Weighted Fuzzy Similarity Measure Based on Tangent Function and its Application to Medical Diagnosis

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2 ISSNOle : ISSN rt : Weghted uzzy Smlarty Measure Based o aget ucto ad ts pplcato to Medcal Dagoss Surapat ramak, Kalya Modal ssstat rofessor, Departmet of Mathematcs, Nadalal Ghosh B.. ollege, apur, O-Narayapur, ad Dstrct: North 4 argaas, West Begal, Ida ssstat eacher, Departmet of Mathematcs, Bragar Hgh School HS, Bragar, aaghat, ad Dstrct: Nada, West Begal, Ida. BS: I ths paper, the weghted taget smlarty measure of fuzzy sets s proposed ad ts propertes are studed. he cocept of the weghted taget smlarty measure of fuzzy sets s a decso makg tool whch s characterzed by the degree of membershp fucto, degree of o-membershp fucto sum of ths two compoets s equal to oe. ally, usg ths proposed method, a applcato o medcal dagoss s gve for the applcablty of the proposed approach. KEYWODS: aget smlarty measure, weghted taget smlarty measure, fuzzy sets, decso makg, ad medcal dagoss. I. INODUION It s recogzed that ucertaty plays a mportat role modellg real world problems. esearchers recogze the eed to brdge the gap betwee mathematcal models ad ucertaty ad ther emprcal terpretatos. he reflecto of ths gap ca be foud problems of operatos research, mathematcs, bologcal, cogtve, ad socal sceces as well as moder techology, medce ad other appled sceces. he eed to brdge the gap betwee a mathematcal model ad eperece s well addressed by Ma Black [] 937. Zadeh [] epressed the same eed 96. I 965, Zadeh [3] proposed the ew paradgm of mathematcs based o the very cocept of fuzzy sets. Whe the ew paradgm was proposed [3], the usual process of a paradgm shft [4]] begs. or detals of scetfc paradgm you may cosult the hghly fluetal book amely, he Structure Of Scetfc evolutos 4]. he paradgm shft s stll cotug. It s reflected the challeges throw by the theory of fuzzy sets to the very foudato of scece.e. the rstotela two-valued logc. he ew paradgm has a greater capablty to deal wth huma decso makg, mache tellgece, etc. he cocept of uzzy set [3] geeralzes the ator set dscovered by Smth [5] 874 ad troduced by Germa mathematca ator [6] 883. I fuzzy set theory, membershp ad o-membershp degrees are complemetary,.e., the sum of membershp ad o-membershp degrees of a elemet belogg a fuzzy set s equal to oe. he fuzzy set theory facltates to solve varous real world problems volvg partally ukow formato. Lterature revew reflects that the studes o fuzzy smlarty measures are mostly theoretc [7], [8], [9], [0], []. uzzy smlarty measures are appled to mage processg. [], [3], fuzzy reasog. [4], medcal dagoss [5], etc. Kakat has establshed a ew smlarty measure [6] for fuzzy sets usg the eteded defto of complemetato [7] based o referece fucto.. Kakat has proved the valdty [8] of the ew smlarty measure [6] wth the help of tradtoal Hammg Dstace ad Eucldea Dstace measures ad appled t to medcal reasog.

3 ISSNOle : ISSN rt : I ths paper the authors have proposed a ew smlarty measure of fuzzy sets amely; weghted fuzzy smlarty measure based o taget fucto ad studed ts basc propertes ad appled t to medcal dagoss. he rest of the paper s orgazed as follow: Secto II presets the cocepts of Ss, taget smlarty measures ad weghted taget smlarty measures for fuzzy sets. Secto III presets decso makg methodology based o weghted taget smlarty measures. Secto IV s devotes to preset a eample o medcal dagoss usg the proposed approach. ally, the cocluso of the paper ad scope of future work are preseted Secto V. II. MHEMIL ELIMINIES uzzy set: I 965, Zadeh [3] troduced the cocept of fuzzy sets as a mathematcal form for represetg mprecseess. Defto.: uzzy set: fuzzy set a uverse of dscourse X s defed as the followg set of pars, : X. Here, : [0,] s a mappg called the membershp value of X a fuzzy set Defto.: he value of s called the degree of o membershp of the elemet X to the fuzzy set. Defto.3: uzzy Number: fuzzy umber s a eteso of a regular umber such that t does ot refer to oe sgle value but related to a coected set of possble values, where each possble value has ts ow weght betwee 0 ad. hs weght s called the membershp fucto. hus a fuzzy umber s a ormal fuzzy set ad cove set. Defto.4: Hammg dstace betwee two fuzzy sets ad B s defed as H, B = B B Defto.5: aget smlarty measure for fuzzy sets he authors propose the taget smlarty measure for fuzzy sets the followg way. Let =, ad =, be two fuzzy umbers. Now taget smlarty fucto S, whch measures the smlarty betwee two vectors ad based oly o the drecto, gorg the mpact of the dstace betwee them ca be preseted as: S, = ta 8 roposto. he defed taget smlarty measure S, betwee two fuzzy umbers ad satsfes the followg propertes:. 0 S,. S, = f ad oly f = 3. S, = S, 4. If s a S X ad the S, S, ad S, S, roofs: s the membershp, o-membershp fucto of the fuzzy set are [0, ] ad the value of the taget fucto are wth [0,], the smlarty measure based o taget fucto also s wth [ 0,]. Hece 0 S, or ay two fuzzy sets ad f = ths mples =, =. Hece 0, 0, hus S, = oversely, If S, = the 0, 0, sce ta0 = 0. So we ca wrte,. Hece =. 3

4 ISSNOle : ISSN rt : hs proof s obvous. 4 If the, ad for X. Now we have the followg equaltes:, ;,. hus S, S, ad S, S,. Sce taget fucto s creasg the terval 4, 0. Defto.6: Weghted taget smlarty measure for fuzzy sets he authors propose weghted taget smlarty measure for fuzzy sets the followg way. Let =, ad =, be two fuzzy umbers. Now we preset weghted taget smlarty fucto WS, whch measures the smlarty betwee two vectors ad based oly o the drecto, gorg the mpact of the dstace betwee them ca be preseted as: WS, = w 8 ta Where, 0 w ad w. Defto.6 coverts to defto.5 f w roposto. he defed taget smlarty measure WS, betwee two fuzzy umbers ad satsfes the followg propertes: 5. 0 WS, 6. WS, = f ad oly f = 7. WS, = S, 8. If s a S X ad the WS, WS, ad WS, WS, roofs: 5 s the membershp, o-membershp fucto of the fuzzy set are [0, ] ad the value of the taget fucto are wth [0,], the smlarty measure based o taget fucto also s wth [ 0,]. Hece 0 WS, 6 or ay two fuzzy sets ad f = ths mples =, =. Hece 0, 0, hus WS, = oversely, If WS, = the 0, 0, sce ta0 = 0. So we ca wrte,. Hece =. 7 hs proof s obvous. 8 If the, ad for X. Now we have the followg equaltes:, ;,.

5 ISSNOle : ISSN rt : hus WS, WS, ad WS, WS,. Sce taget fucto s creasg the terval 0,. 4 III. UZZYY DEISION MKING BSED ON NGEN UNION Let,,, m be a dscrete set of caddates,,,, be the set of crtera of each caddate, ad D, D,, D k are the alteratves of each caddate. he decso-maker provdes the rakg of alteratves wth respect to each caddate. he rakg presets the performaces of caddates =,,, m agast the crtera j j =,,,. he values assocated wth the alteratves for MDM problem ca be preseted the followg two decso matrces see able ad able. able : he relato betwee caddates ad attrbutes m m able : he relato betwee attrbutes ad alteratves D D m D k k k Here [] j ad [D] jk ad are all fuzzy umbers. he steps of decso makg correspodg to fuzzy umber based o taget fucto are preseted as followg steps. Step : Determato of the relato betwee caddates ad attrbutes: Each caddate =,,, m havg the attrbute j j =,,,. he correspodg relatoal values betwee caddates ad ther attrbutes are preseted terms of fuzzy umbers as follows see able 3: able 3: elato betwee caddates ad attrbutes terms of fuzzy umbers m,, m, m, m,, m k m m, Step : Determato of the relato betwee attrbutes ad alteratves: he relato betwee attrbutes =,,, ad alteratves D t t =,,, k s preseted as follows see able 4: able4: he relato betwee attrbutes ad alteratves terms of fuzzy umbers D D D,,,,,, k k k,, k, Step 3: Determato of the crtera weght structure of smlarty measure: I the dagoss process, decso maker may ofte ecouter wth ukow crtera weghts. It may happe that the mportace of the crtera s dfferet. herefore t s ecessary to determe reasoable crtera weght for smlarty measures. Step 4: Determato of the co-relato measure betwee two relatos: Determe the correlato measure betwee the able 3 ad the able 4 usg WS, from the equato. Step 5: akg the alteratves:,, m k k k

6 ISSNOle : ISSN rt : akg the alteratves correspodg to each caddate s prepared as the descedg order of correlato measures. Hghest value dcates the best alteratve for correspodg caddate. Step 6: Ed IV. NUMEIL EXMLE ON MEDIL DIGNOSIS BSED ON NGEN UNION Let us cosder a umercal eample o medcal dagoss. Medcal dagoss cossts of a large amout of ucertates ad creased volume of formato avalable to physcas from ew medcal techologes. he process of classfyg dfferet set of symptoms s uder a sgle ame of a dsease. he proposed smlarty measure amog the patets Vs symptoms ad symptoms Vs dseases gves proper medcal dagoss. he ma feature of ths proposed method s that t cosders membershp, o-membershp degree by takg oe tme specto for dagoss. Now, a eample of a medcal dagoss wll be preseted. Eample: Let = {,,, 4 } be a set of patets, D = {Vral ever, Malara, yphod, Stomach problem, hest problem} be a set of dseases ad S= {emperature, Headache, Stomach pa, cough, hroat pa.} be a set of symptoms. Our soluto s to eame the patets whch tur gves arse to membershp ad o-membershp fucto for each patet. Step : Determato of the relato betwee caddates ad attrbutes: our patets,,, ad 4 have the symptoms temperature, Headache, Stomach pa, cough, ad hroat pa. hey feel lless. Wth the help of epert assessmets, we tabulate the relatoal values betwee patets ad ther symptoms as follows see able 5. able 5: elato- - he relato betwee atet ad Symptoms elato- emperature Headache Stomach pa cough hroat pa 0.8, , , , , , , , , , , , , , , , , , , , 0.3 Step : Determato of the relato betwee attrbutes ad alteratves: Every dsease has some symptoms. here are some dseases whose symptoms are more or less same. So, medcal t s mportat to set up the relatos betwee symptoms ad dseases ucerta evromet. Here, the relatos are preseted fuzzy umbers as follows see able 6. able 6: elato- -he relato amog Symptoms ad Dseases elato- Vral ever Malara yphod Stomach problem hest problem emperature 0.7, , , , , 0.4 Headache 0.5, , , , , 0.7 Stomach pa 0.3, , , , 0. 0., 0.8 ough 0.8, , , , , 0. hroat pa 0.6, , , , , 0.4 Step 3: Determato of the weght structure of each smlarty measure: Weght structure of each crtero for proposed smlarty measure s determed by epert doctor/ medcal practtoer as follows: w = 0.5, w = 0.95, w 3 = 0.00, w 4 = 0.90, w 5 = 0.90 Step 4: Determato of the co-relato measure betwee two relatos: Usg equato taget fucto we calculate correlato measures betwee elato- ad elato- as follows see able 7.

7 ISSNOle : ISSN rt : able 7: he orrelato Measure betwee elato- ad elato- Weghted aget Vral ever Malara yphod Stomach problem hest problem smlarty measure Step 5: akg the alteratves: he hghest correlato measure from the able 7 gves proper medcal dagoss. herefore, patet suffers from Malara, suffers from Vral fever, suffers from Vral fever ad 4 suffers from vral fever. V. ONLUSION I ths paper, we have proposed a taget smlarty measure approach of fuzzy sets ad proved some of ther basc propertes. We have preseted a applcato of weghted taget smlarty measure of fuzzy sets medcal dagoss problem. I the future work, we wll eted ths taget smlarty measure to fuzzy mult sets. EEENES. Black, M., Vagueess: a eercse logcal aalyss hlosophy of Scece, Vol.44 4, pp , Zadeh, L.., rom crcut theory to systems theory, I roc. IE, Vol.50, pp , Zadeh, L.., ormato ad omputato, Vol.83, pp , Kuh,. S., he Structure of Scetfc evolutos, Uversty of hcago ress; 3rd edto December 5, Smth, H. J. S., O the tegrato of dscotuous fuctos", roceedgs of the Lodo Mathematcal Socety, Vol.6, pp.40-53, ator, G., Über uedlche, leare uktmagfaltgkete V, Mathematsche ale, Vol., pp , he, S. M., Yeh, M. S., ad Hsao,. Y., uzzy va 79-89, a, J., ad Xe, W., ol.0, pp.403 4, Hyug, Y. S., Sog, L. K., ad Lee, K. M., measures betwee fuzzy sets ad betwee ele 6, pp.9 93, apps,.., ad Karacaplds, I., 56, pp. 7 74, Wag, X. Z., Baets, B. D., ad Kerre, E., 59 68, Weke, D. V. D., Nachtegael, M., De, W. V., Schulte, S., ad Kerre, E. E., costructo of fuzzy smlarty -IEEE Iteratoal oferece o omputatoal Itellgece for Measuremets Systems ad pplcatos, Weke, D. V. D., Nachtegael, M., ad Kerre, E. E., measures ad homogeety for the comparso of m computg, Vol., pp , Wag, D. G., Meg, Y.., ad L, H. X., mathematcs wth applcatos, Vol.56, pp , he ew smlarty measure for fuzzy sets ad ts applcato to medca Iteratoal Joural of omputer pplcatos, Vol.805, 3-7, Kakat,., uzzy Sets wth the Eteded Defto of omplemetato, Iteratoal Joural of Soft omputg ad Egeerg, Vol.34, pp.03-07, Baruah, H. K., owards ormg eld Of uzzy Sets, Iteratoal Joural of Eergy, Iformato ad ommucatos, Vol., pp.6-0, Kakat,., Note o the New Smlarty Measure for uzzy sets, Iteratoal Joural of omputer pplcatos echology ad esearch, Vol.5, pp , 03. BIOGHY Dr. Surapat ramak h. D., M.S., M. Ed. dd hs B. Sc. ad M. Sc. Mathematcs from Uversty of Kalya. He receved h. D. Mathematcs 00 from Begal Egeerg ad Scece Uversty BESU Shbpur, Ida. He s curretly a ssstat rofessor of Mathematcs at the Nadalal Ghosh B.. ollege, apur,.o.-narayapur, West Begal, Ida. He has authored/co-authored more tha 60 research papers teratoal peer revewed jourals; He authored mathematcs method book for B. Ed. ourses from hel ublsher, Kolkata, Ida. Hs research terests

8 ISSNOle : ISSN rt : clude optmzato, soft computg, grey system theory, eutrosophc decso makg, rough sets, mathematcs educato, comparatve educato, teratoal relato. Kalya Modal M.Sc., B. Ed. passed B. Sc. Hoours ad M. Sc Mathematcs 00 ad 003 respectvely from the Uversty of alcutta ad Uversty of Kalya. urretly, he s a assstat eacher of mathematcs at Bragar Hgh School HS, Bragar, aaghat, Nada, ode: 747, West Begal, Ida. He has co-authored more tha 0 research papers teratoal jourals. Hs feld of research terests cludes fuzzy goal programmg, grey system theory, uzzy decso makg, tutostc fuzzy sets, eutrosophc sets, rough eutrosophc sets, ad eutrosophc decso makg.

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