Comparison of the COG Defuzzification Technique and Its Variations to the GPA Index

Transcription

1 Amercan Journal of Computatonal and Appled Mathematcs 06, 6(): DOI: 0.93/.acam Comparson of the COG Defuzzfcaton Technque and Its Varatons to the GPA Index Mchael Gr. Voskoglou Department of Mathematcal Scences, School of Technologcal Applcatons, Graduate Technologcal Educatonal Insttute (T. E. I.) of Western Greece, Patras, Greece Abstract The Center of Gravty (COG) method s one of the most popular defuzzfcaton technques of fuzzy mathematcs. In earler works the COG technque was properly adapted to be used as an assessment model (RAM) and several varatons of t (GRAM, TAM and TpAM) were also constructed for the same purpose. In ths paper the outcomes of all these models are compared to the correspondng outcomes of a tradtonal assessment method of the b-valued logc, the Grade Pont Average (GPA) Index. Examples are also presented llustratng our results. Keywords Grade Pont Average (GPA) Index, Center of Gravty (COG) Defuzzfcaton Technque. Rectangular uzzy Assessment Model (RAM), Generalzed RAM (GRAM), Trangular (TAM) and Trapezodal (TpAM) uzzy Assessment Models. Introducton uzzy Logc (L), due to ts nature of characterzng the ambguous stuatons of our day to day lfe by multple values, offers rch resources for the assessment of such knd of stuatons. A characterstc example s the process of learnng a subect matter, where the new knowledge s frequently connected to a degree of vagueness and/or uncertanty from the learner s, as well as the teacher s pont of vew. In 999 Voskoglou [0] developed a fuzzy model for the descrpton of the process of learnng a subect matter n the classroom n terms of the possbltes of the student profles and later he assessed the student learnng sklls by calculatng the correspondng system s total possblstc uncertanty []. Meanwhle, Subbotn et al. [], based on Voskoglou s model [0], 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. Snce then, Voskoglou and Subbotn, workng ether ontly or ndependently, appled the COG technque and a number of varatons of t for assessng several human or machne (Decson Makng, Case-Based Reasonng, etc.) sklls, e.g. see [-7, -6], etc. In the present paper the outcomes of the COG technque and ts varatons are compared to the correspondng * Correspondng author: mvosk@hol.gr (Mchael Gr. Voskoglou) Publshed onlne at Copyrght 06 Scentfc & Academc Publshng. All Rghts Reserved 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 we descrbe the classcal GPA assessment method. In Secton 3 we sketch the use of the COG technque as an assessment method, whle n Secton 4 we brefly descrbe the varatons of the COG technque constructed n earler papers and the reasons who led to the development of these varatons. In Secton the outcomes of the COG technque and ts varatons are compared to the outcomes of the GPA ndex and examples are presented to llustrate our results. The last Secton 6 s devoted to our concluson and a dscusson on the perspectves for future research on the subect.. Tradtonal Assessment Methods The assessment methods whch are commonly used n practce are based on prncples of the b-valued logc. The calculaton of the mean value of the scores acheved by each one of ts members s the classcal method for assessng the mean performance of a group of obects (e.g. students, players, machnes, etc.) wth respect to an acton. On the other hand, a very popular n the USA and other Western countres assessment method s the calculaton of the Grade Pont Average (GPA) ndex. Ths ndex s a weghted average n whch greater coeffcents (weghts) are assgned to the hgher scores. GPA, whch s connected to the qualty group s performance, s calculated by the formula GPA = 0 n n n 3 n 4 n n D C B A ()

2 88 Mchael Gr. Voskoglou: Comparson of the COG Defuzzfcaton Technque and Its Varatons to the GPA Index where n s the total number of the group s members and n A, n B, n C, n D and n denote the numbers of the group s members that demonstrated excellent (A), very good (B), good (C), far (D) and unsatsfactory () performance respectvely [8]. In case of the worst performance (n = 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 ndcate a more than satsfactory performance. nally note that formula () can be also wrtten n the form GPA = y + y 3 +3y 4 + 4y () n where y = n, y n = D n, y 3 = n, y n 4 = B n and y n = A n denote the frequences of the group s members whch demonstrated unsatsfactory, far, good, very good and excellent performance respectvely. 3. The COG Defuzzfcaton Technque as an Assessment Method (RAM) The soluton of a problem n terms of L nvolves n general the followng steps: Choce of the unversal set U of the dscourse. uzzfcaton of the problem s data by defnng the proper membershp functons. Evaluaton of the fuzzy data by applyng rules and prncples of L to obtan a unque fuzzy set, whch determnes the requred soluton. Defuzzfcaton of the fnal outcomes n order to apply the soluton found n terms of L to the orgnal, real world problem. One of the most popular n fuzzy mathematcs defuzzfcaton methods s the Centre of Gravty (COG) technque. or applyng ths method, let us assume that A = {(x, m(x)): x U} s the fnal fuzzy set determnng the problem s soluton. We correspond to each xu an nterval of values from a prefxed numercal dstrbuton, whch actually means that we replace U wth a set of real ntervals. Then, we construct the graph of the membershp functon y=m(x) and we consder the level s area contaned between ths graph and the OX axs. There s a commonly used n L approach (e.g. see [9]) to represent the system s fuzzy data by the coordnates (x c, y c ) of the COG, say c, of the area, whch we calculate usng the followng well-known [9] from Mechancs formulas: x c xdxdy n C, yc dxdy ydxdy dxdy Consder now the specal case where one deals wth the assessment of a group s performance Then, we choose as set of the dscourse the set U = {A, B, C, D, } of the fuzzy lngustc labels (characterzatons) of excellent (A), very (3) good (B), good (C), far (D) and unsatsfactory () performance respectvely of the group s members. When a score, say y, s assgned to a group s member (e.g. a mark n case of a student), then ts performance s characterzed by, f y [0, ), by D, f y [, ), by C, f y [, 3), by B f y [3, 4) and by A f y [4, ] respectvely. Consequently, we have that y = m(x) = m() for all x n [0,), y = m(x) = m(d) for all x n [,), y 3 = m(x) = m(c) for all x n [, 3), y 4 = m(x) = m(b) for all x n [3, 4) and y = m(x) = m(a) for all x n [4, ]. Therefore, the graph of the membershp functon y = m(x), takes the form of gure, where the area of the level s secton contaned between the graph and the OX axs s equal to the sum of the areas of the rectangles S, =,, 3, 4,. y m(b) m(d) m() m(c) m(a) O s s s 3 gure. The graph of the COG method It s straghtforward then to check (e.g. see Secton 3 of []) that n ths case formulas (3) take the form: x c = (y +3y +y 3 +7y 4 +9y ), y c = (y +y +y 3 +y 4 +y ) wth x =, x =D, x 3 =C, x 4 =B, x =A and y = mx ( ) (4) mx ( ), =,, 3, 4,. Note that the membershp functon y = m(x), as t usually happens wth fuzzy sets, can be defned, accordng to the user s choce, n any compatble to the common logc way. However, n order to obtan assessment results compatble to the correspondng results of the GPA ndex, we defne here y = m(x) n terms of the frequences, as n formula () of Secton. Then s 4 mx ( ) = (00%). Usng elementary algebrac nequaltes and performng elementary geometrc observatons (e.g. Secton 3 of []) one obtans the followng assessment crteron: Among two or more groups the group wth the bggest x c performs better. If two or more groups have the same x c., then the group wth the hgher y c performs better. If two or more groups have the same x c <., then the s D C B A 3 4 x

3 Amercan Journal of Computatonal and Appled Mathematcs 06, 6(): group wth the lower y c performs better. As t becomes evdent from the above statement, 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 (4). 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 lengths equal to the correspondng frequences. Also, wthout loss of generalty we allow the sdes of the adacent rectangles lyng on the OX axs to share common parts wth length equal to the 30% of ther lengths,.e. 0.3 unts. frst of formulas (4) gves that x c = 9. Therefore, values 9 of x c greater than =. demonstrate a more than 4 satsfactory performance. Due to the shape of the correspondng graph (gure ) the above method was named as the Rectangular uzzy Assessment Model (RAM). 4. Varatons of the COG Technque (GRAM, TAM and TpAM) A group s performance s frequently represented by numercal scores n a clmax from These scores can be connected to the lngustc labels of U as follows: A (8-00), B(7-84), C (60-74), D(0-9) and (0-49). Ambguous cases appear 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 [4] moved the rectangles of gure to the left, so that to share common parts (see gure ). 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 gure. However, the method used n Secton 3 for calculatng the coordnates of the COG of the area contaned between the graph and the OX-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 [7] and the resultng framework was called the Generalzed Rectangular uzzy Assessment Model (GRAM). The development of GRAM nvolves the followng steps:. Let y, y, y 3, y 4, y 3 be the frequences a group s members who obtaned the grades, D, C, B, A respectvely. Then y = (00%).. We take the heghts of the rectangles n gure to have gure. Graphcal representaton of the GRAM 3. We calculate the coordnates ( x, c y ) of the COG, say, 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 s equal to the x- coordnate of the mddle of the sde of the correspondng rectangle lyng on the OX axs, from gure t s easy to observe that x = We consder the system of the COGs and we calculate the coordnates (X c, Y c ) of the COG of the whole area consdered n gure as the resultant of the system of the GOCs of the fve rectangles from the followng well known [0] formulas S Sx c, Y c = S c Sy In the above formulas S, =,, 3, 4, denote the areas of the correspondng rectangles, whch are equal to y. Therefore S = S = y (0.7 0.), Y c = c c () y = and formulas () gve that y ( y ) or (0.7 y ) 0., Y c = y (6). We determne the area n whch the COG les as follows: or, =,, 3, 4,, we have that 0 (y - y ) = y + y - y y, therefore y + y y y, wth the equalty holdng f, and only f, y = y. Therefore Ths way of connecton, 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 (0-9), etc. Snce the ambguous assessment cases are stuated at the boundares between the adacent grades, t s logcal to accept a percentage for the common lengths of less than 0%.

4 90 Mchael Gr. Voskoglou: Comparson of the COG Defuzzfcaton Technque and Its Varatons to the GPA Index = ( + y ) = =,, ( y y ) = y +,, yy y or y y (7) wth the equalty holdng f, and only f, y = y = y 3 = y 4 = y =. In case of the equalty the frst of formulas (6) gves values of X c greater than 3.3 =.6 ndcate 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? or example, what wll happen f the rectangles of the GRAM wll be replaced by sosceles trangles? The effort to answer ths queston led to the constructon of the Trangular uzzy Assessment Model (TAM), created by Subbotn & Blotsk [] and fully developed by Subbotn & Voskoglou [3]. that 0.7( ) =.9. urther, combnng the nequalty (7) wth the second of formulas (6), one fnds that Y c Therefore the unque mnmum for Y c 0 corresponds to the COG m (.9, 0.). The deal case s when y = y = y 3 = y 4 = 0 and y =. Then formulas () gve that 3.3 and Y c =. Therefore the COG n ths case s the pont (3.3, 0.). On the other hand, the worst case s when y = and y = y 3 = y 4 = y = 0. Then from formulas () we fnd that the COG s the pont w (0., 0.). Therefore, the area n whch the COG les s the area of the trangle w m (gure 3). gure 3. The trangle where the COG les 6. rom elementary geometrc observatons on gure 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 rom the frst of formulas (6) t becomes evdent that the GRAM measures the qualty group s performance. Also, snce the deal performance corresponds to the value 3.3, gure 4. Graphcal Representaton of the TAM The graphcal representaton of TAM s shown n gure 4 and the steps followed for ts development are the same wth the correspondng steps of GRAM 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 TAM are: (0.7 y ) 0., Y c = y (8) and the correspondng assessment crteron s the same wth the crteron obtaned for GRAM. An alternatve to the TAM approach s to consder sosceles trapezods nstead of trangles [4, ]. In ths case we called the resultng framework Trapezodal uzzy Assessment Model (TpAM). The correspondng scheme s that shown n gure. In ths case the y - coordnate of the COG, =,, 3, 4,, of each trapezod s calculated n terms of the fact that the COG of a trapezod les on the lne segment onng the mdponts of ts parallel sdes a and b at a dstance d from the h longer sde b gven by d= ( a b ), where h s ts heght 3( a b ) [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 gure that x = One fnally obtans from formulas () that (0.7 y ) 0., Y c = 3 y (9) 7

5 Amercan Journal of Computatonal and Appled Mathematcs 06, 6(): and the assessment crteron s the same agan. y y y 3 y 4 y y 0=A gure. The TpAM s scheme. Comparson of the Assessment Methods One can wrte formulas (6), (8) and (9) of Secton 4 n the sngle form: (0.7 y ) 0., Y c = a y (0) where a = for the GRAM, a = for the TAM and a = 3 for the TpAM. Combnng formulas (0) wth the 7 common assessment crteron stated n Secton 4 one obtans the followng result:.. Theorem The three varatons of the COG technque,.e. the GRAM, the TAM and the TpAM are equvalent assessment models. urther, the frst of formulas (0) can be wrtten as 0.7(y + y + 3y3 + 4y4 + y) 0. = 0.7 [(y + y3 + 3y4 + 4y) + y ] 0.. Therefore, by formula () of Secton 3, one fnally gets that 0.7(GPA + ) 0., or 0.7GPA + 0. () In the same way, the frst of formulas (4) of Secton 3 for RAM can be wrtten as x c = (y + 3y + y 3 +7y 4 + 9y ) = (GPA + ), or x c = GPA + 0. () We are ready now to prove:.. Theorem B H C H B 3 C 3 H3 A D A 3 D A B 4 C 4 B C If the values of the GPA ndex are dfferent for two groups, then the GPA ndex, the RAM and ts varatons (GRAM, H 3 D 3 A D 4 D x TAM and TpAM) 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 RAM. 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 3, shows that the frst group performs also better accordng to the RAM. In the same way, from equaton () and the frst case of the assessment crteron of Secton 4, one fnds that the frst group performs better too accordng to the equvalent assessment models GRAM, TAM and TpAM. In case of the same GPA ndex we shall show the followng result:.3. Theorem If the GPA ndex s the same for two groups then the RAM and ts varatons (GRAM, TAM and TpAM) provde the same assessment outcomes on comparng the performance of these groups. Proof: Snce the two groups possess the same value of the GPA ndex, equatons () 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 3 and 4 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.9 respectvely. Obvously, f both values of x are n [0,.9), or 9 n [., ], 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 (),.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:.4. Corollary The RAM and ts varatons GRAM, TAM and TpAM provde always the same assessment results on comparng the performance of two groups. The followng example shows that n case of the same GPA values the applcaton of the GPA ndex could not lead to logcally based conclusons (see also paragraph (v) of Secton 4 of [7]). 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

6 9 Mchael Gr. Voskoglou: Comparson of the COG Defuzzfcaton Technque and Its Varatons to the GPA Index 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*0 4* , whch means that the two Classes demonstrate the same performance n terms of the GPA ndex. Therefore equaton () gves that 0.7* , whle equaton () gves that x c = 4.7 for both Classes. But the frst and 4 y = ( ) ( ) = y = ( ) ( ) for for the second 36 Class. Therefore, accordng to the assessment crtera of Sectons 3 and 4 the frst Class demonstrates a better performance n terms of the RAM and ts varatons. Now whch one of the above two conclusons s closer to the realty? or 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 6 for the frst and for the second Class. Therefore, from the common pont of vew, the stuaton n Class II s better. However, 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 the RAM, GRAM, TAM and TpAM provde always the same assessment results on comparng the performance of two groups (Corollary.4), they are not equvalent assessment models..6. Example 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 Total No. of students 30 3 The GPA ndex s equal to *9 * 3*3 4* for D and *0 *3 3*6 4*.66 for D. Therefore, 3 the two Departments demonstrated a less than satsfactory performance (snce GPA < ), wth the performance of D beng better. urther, equaton () 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 4, D demonstrated (wth respect to GRAM, TAM and TpAM) 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 3, D demonstrated (wth respect to RAM) a better performance than D. But n ths case, snce for both Departments X c < 4. =., the two Departments demonstrated a less than satsfactory performance. REMARK: Note that, f GPA > (more than satsfactory performance), then 0.7GPA + 0. > 0.7 * + 0. =.9 >.6 and x c = GPA + 0. > =.>.. Therefore the correspondng group s performance s also more than satsfactory wth respect to GRAM, TAM, TpAM and RAM. However, f GPA < (less than satsfactory performance), then X c <.9 and x c <., whch do not guarantee that X c <.6 and x c <.. Therefore the assessment characterzatons of RAM and the equvalent GRAM, TAM, TpAM can be dfferent only when GPA <. 6. Conclusons and Perspectves for uture Research rom the dscusson performed n ths paper t becomes evdent that the RAM and ts varatons GRAM, TAM and TpAM, although they provde always the same assessment outcomes on comparng the performance of two groups, they are not equvalent assessment methods. The above assessment outcomes are also the same wth those of the GPA ndex, unless f the groups under assessment possess the same values. In the last case the GPA ndex could not lead to logcally based conclusons. Therefore, n ths case ether the use of RAM 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 [] and the applcaton of the fuzzy numbers [7]. These methods, n contrast to the prevous ones whch

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