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

Transcription

1 Varatons of the Rectangular Fuzzy Assessment Model and Applcatons to Human Actvtes MICHAEl GR. VOSKOGLOU Department of Mathematcal Scence3s Graduate Technologcal Educatonal Insttute of Western Greece Meg. Alexandrou Patras GREECE mvosk@hol.gr ; Abstract: - The Rectangular Fuzzy Assessment Model (RFAM) s based on the popular n fuzzy mathematcs COG defuzzfcaton technque. In the paper at hands the RFAM and ts varatons GRFAF, TFAM and TpFAM, developed for treatng better the ambguous assessment stuatons beng at the boundares between two successve assessment grades, are presented and ther outcomes are compared to each other through the correspondng outcomes of a tradtonal assessment method of the b-valued logc, the Grade Pont Average (GPA) ndex. Examples are also presented llustratng our results. Keywords: - Grade Pont Average (GPA) ndex, Centre of Gravty (COG) defuzzfcaton technque, Rectangular Fuzzy Assessment Model (RFAM), Generalzed RFAM (GRFAM), Trangular FAM (TFAM), Trapezodal FAM (TpFAM).. Introducton In 999 Voskoglou [] developed a fuzzy model for the descrpton of the process of learnng a subject matter n classroom wth the help of the possbltes of student profles and later he assessed the student learnng sklls by calculatng the correspondng system s total possblstc uncertanty []. Meanwhle, Subbotn et al. [3], based on Voskoglou s model [], adapted properly the frequently used n fuzzy mathematcs Center of Gravty (COG) defuzzfcaton technque and used t as an alternatve assessment method of student learnng sklls, later named as the Rectangular Fuzzy Assessment Model (RFAM). Snce then, Voskoglou and Subbotn, workng ether jontly or ndependently, appled the RFAM and three varatons of t, the GRFAM, the TFAM and the TpFAM, for assessng several human or machne (Decson Makng and Case-Based Reasonng wth the help of computers, etc.) actvtes, e.g. see [4 - ],, etc. In the present work the outcomes of the RFAM and of ts varatons are compared to each other through the correspondng outcomes of a tradtonal assessment method of the b-valued logc, the Grade Pont Average (GPA) ndex. The rest of the paper s formulated as follows: In Secton II we present the GPA method. In Secton II we sketch the RFAM, whle n Secton IV we brefly descrbe the three varatons of the TRFAM, developed for treatng better the ambguous assessment stuatons beng at the boundares between two successve assessment grades. In Secton V the outcomes of the RFAM and ts varatons are compared to each other through the outcomes of the GPA ndex and examples are presented llustratng our results. The last Secton VI s devoted to our concluson and to some hnts for future research on the subject.. The GPA Index The calculaton of the mean value of the ndvdual scores of a group of objects (e.g. students, players, machnes, etc.) characterzng ther performance wth respect to an acton s the tradtonal method for assessng the group s mean performance. On the other hand, the GPA ndex s a very popular n the USA and other Western countres assessment method calculatng the group s qualty performance. The GPA ndex, a weghted average n whch greater coeffcents (weghts) are assgned to the hgher scores, s calculated by the formula ISSN: Volume, 07

2 GPA = 0 nf + n D + nc + 3nB + 4n A (), n where n s the total number of the group s members and n 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), far (D) and unsatsfactory (F) performance respectvely [6]. Formula () can be also wrtten n the form GPA = y + y 3 +3y 4 + 4y (), where y = n F, y = n D, y 3 = n C, y 4 = n B and y = n n n n n A denote the frequences of the group s members n whch demonstrated unsatsfactory, far, good, very good and excellent performance respectvely. In case of the worst performance (n F = n) formula () gves that GPA = 0, whle n case of the deal performance (n A = n) t gves GPA = 4. Therefore we have n general that 0 GPA 4. Consequently, values of GPA greater than could be consdered as ndcatng a more than satsfactory performance. 3. The Rectangular Fuzzy Assessment Model (RFAM) A commonly used n fuzzy mathematcs technque s the defuzzfcaton of a gven fuzzy set (FS) by calculatng the coordnates of the COG of the level s secton contaned between the graph of the FS s membershp functon and the X - axs [7]. In [] we have descrbed n detal the use of the COG technque as an assessment method and we have appled t for evaluatng the student understandng of the polar coordnates n the plane. The graph of the membershp functon n ths case s that presented n Fgure. It s recalled here that the desgn of ths graph was acheved by replacng the elements of the unversal set U = {A, B, C, D, F} of the lngustc characterzatons ntroduced n Secton II by the real ntervals [4, ], [3, 4), [, 3), [, ) and [0, ) respectvely, correspondng to a scale of numercal scores assgned to each lngustc characterzaton.. y m(b) m(d) m(f) m(c) m(a) O s s s 3 s 4 s F D C B A 3 4 Fgure : The graph of the COG technque In Fgure the area of the level s secton contaned between the graph and the X - axs s equal to the sum of the areas of the rectangles S,,, 3, 4,. Due to the shape of ths graph we have named the above method as the Rectangular Fuzzy Assessment Model (RFAM). Note that the membershp functon y = m(x) can be defned, accordng to the user s personal goals, n any compatble to the common sense way. However, n order to obtan assessment results compatble to the correspondng results of the GPA ndex, we have defned y = m(x) n terms of the frequences ntroduced n Secton II. Then, usng the well known from Mechancs formulas for calculatng the coordnates of the COG t s straghtforward to check [] that the coordnates of the COG n ths case are gven by the formulas x c = (y +3y +y 3 +7y 4 +9y ), y c = (y +y +y 3 +y 4 +y ) (3), wth x = F, x = D, x 3 = C, x 4 = B, x =A and n x y = m(x ) = n, =,, 3, 4,, where obvously m ( ) =. x Further, usng elementary algebrac nequaltes and performng elementary geometrc observatons [] one obtans the followng assessment crteron: Among two or more groups the group wth the greater x c performs better. x ISSN: Volume, 07

3 If two or more groups have the same x c., then the group wth the greater y c performs better. If two or more groups have the same x c <., then the group wth the smaller y c performs better. As t becomes evdent from the above crteron, a group s performance depends manly on the value of the x-coordnate of the COG of the correspondng level s area, whch s calculated by the frst of formulas (3). In ths formula, greater coeffcents (weghts) are assgned to the hgher grades. Therefore, the COG method focuses, smlarly to the GPA ndex, on the group s qualty performance. In case of the deal performance (y = and y = 0 for ) the frst of formulas (3) gves that x c = 9. Therefore, values of x c greater than 4 9 =. could be consdered as demonstratng a more than satsfactory performance. the X-axs s not the proper one here, because n ths way the common rectangle parts are calculated only once. The rght method for calculatng the coordnates of the COG n ths case was fully developed by Subbotn & Voskoglou [9] and the resultng framework was called the Generalzed Rectangular Fuzzy Assessment Model (GRFAM). The development of GRFAM nvolves the followng steps:. Let y, y, y 3, y 4, y 3 be the frequences of a group s members who obtaned the grades F, D, C, B, A respectvely. Then y = (00%).. We take the heghts of the rectangles n Fgure to have lengths equal to the values of correspondng frequences. Also, wthout loss of generalty we allow the sdes of the adjacent rectangles lyng on the OX axs to share common parts wth length equal to the 30% of ther lengths,.e. 0.3 unts. 4. The Varatons GRFAM, TFAM and TpFAM of the RFAM A group s performance s frequently represented by numercal scores n a clmax from These scores can be assgned to the lngustc labels of U as follows: A (8-00), B(7-84), C (60-74), D(0-9) and F (0-49). Nevertheless, ambguous cases appear frequently n practce, beng at the boundares between two successve assessment grades; e.g. somethng lke 84-8%, beng at the boundares between A and B. In an effort to treat better such knd of cases, Subbotn [8] moved the rectangles of Fgure to the left, so that to share common parts (see Fgure ). In ths way, the ambguous cases, beng at the common rectangle parts, belong to both of the successve grades, whch means that these parts must be consdered twce n the correspondng calculatons. The graph of the resultng fuzzy set s now the bold lne of Fgure. However, the method mentoned n Secton II for calculatng the coordnates of the COG of the area contaned between the graph and Fgure : Graphcal representaton of the GRFAM 3. We calculate the coordnates ( xc, y c ) of the COG, say F, of each rectangle, =,, 3, 4, as follows: Snce the COG of a rectangle s the pont of the ntersecton of ts dagonals, we have that yc = y. Also, snce the x-coordnate of each COG F s equal to the x- coordnate of the mddle of the sde of the correspondng rectangle lyng on the OX axs, from Fgure t s easy to observe that Ths way of assgnment, although t satsfes the common sense, t s not unque; n a more strct assessment, for example, one could take A(90-00), B(80-89), C(70-79), D(60-69) and F (0-9), etc. Snce the ambguous assessment cases are stuated at the boundares between the adjacent grades, t s logcal to accept a percentage for the common lengths of less than 0%. ISSN: Volume, 07

4 x c = We calculate the coordnates (X c, Y c ) of the COG F of the whole area consdered n Fgure as the resultant of the system of the GOCs F of the fve rectangles from the followng well known [0] formulas In case of the equalty the frst of formulas () gves that X c = 0.7( ) =.9. Further, combnng the nequalty (6) wth the second of formulas (), one fnds that X c = Sx c S, Y c = = Sy c S (4). = Y c. 0 In the above formulas S,,, 3, 4, denote the areas of the correspondng rectangles, whch are equal to y. Therefore S = S = and formulas (4) gve that X c = y = y (0.7 0.), Y c = y( y) or X c = (0.7 y ) 0., Y c = y ().. We determne the area n whch the COG F les as follows: For, j =,, 3, 4,, we have that 0 (y - y j ) = y + y j - y y j, therefore y + y j y y j, wth the equalty holdng f, and only f, y = y j. Therefore =( y ) = y + yy j y +, j=, or j ( y + yj ), j=, j = y y (6), wth the equalty holdng f, and only f, y = y = y 3 = y 4 = y =. Therefore the unque mnmum for Y c corresponds to the COG F m (.9, 0.). The deal case s when y = y = y 3 = y 4 = 0 and y =. Then formulas () gve that X c = 3.3 and Y c. Therefore the COG n ths case s the pont F = (3.3, 0.). On the other hand, the worst case s when y = and y = y 3 = y 4 = y = 0. Then from formulas () one fnds that the COG s the pont F w (0., 0.). Therefore, the area n whch the COG F les s the area of the trangle F w F m F (Fgure 3). Fgure 3: The trangle where the COG les 6. From elementary geometrc observatons on Fgure 3 one obtans the followng assessment crteron: Between two groups, the group wth the greater X c performs better. If two groups have the same X c.9, then the group wth the greater Y c performs better. If two groups have the same X c <.9, then the group wth the lower Y c performs better ISSN: Volume, 07

5 From the frst of formulas () t becomes evdent that the GRFAM measures the qualty group s performance. Also, snce the deal performance corresponds to the value X c = 3.3, values of X c greater than 3.3 =.6 could be consdered as ndcatng a more than satsfactory performance. At ths pont one could rase the followng queston: Does the shape of the membershp functon s graph of the assessment model affect the assessment s conclusons? For example, what wll happen f the rectangles of the GRFAM wll be replaced by sosceles trangles? The effort to answer ths queston led to the development of the Trangular Fuzzy Assessment Model (TFAM), created by Subbotn & Blotsk [4] and fully developed by Subbotn & Voskoglou [6]. Fuzzy Assessment Model (TpFAM). The correspondng scheme s that shown n Fgure. In ths case the y - coordnate of the COG F,,, 3, 4,, of each trapezod s calculated n terms of the fact that the COG of a trapezod les on the lne segment jonng the mdponts of ts parallel sdes a and b at a dstance d from the longer sde b gven by h d= ( a+ b ), 3( a+ b) where h s ts heght [8]. Also, snce the x- coordnate of the COG of each trapezod s equal to the x-coordnate of the mdpont of ts base, t s easy to observe from Fgure that y x = y B C y 3 B 3 C 3 y 4 H B 4 C 4 y y H H3 H 3 B C 0=A A D A 3 D A D 3 A D 4 D x Fgure : The TpFAM s scheme Fgure 4: Graphcal Representaton of the TFAM The graphcal representaton of the TFAM s shown n Fgure 4 and the steps followed for ts development are the same wth the correspondng steps of GRFAM presented above. The only dfference s that one works wth sosceles trangles nstead of rectangles. The fnal formulas calculatng the coordnates of the COG of TFAM are: X c = (0.7 y ) 0., Y c = y (7) and the correspondng assessment crteron s the same wth the crteron obtaned for GRFAM. An alternatve to the TFAM approach s to consder sosceles trapezods nstead of trangles [6, 7]. In ths case we called the resultng framework Trapezodal One fnally obtans from formulas (4) that 3 X c = (0.7 y ) 0., Y c = y (8) 7 and the assessment crteron remans the same agan.. Comparson of the Assessment Models One can wrte formulas (), (6) and (7) n the unfed form: X c = (0.7 y ) 0., Y c = a y (9), where a = for the GRFAM, a = TFAM and for the ISSN: Volume, 07

6 a = 3 for the TpFAM. Combnng formulas (9) 7 wth the common assessment crteron stated n Secton 4 one obtans the followng result: Theorem : The three varatons of the COG technque,.e. the GRFAM, the TFAM and the TpFAM are equvalent to each other assessment models. Further, the frst of formulas (9) can be wrtten as X c = 0.7(y + y + 3y 3 + 4y 4 + y ) 0. = 0.7 [(y + y 3 + 3y 4 + 4y ) + y ] 0.. Therefore, by formula () one fnally gets that X c = 0.7(GPA + ) 0. = 0.7GPA + 0. (0). In the same way, the frst of formulas (3) for RFAM can be wrtten as or x c = (y + 3y + y 3 +7y 4 + 9y ) = (GPA + ), x c = GPA + 0. (). We are ready now to prove: Theorem : If the values of the GPA ndex are dfferent for two groups, then the GPA ndex, the RFAM and ts varatons (GRFAM, TFAM and TpFAM) provde the same assessment outcomes on comparng the performance of these groups. Proof: Let G and G be the values of the GPA ndex for the two groups and let x c, x c be the correspondng values of the x-coordnate of the COG for the RFAM. Assume wthout loss of generalty that G>G,.e. that the frst group performs better accordng to the GPA ndex. Then, equaton () gves that x c > x c, whch, accordng to the frst case of the assessment crteron of Secton III, shows that the frst group performs also better accordng to the RFAM. In the same way, from equaton (0) and the frst case of the assessment crteron of Secton IV, one fnds that the frst group performs better too accordng to the equvalent assessment models GRFAM, TFAM and TpFAM.- The followng result shows that Theorem remans true even n case of the same GPA ndex. Theorem 3: If the values of the GPA ndex are the same for two groups, then the RFAM and ts varatons GRFAM, TFAM and TpFAM provde the same assessment outcomes on comparng the performance of these groups. Proof: Snce the two groups have the same value of the GPA ndex, equatons (0) and () show that the values of X c and x c are also the same. Therefore, one of the last two cases of the assessment crtera of Sectons III and IV could happen. The possble values of x n these crtera le n the ntervals [0, 9 ] and [0, 3.3] respectvely, whle the crtcal ponts correspond to the values x c =. and X c =.9 respectvely. Obvously, f both values of x are n [0,.9), or n [., 9 ], then the two crtera provde the same assessment outcomes on comparng the performance of the two groups. Assume therefore that.9 < X c and x c <.. Then, due to equaton (0),.9 < X c.9< 0.7GPA <0.7GPA GPA >. Also, due to equaton (), x c <. GPA + 0. <. GPA <. Therefore, the nequaltes.9 < X c and x c <. cannot hold smultaneously and the result follows.- Combnng Theorems and 3 one obtans the followng corollary: Corollary 4: The RFAM and ts varatons GRFAM, TFAM and TpFAM provde always the same assessment results on comparng the performance of two groups. The followng example ([9], Secton 4, paragraph v) shows that n case of the same GPA values the applcaton of the GPA ndex could not lead to logcally based conclusons. Therefore, n such stuatons, our crtera of Sectons 3 and 4 become useful due to ther logcal nature. Example : The student grades of two Classes wth 60 students n each Class are presented n Table ISSN: Volume, 07

7 Table : Student Grades Grades Class I Class II C 0 0 B 0 0 A 0 40 The GPA ndex for the two classes s equal to *0 + 4 *0 3* * 40 = 3.67, whch means that the two Classes demonstrate the same performance n terms of the GPA ndex. Therefore equaton (0) gves that X c = 0.7* , whle equaton () gves that x c 4.7 for both Classes. But 6 y = ( ) + ( ) = for the frst and 4 0 y = ( ) + ( ) = for the second Class. Therefore, accordng to the assessment crtera of Sectons 3 and 4 the frst Class demonstrates a better performance n terms of the RFAM and ts varatons. Now whch one of the above two conclusons s closer to the realty? For answerng ths queston, let us consder the qualty of knowledge,.e. the rato of the students receved B or better to the total number of students, whch s equal to for the 6 frst and for the second Class. Therefore, from the common pont of vew, the stuaton n Class II s better. Nevertheless, many educators could prefer the stuaton n Class I havng a greater number of excellent students. Conclusvely, n no case t s logcal to accept that the two Classes demonstrated the same performance, as the calculaton of the GPA ndex suggests. The next example shows that, although by Corollary 4 the RFAM, GRFAM, TFAM and TpFAM provde always the same assessment results on comparng the performance of two groups, they are not equvalent assessment models. Example 6: Table depcts the results of the fnal exams of the frst term mathematcal courses of two dfferent Departments, say D and D, of the School of Technologcal Applcatons (future engneers) of the Graduate T. E. I. of Western Greece. Note that the contents of the two courses and the nstructor were the same for the two Departments. Table : Results of the two Departments Grade D D A B 3 6 C 3 D 9 0 F 6 No. of students 30 3 The GPA ndex s equal to *9 + *+ 3*3 + 4* for D and *0 + *3 + 3*6 + 4*.66 3 for D. Therefore, the two Departments demonstrated a less than satsfactory performance (snce GPA < ), wth the performance of D beng better. Further, equaton (0) gves that X c.3 for D and X c.66 for D. Therefore, accordng to the frst case of the assessment crteron of Secton IV, D demonstrated, wth respect to GRFAM, TFAM and TpFAM, a better performance than D. Moreover, snce.3 < 3.3 =.6 <.66, D demonstrated a less than satsfactory performance, whle D demonstrated a more than satsfactory performance. In the same way equaton () gves that x c.97 for D and x c.6 for D. Therefore, accordng to the frst case of the assessment crteron of Secton III, D demonstrated, wth respect to RFAM, a better performance than D. But n ths case, snce for both Departments X c < ISSN: Volume, 07

8 4. =., both Departments demonstrated a less than satsfactory performance. Remark: Note that, f GPA >, then and X c = 0.7GPA + 0. > 0.7 * + 0. =.9 >.6 x c = GPA + 0. > =.>.. Therefore, the correspondng group s performance s more than satsfactory wth respect to GRFAM, TFAM, TpFAM and RFAM. However, f GPA <, then X c <.9 and x c <., whch do not guarantee that X c <.6 and x c <.. Therefore the assessment characterzatons of RFAM and the equvalent to each other GRFAM, TFAM, TpFAM can be dfferent only when GPA <. 6. Concluson From the dscusson performed n ths paper t becomes evdent that the RFAM and the equvalent to each other GRFAM, TFAM and TpFAM, although they provde always the same assessment outcomes on comparng the performance of two groups, they are not equvalent assessment models. Further, the assessment outcomes of the above models are also the same wth those of the GPA ndex, unless f the value of the GPA ndex s the same for both groups. In the last case the GPA ndex could not lead to logcally based conclusons. Therefore, n ths case ether the use of RFAM or of ts varatons must be preferred. Other fuzzy assessment methods have been also used n earler author s works lke the measurement of a system s uncertanty [9] and the applcaton of the fuzzy numbers [0]. These methods, n contrast to the prevous ones whch focus on the correspondng group s qualty performance, they measure ts mean performance. The plans for our future research nclude the effort to compare all these methods to each other n order to obtan the analogous conclusons. References: [] Voskoglou, M. Gr., An Applcaton of Fuzzy Sets to the Process of Learnng, Heurstcs and Ddactcs of Exact Scences, 0, 999, pp [] Voskoglou, M. Gr., Transton across levels n the process of learnng: A fuzzy Model, Internatonal Journal of Modellng and Applcaton, Unv. Blumenau,, 009, pp [3] Subbotn, I. Ya., Badkoobeh, H., Blotck, N. N., Applcaton of fuzzy logc to learnng assessment, Ddactcs of Mathematcs: Problems and Investgatons,, 004, pp [4] Subbotn, I. Ya., Blotsk, N. N., Trangular fuzzy logc model for learnng assessment, Ddactcs of Mathematcs: Problems and Investgatons, 4, 04, pp [] Subbotn, I. Ya, Voskoglou, M. Gr., A Trangular Fuzzy Model for Assessng Crtcal Thnkng Sklls, Internatonal Journal of Applcatons of Fuzzy Sets and Artfcal Intellgence, 4, 04, pp [6] Subbotn, I. Ya., Trapezodal Fuzzy Logc Model for Learnng Assessment, 04, arxv [math..GM] [7] Subbotn, I. Ya, Voskoglou, M. Gr., Fuzzy Assessment Methods, Unversal Journal of Appled Mathematcs, (9), 04, pp [8] Subbotn, I. Ya., On Generalzed Rectangular Fuzzy Model for Assessment, Global Journal of Mathematcs, (), 0, pp [9] Subbotn, I.Ya. & Voskoglou, M.Gr., An Applcaton of the Generalzed Rectangular Fuzzy Model to Crtcal Thnkng Assessment, Amercan Journal of Educatonal Research, 4(), 06, pp [0] Voskoglou, M. Gr., A Study on Fuzzy Systems, Amercan Journal of Computatonal and Appled Mathematcs, (), 0, pp [] Voskoglou, M. Gr., Subbotn, I. Ya., Dealng wth the Fuzzness of Human Reasonng, Internatonal Journal of Applcatons of Fuzzy Sets and Artfcal Intellgence, 3, 03, pp [] Voskoglou, M. Gr., Case-Based Reasonng n Computers and Human Cognton: A Mathematcal Framework, Internatonal Journal of Machne Intellgence and Sensory Sgnal Processng,, 03, pp. 3-. [3] Voskoglou, M. Gr, Assessng the Players Performance n the Game of Brdge, Amercan Journal of Appled Mathematcs and Statstcs, (3), 04, pp. -0 [4] Voskoglou, M. Gr., A Trangular Fuzzy Model for Decson Makng, Amercan Journal of Computatonal and Appled Mathematcs, 4(6), 04, pp [] Voskoglou, M. Gr., An applcaton of the COG technque to assessment of student ISSN: Volume, 07

9 understandng of polar coordnates, NAUN Internatonal Journal of Educaton and Informaton Technologes,, 07, pp. -. [6] Swnburne.edu.au, Grade Pont Average Assessment, retreved on October, 04 from: /assessment/gpa.html [7] van Broekhoven, E. & De Baets, B., Fast and accurate centre of gravty defuzzfcaton of fuzzy system outputs defned on trapezodal fuzzy parttons, Fuzzy Sets and Systems, 7(7), 006, pp [8] Wkpeda, Trapezod: Other propertes, retreved on October 0, 04 from rtes [9] Voskoglou, M. Gr., Managng the uncertanty for student understandng of the nfnty, NAUN Internatonal Journal of Fuzzy Systems and Advanced Applcatons, 4, 07, pp [0] Voskoglou, M. Gr., Use of TFNs and TpFNs for evaluatng the effectveness of CBR systems, NAUN Internatonal Journal of Fuzzy Systems and Advanced Applcatons, 4, 07, pp. -7. ISSN: Volume, 07

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