DPGWO based feature selection machine learning model for prediction of crack dimensions in steam generator tubes

Ci a ion: William, M.V.A.; Ramesh, S.; Cep, R.; Mahalingam, S.K.; Elango an, M. DPGWO Based Fea u e Selec ion Machine Lea ning Model o P edic ion o C ack Dimensions in S eam Gene a o Tubes. Appl. Sci. 2023,13, 8206. h ps://doi.o g/10.3390/ app13148206 Academic Edi o : Ki-Yong Oh Recei ed: 21 June 2023 Re ised: 11 July 2023 Accep ed: 13 July 2023 Published: 14 July 2023 Copy igh : © 2023 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). applied sciences A icle DPGWO Based Fea u e Selec ion Machine Lea ning Model o P edic ion o C ack Dimensions in S eam Gene a o Tubes Ma hias Vijay Albe William 1, Sub amanian Ramesh 2, Robe Cep 3,* , Si a Kuma Mahalingam 4 and Muniyandy Elango an 5,6,* 1Depa men o Elec onics and Communica ion Enginee ing, Vel Tech Ranga ajan D . Sagun hala R&D Ins i u e o Science and Technology, A adi 600062, India 2Depa men o Elec ical and Elec onics Enginee ing, Vel Tech Ranga ajan D . Sagun hala R&D Ins i u e o Science and Technology, A adi 600062, India 3Depa men o Machining, Assembly and Enginee ing Me ology, Facul y o Mechanical Enginee ing, VSB-Technical Uni e si y o Os a a, 70800 Os a a, Czech Republic 4Depa men o Mechanical Enginee ing, Vel Tech Ranga ajan D . Sagun hala R&D Ins i u e o Science and Technology, A adi 600062, India 5Depa men o Biosciences, Sa ee ha School o Enginee ing, Sa ee ha Naga , Thandalam 602105, India 6Depa men o R&D, Bond Ma ine Consul ancy, London EC1V 2NX, UK *Co espondence: [email p o ec ed] (R.C.); muniyandy[email p o ec ed] (M.E.) Abs ac : The selec ion o an app op ia e numbe o ea u es and hei combina ions will play a majo ole in imp o ing he lea ning accu acy, compu a ion cos , and unde s anding o machine lea ning models. In his p esen wo k, 22 g ay-le el co-occu ence ma ix ea u es ex ac ed om magne ic lux leakage images cap u ed in s eam gene a o ubes’ c acks a e conside ed o de el- oping a machine lea ning model o p edic and analyze c ack dimensions in e ms o hei leng h, dep h, and wid h. The pe o mance o he models is examined by conside ing R 2 and RMSE alues calcula ed using bo h aining and es ing da a se s. The F Sco e and Mu ual In o ma ion Sco e me hods ha e been applied o p io i ize he ea u es. To analyze he e ec o di e en machine lea ning models, hei numbe o ea u es, and hei selec ion me hods, a Taguchi expe imen al design has been implemen ed and an analysis o a iance es has been conduc ed. The dynamic popula ion g ay wol algo i hm (DPGWO) has been adop ed o selec he bes ea u es and hei combina ions. Due o he wo con adic o y na u es o pe o mance me ics, Pa e o op imal solu ions a e conside ed, and he bes one is ob ained using Deng’s me hod. The e ec i eness o DPGWO is p o ed by compa ing i s pe o mance wi h G ey Wol Op imiza ion and Mo h Flame Op imiza ion al- go i hms using he F iedman es and pe o mance indica o s, namely in e ed gene a ional dis ance and spacing. Keywo ds: machine lea ning model; ea u e selec ion me hods; op imiza ion algo i hms; F iedman es ; Deng’s me hods; pe o mance indica o s 1. In oduc ion In nuclea powe plan s, c i ical componen s such as s eam gene a o ubes (SGT), eed wa e hea e s, and p essu e essels ha e s ingen design equi emen s due o he high empe a u e, p essu e, and adia ion-exposing en i onmen , which induce s ess co osion c acking, pi ing, ouling, and mechanical e ing [ 1 ]. Apa om ha , s eam gene a o ube up u e (SGTR) leads o he ine i able elease o adia ion in o he en i onmen [ 2 ]. Fo he sa e y and eliable ope a ion o he plan , i is necessa y o go o online moni o ing and pe iodic inspec ion [ 3 , 4 ]. Non-des uc i e es ing (NDT) is a mode o es ing ha has been adop ed by indus ies o mo e han a decade o es mass-manu ac u ed p oduc s o anomalies. This p ocess is inc easingly being used by sec o s such as ae ospace, oil and gas, pe oleum, nuclea , and cons uc ion indus ies [ 5 – 7 ]. Mo e pa ailu es occu due Appl. Sci. 2023,13, 8206. h ps://doi.o g/10.3390/app13148206 h ps://www.mdpi.com/jou nal/applsci Appl. Sci. 2023,13, 8206 2 o 33 o he complexi y o he de ices p oduced. To a oid such hings, NDT indings o ecas ailu e and enhance he sa e y and economy o en e p ises. Sump uous non-des uc i e e alua ion echniques ha e been es ablished o majo sys ems such as powe plan s and ai planes o con i m he du abili y o he NDT es and hei sa e y. Singh e al. [ 8 ] p oposed he magne ic lux leakage (MFL) echnique o iden i y he localized aul s in he SGT. Zhang e al. [ 9 ] implemen ed he MFL echnique o de ec bo h shallow su ace and deep sub-su ace de ec s in e omagne ic ma e ials. Su esh e al. [ 10 , 11 ] sugges ed he MFL app oach o he de ec ion o de ec s and subsu ace c acks in small-diame e SGT. The expe imen al se up o he measu emen o MFL is de ailed in Su esh e al. [ 10 ]. Daniel e al. [ 12 ] designed an ANN model o o ecas he SGT’s de ec in e ms o he leng h, b ead h, and dep h o he c ack by p o iding he g ay-le el co-occu ence ma ix (GLCM) in o ma ion ex ac ed om he MFL images. Wang e al. [ 13 ] s udied a de ailed e iew o he applica ion o ML models in p edic ing he ou comes o s oke wi h s uc u ed da a. The andom o es (RF) ML algo i hm has been implemen ed o p edic he ou come o endo ascula ea men [ 14 ] and acu e s oke [ 15 ] and imp o e p edic ion accu acy in Ischemic s oke pa ien s. Decision ee (DT) and ex eme g adien boos ing (XGB) ML algo i hms we e implemen ed o imp o e he accu acy o he p edic ion o Ischemic s oke pa ien s [16]. In ML, ea u e selec ion, a me hod o selec ing independen pa ame e s, educes he compu a ion ime and complexi y o he p oblem by elimina ing i ele an , no -use ul, and edundan ea u es. Due o i s combina o ial na u e, ea u e selec ion is ea ed as a non-polynomial ha d p oblem, which c ea es he oppo uni y o he esea che s o implemen me a-heu is ic algo i hms. De i e al. [ 17 ] p oposed a new a ian o he Golden Jackel Op imiza ion (GJO) algo i hm called Imp o ed GJO (IGJO) o sol e he ea u e selec ion p oblem in ML. Qa aad e al. [ 18 ] in oduced quad a ic in e pola ion wi h a salp swa m-based local escape ope a o in he ea u e selec ion o 19 da ase s and conduc ed F iedman and Wilcoxon es s o analyze he esul s. Houssein e al. [ 19 ] imple- men ed a cen oid mu a ion-based sea ch and escue op imiza ion algo i hm o ea u e selec ion in 15 disease da a se s wi h di e en ea u e sizes ex ac ed om he UCI ma- chine lea ning eposi o y and compa ed he pe o mance wi h six exis ing me a-heu is ic algo i hms. Fil e -based and w appe -based echniques a e he wo basic ca ego ies un- de which ea u e selec ion may be classi ied [ 20 , 21 ]. While he w appe -based s a egy bases i s e alua ion o he solu ion h oughou he sea ching and op imiza ion p oce- du es on he lea ning algo i hm, he il e -based app oach uses he co ela ion be ween he da a and he ele an class label wi hou consul ing he lea ning algo i hm. The w appe -based echnique is he mos equen ly employed ea u e selec ion me hod as compa ed o he less compu a ionally expensi e il e -based app oach due o i s highe pe o mance accu acy [22,23]. In his wo k, he ML model is o be de eloped o p edic he SGT’s de ec in e ms o leng h, wid h, and dep h o c ack om he gi en g ay le el co-occu ence ma ix ea u e om he MFL image. The ea u e selec ion me hod and he numbe o selec ed ea u es will a ec he pe o mance o he ML models. The R 2 and oo mean squa e e o (RMSE) alues a e conside ed me ics o measu e he pe o mance o he ML models. Mul iple con- adic o y pe o mance measu es equi e he con e sion o mul iple objec i es in o a single objec i e, which ini ia ed he implemen a ion o a mul i-c i e ion decision-making me hod, namely Deng’s me hod. The dynamic popula ion g ay wol op imiza ion (DPGWO) algo- i hm is implemen ed o selec he numbe o ea u es and hei combina ions o minimize he p edic ion e o . The e ec i eness o he DPGWO algo i hm is p o ed by compa ing i s pe o mance wi h he g ay wol op imiza ion (GWO) Algo i hm [ 24 ] and mo h lame op imiza ion algo i hm (MFO) [ 25 ] by implemen ing a non-pa ame ic F iedman es [ 25 ] and pe o mance indica o s. The pape is o ganized as ollows. Sec ion 2desc ibes he p oblem s a emen , and Sec ion 3desc ibes he p oposed me hodology and he a ious s ages, namely p io i izing he ea u es, Taguchi o hogonal a ay expe imen al design [ 25 ], analysis o Va iance Appl. Sci. 2023,13, 8206 3 o 33 (ANOVA) Tes , and me a-heu is ic algo i hms o ea u e selec ion in ol ed in sol ing he p oblem. Sec ion 4deals wi h he esul s and discussion, including a compa ison o he pe o mance o he DPGWO algo i hm wi h he MFO and GWO algo i hms. Finally, he conclusion pa o he pape gi es u u e scope. 2. P oblem S a emen Any model mus be app op ia ely selec ed o ma ch he equi emen s o he applica- ion. Decision-make s analyze he beha io o da a using p edic ion models. Reg ession models, a i icial neu al ne wo ks, and suppo ec o machines a e a ew echniques used o build p edic ion models. Each echnique has i s own p os and cons based on he size o he da ase . Recen ly, machine lea ning echniques ha e assis ed esea che s in de el- oping mo e accu a e p edic ion models. The ype o machine lea ning (ML) model will di e om p oblem o p oblem. The same ML model may no gi e simila pe o mance ac oss applica ions. Hence, he selec ion o a sui able ML model o he gi en p oblem is a challenging ask. Apa om ha , he selec ion o app op ia e ea u es will play a majo ole in imp o - ing he lea ning accu acy, compu a ion cos , and unde s anding o he model. O e i ing o models is one o he majo p oblems ha will educe he applica ion o ML in a ious ields. The p oblem conside ed in his wo k will be de eloping an accu a e ML model wi h a smalle numbe o ea u es o p edic he c ack dimensions om 22 GLCM ea u es ex ac ed om 105 MFL c ack images p esen ed by Daniel e al. [12]. 3. Me hodology The main ocus o he p esen wo k is o implemen a sui able ML Model wi h an app op ia e selec ion o ea u es and hei combina ions o p edic he c ack dimensions mo e accu a ely as compa ed wi h he exis ing ANN model. Daniel e al. [ 12 ] om he gi en 22 GLCM ea u es ex ac ed om 115 MFL c ack images. This has been ca ied ou in ou s ages. In he i s s age, he ea u es a e p io i ized based on F Sco e (FS) and Mu ual In o ma ion Sco e (MIS) alues. L16 Taguchi o hogonal a ay design [ 25 ] has been cons uc ed by conside ing ea u e selec ion me hods (FSM), machine lea ning models (MLM), and numbe o ea u es as pa ame e s and R 2 and RMSE alues as esponse alues. Fo each expe imen , py hon codes a e execu ed using he co esponding MLM, FSM, and a ge numbe o ea u es as de e mined by he Taguchi a ay o de e mine he co esponding R 2 and RMSE alues o bo h aining and es da a se s. To analyze he e ec o pa ame e s and hei in e ac ion on he esponse alues, an ANOVA es has been ca ied ou in he hi d s age. The esponses, namely R 2 and RMSE alues, o he aining and es ing da a se s, a e di e en o each expe imen , which is simila o di e en al e na i es among mul iple design p oblems. To selec he bes design, hese mul iple esponses a e o be con e ed in o a single alue. Hence, in his wo k, a simila i y- based mul i-c i e ion anking me hod called Deng’s me hod [ 26 ] is in oduced o selec he bes ML model in he hi d s age. In he ou h s age, me a-heu is ic algo i hms, namely MFO, GWO, and DPGWO, a new a ian o GWO p oposed in his wo k, ha e been implemen ed o selec he numbe o ea u es and hei combina ions. Also, Pe o mance indica o s and F iedman’s es ha e been conduc ed in he ou h s age o p o e he e ec- i eness o he DPGWO, along wi h s a is ical analysis o con i m he esul s. The p oposed me hodology is shown in Figu e 1. The s ep-by-s ep algo i hm o he p oposed wo k is gi en below. Appl. Sci. 2023,13, 8206 4 o 33 Appl. Sci. 2023, 13, x FOR PEER REVIEW 4 o 35 Figu e 1. P oposed Me hodology. Figu e 1. P oposed Me hodology. Appl. Sci. 2023,13, 8206 5 o 33 S ep 1: Read he GLCM ea u es and hei co esponding c ack dimension da ase s. S ep 2: P io i ize he ea u es’ o de based on (a) Mu ual In o ma ion Sco e (MIS) (b) F Sco e (FS) S ep 3: A ange he ea u es based on MIS. S ep 4: Selec he i s 15 ea u es and ix one o he c ack’s dimensions as he a ge alue. S ep 5: Se a machine lea ning model and selec R 2 and RMSE as pe o mance me ics o he machine lea ning model. S ep 6: Se he size o a aining da a se , and based on ha , sepa a e he aining and es ing da a se s along hei a ge alues. S ep 7: Fi he ML model o he da a se and p edic bo h aining and es ing a ge alues. S ep 8: Compu e he esponse alues (pe o mance me ics, namely R 2 and RMSE) o he ML model o bo h he aining and es ing da a se s. S ep 9: Repea s eps 5 o 8 by changing he di e en ML models. S ep 10: Repea s eps 4 o 9 by changing he numbe o ea u es o 17, 19, and 21. S ep 11: Repea s eps 3 o 10 by a anging he ea u es based on he F Sco e. S ep 12: Conduc an ANOVA and s a is ical es o es he signi icance o he pa ame e s on he esponse alues. S ep 10: Implemen Deng’s me hod o selec he bes machine lea ning model based on R2 and RMSE alues. S ep 11: Tune he hype -pa ame e s o he bes machine lea ning model. S ep 12: Implemen he MFO, GWO, and DPGWO algo i hms by andomly selec ing he gi en ea u es along wi h hei a ge alues be ween 15 and 21. S ep 13: Compa e he pe o mance o algo i hms using pe o mance indica o s (In e ed Gene a ional Dis ance and Spacing) and F iedman’s Tes . Selec he bes numbe o ea u es and hei combina ions. S ep 14: Repea he s eps om 3 o 15 o o he dimensions o he c ack, like c ack dep h and c ack wid h. 3.1. S age 1: P io i izing he Fea u es Conside a ion o all ea u es in machine lea ning may lead o poo pe o mance, excess compu a ion ime, and o e i ing o he p edic ion model. To a oid his, ea u e selec ion echniques ha e been used in ML. Since he w appe me hod o ea u e selec ion is an i e a i e p ocess and slow in na u e, in his wo k wo il e me hods, namely, he F sco e (FS) and he mu ual in o ma ion sco e (MIS), ha e been in oduced o selec ion. In he F sco e me hod, ea u es a e selec ed s a is ically by iden i ying he ela ionship be ween pa ame ic ea u es and a ge ea u es, whe eas in he mu ual in o ma ion sco e me hod, ea u es a e selec ed based on hei en opy. Table 1 ep esen s he lis o 22 GLCM ea u es ex ac ed om MFL images o c ack dimensions. Figu e 2 ep esen s he p io i ized ea u es o c ack dimensions. I is unde s ood ha in Figu e 2a, he 3 d ea u e is gi en p io i y and he 2nd ea u e is gi en leas he p io i y in he F-Sco e-based p io i ized me hod, whe eas in he MIS me hod, he 1s ea u e has i s p io i y and he 19 h ea u e has he leas p io i y in c ack leng h p edic ion. In c ack wid h p edic ion, he 7 h ea u e has i s p io i y in bo h p io i ized me hods. Di e en ea u es, namely he 20 h and 6 h ea u es, ha e he leas p io i y in c ack dep h p edic ion [27]. Table 1. GLSM Fea u e De ails (Daniel e al. [12]). FNo. FName Fea u e Name 0 UNF Ene gy/Uni o mi y 1 ETR En opy 2 DSL Dissimila i y Appl. Sci. 2023,13, 8206 6 o 33 Table 1. Con . FNo. FName Fea u e Name 3 CST Con as 4 ID In e se Di e ence 5 CN Co ela ion 6 H Homogenei y 7 AC Au o co ela ion 8 CS Clus e shade 9 CP Clus e p ominence 10 MP Maximum P obabili y 11 SS Sum o squa es 12 SA Sum a e age 13 SV Sum Va iance 14 SE Sum en opy 15 DV Di e ence Va iance 16 DE Di e ence en opy 17 IMC (1) In o ma ion measu e o co ela ion1 18 IMC (2) In o ma ion measu e o co ela ion 2 19 MCC Maximal Co ela ion Coe icien 20 INN In e se Di e ence No malized 21 IDN In e se di e en momen no malized FNo.—Fea u e Numbe and Fname—Fea u e Name. Appl. Sci. 2023, 13, x FOR PEER REVIEW 6 o 35 3 CST Con as 4 ID In e se Di e ence 5 CN Co ela ion 6 H Homogenei y 7 AC Au o co ela ion 8 CS Clus e shade 9 CP Clus e p ominence 10 MP Maximum P obabili y 11 SS Sum o squa es 12 SA Sum a e age 13 SV Sum Va iance 14 SE Sum en opy 15 DV Di e ence Va iance 16 DE Di e ence en opy 17 IMC (1) In o ma ion measu e o co ela ion1 18 IMC (2) In o ma ion measu e o co ela ion 2 19 MCC Maximal Co ela ion Coe icien 20 INN In e se Di e ence No malized 21 IDN In e se di e en momen no malized FNo.—Fea u e Numbe and Fname—Fea u e Name. (a) (b) (c) (d) Figu e 2. Con . Appl. Sci. 2023,13, 8206 7 o 33 Appl. Sci. 2023, 13, x FOR PEER REVIEW 7 o 35 (e) ( ) Figu e 2. P io i ized ea u es o c ack dimensions. (a) F Sco e-based p io i ized ea u es o C ack Leng h. (b) MIS-based p io i ized ea u es o C ack Leng h. (c) F Sco e-based p io i ized Fea u es o C ack Dep h. (d) MIS-based p io i ized ea u es o C ack Dep h. (e) F Sco e-based p io i ized Fea u es o C ack Wid h. ( ) MIS-based p io i ized ea u es o C ack Wid h. 3.2. S age 2: Taguchi O hogonal A ay Expe imen al Design o ML Model Selec ion In his wo k, a o al o six di e en ML models ha e been used, in which a di e en combina ion o ou models is conside ed o each c ack dimension. The ML models’ names a e desc ibed in Table 2. The pa ame e s and hei le els conside ed o c ack leng h a e ep esen ed in Table 3. The ML models conside ed o c ack dep h and c ack wid h a e ep esen ed in Tables A1 and A4. Table 4 ep esen s he expe imen al pa ame e se ings o he L16 Taguchi o hogonal a ay o he gi en combina ion o le els o pa am- e e s men ioned in Table 3, ob ained using Mini ab 19 So wa e, whe e 16 ep esen s he minimum numbe o expe imen al uns equi ed o conduc he expe imen s and o ind he bes pa ame e se ings [28]. Fo each expe imen al un co esponding o i s pa ame e se ings, Py hon code has been execu ed o he aining da a se wi h a sample size o 95 images’ ea u es and es ed wi h a da a size o 20 images. The pe o mance o he ML model is eco ded o each expe imen based on he R2 and RMSE alues o bo h he ain- ing and es ing da a se s. Deng’s me hod has been implemen ed o selec he bes ML model based on he o e all pe o mance index calcula ed using R2 and RMSE alues, which a e lis ed in Table 4. L16 Taguchi o hogonal a ay and Deng’s alues o c ack dep h and c ack wid h a e p esen ed in Tables A2 and A5. Table 2. Lis o Machine Lea ning (ML) Models Used in C ack Leng h. Model Name DT Decision T ee Reg esso LoR Logis ic Reg esso LiR Linea Reg esso XGB Ex eme G adien Boos e ABR Adap i e Boos e Reg esso RF Random Fo es Reg esso Table 3. Pa ame e s and I s Le el o C ack Leng h. Pa ame e No. o Le els Le els 1 2 3 4 FSM 2 FS MIS MLM 4 DT LiR LoR XGB NoF 4 15 17 19 21 Figu e 2. P io i ized ea u es o c ack dimensions. ( a ) F Sco e-based p io i ized ea u es o C ack Leng h. ( b ) MIS-based p io i ized ea u es o C ack Leng h. ( c ) F Sco e-based p io i ized Fea u es o C ack Dep h. ( d ) MIS-based p io i ized ea u es o C ack Dep h. ( e ) F Sco e-based p io i ized Fea u es o C ack Wid h. ( ) MIS-based p io i ized ea u es o C ack Wid h. 3.2. S age 2: Taguchi O hogonal A ay Expe imen al Design o ML Model Selec ion In his wo k, a o al o six di e en ML models ha e been used, in which a di e en combina ion o ou models is conside ed o each c ack dimension. The ML models’ names a e desc ibed in Table 2. The pa ame e s and hei le els conside ed o c ack leng h a e ep esen ed in Table 3. The ML models conside ed o c ack dep h and c ack wid h a e ep- esen ed in Tables A1 and A4. Table 4 ep esen s he expe imen al pa ame e se ings o he L16 Taguchi o hogonal a ay o he gi en combina ion o le els o pa ame e s men ioned in Table 3, ob ained using Mini ab 19 So wa e, whe e 16 ep esen s he minimum numbe o expe imen al uns equi ed o conduc he expe imen s and o ind he bes pa ame e se ings [ 28 ]. Fo each expe imen al un co esponding o i s pa ame e se ings, Py hon code has been execu ed o he aining da a se wi h a sample size o 95 images’ ea u es and es ed wi h a da a size o 20 images. The pe o mance o he ML model is eco ded o each expe imen based on he R 2 and RMSE alues o bo h he aining and es ing da a se s. Deng’s me hod has been implemen ed o selec he bes ML model based on he o e all pe o mance index calcula ed using R 2 and RMSE alues, which a e lis ed in Table 4. L16 Taguchi o hogonal a ay and Deng’s alues o c ack dep h and c ack wid h a e p esen ed in Tables A2 and A5. Table 2. Lis o Machine Lea ning (ML) Models Used in C ack Leng h. Model Name DT Decision T ee Reg esso LoR Logis ic Reg esso LiR Linea Reg esso XGB Ex eme G adien Boos e ABR Adap i e Boos e Reg esso RF Random Fo es Reg esso Table 3. Pa ame e s and I s Le el o C ack Leng h. Pa ame e No. o Le els Le els 1 2 3 4 FSM 2 FS MIS MLM 4 DT LiR LoR XGB NoF 4 15 17 19 21 Appl. Sci. 2023,13, 8206 8 o 33 Table 4. L16 OA—Pe o mance o ML o C ack Leng h. E.No. FSM MLM NoF R2T R2T RMSET RMSET Deng’s Value 1 FS DT 15 0.2619 0.3737 0.9684 1.4000 0.4569 2 FS DT 17 0.1998 0.4572 1.0000 1.3000 0.4771 3 MIS DT 19 0.6100 0.1434 1.0278 1.6440 0.4879 4 MIS DT 21 0.5940 0.1429 1.0282 1.6446 0.5040 5 FS LiR 15 0.9596 0.6091 0.2346 1.2090 0.4434 6 FS LiR 17 0.9592 0.6209 0.2389 1.1815 0.4616 7 MIS LiR 19 0.9221 0.6109 0.3917 1.1977 0.4643 8 MIS LiR 21 0.9078 0.5437 0.4225 1.2860 0.4248 9 MIS LoR 15 0.6002 0.1449 0.8200 1.6425 0.5484 10 MIS LoR 17 0.6120 0.1441 0.8150 1.6433 0.5360 11 FS LoR 19 0.5860 0.1434 0.8008 1.6441 0.5324 12 FS LoR 21 0.3960 0.1430 0.8400 1.6445 0.5408 13 MIS XGB 15 0.8652 0.8646 0.4410 0.6942 0.6073 14 MIS XGB 17 0.8757 0.8705 0.4055 0.6869 0.6444 15 FS XGB 19 1.0000 0.5988 0.0005 1.1801 0.6263 16 FS XGB 21 1.0000 0.5841 0.0006 1.2013 0.6326 3.3. S age 3: ANOVA Tes The ANOVA Tes is used o analyze he di e ences among he means o di e en g oups using a iance and o p o e which g oups a e s a is ically signi ican . To p o e ha he pe o mance is signi ican ly di e en o a ious ML models, an ANOVA es [ 24 ] is implemen ed in his wo k. Figu e 3 ep esen s he p obabili y plo o pe o mance measu es o he ML model ob ained by execu ing he Py hon codes. I is unde s ood ha he alues o R 2 and RMSE o bo h aining and es ing a e wi hin he 95% con idence in e al and ha hei p obabili y alues a e g ea e han 0.005. This shows ha he expe imen a ion esul s a e accep able o c ack leng h p edic ion. Simila ly, o c ack dep h and c ack wid h, he p- alue shown in Figu e A1 and Table A6 is g ea e han 0.005, which e eals ha he expe imen al esul s ob ained a e accep able. The main and in e ac ion e ec plo s o pa ame e s a e depic ed in Figu e 4a–h o c ack leng h p edic ion. I is in e ed om Figu e 4a,b ha he mean alue o R2T is highe when he pa ame e s a e MIS, 15, and XGB, whe eas in R2T he pa ame e s a e FS and 18, espec i ely. I is unde s ood om Figu e 4c,d, ha o ge ing he lowe mean alue o RMSET and RMSET , he pa ame e s a e o be se in a di e en way han R2T and R2T . F om Figu e 4e, , i is no ed ha inc easing he NoF alue dec eases he R2T alue o all ML models, whe eas in R2T excep MLM and DT in all o he models, i inc eases. Using XGB, a lowe RMSE is epo ed in Figu e 4g,h. Appl. Sci. 2023, 13, x FOR PEER REVIEW 8 o 35 Table 4. L16 OA—Pe o mance o ML o C ack Leng h. E.No. FSM MLM NoF R2T R2T RMSET RMSET Deng’s Value 1 FS DT 15 0.2619 0.3737 0.9684 1.4000 0.4569 2 FS DT 17 0.1998 0.4572 1.0000 1.3000 0.4771 3 MIS DT 19 0.6100 0.1434 1.0278 1.6440 0.4879 4 MIS DT 21 0.5940 0.1429 1.0282 1.6446 0.5040 5 FS LiR 15 0.9596 0.6091 0.2346 1.2090 0.4434 6 FS LiR 17 0.9592 0.6209 0.2389 1.1815 0.4616 7 MIS LiR 19 0.9221 0.6109 0.3917 1.1977 0.4643 8 MIS LiR 21 0.9078 0.5437 0.4225 1.2860 0.4248 9 MIS LoR 15 0.6002 0.1449 0.8200 1.6425 0.5484 10 MIS LoR 17 0.6120 0.1441 0.8150 1.6433 0.5360 11 FS LoR 19 0.5860 0.1434 0.8008 1.6441 0.5324 12 FS LoR 21 0.3960 0.1430 0.8400 1.6445 0.5408 13 MIS XGB 15 0.8652 0.8646 0.4410 0.6942 0.6073 14 MIS XGB 17 0.8757 0.8705 0.4055 0.6869 0.6444 15 FS XGB 19 1.0000 0.5988 0.0005 1.1801 0.6263 16 FS XGB 21 1.0000 0.5841 0.0006 1.2013 0.6326 3.3. S age 3: ANOVA Tes The ANOVA Tes is used o analyze he di e ences among he means o di e en g oups using a iance and o p o e which g oups a e s a is ically signi ican . To p o e ha he pe o mance is signi ican ly di e en o a ious ML models, an ANOVA es [24] is implemen ed in his wo k. Figu e 3 ep esen s he p obabili y plo o pe o mance measu es o he ML model ob ained by execu ing he Py hon codes. I is unde s ood ha he alues o R2 and RMSE o bo h aining and es ing a e wi hin he 95% con idence in e al and ha hei p obabili y alues a e g ea e han 0.005. This shows ha he expe - imen a ion esul s a e accep able o c ack leng h p edic ion. Simila ly, o c ack dep h and c ack wid h, he p- alue shown in Figu e A1 and Table A6 is g ea e han 0.005, which e eals ha he expe imen al esul s ob ained a e accep able. (a) (b) Figu e 3. Con . Appl. Sci. 2023,13, 8206 9 o 33 Appl. Sci. 2023, 13, x FOR PEER REVIEW 9 o 35 (c) (d) Figu e 3. P obabili y Plo o Pe o mance Measu es o ML Models in C ack Leng h. (a) P obabili y Plo o C ack Leng h-R2T . (b) P obabili y Plo o C ack Leng h-R2T . (c) P obabili y Plo o C ack Leng h-RMSET . (d) P obabili y Plo o C ack Leng h-RMSET . The main and in e ac ion e ec plo s o pa ame e s a e depic ed in Figu e 4a–h o c ack leng h p edic ion. I is in e ed om Figu e 4a,b ha he mean alue o R2T is highe when he pa ame e s a e MIS, 15, and XGB, whe eas in R2T he pa ame e s a e FS and 18, espec i ely. I is unde s ood om Figu e 4c,d, ha o ge ing he lowe mean alue o RMSET and RMSET , he pa ame e s a e o be se in a di e en way han R2T and R2T . F om Figu e 4e, , i is no ed ha inc easing he NoF alue dec eases he R2T alue o all ML models, whe eas in R2T excep MLM and DT in all o he models, i in- c eases. Using XGB, a lowe RMSE is epo ed in Figu e 4g,h. (a) (b) (c) (d) Figu e 3. P obabili y Plo o Pe o mance Measu es o ML Models in C ack Leng h. ( a ) P obabili y Plo o C ack Leng h-R 2 T . ( b ) P obabili y Plo o C ack Leng h-R 2 T . ( c ) P obabili y Plo o C ack Leng h-RMSET . (d) P obabili y Plo o C ack Leng h-RMSET . Appl. Sci. 2023, 13, x FOR PEER REVIEW 9 o 35 (c) (d) Figu e 3. P obabili y Plo o Pe o mance Measu es o ML Models in C ack Leng h. (a) P obabili y Plo o C ack Leng h-R2T . (b) P obabili y Plo o C ack Leng h-R2T . (c) P obabili y Plo o C ack Leng h-RMSET . (d) P obabili y Plo o C ack Leng h-RMSET . The main and in e ac ion e ec plo s o pa ame e s a e depic ed in Figu e 4a–h o c ack leng h p edic ion. I is in e ed om Figu e 4a,b ha he mean alue o R2T is highe when he pa ame e s a e MIS, 15, and XGB, whe eas in R2T he pa ame e s a e FS and 18, espec i ely. I is unde s ood om Figu e 4c,d, ha o ge ing he lowe mean alue o RMSET and RMSET , he pa ame e s a e o be se in a di e en way han R2T and R2T . F om Figu e 4e, , i is no ed ha inc easing he NoF alue dec eases he R2T alue o all ML models, whe eas in R2T excep MLM and DT in all o he models, i in- c eases. Using XGB, a lowe RMSE is epo ed in Figu e 4g,h. (a) (b) (c) (d) Figu e 4. Con . Appl. Sci. 2023,13, 8206 16 o 33 se s. Ea ly con e gence is eco ded in i e a ion numbe s 16, 18, 30, and 28 in he cases o R 2 T , RMSET , R 2 T , and RMSET , espec i ely, in Figu e A5b,c,e, . Simila kinds o Pa e o solu ion and con e gence plo s a e ob ained in he case o c ack wid h, as shown in Figu e A6a– . I is also obse ed ha a majo di e ence in RMSE alue is achie ed in DPGWO as compa ed o MFO and GWO in Figu e A6c, . The p- alue shown in Figu e A4 con i ms ha he esul s ob ained o 25 uns a e no mally dis ibu ed and accep able. Pe - o mance indica o s a e used o compa e he pa e o solu ions gene a ed by he algo i hms. In his wo k, wo di e en pe o mance indica o s, namely in e ed gene a ional dis ance (IGD) and spacing (SP), a e implemen ed o check he e ec i eness o he algo i hms. Fo p oblems wi h mo e han h ee objec i es, IGD is used o measu e he quali y o pa e o op imal solu ions in e ms o dis ibu ion and con e gence, whe eas SP is used o measu e he dis ibu ion and sp ead o pa e o op imal solu ions. In bo h cases, he lowe he alue, he be e he pe o mance o he algo i hm. F om Table 12, i is con i med ha he DPGWO algo i hm has lowe alues o IGD and SP as compa ed o he o he wo algo i hms, hence i is ou pe o med. Table 12. Pe o mance Indica o s o Algo i hms [30]. C ack Dimension Algo i hm IGD SP C ack Leng h MFO 0.10066 0.05977 GWO 0.10413 0.04707 DPGWO 0.09652 0.04558 C ack Leng h MFO 0.23810 0.49592 GWO 0.21844 0.24535 DPGWO 0.19339 0.08908 C ack Leng h MFO 0.20932 0.15989 GWO 0.34371 0.13514 DPGWO 0.08245 0.06335 To u he suppo he pe o mance o he algo i hms, F iedman’s es , a non-pa ame ic es , has been implemen ed in his wo k. This es is used o de e mine whe he s a is ically signi ican di e ences exis be ween he means o h ee o mo e g oups. In his wo k, he Deng’s alues ob ained by 25 uns using h ee algo i hms a e conside ed o anking and he calcula ion o mean alues. Using he “F iedman ()” unc ion in Ma lab, his es is conduc ed. Table 13 ep esen s F iedman’s Tes alues and he mean anking o each algo- i hm. All he p obabili y alues o c ack dimensions a e less han 0.05, which shows ha he algo i hm’s pe o mance di e s om each o he , i.e., a signi ican e ec will be he e on he selec ion o algo i hms. Usually, a lowe alue will be allo ed ank 1 in F iedman’s Tes . Bu in his wo k, he highe he Deng’s alue, he be e he solu ion; he highe he mean ank alue, he highe he pe o mance o he algo i hm; hence, i is p o ed ha he mean ank alue o 2.24 in he case o c ack leng h p edic ion using DPGWO ou pe o med as compa ed o he mean ank alues o 2.2 and 1.56 o he MFO and GWO algo i hms, espec i ely. Simila ly, in he case o c ack dep h and wid h p edic ion, he mean ank o DPGWO will be 2.23 and 2.12, which a e highe han he mean ank o MFO and GWO. Table 13. F iedman’s Tes Values and Mean Ranking o Algo i hms [25]. C ack Dimensions Mean Rank P obabili y MFO GWO DPGWO C ack Leng h 2.2 1.56 2.24 0.0263 C ack Dep h 1.98 1.78 2.23 0.0049 C ack Wid h 1.84 2.04 2.12 0.0093 Appl. Sci. 2023,13, 8206 17 o 33 Appl. Sci. 2023, 13, x FOR PEER REVIEW 17 o 35 (a) (b) (c) (d) (e) ( ) Figu e 7. Pa e o and Con e gence Plo o C ack Leng h. (a) Pa e o Plo o T aining Da a Se —C ack Leng h. (b) R2 Con e gence Plo o T aining Da a Se — C ack Leng h. (c) RMSE Con e gence Plo o T aining Da a Se —C ack Leng h. (d) Pa e o Plo o Tes ing Da a Se —C ack Leng h. (e) R2 Con e gence Plo o Tes ing Da a Se —C ack Leng h. ( ) RMSE Con e gence Plo o Tes ing Da a Se —C ack Leng h. Figu e 7. Pa e o and Con e gence Plo o C ack Leng h. ( a ) Pa e o Plo o T aining Da a Se —C ack Leng h. ( b ) R 2 Con e gence Plo o T aining Da a Se —C ack Leng h. ( c ) RMSE Con e gence Plo o T aining Da a Se —C ack Leng h. ( d ) Pa e o Plo o Tes ing Da a Se —C ack Leng h. ( e ) R 2 Con e gence Plo o Tes ing Da a Se —C ack Leng h. ( ) RMSE Con e gence Plo o Tes ing Da a Se —C ack Leng h. Appl. Sci. 2023,13, 8206 18 o 33 The numbe o ea u es and hei combina ions ob ained by implemen ing he MFO, GWO, and DPGWO o each c ack dimension a e p esen ed in Table 14, along wi h ea- u es conside ed in he Taguchi O hogonal A ay Me hod (TM) and Desi abili y Me hod (DM). In Table 14, ‘N’ ep esen s ha he pa icula ea u e is no selec ed by he me h- ods/algo i hms. Table 14. Lis o Fea u es Conside ed in Va ious Me hods. FNo. Fname C ack Leng h C ack Dep h C ack Wid h TM DM OT TM DM OT TM DM OT MFO GWO DPGWO MFO GWO DPGWO MFO GWO DPGWO 0 UNF N N N 1 ETR N N N N N N N 2 DSL N N N N N 3 CST N N N N N N N 4 ID N N N N N N N 5 CN N N N N N N 6 H N N 7 AC N N N N N N N N 8 CS 9 CP 10 MP 11 SS 12 SA 13 SV 14 SE 15 DV 16 DE N N N N 17 IMC(1) N N N N 18 IMC(2) N 19 MCC 20 INN N 21 IDN N N N N N N N N N N No. o Fea u es 19 17 17 17 17 19 21 17 17 17 17 19 17 17 17 The ea u e ha has a highe coun o ‘N’ is decided o be an impo an ea u e. The lis o no -impo an ea u es compa ed by a ious me hods is p esen ed in Table 15. A o al o 5 ou o 22 ea u es a e no conside ed impo an ea u es. Table 15. No Impo ance Fea u es. FNo. FName 1 ETR 3 CST 4 ID 7 AC 21 IDN The pe o mance me ics o he sample conside ed in he exis ing li e a u e by Daniel e al. [ 12 ] a e calcula ed based on he numbe o ea u es and hei combina ions gi en in Table 16. Figu es 8–10 ep esen he ac ual c ack dimensions o he sample along wi h he calcula ed c ack dimensions using he exis ing me hods, MFO, GWO, and DPGWO. I is clea ha all he dimensions calcula ed by DPGWO a e e y close o he ac ual di- mensions. Table 16 ep esen s he compa ison o he pe o mance me ics wi h exis ing and p oposed algo i hms o he sample da a se conside ed by Daniel e al. I is obse ed ha he R 2 alue is highe in DPGWO as compa ed wi h he exis ing me hods, MFO, and GWO. In he mean ime, o he me ics like RMSE, MAE, and MAPE a e also lowe in DPGWO as compa ed wi h o he s. I shows ha he DPGWO pe o med well. The s a is ical compa ison o compu a ion ime o he p oposed DPGWO wi h MFO and GWO is shown in Table 17. I is unde s ood ha he minimum compu a ion imes o DPGWO, MFO, and GWO will be 32.1 s, 29.6 s, and 89.6 s, espec i ely. Due o he in oduc ion o li e, ep oduc ion, and disease s a egies in DPGWO, he compu a ion ime is 8.4% highe han he GWO algo i hm and 64.2% lowe han he MFO algo i hm. Appl. Sci. 2023,13, 8206 19 o 33 Appl. Sci. 2023, 13, x FOR PEER REVIEW 19 o 35 GWO. In he mean ime, o he me ics like RMSE, MAE, and MAPE a e also lowe in DPGWO as compa ed wi h o he s. I shows ha he DPGWO pe o med well. The s a is- ical compa ison o compu a ion ime o he p oposed DPGWO wi h MFO and GWO is shown in Table 17. I is unde s ood ha he minimum compu a ion imes o DPGWO, MFO, and GWO will be 32.1 s, 29.6 s, and 89.6 s, espec i ely. Due o he in oduc ion o li e, ep oduc ion, and disease s a egies in DPGWO, he compu a ion ime is 8.4% highe han he GWO algo i hm and 64.2% lowe han he MFO algo i hm. Figu e 8. Expe imen al VS P edic ed Value o C ack Leng h Using Di e en Algo i hms (EV-Expe - imen al Value; P_EM-P edic ed alue using he exis ing me hod; P_MFO, P_GWO and P_DPGWO- P edic ed alues based on he numbe o ea u es and i s combina ion ob ained using MFO, GWO, and DPGWO algo i hms. Figu e 9. Expe imen al VS P edic ed Value o C ack Dep h Using Di e en Algo i hms. 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 12345678910 11 12 CRACK LENGTH IN MM SAMPLE NO. EV P_EV P_MFO P_GWO P_DPGWO 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 12345678910 11 12 CRACK DEPTH IN MM SAMPLE NO. EV P_EM P_MFO P_GWO P_DPGWO Figu e 8. Expe imen al VS P edic ed Value o C ack Leng h Using Di e en Algo i hms (EV- Expe imen al Value; P_EM-P edic ed alue using he exis ing me hod; P_MFO, P_GWO and P_DPGWO-P edic ed alues based on he numbe o ea u es and i s combina ion ob ained us- ing MFO, GWO, and DPGWO algo i hms. Appl. Sci. 2023, 13, x FOR PEER REVIEW 19 o 35 GWO. In he mean ime, o he me ics like RMSE, MAE, and MAPE a e also lowe in DPGWO as compa ed wi h o he s. I shows ha he DPGWO pe o med well. The s a is- ical compa ison o compu a ion ime o he p oposed DPGWO wi h MFO and GWO is shown in Table 17. I is unde s ood ha he minimum compu a ion imes o DPGWO, MFO, and GWO will be 32.1 s, 29.6 s, and 89.6 s, espec i ely. Due o he in oduc ion o li e, ep oduc ion, and disease s a egies in DPGWO, he compu a ion ime is 8.4% highe han he GWO algo i hm and 64.2% lowe han he MFO algo i hm. Figu e 8. Expe imen al VS P edic ed Value o C ack Leng h Using Di e en Algo i hms (EV-Expe - imen al Value; P_EM-P edic ed alue using he exis ing me hod; P_MFO, P_GWO and P_DPGWO- P edic ed alues based on he numbe o ea u es and i s combina ion ob ained using MFO, GWO, and DPGWO algo i hms. Figu e 9. Expe imen al VS P edic ed Value o C ack Dep h Using Di e en Algo i hms. 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 12345678910 11 12 CRACK LENGTH IN MM SAMPLE NO. EV P_EV P_MFO P_GWO P_DPGWO 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 12345678910 11 12 CRACK DEPTH IN MM SAMPLE NO. EV P_EM P_MFO P_GWO P_DPGWO Figu e 9. Expe imen al VS P edic ed Value o C ack Dep h Using Di e en Algo i hms. Appl. Sci. 2023,13, 8206 20 o 33 Appl. Sci. 2023, 13, x FOR PEER REVIEW 20 o 35 Figu e 10. Expe imen al VS P edic ed Value o C ack Wid h Using Di e en Algo i hms. Table 16. Compa ison o Pe o mance wi h exis ing and p oposed algo i hms. Pe o mance Me ics EM MFO GWO DPGWO R2 0.9970 0.9987 0.9981 0.9992 0.9938 0.9975 0.9966 0.9989 0.9969 0.9993 0.9990 0.9994 RMSE 0.101 0.066 0.078 0.050 0.032 0.020 0.023 0.014 0.208 0.099 0.126 0.089 MAE 0.0872 0.0563 0.0568 0.0417 0.0247 0.0147 0.0135 0.0099 0.1922 0.0792 0.1028 0.0727 MAPE 2.93 1.88 1.60 1.38 3.09 1.72 1.60 1.16 2.25 0.91 1.09 0.91 Table 17. Compa ison o compu a ion ime o algo i hms. Va iable Samples Mean S De Va iance Minimum Q1 Median Q3 Maximum Skewness Ku osis DPGWO 12 45 8.23 67.68 32.1 39.95 43.7 52.77 59.6 0.22 −0.65 GWO 12 38.4 6.98 48.75 29.6 31.62 39.25 41.38 51.8 0.57 −0.09 MFO 12 108.42 15.64 244.77 89.6 95.93 103.75 126 133.7 0.42 −1.36 5. Conclusions In his wo k, he bes ML model sui able o p edic he c ack dimensions om he gi en 22 GLCM ea u es ex ac ed om MFL images epo ed by Daniel e al. [12], was de e mined using he Taguchi L16 o hogonal a ay expe imen al design. Ou o six ML models, he XGB model is iden i ied as he bes sui ed o he p oposed p oblem. Apa om ha , wo- il e -based ea u e selec ion me hods a e implemen ed o p io i izing he ea u es. The pe o mance o he ML models is examined based on he numbe o ea u es and hei combina ions. Me a-heu is ic algo i hms, namely MFO, GWO, and a new a i- an o GWO called DPGWO, a e implemen ed o iden i y he numbe o ea u es and hei combina ions o achie e be e pe o mance o he selec ed XGB ML model. The p oposed me hod p o ed ha by elimina ing 23% o he numbe o ea u es as compa ed wi h he 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 12345678910 11 12 CRACK WIDTH IN MM SAMPLE NO. EV P_EM P_MFO P_GWO P_DPGWO Figu e 10. Expe imen al VS P edic ed Value o C ack Wid h Using Di e en Algo i hms. Table 16. Compa ison o Pe o mance wi h exis ing and p oposed algo i hms. Pe o mance Me ics EM MFO GWO DPGWO R2 0.9970 0.9987 0.9981 0.9992 0.9938 0.9975 0.9966 0.9989 0.9969 0.9993 0.9990 0.9994 RMSE 0.101 0.066 0.078 0.050 0.032 0.020 0.023 0.014 0.208 0.099 0.126 0.089 MAE 0.0872 0.0563 0.0568 0.0417 0.0247 0.0147 0.0135 0.0099 0.1922 0.0792 0.1028 0.0727 MAPE 2.93 1.88 1.60 1.38 3.09 1.72 1.60 1.16 2.25 0.91 1.09 0.91 Table 17. Compa ison o compu a ion ime o algo i hms. Va iable Samples Mean S De Va iance Minimum Q1 Median Q3 Maximum Skewness Ku osis DPGWO 12 45 8.23 67.68 32.1 39.95 43.7 52.77 59.6 0.22 −0.65 GWO 12 38.4 6.98 48.75 29.6 31.62 39.25 41.38 51.8 0.57 −0.09 MFO 12 108.42 15.64 244.77 89.6 95.93 103.75 126 133.7 0.42 −1.36 5. Conclusions In his wo k, he bes ML model sui able o p edic he c ack dimensions om he gi en 22 GLCM ea u es ex ac ed om MFL images epo ed by Daniel e al. [ 12 ], was de e mined using he Taguchi L16 o hogonal a ay expe imen al design. Ou o six ML models, he XGB model is iden i ied as he bes sui ed o he p oposed p oblem. Apa Appl. Sci. 2023,13, 8206 21 o 33 om ha , wo- il e -based ea u e selec ion me hods a e implemen ed o p io i izing he ea u es. The pe o mance o he ML models is examined based on he numbe o ea u es and hei combina ions. Me a-heu is ic algo i hms, namely MFO, GWO, and a new a ian o GWO called DPGWO, a e implemen ed o iden i y he numbe o ea u es and hei combina ions o achie e be e pe o mance o he selec ed XGB ML model. The p oposed me hod p o ed ha by elimina ing 23% o he numbe o ea u es as compa ed wi h he exis ing me hod, i was possible o ge good pe o mance using he XGB ML model. Also, in his wo k, i is iden i ied ha he ea u es ETR, CST, ID, AC, and IDN a e no impo an ea u es, i.e., hese ea u es a e elimina ed du ing he p edic ion o c ack dimensions. The e ec i eness o he p oposed DPGWO is p o ed by compa ing F iedman’s Tes and pe o mance indica o s’ alues, iz., IGD and SP wi h MFO and GWO. Also, he p oposed me hod is implemen ed on he 12- es da a se ca ied ou by Daniel e al. [ 12 ]. As compa ed o he exis ing me hod, he p oposed DPGWO pe o med well in calcula ing pe o mance me ics like R 2 , RMSE, MAE, and MAPE. App oxima ely 0.2 o 0.5% imp o emen s in R 2 alue, 50% o 57% imp o emen in RMSE alue, 52% o 62% imp o emen in MAE alue, and 53% o 63% imp o emen in MAPE alues a e epo ed by using a smalle numbe o ea u es, i.e., 17 and i s combina ions, and implemen ing a new a ian o GWO called DPGWO. As a u he scope, he w appe me hod o ea u es selec ion along wi h he mul i- eg esso concep o ML models may be conside ed o imp o e pe o mance. Apa om ha , he c ack dimensions may be changed om nume ical o ca ego ical da a ype, and ML classi ie s can be in oduced ins ead o ML eg esso s. The p oposed DPGWO may be implemen ed o cons ained enginee ing p oblems. Au ho Con ibu ions: Concep ualiza ion, M.V.A.W., S.R., R.C., S.K.M. and M.E.; da a cu a ion, M.V.A.W.; Fo mal analysis, M.V.A.W.; in es iga ion, M.V.A.W.; me hodology, S.R. and M.E.; esou ces, R.C. and S.K.M.; so wa e, S.R., R.C., S.K.M. and M.E.; isualiza ion, M.V.A.W. and S.R.; w i ing— o iginal d a , M.V.A.W. and S.R.; w i ing— e iew and edi ing, R.C., S.K.M. and M.E. All au ho s ha e ead and ag eed o he published e sion o he manusc ip . Funding: This wo k was suppo ed by he p ojec SP2023/088 suppo ed by he Minis y o Educa- ion, You h, and Spo s, Czech Republic. Ins i u ional Re iew Boa d S a emen : No applicable. In o med Consen S a emen : No applicable. Da a A ailabili y S a emen : The da a p esen ed in his s udy a e a ailable upon eques h ough email o he co esponding au ho . Con lic s o In e es : The au ho s decla e no con lic o in e es . Appendix A Gi Hub Link: h ps://gi hub.com/lawansisa/ML_MFO_GWO_DPGWO (accessed on 1 June 2023). Table A1. Pa ame e s and I s Le el—C ack Dep h. Pa ame e No. o Le els Le els 1 2 3 4 FSM 2 FS MIS MLM 4 ABR LiR RF XGB NoF 4 15 17 19 21 Appl. Sci. 2023,13, 8206 22 o 33 Table A2. L16 OA—Pe o mance o ML o C ack Dep h. E.No. FSM MLM NoF R2T R2T RmseT RmseT Deng’s Value 1 FS ABR 15 0.7741 0.0441 1.3924 3.0091 0.3319 2 FS ABR 17 0.8537 0.1680 1.2802 3.3972 0.4154 3 MIS ABR 19 0.8699 0.0221 1.1833 3.1780 0.3480 4 MIS ABR 21 0.9154 0.0446 0.9110 3.2148 0.3941 5 FS LiR 15 0.5956 0.3609 1.8665 3.5540 0.4292 6 FS LiR 17 0.6049 0.2698 1.8390 3.3689 0.3965 7 MIS LiR 19 0.6883 0.4080 1.6204 3.6648 0.4693 8 MIS LiR 21 0.7274 0.2882 1.5332 3.5175 0.4361 9 MIS RF 15 0.9207 0.0444 0.8105 3.1208 0.4061 10 MIS RF 17 0.9228 0.0584 0.7603 3.0285 0.4208 11 FS RF 19 0.9330 0.0469 0.6982 3.2378 0.4224 12 FS RF 21 0.9412 0.0455 0.6224 3.0668 0.4325 13 MIS XGB 15 1.0000 0.4452 0.0006 3.6458 0.6737 14 MIS XGB 17 1.0000 0.4829 0.0005 3.7246 0.6806 15 FS XGB 19 1.0000 0.5153 0.0006 3.9049 0.6836 16 FS XGB 21 1.0000 0.4663 0.0004 3.7602 0.6761 Appl. Sci. 2023, 13, x FOR PEER REVIEW 22 o 35 9 MIS RF 15 0.9207 0.0444 0.8105 3.1208 0.4061 10 MIS RF 17 0.9228 0.0584 0.7603 3.0285 0.4208 11 FS RF 19 0.9330 0.0469 0.6982 3.2378 0.4224 12 FS RF 21 0.9412 0.0455 0.6224 3.0668 0.4325 13 MIS XGB 15 1.0000 0.4452 0.0006 3.6458 0.6737 14 MIS XGB 17 1.0000 0.4829 0.0005 3.7246 0.6806 15 FS XGB 19 1.0000 0.5153 0.0006 3.9049 0.6836 16 FS XGB 21 1.0000 0.4663 0.0004 3.7602 0.6761 (a) (b) (c) (d) Figu e A1. S a is ical In e ence o ML Models’ Pe o mance o C ack Dep h P edic ion. (a) P oba- bili y Plo o C ack Leng h-R2T . (b) P obabili y Plo o C ack Leng h-R2T . (c) P obabili y Plo o C ack Leng h-RMSET . (d) P obabili y Plo o C ack Leng h-RMSET Figu e A1. S a is ical In e ence o ML Models’ Pe o mance o C ack Dep h P edic ion. ( a ) P obabil- i y Plo o C ack Leng h-R2T . ( b ) P obabili y Plo o C ack Leng h-R2T . ( c ) P obabili y Plo o C ack Leng h-RMSET . (d) P obabili y Plo o C ack Leng h-RMSET . Appl. Sci. 2023,13, 8206 23 o 33 Appl. Sci. 2023, 13, x FOR PEER REVIEW 23 o 35 (a) (b) (c) (d) (e) ( ) Appl. Sci. 2023, 13, x FOR PEER REVIEW 24 o 35 (g) (h) Figu e A2. Fac o ial and In e ac ion Plo s o Pe o mance o ML Models o C ack Dep h P edic- ion. (a) R2 Fac o ial Plo o T aining Da ase —D. (b) R2 Fac o ial Plo o Tes ing Da ase —D. (c) RMSEFac o ial Plo o T aining Da ase —D. (d) RMSEFac o ial Plo o Tes ing Da ase —D. (e) R2 In e ac ion Plo o T aining Da ase —D. ( ) R2 In e ac ion Plo o Tes ing Da ase —D. (g) RMSE In e ac ion Plo o T aining Da ase —D. (h) RMSE In e ac ion Plo o Tes ing Da ase —D. Figu e A2. Fac o ial and In e ac ion Plo s o Pe o mance o ML Models o C ack Dep h P e- dic ion. ( a ) R 2 Fac o ial Plo o T aining Da ase —D. ( b ) R 2 Fac o ial Plo o Tes ing Da ase —D. ( c ) RMSEFac o ial Plo o T aining Da ase —D. ( d ) RMSEFac o ial Plo o Tes ing Da ase —D. ( e ) R 2 In e ac ion Plo o T aining Da ase —D. ( ) R 2 In e ac ion Plo o Tes ing Da ase —D. ( g ) RMSE In e ac ion Plo o T aining Da ase —D. (h) RMSE In e ac ion Plo o Tes ing Da ase —D. Appl. Sci. 2023,13, 8206 24 o 33 Table A3. ANOVA o C ack Dep h P edic ion. Sou ce DF R2T R2T RMSET RMSET Adj SS Adj MS F-Value p-Value Adj SS Adj MS F-Value p-Value Adj SS Adj MS F-Value p-Value Adj SS Adj MS F-Value p-Value Reg ession 13 0.29005 0.02231 87.17000 0.01100 0.54741 0.04211 65.07000 0.01500 6.58227 0.50633 753.53000 0.00100 1.22728 0.09441 8.76000 0.10700 NoF 1 0.00018 0.00018 0.70000 0.49100 0.00530 0.00530 8.20000 0.10300 0.00054 0.00054 0.81000 0.46300 0.03826 0.03826 3.55000 0.20000 FSM 1 0.00001 0.00001 0.02000 0.89300 0.00028 0.00028 0.44000 0.57700 0.00329 0.00329 4.90000 0.15700 0.00192 0.00192 0.18000 0.71400 MLM 3 0.00416 0.00139 5.42000 0.16000 0.02247 0.00749 11.57000 0.08100 0.05840 0.01947 28.97000 0.03400 0.10052 0.03351 3.11000 0.25300 NoF*NoF 1 0.00000 0.00000 0.00000 0.98300 0.00336 0.00336 5.20000 0.15000 0.00377 0.00377 5.60000 0.14200 0.02372 0.02372 2.20000 0.27600 NoF*FSM 1 0.00001 0.00001 0.03000 0.88400 0.00005 0.00005 0.08000 0.80900 0.00236 0.00236 3.51000 0.20200 0.00001 0.00001 0.00000 0.98100 NoF*MLM 3 0.00241 0.00080 3.14000 0.25100 0.01630 0.00543 8.39000 0.10800 0.01977 0.00659 9.81000 0.09400 0.08768 0.02923 2.71000 0.28100 FSM*MLM 3 0.00113 0.00038 1.47000 0.42900 0.02182 0.00727 11.24000 0.08300 0.00680 0.00227 3.37000 0.23700 0.09859 0.03287 3.05000 0.25700 E o 2 0.00051 0.00026 0.00129 0.00065 0.00134 0.00067 0.02154 0.01077 To al 15 0.29057 0.54870 6.58361 1.24882 Appl. Sci. 2023,13, 8206 25 o 33 Table A4. Pa ame e s and I s Le el—C ack Wid h. Pa ame e No. o Le els Le els 1234 FSM 2 FS MIS MLM 4 ABR LiR RF XGB NoF 4 15 17 19 21 Table A5. L16 OA—Pe o mance o ML o C ack Wid h. E.No. FSM MLM NoF R2T R2T RMSET RmseT Deng’s Value 1 FS ABR 15 0.8447 0.2031 0.1349 0.3008 0.4569 2 FS ABR 17 0.8633 0.2435 0.1240 0.2896 0.4771 3 MIS ABR 19 0.8573 0.3026 0.1271 0.2798 0.4879 4 MIS ABR 21 0.8789 0.3411 0.1191 0.2724 0.5040 5 FS LiR 15 0.6983 0.3227 0.1795 0.2604 0.4434 6 FS LiR 17 0.7047 0.3961 0.1781 0.2391 0.4616 7 MIS LiR 19 0.7382 0.3611 0.1663 0.2561 0.4643 8 MIS LiR 21 0.7632 0.1582 0.1518 0.2909 0.4248 9 MIS RF 17 0.9351 0.3889 0.0820 0.2648 0.5484 10 MIS RF 19 0.9361 0.3163 0.0800 0.2822 0.5360 11 FS RF 19 0.9328 0.2949 0.0800 0.2665 0.5324 12 FS RF 21 0.9337 0.3332 0.0793 0.2635 0.5408 13 MIS XGB 15 1.0000 0.1967 0.0006 0.2825 0.6073 14 MIS XGB 17 1.0000 0.4196 0.0006 0.2440 0.6444 15 FS XGB 19 1.0000 0.3050 0.0005 0.2688 0.6263 16 FS XGB 21 1.0000 0.3386 0.0006 0.2570 0.6326 Table A6. S a is ical Analysis o ML Models’ Pe o mance o C ack Wid h P edic ion. Va iable Mean S De Va iance Minimum Q1 Median Q3 Maximum Range p-Value R2T 0.8804 0.1061 0.0113 0.6983 0.7836 0.9058 0.9840 1.0000 0.3017 0.0950 R2T 0.3076 0.0743 0.0055 0.1582 0.2564 0.3195 0.3561 0.4196 0.2614 0.3030 RmseT 0.0940 0.0648 0.0042 0.0005 0.0203 0.1006 0.1476 0.1795 0.1790 0.0750 RmseT 0.2699 0.0170 0.0003 0.2391 0.2579 0.2676 0.2825 0.3008 0.0617 0.9400 Appl. Sci. 2023, 13, x FOR PEER REVIEW 26 o 35 Table A4. Pa ame e s and I s Le el—C ack Wid h. Pa ame e No. o Le els Le els 1 2 3 4 FSM 2 FS MIS MLM 4 ABR LiR RF XGB NoF 4 15 17 19 21 Table A5. L16 OA—Pe o mance o ML o C ack Wid h. E.No. FSM MLM NoF R2T R2T RMSET RmseT Deng’s Value 1 FS ABR 15 0.8447 0.2031 0.1349 0.3008 0.4569 2 FS ABR 17 0.8633 0.2435 0.1240 0.2896 0.4771 3 MIS ABR 19 0.8573 0.3026 0.1271 0.2798 0.4879 4 MIS ABR 21 0.8789 0.3411 0.1191 0.2724 0.5040 5 FS LiR 15 0.6983 0.3227 0.1795 0.2604 0.4434 6 FS LiR 17 0.7047 0.3961 0.1781 0.2391 0.4616 7 MIS LiR 19 0.7382 0.3611 0.1663 0.2561 0.4643 8 MIS LiR 21 0.7632 0.1582 0.1518 0.2909 0.4248 9 MIS RF 17 0.9351 0.3889 0.0820 0.2648 0.5484 10 MIS RF 19 0.9361 0.3163 0.0800 0.2822 0.5360 11 FS RF 19 0.9328 0.2949 0.0800 0.2665 0.5324 12 FS RF 21 0.9337 0.3332 0.0793 0.2635 0.5408 13 MIS XGB 15 1.0000 0.1967 0.0006 0.2825 0.6073 14 MIS XGB 17 1.0000 0.4196 0.0006 0.2440 0.6444 15 FS XGB 19 1.0000 0.3050 0.0005 0.2688 0.6263 16 FS XGB 21 1.0000 0.3386 0.0006 0.2570 0.6326 Table A6. S a is ical Analysis o ML Models’ Pe o mance o C ack Wid h P edic ion. 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MDPI and/o he edi o (s) disclaim esponsibili y o any inju y o people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .