SINGLE VALUED NEUTROSOPHIC EXPONENTIAL SIMILARITY MEASURE FOR MEDICAL DIAGNOSIS AND MULTI ATTRIBUTE DECISION MAKING

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1 teratioal Joural of Pure ad pplied Mathematics Volume 6 No SSN: (prited versio); SSN: (o-lie versio) url: doi: 0.73/ipam.v6i.7 Special ssue ipam.eu SNGLE VLUED NEUROSOPHC EXPONENL SMLRY MESURE OR MEDCL DGNOSS ND MUL RUE DECSON MKNG J.Dhivya ad.sridevi ssistat Professor Departmet of Mathematics is with Kumaraguru College of echology Coimbatore amil Nadu dia (Correspodig author phoe: dhivyamaths@gmail.com). ssociate Professor Departmet of Mathematics is with PSG College of echology Coimbatore amil Nadu dia sriichu@yahoo.com bstract Neutrosophic set (NS) is very useful to epress icomplete ucertaity ad icosistet iformatio i a more geeral way. the moder medical techologies each elemet ca be epressed as NS havig differet truth membership idetermiacy membership ad falsity membership degrees. hus the similarity measures for NS are importat tool to deal with the decisio makig problems with eutrosophic iformatios. However to overcome some drawbacks of eistig similarity measures this paper focus o itroducig Sigle Valued Neutrosophic Epoetial Similarity Measure (SVNESM) ad weighted SVNESM. he we compare the proposed SVNESM with the similarity measures available i the literature ad show their efficiecy usig some umerical eamples for overcomig the drawbacks. ially the similarity measure is applied for a medical diagosis problem ad Multi ttribute Decisio Makig (MDM) problem. Keywords Neutrosophic Set (NS) Sigle Valued Neutrosophic Set (SVNS) Similarity measure (SM) Sigle Valued Neutrosophic Epoetial Similarity Measure (SVNESM) Medical Diagosis Multi ttribute Decisio Makig (MDM). troductio Smaradache [0] itroduced the cocept of eutrosophic set (NS) to deal with imprecise idetermiate ad icosistet data. determiacy plays a importat role i may real world decisio makig problems. NS truth-membership 57

2 teratioal Joural of Pure ad pplied Mathematics Special ssue idetermiacy-membership ad falsity-membership are idepedet. he cocept of NSs geeralizes the fuzzy set itroduced by Zadeh [8] i 965 ad the ituitioistic fuzzy set proposed by taassov [] i 986. rom philosophical poit of view truth-membership idetermiacy-membership ad falsitymembership of the NS assume the value from real stadard or o-stadard subsets of ] 0 + [. Realizig the difficulty i applyig the eutrosophic sets i realistic problems Wag [] itroduced the cocept of a sigle valued eutrosophic set (SVNS) which is a subclass of the eutrosophic set. Neutrosophic set have bee applied i the field of medical diagosis [4] decisio makig problems [6 3 4] etc. ecause of the icreased volume of iformatio available i medical diagosis it cotais a lot of icomplete ucertaity ad icosistet iformatio to physicias. the multi attribute decisio makig (MDM) problems similarity measure approach ca be used i rakig the alteratives ad determiig the best amog them i eutrosophic form [ ].. Literature Review Let X ad X be two SVNSs i the uiverse of discourse X.... he Ye [5] Ye [6] ad Ye [7] preseted the Jaccard Dice cosie mproved Cosie ad aget similarity measures betwee SVNSs ad i vector space respectively as follows: Jaccard Similarity measure [5] S J () Dice Similarity measure [5] 58

3 D S () Cosie Similarity measure [5] i C S (3) mproved Cosie Similarity measure [6] SC cos (4) SC 6 cos (5) aget Similarity measure [7] ma 4 ta (6) ta (7) 3. Drawbacks of the eistig similarity measure Some drawbacks of Eqs. () to (7) as follows: (i) or two SVNSs ad if 0 ad/or 0 for ay i X (= ) Eqs. () () ad (3) is udefied or umeaigful. (ii) f ad or ad for ay i X (= ) applyig Eq. (3) we have teratioal Joural of Pure ad pplied Mathematics Special ssue 59

4 teratioal Joural of Pure ad pplied Mathematics Special ssue SC i i Sice. his meas that it fails to satisfy property S C (S3) i Defiitio. (iii) or the SVNSs C ad D y applyig Eq. (4) we get SC C D SC. this case improved cosie similarity measure produce a ureasoable pheomeo ad also ot carry out the recogitio for the two SVNSs C ad D as well fails to satisfy property (S3) i Defiitio. (iv) or the SVNSs ad 00 C ; C C C 0 C implies i Eq. (6) carry out higher similarity value ad produces a ureasoable pheomeo for the similarity measures of two SVNSs ad. Hece the similarity measures i Eq. (6) are ot suitable for hadlig patter recogitio ad medical diagosis problems. 3. Compariso of the similarity measures this sectio to eamie the performace of the proposed similarity measure o precisio ad discrimiatory ability a comparative study is coducted o differet sets used i the literature. Let ad C be three eutrosophic sets i the uiverse of discourse. able is used to compare the proposed similarity measure SNS with the eistig similarity measures[5-7]. table five differet patters case to case 5 are take ad the result obtaied by [5-7] ad SNS are listed. t is see that the proposed similarity measure SNS ca overcome the drawbacks of gettig the ureasoable results of the eistig 60

5 teratioal Joural of Pure ad pplied Mathematics Special ssue similarity measures S J [5] S D [5] S C [5] C [6] [6] [7] ad [7]. C rom able S C [5] caot carry out the recogitio betwee case ad case 5 also produces the ureasoable ad udefied pheomeo for case 5 ad case 4. he we ca see that the similarity measures S J S D C ad caot carry out the recogitio betwee case 3 ad case 4 while the similarity measures C ad ca carry out all recogitios. Hece the similarity measure C ad demostrate stroger discrimiatio amog them ad are superior to other similarity measures while the cosie similarity measure S C i vector space demostrates bad discrimiatio amog them ad is iferior to other similarity measures. Eve though C ad reveal stroger discrimiatio it gives the higher similarity value for all the cases discussed i able. Hece the similarity measure C ad are ot suitable for hadlig patter recogitio ad medical diagosis problems. he proposed measure overcomes the shortcomigs of the eistig measure by defiig the similarity measure usig the epoetial operatio. herefore the similarity measure S NS medical diagosis problems. able : Similarity measure values of Eqs. ()-(7) (9) will be applied to Case Case Case 3 Case 4 Case 5 < > < > <00> <00> < > < > < > <0> <000> < > S J [5] S D [5] S C [5] Null C [6] C [6] [7] [7] S NS pplicatios 6

6 teratioal Joural of Pure ad pplied Mathematics Special ssue this sectio the proposed SVNESM is applied to medical diagosis problems ad multi attribute decisio makig problems as discussed below. 6. Medical Diagosis usig Neutrosophic epoetial operatio Let us cosider the medical diagosis problem adopted from [7]. ssume that the set of diagosis Viral ever Q Malaria Q yphoid Q gastritis Q Steocardia ad a set of Q Q symptoms S ever S Headache S Stomach pai S Cough S Chest pai S he the attribute values of the cosidered diseases are represeted by SVNSs show i able. the medical diagosis assume that we take a sample from a patiet P with all the symptoms which is represeted by the followig NSs iformatio: Patiet S S S S P 3 4 S5. or compariso we ca utilize the eistig similarity measure [5-7] for hadlig the diagosis problem. Usig Eqs. () to (7) ad (9) we ca obtai the results of the three similarity measures betwee the patiet P ad cosidered disease Q i i as show i able 3. able : ttribute values of the cosidered diseases represeted by NSs S (ever) S (Headache) S 3 (Stomachpai) S 4 (Cough) S 5 (Chestpai) Q (Viral fever) <s 0.4 <s <s <s <s > 0.5> 0.7> > 0.7> Q (Malaria) <s 0.7 <s <s > <s <s > 0.6> 0.3 0> 0.8> Q 3 (yphoid) <s 0.3 <s <s <s 4 0. <s > 0.> 0.7> > 0.9> Q 4 (Gastritis) <s 0. <s <s > <s 4 0. <s > 0.4> > 0.7> Q 5 (Steocardia) <s 0. <s 0 0. <s > <s <s > 0.8> 0.8> 0.> able 3: Similarity measure values for NSs iformatio Viral ever Malaria (Q ) yphoid (Q 3 ) Gastritis (Q 4 ) Steocardia (Q ) (Q 5 ) S J [5] S D [5]

7 teratioal Joural of Pure ad pplied Mathematics Special ssue S C [5] C [6] C [6] [7] [7] S NS rom able 3 the largest similarity values idicate the proper diagosis. Hece the patiet P suffers from malaria. Evidetly the medical diagosis usig various similarity measures idicate the same diagosis results ad reveal the efficiecy of these diagosis. 6. Multi ttribute Decisio Makig Problems or coveiet compariso a illustrative eample about the selectio problem of ivestmet alteratives adopted from [6] is provided to demostrate the applicatios ad effectiveess of the proposed SVNESM. ivestmet compay eeds to ivest a sum of moey to the best idustry. he four possible alteratives are cosidered as four potetial idustries: (i) P - a car compay; (ii) P - a food compay; (iii) P3 - a computer compay; (iv) P4 - a arms compay. he four possible alteratives i the decisio-makig process must satisfy the requiremets of the three attributes: (i) R is the risk; (ii) R is the growth; (iii) R3 is the eviromet where the attributes R ad R are beefit types ad the attribute R3 is a cost type. ssume that the weightig vector of the attributes is give by w= ( ). y the statistical aalysis ad the evaluatio of ivestmet data regardig the four possible alteratives of Pi (i= 3 4) over the three attributes of R (= 3) we ca establish the followig decisio matri [6]: M p i

8 teratioal Joural of Pure ad pplied Mathematics Special ssue he we use Equatio (9) with eutrosophic iformatio which is described by the followig procedures: Step: Establish a ideal solutio (a ideal alterative) p p... p P epressed by the ideal NS P types ad cost types of attributes. correspodig to the beefit Step : Calculated the weighted SVNESM betwee the alterative P ad the ideal solutio P by usig Equatio (8) he weighted SVNESM betwee each alterative Pi (i= 3 4) ad the ideal solutio P ca be give as the followig values of S NS P P S NS P P S NS P P S NS P4 P the rakig order of the alteratives is P> P4> P3> P ad the best oe is P. hese results are the same as i [6]. Hece the proposed eutrosophic decisio-makig method based o the epoetial fuctio illustrates its feasibility ad effectiveess. Compared with eistig decisio-makig methods based o the NSs the proposed decisio-makig method based o the Neutrosophic epoetial similarity measure shows that it is simpler to employ tha eistig eutrosophic decisio-makig methods i [6 7 9] uder eutrosophic eviromets because the decisio-makig method proposed i this paper implies its simple algorithms ad decisio steps i the eutrosophic decisiomakig problems. 5. Coclusios his paper preseted a ew similarity measure of SVNESM based o the epoetial fuctio. he by cosiderig the importace of each elemet the weighted similarity measures of SVNESM are itroduced. Several ovel measures are available i the literature to access the similarity measures of NSs but the proposed measure correlates better tha the other measures. he proposed similarity measure effectively deals with some 64

9 teratioal Joural of Pure ad pplied Mathematics Special ssue demadig situatios by satisfyig the basic properties as well overcome the drawbacks of the eistig similarity measures. this study the proposed similarity measure ca be effectively used i real applicatios of medical diagosis ad decisio makig. Refereces [] taassov K. tuitioistic fuzzy sets. uzzy Sets ad Syst 986; 0: [] ttaasov K Gargov G. terval valued ituitioistic fuzzy sets. uzzy Sets Syst 989; 3: [3] roumi S Smaradache. Several similarity measures of eutrosophic sets. Neutrosophioc Sets syst 03; (): [4] De. S. K iswas R Roy.R applicatio of ituitioistic fuzzy sets i medical diagosis uzzy Sets ad Syst. 7 () (00) [5] Kou G Li C. Cosie maimizatio method for the priority vector deviatio i HP. Eur J Oper Res 04; 35: 5-3. [6] Liu P.D. eg. Multiple attribute decisio-makig method based o ormal eutrosophic geeralized weighted power averagig operator t. J. Mach. Lear. Cyber [7] Liu P.D Li H.G. Multiple attribute decisio-makig method based o some ormal eutrosophic oferroi mea operators. Neural Comput. pplic [8] Maumdar P Samata SK. O similarity ad etropy of eutrosophic sets. J tell uzzy syst 04; 6(3):