Application of Vague Soft Sets in students evaluation

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1 Availabl onlin at Advancs in Applid Scinc Rsarch, 0, (6):48-43 ISSN: CODEN (USA): AASRFC Application of Vagu Soft Sts in studnts valuation B. Chtia*and P. K. Das Dpartmnt of Mathmatics, Brahmaputra Vally Acadmy, North Lakhimpur, Assam, India Dpartmnt of Mathmatics, NERIST, Nirjuli, Itanagar, Arunachal Pradsh, India _ ABSTRACT In this papr, vagu soft sts concpt is applid to xtnd Biswas s mthod for studnts answr scripts valuation and a hypothtical cas study has bn takn as an xampl. Kywords: Soft sts, Vagu soft sts, and Soft Evaluation Knowldg. INTRODUCTION Out of svral gnralizations of fuzzy st thory for various objctivs, th notion introducd by Gau and Buhrr [] in dfining vagu sts (VSs) is intrsting and usful. Molodtsov (999) pointd out that th xisting thoris, viz., thory of probability, thory of fuzzy sts, thory of intuitionistic fuzzy sts, thory of vagu st, thory of intrval mathmatics and thory of rough sts can b considrd as mathmatical tools for daling with uncrtaintis but all ths thoris hav thir own limitations. Th rason for this is most possibly th inadquacy of th paramtrization tool of th thoris. So h dvlopd a nw mathmatical thory calld Soft St for daling with uncrtaintis which is fr from th abov limitations. Th absnc of any rstrictions on th approximat dscription in soft st thory maks this thory vry convnint and asily applicabl in practic. Xu t al.[ ] hav dvlopd a thortical study of th Vagu Soft St (VSS).Th combination of Vagu St and Soft St will b mor usful in th fild of applications whrvr uncrtainty appar. In [3], Biswas pointd out that on of th chif aim of ducational institutions is to provid studnts with th valuation rports rgarding thir tst/xamination as sufficint as possibl and with th unavoidabl rror as small as possibl and prsntd a fuzzy valuation mthod(fm) for applying fuzzy sts in studnts answr scripts valuation. H also modifid th fuzzy valuation mthod to propos a gnralizd fuzzy valuation mthod (gfm) for studnts answr scripts valuation. In [4], Chn and L pointd out that th mthods prsntd in [3] hav two 48

2 B. Chtia Adv. Appl. Sci. Rs., 0, (6):48-43 drawbacks, (i) it would tak a larg amount of tim to dal with th matching oprations of th matching function and (ii) two diffrnt fuzzy marks might b translatd into th sam awardd lttr grad which would b unfair for studnts valuation. Thus, thy prsntd two mthods for valuating studnts answr scripts using fuzzy sts. A soft st is a paramtrizd family of substs of th univrsal st. W can say that soft sts ar nighborhood systms, and that thy ar a spcial cas of contxt-dpndnt fuzzy sts. In soft st thory th problm of stting th mmbrship function, among othr rlatd problms, simply dos not aris. This maks th thory vry convnint and asy to apply in practic. Soft st thory has potntial applications in many diffrnt filds, including th smoothnss of functions, gam thory, oprations rsarch, Rimann intgration, Prron intgration, probability thory, and masurmnt thory. Most of ths applications hav alrady bn dmonstratd in Molodtsov s papr. In th first sction of this papr, w prsnt a nw mthod for studnts answr script valuation using VSS and th scond sction contains an algorithm of th mthod. Thn in th last sction a hypothtical cas study is discussd using th proposd mthod. Th proposd mthod can valuat studnts answr script in a mor flxibl and mor intllignt mannr.. PRELIMINARIES. Soft sts and Fuzzy soft sts Dfinition.[8] Lt U b a univrsal st, E a st of paramtrs and A E. Thn a pair (F,A) is calld soft st ovr U, whr F is a mapping from A to U, th powr st of U. Exampl. Lt X= {c, c, c 3 } b th st of thr cars and E ={costly( ), mtallic colour ( ), chap ( 3 )} b th st of paramtrs,whr A={, } E. Thn (F,A)={F( )={c,c,c 3 },F( )={c,c 3 }}is th crisp soft st ovr X which dscribs th Attractivnss of th cars which Mr. S (say) is going to buy. Dfinition.[6] Lt Ub a univrsal st, E a st of paramtrs and A E. Lt F (X) dnots th st of all fuzzy substs of U. Thn a pair (F,A) is calld fuzzy soft st ovr U, whr F is a mapping from A to F (U). Exampl. Lt U={c,c,c 3 } b th st of thr cars and E ={costly( ), mtallic colour( ) chap( 3 )} b th st of paramtrs,whr A={, ) E. Thn (F,A)={F( )={c /.6,c /.4,c 3 /.3},F( )={c /.5,c /.7,c 3 /.8}} is th fuzzy soft st ovr U dscribs th attractivnss of th cars which Mr. S(say) is going to buy.. Vagu sts and vagu soft sts. Dfinition.3[] Lt U b an initial univrs st, U ={u,u,.,u n }. A vagu st ovr U is charactrizd by truthmmbrship function t v and a fals-mmbrship function f v, t v : U [0,], 49

3 B. Chtia Adv. Appl. Sci. Rs., 0, (6):48-43 f v :U [0,], whr t v (u i ) is a lowr bound on th on th grad of mmbrship of u i drivd from th vidnc for u i, f v (u i ) is a lowr bound on th ngation of u i drivd from th vidnc against u i, and t v (u i )+ f v (u i ). Th grad of mmbrship of u i in th vagu st is boundd to a subintrval [t v (u i ),- f v (u i )]of [0, ]. Th vagu valu [t v (u i ),- f v (u i )] indicats that th xact grad of mmbrship µ v ( u i ) of u i may b unknown, but it is boundd by t v (u i ) µ v ( u i ) - f v (u i ), whr t v (u i )+ f v (u i ). Dfinition.4[] Lt U b a univrs, E a st of paramtrs, V(U) th powr st of vagu sts on U, and A E. A pair (F, A) is calld a vagu soft st ovr U, whr F is a mapping givn by F:A V(U). Exampl.3 Lt U= {c, c, c 3 } b th st of thr cars and E = {costly ( ), mtallic colour( ), chap( 3 )} b th st of paramtrs,whr A={, } E. Suppos that F( )={ [.6,.7]/ c, [.4,.6]/ c,[.3,.5]/ c 3 }, F( )= { [.5,.7]/ c, [.7,.8]/ c,[.8, ]/ c 3 } thn th vagu soft sts (F,A) is a paramtrizd family{ F( ), F( )} of vagu sts on U dscribs th attractivnss of th cars which Mr. S(say) is going to buy. 3. Application of vagu soft st in studnts valuation In this sction, w prsnt an application of vagu soft st (VSS) thory in studnts answr scripts valuation following Biswas approach [4]. Assum that thr ar fiv satisfaction lvls to valuat th studnts answr scripts rgarding a qustion of an xamination i.. xcllnt ( ), vry good ( ), good ( 3 ), satisfactory ( 4 ) and unsatisfactory ( 5 ). Lt X b a st of satisfaction lvl, X = {xcllnt ( ), vry good ( ), good ( 3 ), satisfactory ( 4 ) and unsatisfactory ( 5 )} and again lt S = {0%. 0%, 40%, 60%, 80%, 00%} b th dgr of satisfaction of th valuator for a particular qustion of th studnt s answr script. Suppos is a st of qustions for a particular papr of 00 marks. W first assum X as a univrsal st and S th st of paramtrs. Thn a VSS is constructd ovr th X, whr F is a mapping F:S V(X) and V(X) is th powr st of vagu sts on X. This VSS givs a rlation matrix, say, R, calld xprt studnts valuation matrix. W rfr to th matrix R as Soft Evaluation Knowldg. Again w construct anothr VSS (F, X) ovr, whr F is a mapping givn by F :X V() and V() is th powr st of vagu sts on. This VSS givs a rlation matrix R, calld xamination knowldg matrix. Thn, w obtain a nw rlation T= R o R calld satisfaction qustion matrix in which th mmbrship valus ar givn by t (,S ) = {t (, ) t (,S )} T i k R i j R j k (-f )(,S ) = {(-f )(, ) (-f )(,S )}, whr =max and = min. T i k R i j R j k Thn comput th matrix S T whr S T =[( λ) t + λ ( f )], whr λ [0,] is th dgr of optimism of th valuator dtrmind by th valuator for valuating studnts answr script of [t, f ] of th matrix T. 40

4 B. Chtia Adv. Appl. Sci. Rs., 0, (6):48-43 Corrsponding to ach qustion i of th papr for th matrix T w tak th highst valu 0.x i = ( λ) t + λ ( f ) (say) which indicats that th dgr of satisfaction of th qustion i is 00x i %. Thn th highst scor of th qustion i is H( i ) =00x i %. If M( i ) is th mark allottd to th qustion i thn th total scor of th studnt is calculatd by th formula = { H( i) M( i )} Algorithm: input th VSS (F,S) ovr th st X of satisfaction lvls, whr S is th st of dgr of satisfaction of th particular qustion papr and also writ th soft valuation knowldg R rprsnting th rlation matrix of th VSS (F,S). input th VSS (F,X) ovr th st of qustions of th papr and writ its rlation matrix R. comput th rlation matrix T = R o R comput S T from th matrix T. comput th highst scor for ach qustion for th matrix T. calculat th total scor for th studnt for ach papr. 3. Cas Study: Considr a candidat answr scripts to papr of 00 marks. Assum that in total thr wr four qustions to b answrd. Lt X b a st of satisfaction lvl and lt X={,, 3, 4, 5 }whr,, 3, 4 and 5 rprsnts xcllnt, vry good, good, satisfactory and unsatisfactory rspctivly. Suppos an valuator is using vagu soft grad sht. Considr X b as th univrsal st and S = {0%. 0%, 40%, 60%, 80%, 00%} b th st of dgr of satisfaction of th valuator s as th st of paramtrs. Suppos that F(0%) = { [0,.]/, [0,.]/, [0,.]/ 3, [.4,.5]/ 4, [, ]/ 5 } F(0%) = { [0,0] /, [0,0] /, [.,.] / 3, [.4,.5] / 4, [,] / 5 } F(40%) = { [.6,.6] /, [.5,.6] /, [.5,.6] / 3, [.4,.5] / 4, [.4,.5] / 5 } F(60%) = { [.8,.9] /, [.8,.8] /, [.7,.9] / 3, [.6,.7] / 4, [.,.3] / 5 } F(80%) = { [,] /, [.9,.9] /, [.4,.5] / 3, [.,.3] / 4, [0,0] / 5 } F(00%) = { [,] /, [.8,.9] /, [.,.3] / 3, [0,] / 4, [0,0] / 5 } Than th VSS (F,S) is a paramtrizd family {F(0%), F(0%), F(40%), F(60%), F(80%), F(00%)} of vagu soft sts ovr th st X and ar dtrmind from xprt studnt valuation documntation. Thus th VSS (F,S) givs an approximat dscription of th vagu soft xamination knowldg of th four qustions and thir lvl of satisfaction. This VSS (F,S) is rprsntd by matrix R, calld xprt studnts valuation matrix and is givn by 0% 0% 40% 60% 80% 00% [0,.] [0,0] [.6,.6] [.8,.9] [,] [,] [0,.] [0,0] [.5,.6] [.8,.8] [.9,.9] [.8,.9] R = 3[0,.] [.,.] [.5,.6] [.7,.9] [.4,.5] [.,.3] 4[.4,.5] [.4,.5] [.4,.5] [.6,.7] [.,.3] [0,0] 5 [,] [,] [.4,.5] [.,.3] [0,0] [0,0] 4

5 B. Chtia Adv. Appl. Sci. Rs., 0, (6):48-43 Suppos an valuator is using vagu soft grad sht. Suppos thr ar four qustions,, 3 and 4 in th qustion papr and w considr th st = {,, 3, 4 } as univrsal st and S = {,, 3, 4 } as th st of paramtrs rspctivly. Th valuator s satisfaction lvl for th studnt for qustion with rspct to paramtrs is rspctivly F ( ) = { [.5,.7]/,[,] /,[.5,.7] / 3, [.8,.9] / 4 } F ( ) = { [.8,.8]/, [.8,.9] /, [.6,.7] / 3, [.5,.7] / 4 } F ( 3 ) = { [.6,.7]/,[.4,.5] /, [.4,.5] / 3, [.,.3] / 4 } F ( 4 ) = { [0,0]/, [0,0] /, [.,.3] / 3, [0,0] / 4 } F ( 5 ) = { [0,0]/, [0,0] /, [0,0] / 3, [.5,.6] / 4 } Thn th VSS (F,X) is a paramtrizd family { F ( ), F ( ), F ( 3 ), F ( 4 ), F ( 5 )} of all vagu st ovr th st S and ar dtrmind from valuatd satisfaction for a particular studnt. This VSS (F,X) givs approximat dscription of th vagu soft xamination knowldg of th four qustion and thir lvl of satisfaction. This VSS (F, X) is rprsntd by rlation matrix R, calld xamination knowldg matrix and givn by R [.5,.7] [.8,.8] [.6,.7] [0,0] [0,0] [,] [.8,.9] [.4,.5] [0,0] [0,0] = 3 [.5,.7] [.6,.7] [.4,.5] [.,.3] [0,0] 4 [.8,.9] [.5,.7] [.,.3] [0,0] [.5,.6] Thn combining th rlation matrics 0% 0% 40% 60% 80 % 00% [0,.] [.,.] [.5,.6] [.8,.8] [.8,.8] [.8,.8] [0,.] [.,.] [.6,.6] [.8,.9] [,] [,] T= Ro R = 3 [.,.3] [.,.3] [.5,.6] [.6,.7] [.6,.7] [.6,.7] 4 [.5,.6] [.5,.6] [.6,.6] [.8,.9] [.8,.9] [.8,.9] Suppos that th indx of optimism λ dtrmind by th valuator is 0.60 [0,] thn S T can b calculatd in th following way, i.. 0% 0% 40% 6 0% 80% 00% S T = Hnc th highst scor for is.8 i.. it indicats that th dgr of satisfaction of th qustion of th studnt s answr script valuation by th valuator is 80%. Similarly for is 00%, 3 is 66% and 4 is 86%. Thrfor H( )=80, H( )=86, H( 3 )=66 and H( 4 ) = 86. Again suppos that carris 0 marks, carris 30 marks, 3 carris 5 marks and 4 carris 5 marks. 4

6 B. Chtia Adv. Appl. Sci. Rs., 0, (6):48-43 Thrfor th total scor of th studnt = { H( i ) x M ( i )} 00 = /00 {80 x x x x 5} = /00 { } = 84. CONCLUSION In this papr, w hav applid th notion of vagu soft sts in valuating studnts answr scripts. A cas study has bn takn to xhibit th simplicity of th tchniqu. Acknowldgmnts Th authors would lik to thank Dr. B.K.Saikia, Dpartmnt of Mathmatics, Lakhimpur Girls Collg, Assam, India for providing vry hlpful suggstions. REFERENCES [] S. M. Bai, S.M. Chn, Exprt Systms with Applications, 008, 38, [] S. M. Bai, S.M. Chn, Exprt Systms with Applications, 008, 38, [3] R. Biswas, Fuzzy Sts and Systms, 999, 04, [4] S. M. Chn, C.H. L, Fuzzy Sts and Systms, 999, 04, [5] S. M. Chn, H.Y. Wang, Exprt Systms with Applications, 009, 36, [6] P. K. Maji, R. Biswas, Roy, A.R., Th Journal of Fuzzy Mathmatics, 00, 9(3), [7] P. K. Maji, R. Biswas, A.R. Roy, Computrs & Mathmatics with Applications, 00, 44, [8] D. Molodtsov, Computrs and Mathmatics with Application, 999, 37, 9-3. [9] B. K. Saikia, Int. Journal of Mathmatical Archiv, 0, (0), [0] W. H.Y., S. M. Chn, Educational Tchnology & Socity, 0(4), 4-4. [] W. L. Gau, D. J. Buhrr, IEEE Transactions on Systms, Man and Cybrntics, 993, 39, [] W. Xu,J.Ma,S.Wang,G.Hao, Computrs & Mathmatics with Applications, 00, 59(),