the assessment criteria of Sections 3 and 4 the first Class demonstrates a better performance in terms of the RFAM and its variations. Now which one of the above two conclusions is closer to the reality? For answering this question, let us consider the quality of knowledge, i.e. the ratio of the students received B or better to the total number of students, which is equal to for the first and 1 for the second Class. Therefore, from the common point of view, the situation in Class II is better. Nevertheless, many educators could prefer the situation in Class I having a greater number of excellent students. Conclusively, in no case it is logical to accept that the two Classes demonstrated the same performance, as the calculation of the GPA index suggests. The next example shows that, although by Corollary 4 the RFAM, GRFAM, TFAM and TpFAM provide always the same assessment results on comparing the performance of two groups, theyare not equivalent assessment models .
Example 6: Table 2 depicts the results of the final exams of the first term mathematical courses of two different Departments, say D_{1} and D_{2}, of the School of Technological Applications (future engineers) of the Graduate T. E. I. of Western Greece. Note that the contents of the two courses and the instructor were the same for the two Departments.
Table 2 : Results of the two Departments
The GPA index is equal to for D_{1} and for D_{2}. Therefore, the two Departments demonstrated a less than satisfactory performance (since GPA < 2), with the performance of D_{2} being better. Further, equation (10) gives that X_{c} 1.53 for D_{1} and X_{c} 1.66 for D_{2}. Therefore, according to the first case of the assessment criterion of Section IV, D_{2} demonstrated, with respect to GRFAM, TFAM and TpFAM, a better performance than D _{1}. Moreover, since 1.53_{ }< = 1.65 < 1.66, D_{1} demonstrated a less than satisfactory performance, while D_{2} demonstrated a more than satisfactory performance. In the same way equation (11) gives that x_{c} 1.97 for D_{1} and x_{c} 2.16 for D_{2}. Therefore, according to the first case of the assessment criterion of Section III, D_{2} demonstrated, with respect to RFAM, a better performance than D_{1}. But in this case, since for both Departments X_{c}_{ }< = 2.25, both Departments demonstrated a less than satisfactory performance.
Remark: Note that, if GPA > 2, then
X_{c} = 0.7GPA + 0.5 > 0.7 * 2 + 0.5 = 1.9 > 1.65 and x_{c} = GPA + 0.5 > 0.2 + 0.5 = 2.5> 2.25. Therefore, the corresponding group’s performance is more than satisfactory with respect to GRFAM, TFAM, TpFAM and RFAM. However, if GPA < 2 , then X_{c} < 1.9 and x_{c} < 2.5, which do not guarantee that X_{c} < 1.65 and x_{c} < 2.25. Therefore the assessment characterizations of RFAM and the equivalent to each other GRFAM, TFAM, TpFAM can be different only when GPA < 2.
6. Conclusion From the discussion performed in this paper it becomes evident that the RFAM and the equivalent to each other GRFAM, TFAM and TpFAM, although they provide always the same assessment outcomes on comparing the performance of two groups, they are not equivalent assessment models. Further, the assessment outcomes of the above models are also the same with those of the GPA index, unless if the value of the GPA index is the same for both groups. In the last case the GPA index could not lead to logically based conclusions. Therefore, in this case either the use of RFAM or of its variations must be preferred. Other fuzzy assessment methods have been also used in earlier author’s works like the measurement of a system’s uncertainty [19] and the application of the fuzzy numbers [20]. These methods, in contrast to the previous ones which focus on the corresponding group’s quality performance, they measure its mean performance. The plans for our future research include the effort to compare all these methods to each other in order to obtain the analogous conclusions.
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Cite this paper Michael Gr. Voskoglou. (2017) Variations of the Rectangular Fuzzy Assessment Model and Applications to Human Activities. International Journal of Education and Learning Systems, 2, 115124 

