Article Simplified Neutrosophic Exponential Similarity Measures for the Initial Evaluation/Diagnosis of Benign Prostatic Hyperplasia Symptoms

Transcription

1 Artcle Smplfed eutrosophc Epoetal Smlarty easures for the tal Evaluato/Dagoss of Beg rostatc Hyperplasa Symptoms Jg u ad Ju Ye 2, * Shaog Secod Hosptal, 23 Yaa Road, Shaog 32000, Zheag, Cha; gsao@63.com 2 Departmet of Electrcal ad formato Egeerg, Shaog versty, 508 Huacheg West Road, Shaog 32000, Zheag, Cha * Correspodece: yehu@alyu.com; el.: Receved: 24 July 207; Accepted: 9 August 207; ublshed: August 207 Abstract: Whe a physca carres out the clcal survey of a patet wth beg prostatc hyperplasa BH symptoms to reach the tal evaluato/dagoss of BH, the estg tal evaluato method of BH based o the teratoal prostate symptom score -SS usually uses the obectve evaluato/dagoss method wth crsp values wthout cosderg fuzzy formato. However, ths commo evaluato/dagoss method may lead to the loss of a great deal of useful complete, ucerta, ad cosstet formato the clcal survey ad tal evaluato process of the BH symptoms for a patet, resultg a ureasoable evaluato ad dagoss dstorto of the BH symptoms. o overcome ths drawback, ths paper ams to propose ew epoetal smlarty measures ESs betwee smplfed eutrosophc sets SSs, cludg sgle-valued eutrosophc ESs ad terval eutrosophc ESs, ad ther tal evaluato/dagoss method of the BH symptoms wth smplfed eutrosophc formato. ally, two evaluato/dagoss eamples of the BH symptoms are provded to demostrate the effectveess ad ratoalty of the proposed method. Keywords: beg prostatc hyperplasa; medcal dagoss; smplfed eutrosophc set; sgle valued eutrosophc set; terval eutrosophc set; epoetal smlarty measure. troducto Beg prostatc hyperplasa BH s a commo medcal problem ecoutered agg me, ad leads to obstructve ad rrtatve vodg symptoms. he Amerca rologcal Assocato AA uses seve questos as the AA symptom dces,2] for BH scored o a scale from 0 to 5 pots. he teratoal prostate symptom score -SS,2] offers a obectve documetato of symptoms: a total score of 0 7 s mldly symptomatc, 8 9 moderately symptomatc, ad severely symptomatc. However, the obectve evaluato s a o-fuzzy evaluato method a commo evaluato method -SS. he tal evaluato of the BH symptoms s obtaed by meas of clcal survey for a patet to select further eamatos e.g., create, traveous urography, urethrogram, urodyamcs, urethrocystoscopy, etc. ad sutable treatmet alteratves e.g., watchful watg, medcal, surgcal, or mmally vasve surgcal treatmets, etc.. he, the choce of treatmet s reached a shared decso-makg process betwee the physca ad the patet. Whe the physca carres out the clcal survey of a patet to reach the tal evaluato of the BH symptoms, the patet gves the resposes to the seve questos whch may cota a grey zoe of the ucertaty for the patet about the BH symptoms. hus, the clcal data of the BH symptoms obtaed by the physca are complete, ucerta, or cotradctory. ths case, fuzzy epresso s a sutable tool. Symmetry 207, 9, 54; do:0.3390/sym

2 Symmetry 207, 9, 54 2 of 0 Zadeh 3] frst troduced the membershp/truth degree 965 ad defed a fuzzy set. Based o a geeralzato of the fuzzy set, Ataassov 4] troduced the omembershp/falsty degree 986 ad defed a tutostc fuzzy set S as a geeralzato of the fuzzy set. he, Ataassov ad Gargov 5] troduced a terval-valued S as a geeralzato of S. urther, Smaradache 6] troduced the degree of determacy/eutralty as a depedet compoet 995 ad defed a eutrosophc set as the geeralzato of S ad the terval-valued S. He coed the terms eutrosophy ad eutrosophc. rom a phlosophcal pot of vew, the eutrosophc set ca represet ucerta, mprecse, complete, ad cosstet formato. ts advatage s that the eutrosophc set ca epress determate ad cosstet formato, but S ad the terval-valued S caot. rom a scece ad egeerg pot of vew, the eutrosophc set wll be dffcult to apply real scece ad egeerg felds 7,8] because the truth, determacy, ad falsty fuctos a eutrosophc set belog to the o-stadard terval ] 0,. herefore, Smaradache 6] ad Wag et al. 7,8] proposed the cocepts of a sgle-valued eutrosophc set SVS ad a terval eutrosophc set S, where the truth, determacy, ad falsty fuctos are costraed the real stadard terval 0, ] as the subclasses of the eutrosophc set. urther, Ye 9] troduced the cocept of a smplfed eutrosophc set SS, whch s a subclass of the eutrosophc set cludg the cocepts of SVS ad S. SSs are very sutable for hadlg medcal dagoss problems, sce a symptom may mply a lot of complete, ucerta, ad cosstet formato for a dsease, whch characterzes a relato betwee symptoms ad a dsease. Recetly, SSs have bee appled to medcal dagoss problems. Ye 0] preseted the mproved cose smlarty measures betwee SSs for medcal dagoss. As a geeralzato of SVS, Ye et al.,2] troduced a sgle-valued eutrosophc multset ad the Dce smlarty measure ad dstace-based smlarty measures of sgle-valued eutrosophc multsets, ad the appled them to medcal dagoss. Broum ad Del 3] preseted a correlato measure of eutrosophc refed sets eutrosophc multsets ad ther applcato medcal dagoss. Broum ad Smaradache 4] troduced a eteded Hausdorff dstace ad ts smlarty measure of refed eutrosophc sets eutrosophc multsets ad appled the smlarty measure to medcal dagoss. urthermore, Ye ad u 5] put forward a sgle-valued eutrosophc smlarty measure based o taget fucto ad the taget smlarty measure-based mult-perod medcal dagoss method a dyamc medcal dagoss method. However, t s dffcult to adapt the above-metoed dagoss methods to deal wth the evaluato problems of BH wth smplfed eutrosophc formato. Geerally, the estg tal evaluato method of BH s commoly based o -SS,2] ad uses the obectve evaluato/dagoss method wth crsp values wthout cosderg fuzzy formato. Hece, ths commo evaluato/dagoss method may lose a lot of complete, ucerta, ad cosstet formato the clcal survey ad tal evaluato process of the BH symptoms for a patet, resultg ureasoable evaluato ad dagoss dstorto of the BH symptoms. o overcome ths drawback, ths paper ams to propose ew epoetal smlarty measures ESs betwee SSs, cludg sgle-valued eutrosophc ESs ad terval eutrosophc ESs, ad ther tal evaluato/dagoss method of the BH symptoms wth smplfed eutrosophc formato. he rest of the artcle s structured as follows. Secto 2, we brefly troduce some basc cocepts of SSs. Secto 3 proposes ESs betwee SSs based o epoetal fucto, cludg sgle-valued eutrosophc ESs ad terval eutrosophc ESs, ad vestgates ther propertes. Secto 4, the tal evaluato/dagoss methods of the BH symptoms are preseted based o the ESs uder a smplfed eutrosophc evromet, ad the two eamples of the evaluato of BH symptoms are gve to show the effectveess ad ratoalty of the proposed evaluato method. Coclusos ad further research are gve Secto Basc Cocepts of SSs he SS troduced by Ye 9] s the geeralzato of a S ad a terval-valued S, ad gves us a addtoal possblty to represet complete, ucerta, ad cosstet formato,

3 Symmetry 207, 9, 54 3 of 0 whch ests real world. herefore, t s more sutable for applcatos a determate ad cosstet evromet. he defto of SS s troduced as follows. Defto. referece 9], let be a SS a uverse of dscourse X, whch s characterzed by a truth-membershp fucto, a determacy-membershp fucto, ad a falsty-membershp fucto. he, the SS ca be epressed as {,,, X }, where,, ad are sgleto subtervals/subsets the real stadard 0, ], such that : X 0, ], : X 0, ], ad : X 0, ]. As a subclass of the eutrosophc set, the SS cotas SVS for,, 0, ], ad 0 3, ad S for,, 0,,] ad 0 sup sup sup 3 for each pot X. Assume that {,,, X} ad {,,, X} are two SSs X. f,, 0, ], 0 3,,, 0, ], ad 0 3 for each pot X, the ad are reduced to two SVSs. hus, the cluso, equato, ad complemet for SSs ad are defed, respectvely, as follows 9]: f ad oly f,, for ay X; 2 f ad oly f ad ; c c,, X,,, X. 3 { }, ad { } Assume that {,,, X } ad {,,, X } are two SSs X. f,, 0, ], 0 sup sup sup 3,,, 0, ], ad 0 sup sup sup 3 for each pot X, the ad are reduced to two Ss. hus, the cluso, equato, ad complemet for SSs ad are defed, respectvely, as follows 9]: f ad oly f f f, f f, f f, sup sup, sup sup, sup sup for ay X; 2 f ad oly f ad ; c {,f,sup ], sup, f ],f,sup ] X } 3 ad c {,f,sup ], sup, f ],f,sup ] X }. Especally whe the upper ad lower eds of the terval umbers,, ad,, are equal, the Ss ad are reduced to the SVSs ad. herefore, SVSs are the specal cases of Ss, ad also SVSs ad Ss are also the specal cases of SSs. 3. ESs of SSs Based o a epoetal fucto, ths secto proposes ESs betwee SSs, cludg sgle-valued eutrosophc ESs ad terval eutrosophc ESs, ad vestgates ther propertes. Defto 2. et {,,, X} ad {,,, X} be ay two SVSs X {, 2,, }. hus, we ca defe a ES betwee ad as follows: E, ep 3 ep ep. Obvously, ES has the followg proposto. roposto. or two SVSs ad X {, 2,, }, the ES E, should satsfy the followg propertes 4:

4 Symmetry 207, 9, 54 4 of 0 0 E, ; 2 E, f ad oly f,.e.,,, ad for X, 2,, ; 3 E, E, ; 4 f s a SVS X ad, the E, E, ad E, E,. roof. Sce there are 0, ] ad 0, ] the two SVSs ad, the dstace 3 / les betwee 0 ad. By applyg Equato, ES also les betwee 0 ad. Hece, there s 0 E,. 2 or the two SVSs ad, f, ths mples,, for X ad, 2,,. Hece, there are 0, 0, ad 0. hus, we ca obta the followg result: ep ep 3, ep ep / ep. E f E,, we have the followg equato: ep ep 3,. ep E he, there ests the followg result: ep ep 3 ep. hs mples ep / ep, ad the there are 0, 0, ad 0. hus, these equaltes dcate that,, ad for X ad, 2,,. Hece. 3 roof s straghtforward. 4 f, the ths mples,, for X ad, 2,,. he, we have,,,,,. Hece, E, E, ad E, E, sce the epoetal fucto 3 ep s a decreasg fucto. herefore, the proofs of these propertes are completed. Geerally, oe takes the weght of each elemet for X to accout ad assumes that the weght of a elemet s w, 2,, wth w 0, ] ad w. Hece, we ca troduce the followg weghted ES betwee ad :

5 Symmetry 207, 9, 54 5 of 0 w W ep ep 3 ep,. 2 Clearly, the ES W, should satsfy the propertes 4 roposto. Especally whe w / for, 2,,, Equato 2 reduces to Equato. Smlarly, we ca eted the ESs of SVSs to propose ESs betwee Ss. et {,,, X} ad {,,, X} be ay two Ss X {, 2,, }, where f, sup ] 0, ], f, sup ] 0, ], ad f, sup ] 0, ] for ay X are deoted by ],, ],, ad ],, respectvely, ad f, sup ] 0, ], f, sup ] 0, ], ad f, sup ] 0, ] for ay X are deoted by ],, ],, ad ],, respectvely, for coveece. he, based o the eteso of the above smlarty measures Equatos ad 2, we ca troduce the followg two ESs betwee ad : E 2 ep ep 6 ep,, 3 w W 2 ep ep 6 ep,, 4 where w s the weght of a elemet, 2,, wth w 0, ] ad w. Obvously, Equatos ad 2 are the specal cases of Equatos 3 ad 4 whe the upper ad lower eds of the terval umbers,, ad,, are equal. herefore, the above ESs of Ss also satsfy propertes 4 roposto. he proof s smlar to that of roposto, ad thus t s ot repeated here. 4. tal Evaluato/Dagoss ethod of BH sg the ESs Accordg to the seve questos the AA symptom dees,2] for BH, we ca cosder a set of the seve questos Q {Q Over the past moth, how ofte have you had a sesato of ot emptyg your bladder completely after you fshed uratg?, Q2 Over the past moth, how ofte have you had to urate aga less tha two hours after you fshed uratg?, Q3 Over the past moth, how ofte have you foud you stopped ad started aga several tmes whe you urated?, Q4 Over the past moth, how ofte have you foud t dffcult to postpoe urato?, Q5 Over the past moth, how ofte have you had a weak urary stream?, Q6 Over the past moth, how ofte have you had to push or stra to beg urato?, Q7 Over the past moth, how may tmes dd you most typcally get up to urate from the tme you wet to bed at ght utl the tme you got up the morg?} for a physca to survey the patets BH symptoms. he clcal survey of the umber of BH symptoms the 5 tmes for a patet k k, 2,, t ca be costructed by able, where,, ad deote truth, determacy, ad falsty, respectvely. able. he umber of beg prostatc hyperplasa BH symptoms the 5 tmes for a patet k. : true; : determate; : false. Questo Q: Over the past moth, how ofte have you had a sesato of ot emptyg your bladder completely after you fshed uratg? Q2: Over the past moth, how ofte have you had to uratg aga less tha two hours after you fshed uratg?

6 Symmetry 207, 9, 54 6 of 0 Q3: Over the past moth, how ofte have you foud you stopped ad started aga several tmes whe you urated? Q4: Over the past moth, how ofte have you foud t dffcult to postpoe urato? Q5: Over the past moth, how ofte have you had a week urary stream? Q6: Over the past moth, how ofte have you had to push or stra to beg urato? Q7: Over the past moth, how may tmes dd you most typcally get up to urate from the tme you wet to bed at ght utl the tme you got up the morg? Based o -SS,2], BH ca be dvded to the four types of symptoms, whch are represeted by a set of the four types of symptoms S {S ormal symptom, S2 ld symptom, S3 oderate symptom, S4 Severe symptom} as the symptom kowledge for the tal evaluato of BH patets, as show able 2. able 2. our types of BH symptoms wth smplfed eutrosophc formato. S Symptom ype Q Q2 Q3 Q4 Q5 Q6 Q7 S ormal symptom <0, 0, > <0, 0, > <0, 0, > <0, 0, > <0, 0, > <0, 0, > <0, 0, > S2 ld symptom <0, 0.2, 0.8> <0, 0.2, 0.8> <0, 0.2, 0.8> <0, 0.2, 0.8> <0, 0.2, 0.8> <0, 0.2, 0.8> <0, 0.2, 0.8> S3 oderate symptom <0.2, 0.4, 0.4> <0.2, 0.4, 0.4> <0.2, 0.4, 0.4> <0.2, 0.4, 0.4> <0.2, 0.4, 0.4> <0.2, 0.4, 0.4> <0.2, 0.4, 0.4> S4 Severe symptom <0.6, 0.4, 0> <0.6, 0.4, 0> <0.6, 0.4, 0> <0.6, 0.4, 0> <0.6, 0.4, 0> <0.6, 0.4, 0> <0.6, 0.4, 0> rom able 2, the BH symptom types of patets wth respect to all the questos ca be represeted by the followg SS formato: S { Q, 0, 0,, Q2, 0, 0,, Q3, 0, 0,, Q4, 0, 0,, Q5, 0, 0,, Q6, 0, 0,, Q7, 0, 0, }, S2 { Q, 0, 0.2, 0.8, Q2, 0, 0.2, 0.8, Q3, 0, 0.2, 0.8, Q4, 0, 0.2, 0.8, Q5, 0, 0.2, 0.8, Q6, 0, 0.2, 0.8, Q7, 0, 0.2, 0.8 }, S3 { Q, 0.2, 0.4, 0.4, Q2, 0.2, 0.4, 0.4, Q3, 0.2, 0.4, 0.4, Q4, 0.2, 0.4, 0.4, Q5, 0.2, 0.4, 0.4, Q6, 0.2, 0.4, 0.4, Q7, 0.2, 0.4, 0.4 }. S4 { Q, 0.6, 0.4, 0, Q2, 0.6, 0.4, 0, Q3, 0.6, 0.4, 0, Q4, 0.6, 0.4, 0, Q5, 0.6, 0.4, 0, Q6, 0.6, 0.4, 0, Q7, 0.6, 0.4, 0 }. Assume that we gve the clcal survey for t BH patets by usg able to obta the t patets resposes of the BH symptom whch are represeted by the form of truth, determacy, ad falsty values. or a patet k k, 2,, t wth SS formato, we ca gve the followg evaluato/dagoss method. o gve a proper evaluato/dagoss for a patet k wth BH symptoms, we ca calculate the smlarty measure Wqk, S for q or 2,, 2, 3, 4 ad k, 2,, t. he proper BH symptom * evaluato S* for patet k s derved by argma{ W, S }. 4 o llustrate the evaluato/dagoss process of the BH symptoms, we provde two evaluato/dagoss eamples of the BH symptoms to demostrate the applcatos ad effectveess of the proposed evaluato/dagoss method uder smplfed eutrosophc sgle-valued eutrosophc ad terval eutrosophc evromets. 4. tal Evaluato of the BH Symptoms der a Sgle-Valued eutrosophc Evromet some cases, we ca obta that data collected from the clcal survey of patets are sgle values rather tha terval values for,, ad. ths case, ES of SVSs s a better tool to gve a proper tal evaluato of a patet s BH symptoms. Eample. Assume that we gve a clcal survey for three BH patets by usg able, ad the we ca obta the three patets resposes of the BH symptoms whch are represeted by the form of truth, determacy, ad falsty values, as show able 3. q k

7 Symmetry 207, 9, 54 7 of 0 able 3. he umber of BH symptoms sgle values the 5 tmes for three patets. 2 3 Questo Q 2/5 /5 2/5 /5 /5 3/5 3/5 0/5 2/5 Q2 2/5 2/5 /5 2/5 /5 2/5 3/5 /5 /5 Q3 2/5 /5 2/5 /5 0/5 4/5 3/5 /5 /5 Q4 2/5 /5 2/5 2/5 /5 2/5 4/5 /5 0/5 Q5 3/5 2/5 0/5 /5 2/5 2/5 3/5 /5 /5 Q6 2/5 0/5 3/5 2/5 0/5 3/5 4/5 /5 0/5 Q7 3/5 0/5 2/5 /5 /5 3/5 2/5 2/5 /5 rom able 3, the BH symptom resposes of the patet k k, 2, 3 wth respect to all the questos ca be represeted by the followg SVS formato: { Q, 0.4, 0.2, 0.4, Q2, 0.4, 0.4, 0.2, Q3, 0.4, 0.2, 0.4, Q4, 0.4, 0.2, 0.4, Q5, 0.6, 0.4, 0.0, Q6, 0.4, 0.0, 0.6, Q7, 0.6, 0.0, 0.4 }, 2 { Q, 0.2, 0.2, 0.6, Q2, 0.4, 0.2, 0.4, Q3, 0.2, 0.0, 0.8, Q4, 0.4, 0.2, 0.4, Q5, 0.2, 0.4, 0.4, Q6, 0.4, 0.0, 0.6, Q7, 0.2, 0.2, 0.6 }, 3 { Q, 0.6, 0.0, 0.4, Q2, 0.6, 0.2, 0.2, Q3, 0.6, 0.2, 0.2, Q4, 0.8, 0.2, 0.0, Q5, 0.6, 0.2, 0.2, Q6, 0.8, 0.2, 0.0, Q7, 0.4, 0.4, 0.2 }. Assume that the weght of each elemet Q s w /7 for, 2,, 7. By applyg Equato 2, we ca obta the results of the smlarty measure betwee the patet k k, 2, 3 ad the cosdered symptom S, 2, 3, 4, as show able 4. able 4. Smlarty measure values betwee k ad S wth sgle-valued eutrosophc sets SVSs. S S2 S3 S4 W, S W2, S W3, S able 4, the largest smlarty measure dcates the proper evaluato/dagoss. herefore, tal clcal evaluatos for the three patets, atets ad 2 have moderate symptoms, ad atet 3 has severe symptoms. 4.2 tal Evaluato of the BH Symptoms der a terval eutrosophc Evromet some cases, we ca obta that data collected from the clcal survey of patets are terval values rather tha sgle values for,, ad, sce patets easly epress real stuatos by usg the terval values. ths case, ES of Ss s a better tool to gve a proper tal evaluato of the BH symptoms. Eample 2. Assume that we gve the clcal survey for three BH patets by usg able, ad the we ca obta the three patets resposes of the BH symptoms, whch are represeted by the terval values of,, ad, as show able 5. able 5. he umber of BH symptoms terval values the 5 tmes for three patets. 2 3 Questo Q 2/5, 3/5] 0/5, /5] /5, 2/5] /5, 2/5] 0/5, /5] 2/5, 3/5] 3/5, 4/5] 0/5, 0/5] /5, 2/5] Q2 2/5, 3/5] 2/5, 3/5] 0/5, /5] 2/5, 2/5] 0/5, /5] /5, 2/5] 3/5, 4/5] /5, /5] 0/5, /5] Q3 2/5, 3/5] /5, 2/5] /5, 2/5] /5, 2/5] 0/5, /5] 3/5, 4/5] 3/5, 4/5] /5, 2/5] 0/5, 0/5] Q4 2/5, 3/5] 0/5, /5] /5, 2/5] 2/5, 3/5] /5, 2/5] 0/5, 2/5] 3/5, 4/5] /5, 2/5] 0/5, 0/5] Q5 3/5, 4/5] /5, 2/5] 0/5, 0/5] /5, 2/5] 2/5, 3/5] 0/5, /5] 3/5, 4/5] /5, 2/5] 0/5, /5] Q6 2/5, 3/5] 0/5, /5] 2/5, 3/5] 2/5, 3/5] 0/5, /5] 2/5, 3/5] 3/5, 4/5] /5, 2/5] 0/5, 0/5] Q7 2/5, 3/5] 0/5, /5] /5, 2/5] /5, 2/5] /5, 2/5] /5, 2/5] 2/5, 3/5] 2/5, 3/5] 0/5, /5]

8 Symmetry 207, 9, 54 8 of 0 rom able 5, the BH symptom resposes of patet k k, 2, 3 wth respect to all the questos ca be represeted by the followg S formato: { Q, 0.4, 0.6], 0, 0.2], 0.2, 0.4], Q2, 0.2, 0.4], 0.4, 0.6], 0, 0.2], Q3, 0.4, 0.6], 0.2, 0.4], 0.2, 0.4], Q4, 0.4, 0.6], 0, 0.2], 0.2, 0.4], Q5, 0.6, 0.8], 0.2, 0.4], 0, 0], Q6, 0.4, 0.6], 0, 0.2], 0.4, 0.6], Q7, 0.4, 0.6], 0, 0.2], 0.2, 0.4] }, 2 { Q, 0.2, 0.4], 0, 0.2],0.4, 0.6], Q2, 0.4, 0.4], 0, 0.2], 0.2, 0.4], Q3, 0.2, 0.4], 0, 0.2], 0.6, 0.8], Q4, 0.4, 0.6], 0.2, 0.4], 0, 0.4], Q5, 0.2, 0.4], 0.4, 0.6], 0, 0.2], Q6, 0.4, 0.6], 0, 0.2], 0.4, 0.6], Q7, 0.2, 0.4], 0.2, 0.4], 0.2, 0.4] }, 3 { Q, 0.6, 0.8], 0, 0], 0.2, 0.4], Q2, 0.6, 0.8], 0.2, 0.2], 0, 0.2], Q3, 0.6, 0.8], 0.2, 0.4], 0, 0], Q4, 0.6, 0.8], 0.2, 0.4], 0, 0], Q5, 0.6, 0.8], 0.2, 0.4], 0, 0.2], Q6, 0.6, 0.8,] 0.2, 0.4], 0, 0], Q7, 0.4, 0.6], 0.4, 0.6], 0, 0.2] }. Assume that the weght of each elemet Q s w /7 for, 2,, 7. By applyg Equato 4, we ca obta the results of the smlarty measure betwee atet k k, 2, 3 ad the cosdered symptom S, 2, 3, 4, as show able 6. able 6. Smlarty measure values of betwee k ad S wth terval eutrosophc sets Ss. S S2 S3 S4 W2, S W22, S W23, S able 6, the largest smlarty measure dcates the proper evaluato/dagoss. herefore, tal clcal evaluatos for the three patets, atets ad 3 have severe symptoms, ad atet 2 has moderate symptoms. 4.3 Comparso ad Aalyss or coveet comparso wth the commo evaluato method,2], based o ables 3 ad 5 we oly gve the BH symptom resposes wth the values of wthout the values of ad -SS of the patet k k, 2, 3 wth respect to all the questos of Eamples ad 2, whch are show ables 7 ad 8, respectvely. Accordg to the commo evaluato method of -SS,2], where oe tme meas oe score -SS,2], we ca gve the clcal tal evaluato results of three patets, 2, 3, whch are also show ables 7 ad 8, respectvely. able 7. he umber of BH symptoms sgle values of the 5 tmes for three patets, where oe tme meas oe score the teratoal prostate symptom score -SS. 2 3 Questo tme tme tme Q 2 / / / / 3 / / Q2 2 / / 2 / / 3 / / Q3 2 / / / / 3 / / Q4 2 / / 2 / / 4 / / Q5 3 / / / / 3 / / Q6 2 / / 2 / / 4 / / Q7 3 / / / / 2 / / otal score BH symptom oderate oderate Severe able 8. he umber of BH symptoms terval values of the 5 tmes for three patets. 2 3 Questo tme tme tme Q 2, 3] / /,2] / / 3, 4] / / Q2 2, 3] / / 2, 2] / / 3, 4] / / Q3 2, 3] / /,2] / / 3, 4] / /

9 Symmetry 207, 9, 54 9 of 0 Q4 2, 3] / / 2, 3] / / 3, 4] / / Q5 3, 4] / /,2] / / 3, 4] / / Q6 2, 3] / / 2, 3] / / 3, 4] / / Q7 2, 3] / /,2] / / 2, 3] / / otal score 5, 22] 0, 6] 20, 27] BH symptom oderate ad/or severe oderate Severe able 7, all the tal evaluato/dagoss results of Eample are the same as the oes of the ew evaluato method proposed ths paper. he, all the tal evaluato/dagoss results of Eample 2 able 8 are almost the same as the oes of the ew evaluato method, but the evaluato/dagoss of s dffcult to determe the moderate ad/or severe symptoms the commo evaluato/dagoss method based o -SS,2]. However, has severe symptoms based o the comprehesve evaluato/dagoss results ths study. herefore, the dagoss results of the two eamples demostrate the effectveess of the proposed dagoss method uder smplfed eutrosophc evromets. Compared wth the estg tal evaluato method based o -SS,2], the proposed evaluato method demostrates ther effectveess ad ratoalty because the developed tal evaluato method wth smplfed eutrosophc formato cotas much more evaluato formato truth, determacy, ad falsty formato tha the estg tal evaluato method based o -SS crsp values wthout determacy ad falsty formato,2]. Obvously, the commo tal evaluato method based o -SS may lose much useful formato determacy ad falsty formato, resultg the ureasoable/dffcult evaluato ad dagoss dstorto for some patets. herefore, the developed dagoss method s more sutable ad more practcal the tal evaluato of BH symptoms, s superor to the estg tal evaluato method,2], ad also shows a advatage terms of the effectveess ad ratoalty of the proposed dagoss method. 5. Coclusos Based o a epoetal fucto, ths paper proposed ESs of SSs, cludg sgle-valued eutrosophc ESs ad terval eutrosophc ESs. he, a tal evaluato/dagoss method for BH symptoms was establshed based o ESs uder a smplfed eutrosophc evromet. ally, two llustratve eamples of the tal evaluatos of BH symptoms are provded to demostrate the applcato ad effectveess of the proposed evaluato method a smplfed eutrosophc settg. he advatage of the evaluato method developed ths paper s that t ca deal wth medcal dagoss problems wth complete, ucerta, ad cosstet formato, whle the estg tal evaluato method,2] caot hadle them ad may lose much useful formato evaluato process. further work, t s ecessary to apply ESs of SSs to other medcal problems, such as medcal mage processg ad medcal clusterg aalyss. Ackowledgmet: hs paper was supported by the atoal atural Scece oudato of Cha o Author Cotrbutos: Ju Ye proposed epoetal smlarty measures ESs betwee smplfed eutrosophc sets SSs, cludg sgle valued eutrosophc ESs ad terval eutrosophc ESs; Jg u establshed a tal evaluato/dagoss method for BH symptom based o ESs uder a smplfed eutrosophc evromet ad provded two llustratve eamples o the tal evaluatos of BH symptom ad comparatve aalyss; we wrote the paper together. Coflct of terest: he authors declare o coflcts of terest. Refereces. AA ractce Gudeles Commttee. Amerca rologcal Assocato Gudele o the aagemet of Beg rostatc Hyperplasa; Amerca rologcal Assocato Educato ad Research, c.: thcum, D, SA, 2003.

10 Symmetry 207, 9, 54 0 of 0 2. AA ractce Gudeles Commttee. Amerca rologcal Assocato Gudele o the aagemet of Beg rostatc Hyperplasa BH Revsed Verso; Amerca rologcal Assocato Educato ad Research, c.: thcum, D, SA, Zadeh,.A. uzzy Sets. f. Cotrol 965, 8, Ataassov, K. tutostc fuzzy sets. uzzy Sets Syst. 986, 20, Ataassov, K.; Gargov, G. terval valued tutostc fuzzy sets. uzzy Sets Syst. 989, 3, Smaradache,. eutrosophy. eutrosophc robablty, Set, ad ogc; Amerca Research ress: Rehoboth,, SA, Wag, H.; Smaradache,.; Zhag, Y.Q.; Suderrama, R. Sgle valued eutrosophc sets. ultsp. ultstr. 200, 4, Wag, H.; Smaradache,.; Zhag, Y.Q.; Suderrama, R. terval eutrosophc Sets ad ogc: heory ad Applcatos Computg; Hes: hoe, AZ, SA, Ye, J. A multcrtera decso-makg method usg aggregato operators for smplfed eutrosophc sets. J. tell. uzzy Syst. 204, 26, Ye, J. mproved cose smlarty measures of smplfed eutrosophc sets for medcal dagoses. Artf. tell. ed. 205, 63, Ye, S.; Ye, J. Dce smlarty measure betwee sgle valued eutrosophc multsets ad ts applcato medcal dagoss. eutrosophc Sets Syst. 204, 6, Ye, S.; u, J.; Y, J. edcal dagoss usg dstace-based smlarty measures of sgle valued eutrosophc multsets. eutrosophc Sets Syst. 205, 7, Broum, S.; Del,. Correlato measure for eutrosophc refed sets ad ts applcato medcal dagoss. alest. J. ath. 204, 3, Broum, S.; Smaradache, S. Eteded Hausdorff dstace ad smlarty measure of refed eutrosophc sets ad ther applcato medcal dagoss. J. ew heor. 205,, Ye, J.; u, J. ult-perod medcal dagoss method usg a sgle valued eutrosophc smlarty measure based o taget fucto. Comput. ethods rogr. Bomed. 206, 23, by the authors. cesee D, Basel, Swtzerlad. hs artcle s a ope access artcle dstrbuted uder the terms ad codtos of the Creatve Commos Attrbuto CC BY lcese