Early Fast Cost Estimates of Sewerage Projects Construction Costs Based on Ensembles of Neural Networks

Ci a ion: Juszczyk, M.; Hanák, T.; Výskala, M.; Pacyno, H.; Siejda, M. Ea ly Fas Cos Es ima es o Sewe age P ojec s Cons uc ion Cos s Based on Ensembles o Neu al Ne wo ks. Appl. Sci. 2023,13, 12744. h ps://doi.o g/ 10.3390/app132312744 Academic Edi o : As e ios Bakolas Recei ed: 7 Oc obe 2023 Re ised: 18 No embe 2023 Accep ed: 24 No embe 2023 Published: 28 No embe 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 Ea ly Fas Cos Es ima es o Sewe age P ojec s Cons uc ion Cos s Based on Ensembles o Neu al Ne wo ks Michał Juszczyk 1,* , Tomáš Hanák2, Milosla Výskala 2, Hanna Pacyno 1,3 and Michał Siejda 1 1Facul y o Ci il Enginee ing, C acow Uni e si y o Technology, 31-155 K aków, Poland; [email p o ec ed] o [email p o ec ed] (H.P.); [email p o ec ed] (M.S.) 2Facul y o Ci il Enginee ing, B no Uni e si y o Technology, 602 00 B no, Czech Republic; hanak. @ u b .cz (T.H.); [email p o ec ed].cz (M.V.) 3Da acomp IT sp. z o.o., 30-532 K aków, Poland *Co espondence: [email p o ec ed] Fea u ed Applica ion: The po en ial applica ions o he esea ch esul s, pa icula ly he models u ilizing ensembles o neu al ne wo ks, o e he easibili y o ea ly cos es ima es o sewe age p ojec s. The cos es ima es o cons uc ion wo ks, de i ed om he de eloped models, can be gene a ed based on he essen ial ea u es o sewe age p ojec s ha a e accessible o analysis p io o he commencemen o de ailed design. Abs ac : This pape p esen s esea ch esul s on he de elopmen o an o iginal cos p edic ion model o cons uc ion cos s in sewe age p ojec s. The ocus is placed on as cos es ima es applicable in he ea ly s ages o a p ojec , based on undamen al in o ma ion a ailable du ing he ini ial design phase o sani a y sewe s p io o he de ailed design. The o iginali y and no el y o his esea ch lie in he applica ion o a i icial neu al ne wo k ensembles, which include a combina ion o se e al indi idual neu al ne wo ks and he use o simple a e aging and gene alized a e aging app oaches. The esea ch esul ed in he de elopmen o wo ensemble-based models, including i e neu al ne wo ks ha we e ained and es ed using da a collec ed om 125 sewe age p ojec s comple ed in he Czech Republic be ween 2018 and 2022. The da a included in o ma ion ele an o a ious aspec s o p ojec s and con ac cos s, upda ed o accoun o changes in cos s o e ime. The de eloped models p esen sa is ac o y p edic i e pe o mance, especially he ensemble model based on simple a e aging, which o e s p edic ion accu acy wi hin he ange o ± 30% (in e ms o pe cen age e o s) o o e 90% o he aining and es ing samples. The de eloped models, based on he ensembles o neu al ne wo ks, ou pe o med he benchma k model based on he classical app oach and he use o mul iple linea eg ession. Keywo ds: sewe age p ojec ; sani a y sewe ne wo ks; cons uc ion cos s; cons uc ion p ojec ; ea ly cos es ima es; as cos es ima es; neu al ne wo ks ensembles; a i icial in elligence 1. In oduc ion Sani a y sewe age sys ems, as a pa o buil in as uc u e, a e unques ionably o high signi icance o mode n socie ies. The ole o sewe ne wo ks is o collec and ans e was ewa e om buildings ( esiden ial, comme cial, o indus ial) and all kinds o public and p i a e es ablishmen s o a poin o ea men and disposal. These sys ems a e he esul s o cons uc ion p ojec s aiming o ei he build new o eno a e exis ing sewe ages. Mos o such p ojec s—one may e en da e o say ha he o e whelming majo i y—a e inanced om public unds. Sewe p ojec s, ega ding hei speci ici y, equi e ca e ul analyses, design, and plan- ning so ha he echnical equi emen s (especially low capaci y) a e me . When planning is conce ned, cos analyses a e o key impo ance as comple ion o a p ojec wi hin a budge Appl. Sci. 2023,13, 12744. h ps://doi.o g/10.3390/app132312744 h ps://www.mdpi.com/jou nal/applsci Appl. Sci. 2023,13, 12744 2 o 24 is one o he “ha d” goals and key measu es o any cons uc ion p ojec success. Thus, he e is a need o ealis ic cos p edic ions, analyses, and es ima es, which make he goal achie able. These p edic ions should e lec he p og ess o he design p ocess and a ail- able in o ma ion, which e ol es om basic and undamen al o de ini e and accu a e. The es ima es ha play a speci ic ole a e called ea ly es ima es. These ely on basic in o ma ion and some essen ial pa ame e s o a p ojec and a e p o ided in he ea ly s age o he design p ocess. On he one hand, he expec ed accu acy is low; on he o he hand, his is when he impac on he cons uc ion cos is g ea , as he essen ial choices and decisions o he p ojec a e made. In he cons uc ion indus y, cos es ima es play a pi o al ole in p ojec planning and execu ion. These es ima es p o ide a closely app oxima ed assessmen o expec ed expenses, empowe ing p ojec s akeholde s o make well-in o med decisions. The accu acy o cos es ima es holds pa amoun impo ance in ac i i ies such as budge ing, secu ing inancial esou ces, and ensu ing he success ul comple ion o cons uc ion p ojec s. The p ecision o cos es ima es can a y, spanning om ough o de o magni ude es ima es a he p ojec ’s concep ual incep ion o highly de ailed assessmen s du ing he design and p e- cons uc ion phases. These es ima es unde go con inual e inemen as addi ional p ojec - speci ic in o ma ion becomes a ailable. I is impe a i e o acknowledge ha e oneous cos es ima es can esul in budge o e uns, p ojec delays, and dispu es wi hin he cons uc ion p ocess. Hence, he de elopmen o p ecise, well-in o med cos es ima es eme ges as a c i ical ac o o he e ec i e and economically iable execu ion o cons uc ion p ojec s. Ad ances and p og ess in da a sciences and a i icial in elligence ools o p ocessing in o ma ion—especially o p edic ion p oblems—opened possibili ies o he de elopmen o cos es ima ion me hods based on he use o collec ed da a, lea ning om expe ience, and knowledge gene aliza ion. Speci ically, a i icial neu al ne wo ks (ANN) a e ools ha ha e signi ican capabili ies ha make hem use ul o ea ly cons uc ion cos es i- ma ion; howe e , hey a e ha dly epo ed in he li e a u e o be applied in he case o sewe age p ojec s. The aim o his wo k is o in oduce a me hod and model o es ima ing cons uc ion cos s o sewe age p ojec s based on a speci ic a i icial in elligence ool—namely ensembles o neu al ne wo ks (la e e e ed o as EoNN). The objec i e o he esea ch, he esul s o which a e p esen ed he ein, was o de elop a model capable o p edic ing cons uc ion cos s in he ea ly s age o a sewe age p ojec wi h sa is ac o y accu acy. Fo he pu pose o model de elopmen , nume ous sewe age cons uc ion p ojec s in ol ing he cons uc ion o new sec ions o he eno a ion o exis ing pa s o he ex e nal g a i y sewage ne wo k comple ed in he Czech Republic be ween 2018 and 2022 we e analyzed. These p ojec s se ed as a sou ce o da a o aining a i icial in elligence ools. The pape ’s con en includes concise li e a u e and s a e-o - he-a e iew; a p esen a- ion o he esea ch assump ions, da a, and me hods applied o he de elopmen o he as cos es ima ion model; in oduc ion o he model i sel along wi h he esul s o he esea ch; discussion o esul s along wi h compa ison wi h a linea eg ession model as a benchma k; summa y and conclusions. 2. Li e a u e Re iew 2.1. Sewe age Cons uc ion P ojec s Managemen When conside ing sewe age cons uc ion p ojec s, managemen p oblems become subjec s o esea ch and s udy, simila o o he ypes o cons uc ion p ojec s. Some no e- wo hy examples o gene al p oblems p esen ed in he li e a u e include he ollowing: an op imiza ion model o sewage ehabili a ion aiming o achie e maximum e ec i eness a he lowes cos , u ilizing gene ic algo i hms [ 1 ]; a me hodology o selec ing and p io i- izing sewe age p ojec s wi hin a ailable unds and sys em capaci y, based on dynamic p og amming p inciples [ 2 ]; a s udy on he isk o cos o e uns in wa e and sewe age sys em cons uc ion p ojec s [ 3 ]; he de elopmen o a new me hod o enhance he accu acy o Mon e Ca lo simula ions and i s alida ion in p edic ing he success likelihood o sewe - Appl. Sci. 2023,13, 12744 3 o 24 age build–ope a e– ans e p ojec s, based on eigh case s udies [ 4 ]; esea ch on cul u ally app op ia e o ganiza ion o p ojec s implemen ed h ough public–p i a e pa ne ships [ 5 ]; heo e ical and empi ical analysis o issues a ising in public–p i a e pa ne ship p ojec s wi hin he sewe age sec o [ 6 ]; and an in es iga ion in o delay ac o s in sewe age p ojec s using simula ions and a dynamic sys ems app oach [7]. Cos - ela ed challenges wi hin sewe age p ojec s cons i u e a dis inc a ea o ocus in he esea ch li e a u e. In one s udy [ 8 ], sewe age p ojec cos es ima es a e examined using wo al e na i e app oaches. The i s app oach in eg a es componen cos anges and p obabili y alues es ablished by a panel o es ima o s. In con as , he second app oach in ol es simula ing cos s based on andom numbe s, whe e componen alues a e selec ed andomly wi hin speci ied anges. The s udy [ 9 ] in oduces a model ha elies on he u iliza ion o p io in o ma ion o he es ima ion o ope a ional cos s wi hin sewe age sys ems. Speci ically, he au ho s del e in o he p ocess o modeling p io in o ma ion o u nish p elimina y assessmen s o in es men equi emen s. To acili a e subsequen es ima ion, he Bayes linea es ima o was employed. Ano he esea ch endea o [ 10 ] cen e s on he in icacies o cos compa ison in was ewa e ea men . The au ho s del e in o equi able me hods o compa ing and alloca ing cos s in municipal sewage ea men conce ning hei s uc u e and o igin. Ano he s udy [ 11 ] conduc s an in-dep h analysis o ac o s esponsible o a ia ions and he esul ing cos s in sewe age cons uc ion p ojec s. The wo k o [ 12 ] discloses esea ch ou comes on benchma king sewe age sys ems, pa - icula ly emphasizing he analysis o in es men cos s. This also in ol es an exhaus i e examina ion o in angible a iables such as economic luc ua ions and ende ing s a egies and how hey in luence cons uc ion cos calcula ions. A no able con ibu ion by [ 13 ] p esen s an Excel-based model capable o e alua ing cos s associa ed wi h sewe age sys em en i onmen al impac . This model comp ehensi ely assesses in es iga ion, in es men , design, ope a ion, main enance, supe ision, and o e all annual cos s. The s udy seeks o p o ide a ool o acili a e en i onmen ally in o med decisions when selec ing was ew- a e sys ems. The de e mina ion o capi al cos s o con en ional sewe age sys ems in de eloping coun ies is sc u inized in [ 14 ]. The analysis in ol es he examina ion o uni cons uc ion cos s exp essed as anges o capi al cos alues. Re . [ 15 ] p esen s a li e a u e e iew on he li ecycle cos s o comple e sani a ion chain sys ems wi hin de eloping ci ies. Mo ing o wa d, e . [ 16 ] conduc s an analysis o ime–cos models ha aid in o ecas ing p ojec du a ions o di e en ypes. The esea ch explo es how cons uc ion echnology in luences he ela ionship be ween ime and cos , pa icula ly ocusing on enchless and open-cu echnologies. In he ealm o public–p i a e pa ne ship sewe age p ojec s, e . [ 17 ] del es in o ansac ion cos s. The s udy employs an explo a o y mul i-case s udy me hod o iden i y po en ial ansac ion cos s wi hin hese p ojec s. Las ly, e . [ 18 ] ad- d esses main enance cos s in sewe sys ems, pa icula ly emphasizing cos es ima ion. The s udy unde sco es he signi icance o main enance cos s wi hin he li ecycle o cons uc ion p ojec s and p oposes a linea eg ession model o acili a e sewe sys em main enance cos es ima ion. The ange o p oblems p esen ed abo e con i ms ha he cos s and cos manage- men o sewe age cons uc ion p ojec s a e o signi ican in e es and impo ance om a esea ch s andpoin . 2.2. Cos -Es ima ing Models o Sewe age Cons uc ion P ojec s In he con ex o he ocus o his pape , he mos c ucial aspec s a e he a emp s o de elop models ha assis in es ima ing cons uc ion cos s o sewe age p ojec s. The ea ly wo k [ 19 ] in es iga ed he applica ion o nonlinea eg ession o sewe cos modeling. The s udy ocused on es ima ing he empi ical pa ame e s wi hin sepa able and gene alized cos unc ions. To model he sewe cos unc ion, he applied echnique equi ed he op imiza ion o he alues o he nonlinea pa ame e s. The wo ypes o analyzed nonlinea pa ame ic cos models a e epo ed o exhibi ela i e insensi i i y o mino e o s in he es ima ed alues o hei model pa ame e s. The de elopmen o cos unc ions o Appl. Sci. 2023,13, 12744 4 o 24 open-cu and jacking me hods in sani a y sewe sys em cons uc ion is he subjec o ano he wo k [ 20 ]. The esea ch esul ed in he o mula ion o cos unc ions applicable o open-cu and jacking me hods, which a e cons uc ion echniques o sewe sys ems. These cos unc ions we e de i ed using linea eg ession and exp essed as unc ions o pipe size and exca a ion dep h. The de i ed unc ions we e alida ed using da a om se e al ac ual sewe sys em cons uc ion p ojec s o e i y he accu acy o cos p edic ions. Ano he wo k [ 21 ] p esen s nonlinea uni cos unc ions o es ima ing cos s associa ed wi h elemen s o wa e bo ne sewe in as uc u e, including g a i y pipes, ising mains, pump s a ions, and was ewa e ea men acili ies. As a esul , a model ha combines se e al cos unc ions was de eloped o p edic he uni cos o a ious sewe elemen s. Modeling he cos s ela ed o sewe sys ems has also been explo ed in he s udy [ 22 ]. The app oach ou lined in his esea ch elies on he u iliza ion o mul iple linea eg ession echniques. The au ho s de ised and alida ed cos unc ions ha pe ain o a ious componen s o sewe sys ems, including g a i y and ising pipes, manholes, and pumping s a ions. The cos s a e delinea ed as unc ions o he p incipal physical a ibu es o hese componen s. The p ocess o es ima ing he cos unc ions in ol ed he applica ion o mul iple linea eg ession analysis. In ano he wo k [ 23 ], a pa ame ic app oach o modeling he cons uc ion cos s o sewe sys ems is p esen ed. The au ho s aimed o es ablish an ini ial cos model on a municipal le el, wi h popula ion size se ing as he p ima y a iable o he cos unc ions. By main aining popula ion size as an independen ac o , an empi ical co ela ion has been deduced be ween popula ion size and he expenses associa ed wi h sewe age sys ems. Va ious o ms o cos unc ions we e expe imen ed wi h, encompassing bo h linea and nonlinea o mula ions. The ma e o ea ly cos es ima es o sewe age lines is also p esen in [ 24 ]. The au ho s employed eg ession analysis o o mula e models o p edic ing ea ly-s age cos s. The s udy de ised models g ounded in linea eg ession, ea u ing he echnical a ibu es o sewe age lines as independen a iables and he es ima ed cos s as he dependen a iable. I can be obse ed ha se e al wo ks a e based on an app oach in which he o m o he cos unc ion is assumed ex-an e. Bo h linea and nonlinea unc ions ha e been employed o model he cos s o sewe sys ems; howe e , linea eg ession appea s o be he mos popula ool among esea che s. None heless, he e a e wo ks ha p esen a emp s o employ a i icial neu al ne wo ks o he pu pose o cos es ima es in sewe age p ojec s. In [ 25 ], a neu al ne wo k is u ilized as he co ne s one o a cos -es ima ing model designed o a budge es ima ion sys em ocused on epai and/o eplacemen cos s o sewe and wa e p ojec s. The model inco po a es 23 p ojec - ela ed ac o s, de i ed h ough Pa e o analysis, as inpu a iables (independen a iables), while he budge es ima e se es as he ou pu (dependen a iable). The au ho s emphasize ha he p oposed model no only sa es ime bu also enhances he p ecision o es ima es, o e ing clien s a means o compa e cos al e na i es and acili a ing decision-making p ocesses in cases in ol ing he ehabili a ion o sewe and wa e sys ems. Simila wo k [ 26 ] deals wi h he p oblems o concep ual cos es ima ing o wa e supply and sewe age p ojec s. This pape p esen s a backp opaga ion neu al ne wo k-based model ha is supposed o assis municipal au ho i ies in he de elopmen o mo e accu a e cos es ima es o hei wa e supply and sewe p ojec s. Cos p edic o s, ep esen ing p ojec echnical pa ame e s and se ing as he model’s inpu , we e iden i ied on he basis o con ac o s’ bids analysis and co e ed 80% o cons uc ion wo k cos s. The bene i s o he p esen ed model include bu a e no limi ed o, be e u iliza ion o inancial esou ces, he p o ision o decision-making guidelines, and he abili y o compa e al e na i es. Addi ionally, he au ho s claim ha he model ul ills he needs o unding en i ies o mo e accu a e cos es ima es. In compa ison o he pa ame ic app oach, modeling cos s using neu al ne wo ks elim- ina es he need o assump ions abou he equa ion ha binds he independen a iables o he models o he cos , which se es as he dependen a iable. Appl. Sci. 2023,13, 12744 5 o 24 2.3. Applica ions o Neu al Ne wo ks and Ensembles o Neu al Ne wo ks o Cons uc ion Managemen P oblems and Cos Es ima ion in Cons uc ion A i icial neu al ne wo ks (ANN) a e a subse o a i icial in elligence ools inspi ed by he lea ning and knowledge s o age pa e ns obse ed in neu obiology. They can be employed o add ess a ious classi ica ion o eg ession challenges. The concep and heo y o neu al ne wo ks ha e been ex ensi ely discussed in nume ous wo ks [ 27 – 30 ]. Neu al ne wo ks possess he capaci y o p ocess da a wi h he aim o unco e ing concealed pa e ns. The p ocedu e o da a p ocessing o acqui e knowledge, e e ed o as aining, is execu ed h ough speci ic algo i hms. Following he aining phase, hese ne wo ks a e an icipa ed o possess he abili y o gene a e p edic ions o no el da a ha we e no u ilized in he aining p ocess. The abili y o gene alize knowledge is a key a ibu e o a i icial neu al ne wo ks, ende ing hem aluable o a ange o enginee ing p oblems. Pa icula ly, ANN has ound applica ion in add essing cos - ela ed challenges wi hin he cons uc ion sec o . An exempla y illus a ion o his is a uzzy neu al ne wo k model aimed a aiding con ac o s in es ima ing and selec ing a sui able ma kup [ 31 ]. In es iga i e e o s in o he wo k [ 32 ] cen e ed on he e alua ion o mul ilaye pe cep on and gene al eg ession neu al ne wo ks o hei po en ial in ea ly cos es ima ion o oad unnel p ojec s. In [ 33 ], he ou comes om he u iliza ion o gene al eg ession neu al ne wo ks o p edic main enance cos s associa ed wi h cons uc ion equipmen a e sha ed. Ano he wo k [ 34 ] del ed in o he u iliza ion o mul ilaye pe cep on neu al ne wo ks o es ima e building cons uc ion cos s du ing he ini ial design phase. In [ 35 ], esea ch esul s on op imizing bo h he cos and imeline o cons uc ion p ojec s h ough he implemen a ion o neu al ne wo ks a e in oduced. A hyb id app oach, combining mul i a ia e eg ession and mul ilaye pe cep on neu al ne wo ks, was employed in ano he esea ch [ 36 ] o es ima e capi al cos s o ea hmo ing, loading, and unloading equipmen . The e a e also some in e es ing wo ks ha explo e he u iliza ion o ANN and machine lea ning in he analysis o wind speed and wind di ec ion [ 37 , 38 ], se lemen p edic ion [ 39 ], and he assessmen o hei impac on exis ing s uc u es, ha is, b idges and me o, espec i ely. Ensembles (also called commi ees) o neu al ne wo ks (EoNN) ha e hei o igins in he ealm o ensemble lea ning sys ems. The ounda ional p inciples o his app oach can be aced back o ea lie e e enced wo ks ha comp ehensi ely del e in o neu al ne wo k concep s [ 28 , 30 ], as well as wo ks dedica ed o he s udy o ensembles [ 40 ]. EoNN consis s o indi idual ained a i icial neu al ne wo ks (ANN), each p o iding p edic ions ha a e subsequen ly agg ega ed, wi h he aim o educing e o s in compa ison o s andalone neu al ne wo ks. The u iliza ion o neu al ne wo k ensembles wi hin classi ica ion and eg ession models, as opposed o employing s andalone neu al ne wo ks, is an icipa ed o yield enhanced pe o mance and p ecision [41]. Enginee ing applica ions ha u ilize EoNN encompass a ange o enginee ing chal- lenges. Some no ewo hy examples include p edic ing he pe o mance o subs an ial cons uc ion equipmen , speci ically unnel bo ing machines [ 42 ], day-ahead elec ici y load o ecas ing o buildings [ 43 ], o o ecas ing hea ing ene gy consump ion [ 44 ]. In he con ex o s uc u al enginee ing, ensemble models a e epo ed o be used o p e- dic ing high-pe o mance conc e e comp essi e s eng h [ 45 ] and iden i ying s uc u al damage [ 46 ]. Finally, an example o an EoNN applica ion o isk analysis in he main ain- abili y o high- ise buildings in speci ic opical condi ions [47] can be p o ided. Due o he dis inc i e capabili ies and ad an ages o EoNN, hei explo a ion wi hin he domain o cons uc ion cos analysis is inc easingly epo ed o di e en ypes o p ojec s. A emp s a de eloping models capable o aiding a ious cos analyses using an ensemble app oach ha e been epo ed in ecen yea s. In he s udy [ 48 ], he de elop- men o a model o assis in p edic ing p ojec cos and schedule success by u ilizing ea ly planning s a us as inpu s is p esen ed. The esul s ob ained alida e ha he p oposed a i icial in elligence models yield sa is ac o y p edic i e ou comes. Ano he publica- ion [ 49 ] explo es he applica ion o EoNN o Mac o BIM cos es ima es. This esea ch de elops es ima ion models o he s uc u al ames o building loo s, demons a ing Appl. Sci. 2023,13, 12744 6 o 24 sa is ac o y accu acy. Au ho s o [ 50 ] cen e hei a en ion on cos p edic ion o a speci ic ca ego y o objec s—spo s ields. The cons uc ion cos o ecas ing model based on EoNN is p o en o ou pe o m linea eg ession and models elying on single neu al ne wo ks. Fu he mo e, an analysis o es ima e e o s and accu acy es ablishes he applicabili y o he p oposed model in he ea ly s ages o cons uc ion p ojec s. In [ 51 ], he ex del es in o p edic ing he cons uc ion cos s o buildings’ s uc u al elemen s wi h he use o a i icial in elligence ools. The in oduced models a e, among o he s, based on mul iple a i icial neu al ne wo ks combined in o an ensemble. The EoNN-based models mee expec a ions o knowledge gene aliza ion and he accu a e p edic ion o buildings’ s uc- u al ames. Fu he mo e, an ensemble algo i hm applica ion [ 52 ] is also employed o p edic he cos o highway cons uc ion p ojec s. The s udy p esen s a model ha employs a i icial in elligence ools—neu al ne wo ks included—in a s acking ensemble model o cos p edic ion. 2.4. Li e a u e Re iew Summa y A li e a u e e iew allows o a jus i iable assump ion ha EoNN, when applied as he co e o a cons uc ion cos es ima ion model o a speci ic ype o cons uc ion objec , will yield be e esul s compa ed o models based on linea eg ession o single neu al ne wo ks. On he o he hand, wo ks epo ing he applica ion o EoNN o p edic ing cons uc ion cos s in he con ex o sewe age p ojec s ha e no been ound hus a . This pape aims o add ess his gap. 3. Me hodology Rega ding public wo ks ende s, he in es o is equi ed o disclose he expec ed alue o he con ac [ 53 ]. Addi ionally, in o ma ion abou he app oxima e alue o he sewe age p ojec is c ucial du ing he design phase o selec he op imal solu ion, no only om a echnical s andpoin bu also om an economic pe spec i e. Es ima ing he alue o cons uc ion wo ks poses a signi ican challenge, pa icula ly when de ailed p ojec documen a ion is una ailable and only basic da a and pa ame e s a e known. These es ima es a e ypically p o ided by cos enginee s and o en ely on he use o echnical–economic indica o s, which may esul in signi ican ly inaccu a e es ima es [ 54 ]. The s a ing poin o he esea ch was he idea o a cos p edic ion model capable o p o iding es ima es o sewe age cons uc ion p ojec s u ilizing in o ma ion abou he p ojec a ailable in he ea ly design phase. The ollowing no e aims o p o ide a concise o e iew o he b oade con ex o he esea ch. The cons uc ion indus y in he Czech Republic is a i al sec o o he coun y’s economy, con ibu ing o in as uc u e de elopmen , esiden ial and comme cial building p ojec s, and employmen oppo uni ies. The indus y has wi nessed s eady g ow h and mode niza ion in ecen yea s. The cons uc ion sec o plays a signi ican ole in he coun y’s economy. I con ibu es o GDP and p o ides jobs o a conside able po ion o he wo k o ce. In he con ex o he esea ch p esen ed he ein, i is wo h men ioning ha in as uc u e de elopmen , which encompasses sewe age cons uc ion p ojec s, in he Czech Republic e lec s he b oade Eu opean end o mode niza ion, sus ainable de elopmen , and ecology. I con inues o play a pi o al ole in he coun y’s de elopmen . In he Czech Republic, he pe cen age o he popula ion supplied wi h wa e om he public wa e supply eached 94.6% in 2020. In he case o connec ion o he sewage sys em, his pe cen age eached a alue o 86.1% [ 55 ]. Al hough his igu e may appea sa is ac o y, in eali y, sewe age sys ems a e eadily a ailable in la ge agglome a ions, and signi ican gaps exis in smalle se lemen s. I is wo h no ing ha cons uc ion wo ks ela ed o sewe s encompass no only he eno a ion o old ne wo ks and he es ablishmen o new ne wo ks o new buildings bu also he expansion o sewe age sys ems in al eady exis ing buil -up a eas. The esea ch’s gene al ideog am is depic ed in Figu e 1, illus a ing he successi e s eps aken by he au ho s. Appl. Sci. 2023,13, 12744 7 o 24 Appl. Sci. 2023, 13, x FOR PEER REVIEW 7 o 25 gaps exis in smalle se lemen s. I is wo h no ing ha cons uc ion wo ks ela ed o sewe s encompass no only he eno a ion o old ne wo ks and he es ablishmen o new ne wo ks o new buildings bu also he expansion o sewe age sys ems in al eady exis - ing buil -up a eas. The esea ch’s gene al ideog am is depic ed in Figu e 1, illus a ing he successi e s eps aken by he au ho s. Figu e 1. Gene al ideog am and scheme o he esea ch. On he basis o he s eps co e ing he s a e-o - he-a and li e a u e e iew, as well as analyses o sewe age cons uc ion p ojec s comple ed in he Czech Republic be ween 2018 and 2022, an ini ial se o cos p edic o s (po en ial independen a iables se ing as inpu o he EoNN-based model o be de eloped) was p oposed. As he esea ch ocused on p ojec s aimed a cons uc ing new sec ions o upg ading exis ing sec ions o ex e nal g a i y sewage ne wo ks, he cos p edic o s we e expec ed o e lec he speci ici y o such cons uc ion p ojec s. The analyzed p ojec s in ol ed sep- a a ed sewe sys ems, wi h was ewa e and s o mwa e unoffs in sepa a e pipes. (An im- po an poin o no e is ha he analyzed p ojec s did no inco po a e combined unoffs; ha is, he unoffs o was ewa e and s o mwa e in a single pipe). Despi e he ac ha was ewa e sys ems a e connec ed o he exis ing sewage ea men plan , he plan s hem- sel es we e no pa o he analyzed p ojec s, and hus, no ela ed cos p edic o s we e conside ed. The men ioned se o cos p edic o s is p esen ed in Table 1, which includes selec ed ypes o cos p edic o s and desc ip ions o he in o ma ion ha is supposed o be inpu in o he model ( his in o ma ion is succinc ly explained in he able). Mo eo e , he able p esen s aw alues o cos p edic o s, as hey we e collec ed be o e p e-p ocessing, o - de ing, and scaling. Figu e 1. Gene al ideog am and scheme o he esea ch. On he basis o he s eps co e ing he s a e-o - he-a and li e a u e e iew, as well as analyses o sewe age cons uc ion p ojec s comple ed in he Czech Republic be ween 2018 and 2022, an ini ial se o cos p edic o s (po en ial independen a iables se ing as inpu o he EoNN-based model o be de eloped) was p oposed. As he esea ch ocused on p ojec s aimed a cons uc ing new sec ions o upg ading exis ing sec ions o ex e nal g a i y sewage ne wo ks, he cos p edic o s we e expec ed o e lec he speci ici y o such cons uc ion p ojec s. The analyzed p ojec s in ol ed sepa a ed sewe sys ems, wi h was ewa e and s o mwa e uno s in sepa a e pipes. (An impo an poin o no e is ha he analyzed p ojec s did no inco po a e combined uno s; ha is, he uno s o was ewa e and s o mwa e in a single pipe). Despi e he ac ha was ewa e sys ems a e connec ed o he exis ing sewage ea men plan , he plan s hemsel es we e no pa o he analyzed p ojec s, and hus, no ela ed cos p edic o s we e conside ed. The men ioned se o cos p edic o s is p esen ed in Table 1, which includes selec ed ypes o cos p edic o s and desc ip ions o he in o ma ion ha is supposed o be inpu in o he model ( his in o ma ion is succinc ly explained in he able). Mo eo e , he a- ble p esen s aw alues o cos p edic o s, as hey we e collec ed be o e p e-p ocessing, o de ing, and scaling. Appl. Sci. 2023,13, 12744 8 o 24 Table 1. Ini ial se o cos p edic o s. Cos P edic o Inpu In o ma ion Desc ip ion Value Type o p ojec New cons uc ion o eno a ion Desc ip i e Sewe pipe’s leng h Size o a p ojec , complexi y o a p ojec , quan i y o wo ks Nume ical (leng h) Type o sewe pipe ma e ial Technical pa ame e , ma e ial pa ame e , applied solu ion Desc ip i e Sewe pipe’s diame e Technical pa ame e , applied solu ion, capaci y o a sewe Desc ip i e A e age dep h o ench Technical pa ame e , empo a y wo ks, sa e y issues Nume ical (dep h) G oundwa e able le el Technical pa ame e , g ound condi ions, empo a y wo ks, sa e y issues Desc ip i e Class o soil Technical pa ame e , g ound condi ions pa ame e , sa e y issues Desc ip i e Numbe o manholes Size o a p ojec , complexi y o a p ojec , sa e y issues Nume ical (coun ) Numbe o c ossings wi h o he se ices Complexi y o a p ojec , empo a y wo ks, sa e y issues Nume ical (coun ) Wo ks on unpa ed su ace Condi ions o wo ks, complexi y o wo ks, soil ype was e p oduc ion, quan i y o wo ks Nume ical (leng h) Wo ks on pa ed su ace Condi ions o wo ks, complexi y o wo ks, ubble ype was e p oduc ion, quan i y o wo ks Nume ical (leng h) Deb is emo al dis ance Condi ions o wo ks, was e managemen pa ame e Nume ical (leng h) In he nex s ep, da a o he pu poses o he cos p edic ion model we e collec ed. This s ep in ol ed he analysis o echnical and design documen a ion, quan i y su eys, cos es ima es, as well as public clien que ies o 135 sewe age p ojec s. This p o ided da a e lec ing he aw alues o cos p edic o s (as p esen ed in Table 1), along wi h eal-li e con ac ne cos s (excluding alue-added ax) o sewe age p ojec cons uc ion wo ks. I is no ewo hy ha , h ough he sys ema ic analysis o sewe age cons uc ion p ojec s (which cons i u ed he second phase o he esea ch), i was obse ed ha he p e iously men ioned ype o uno , whe he was ewa e o s o mwa e , did no impac he cos s. F om bo h echnological and cons uc ion cos pe spec i es, i does no ma e which ype o uno is being cons uc ed. The e o e, his in o ma ion is excluded as a p edic o o cos s. Due o changes in he alue o money and cos a iabili y o e ime, he alues o con ac ne cos s we e upda ed o he end o he i s hal o he yea 2023. The upda ed ule is p o ided below. UCC =ACC · n ∏ i= CIi(1) whe e: UCC—upda ed con ac ne cos o a sewe age p ojec ; ACC —ac ual con ac ne cos o sewe age p ojec s, which was awa ded in he - h hal -yea pe iod be ween he beginning o 2018 and he end o 2022; CI i —cos index o i- h hal -yea pe iod be ween he beginning o 2018 and he i s hal o he 2023 yea ; n—s ands o he i s hal o he 2023 yea . Cos index alues ha we e used o upda e con ac ne cos s a e published pe iodically by a Czech company, RTS ® , a de elope and p o ide o a p ice in o ma ion sys em o Appl. Sci. 2023,13, 12744 9 o 24 he cons uc ion indus y in he Czech Republic. (I is wo h men ioning ha he e a e wo main p icing sys ems ha p o ide a ious cos in o ma ion o cons uc ion cos es ima ion p ac ice in he Czech Republic. These a e RTS ® and URS ® .) Based on he s uc u al and ma e ial cha ac e is ics o sewe age sys ems, he cos indexes used o calcula ions we e de i ed om he RTS ® sys em. Mo e speci ically, p ice indica o s ha e lec he changes in cos s in sewe age cons uc ion p ojec s be ween 2018 and 2023 we e used. The ob ained cos indexes we e also compa ed wi h he second p icing sys em, URS ® , o e i y hei co ec ness and applicabili y. In he Czech Republic, his p ocedu e is commonly used o indexing he p ices o cons uc ion wo ks be ween di e en ime pe iods and is also pe missible o he needs o cou e idence. Ou lie analysis was applied o he upda ed alues o con ac ne cos s o sewe age p ojec cons uc ion wo ks. The concep and undamen als o ou lie analysis can be ound in he s a is ical li e a u e, such as [ 56 – 58 ]. The pu pose was o elimina e da a poin s ha de ia ed signi ican ly om o he s, essen ially excluding unusual cos alues om he da ase . The app oach used in his s udy elied on he concep o he in e qua ile ange (IQR). The IQR was calcula ed as he di e ence be ween he alues o he hi d qua ile (Q3) and he i s qua ile (Q1), ep esen ing he ange o alues be ween hese qua iles: IQR =Q3 − Q1. The ule below allowed us o iden i y and elimina e ou lying alues, as well as en i e eco ds o ce ain p ojec cases om he da ase : • I he j- h alue (in he j- h eco d in he da ase ) o upda ed con ac ne cos s does no belong o he ange: <Q1 − 1.5 · IQR;Q3+1.5 · IQR> → elimina e he j- h eco d om he da ase . The a ionale o his app oach, based on he au ho s’ p io expe iences, is ha ou lie s can be p oblema ic when de eloping cos p edic ion models o a ious cons uc ion p ojec s, acili ies, and s uc u es, as hey o en ep esen speci ic, high-cos p ojec s ha a e spa sely ep esen ed in da ase s. Such da a can dis o he p edic i e pe o mance o a de eloped model. The collec ed da a unde wen u he p e-p ocessing. Nume ical alues o cos p e- dic o s we e linea ly scaled, while desc ip i e alues we e p ocessed di e en ly based on hei na u e; hey we e pseudo- uzzy scaled, coded as one-o -n alues, o con e ed in o bina y alues. De ailed in o ma ion and ou comes o his s ep a e p esen ed in Sec ion 4. The nex s age o he esea ch in ol ed compu a ions and simula ions o a i icial neu al ne wo ks (ANN), as well as he combina ion o hese ne wo ks o c ea e an ensemble. The de elopmen o models based on ensembles o neu al ne wo ks (EoNN) designed o p edic cos s in ol es sol ing eg ession p oblems. Le he dependen a iable yo such models ep esen he cons uc ion cos o a sewe age p ojec . Also, le he independen a iables be he cos p edic o s, wi h he ec o o hese a iables deno ed as x . The EoNN- based models a e expec ed o app oxima e he mapping x→ y. Impo an ly, in he case o employing EoNN, he app oxima ion unc ion his implici ly de ined as ollows: y=h(x)+ε(2) whe e ε co esponds o he p edic ion e o . Consequen ly, he o mal no a ion o p edic - ing cos s ˆ ycan be exp essed as ollows: ˆ y=h(x)(3) The u iliza ion o EoNN as he co e o a cos es ima ion model elies on combining a se o ained neu al ne wo ks o o m an ensemble. As ou lined in [ 28 ], his se migh encompass a ious ypes o ne wo ks o simila ne wo ks ained o di e en local minima. The wo me hods buil on his p emise, which we e applied du ing esea ch p esen ed he ein, a e (1) ensemble a e aging and (2) gene alized a e aging. A summa y o he co e p inciples behind he wo ensemble-based me hods men ioned is p o ided based on [ 28 , 30 ]. The gene al idea is schema ically depic ed in Figu e 2. Appl. Sci. 2023,13, 12744 16 o 24 Fo GAV ENS, compu a ions elied on Equa ions (8) and (9) and equi ed mo e e - o . The weigh s o combining ou pu s p o ided by he membe s o he ensemble a e p esen ed below. α1= 0.109; α2= 0.242; α3=−0.185; α4= 0.170; α5= 0.664 Fo bo h SAV ENS and GAV ENS models, u he compu a ions based on he espec i e weigh s αk and Equa ion (5) we e conduc ed o ob ain le el-1 p edic ions o sewe age p ojec cons uc ion cos s. Figu es 3and 4p esen sca e plo s ha ep esen eal-li e alues o upda ed cons uc ion cos s o sewe p ojec s yand, on he con a y, alues p edic ed by he de eloped models ˆ y. Sca e plo s, which a e al e na i ely e e ed o as sca e g ams o sca e cha s, se e he pu pose o p o iding a isual ep esen a ion o da a poin s wi hin a wo-dimensional Ca esian coo dina e sys em. Each da a poin is depic ed as a poin o do , acili a ing he obse a ion o he ela ionship be ween he ac ual alues yand he p edic ed alues ˆ y. In essence, each da a poin on he plo co esponds o a pai o associa ed alues. Th ough he u iliza ion o sca e plo s, he assessmen o co ela ions be ween eal-wo ld alues and alues p edic ed by a model, as well as he e alua ion o p edic ion quali y, is ca ied ou ; Appl. Sci. 2023, 13, x FOR PEER REVIEW 17 o 25 y and he p edic ed alues ŷ. In essence, each da a poin on he plo co esponds o a pai o associa ed alues. Th ough he u iliza ion o sca e plo s, he assessmen o co ela ions be ween eal-wo ld alues and alues p edic ed by a model, as well as he e alua ion o p edic ion quali y, is ca ied ou ; Figu es 3 and 4 depic esul s o SAV ENS and GAV ENS models, espec i ely. The poin s in he g aphs ep esen he esul s o he aining (L&V subse ) and es ing (T subse ) p o- cesses. The dis ibu ion o poin s indica es ha , in gene al, he quali y o cos p edic ion is compa able o bo h ensemble-based models. The e a e no signi ican de ia ions, and he poin s a e dis ibu ed along he lines o a pe ec i ; On he basis o eal-li e alues y and alues p edic ed by he de eloped models ŷ as well as Equa ion (10), co ela ion coefficien s R we e compu ed; • Fo SAV ENS : R = 0.976 o he L&V subse and R = 0.988 o T subse ; • Fo GAV ENS : R = 0.972 o he L&V subse and R = 0.987 o T subse ; Co ela ion o y and ŷ is e y high, and no signi ican diffe ences be ween he models can be iden i ied. Figu e 3. Sca e plo o eal-li e alues y and p edic ed alues ŷ o cos s o SAV ens model. (a) L&V subse , (b) T subse . Figu e 4. Sca e plo o eal-li e alues y and p edic ed alues ŷ o cos s o GAV ens model. (a) L&V subse , (b) T subse . Figu es 5 and 6 p esen dis ibu ions o pe cen age e o s PE p , compu ed using Equa- ion (15) and ca ego ized wi hin he anges shown on he ho izon al axes o he SAV ENS and GAV ENS models, espec i ely; Figu e 3. Sca e plo o eal-li e alues yand p edic ed alues ˆ yo cos s o SAV ens model. ( a )L&V subse , (b)Tsubse . Appl. Sci. 2023, 13, x FOR PEER REVIEW 17 o 25 y and he p edic ed alues ŷ. In essence, each da a poin on he plo co esponds o a pai o associa ed alues. Th ough he u iliza ion o sca e plo s, he assessmen o co ela ions be ween eal-wo ld alues and alues p edic ed by a model, as well as he e alua ion o p edic ion quali y, is ca ied ou ; Figu es 3 and 4 depic esul s o SAV ENS and GAV ENS models, espec i ely. The poin s in he g aphs ep esen he esul s o he aining (L&V subse ) and es ing (T subse ) p o- cesses. The dis ibu ion o poin s indica es ha , in gene al, he quali y o cos p edic ion is compa able o bo h ensemble-based models. The e a e no signi ican de ia ions, and he poin s a e dis ibu ed along he lines o a pe ec i ; On he basis o eal-li e alues y and alues p edic ed by he de eloped models ŷ as well as Equa ion (10), co ela ion coefficien s R we e compu ed; • Fo SAV ENS : R = 0.976 o he L&V subse and R = 0.988 o T subse ; • Fo GAV ENS : R = 0.972 o he L&V subse and R = 0.987 o T subse ; Co ela ion o y and ŷ is e y high, and no signi ican diffe ences be ween he models can be iden i ied. Figu e 3. Sca e plo o eal-li e alues y and p edic ed alues ŷ o cos s o SAV ens model. (a) L&V subse , (b) T subse . Figu e 4. Sca e plo o eal-li e alues y and p edic ed alues ŷ o cos s o GAV ens model. (a) L&V subse , (b) T subse . Figu es 5 and 6 p esen dis ibu ions o pe cen age e o s PE p , compu ed using Equa- ion (15) and ca ego ized wi hin he anges shown on he ho izon al axes o he SAV ENS and GAV ENS models, espec i ely; Figu e 4. Sca e plo o eal-li e alues yand p edic ed alues ˆ yo cos s o GAV ens model. ( a )L&V subse , (b)Tsubse . Appl. Sci. 2023,13, 12744 17 o 24 Figu es 3and 4depic esul s o SAV ENS and GAV ENS models, espec i ely. The poin s in he g aphs ep esen he esul s o he aining (L&V subse ) and es ing (Tsubse ) p ocesses. The dis ibu ion o poin s indica es ha , in gene al, he quali y o cos p edic ion is compa able o bo h ensemble-based models. The e a e no signi ican de ia ions, and he poin s a e dis ibu ed along he lines o a pe ec i ; On he basis o eal-li e alues yand alues p edic ed by he de eloped models ˆ yas well as Equa ion (10), co ela ion coe icien s Rwe e compu ed; •Fo SAVENS:R= 0.976 o he L&V subse and R= 0.988 o Tsubse ; •Fo GAVENS:R= 0.972 o he L&V subse and R= 0.987 o Tsubse ; Co ela ion o yand ˆ yis e y high, and no signi ican di e ences be ween he models can be iden i ied. Figu es 5and 6p esen dis ibu ions o pe cen age e o s PE p , compu ed using Equa- ion (15) and ca ego ized wi hin he anges shown on he ho izon al axes o he SAV ENS and GAVENS models, espec i ely; Appl. Sci. 2023, 13, x FOR PEER REVIEW 18 o 25 An analysis o he PE p dis ibu ions allows us o selec he SAV ENS model as he one ha is sligh ly mo e s able when compa ing aining and es ing e o s. The sha es o PE p wi hin he ange <−30%; 30%> we e as ollows: • Fo SAV ENS : 90.6% o he L&V subse and 94.7% o he T subse ; • Fo GAV ENS : 84.0% o he L&V subse and 94.7% o he T subse . Table 9 p esen s a summa y and pe o mance measu es o he wo de eloped mod- els, speci ically RMSE and MAPE alues, as well as he maximum alues o APE p . The maximum alues o APE p , which a e lowe o he SAV ENS model, con i m ha i s p edic- i e pe o mance is sligh ly be e han ha o he GAV ENS model. Table 9. Values o gene al p edic i e pe o mance measu es o he EoNN models. EoNN RMSE L&V RMSE T MAPE L&V MAPE T max {APE PL&V } max {APE PT } SAV ENS 436.8 344.5 15.1% 9.3% 57.6% 29.6% GAV ENS 266.3 257.5 14.9% 9.9% 64.6% 31.3% In gene al, i can be concluded ha he ob ained esul s a e sa is ac o y. The de el- oped p edic i e models p o ide cos es ima es wi hin he assumed and p e e ed ange o accu acy o he majo i y o bo h aining and, mos impo an ly, es ing cases. The model based on simple a e aging pe o ms sligh ly be e and offe s highe accu acy. Figu e 5. Dis ibu ion o PE p e o s o SAV ens model. (a) L&V subse , (b) T subse . Figu e 5. Dis ibu ion o PEpe o s o SAVens model. (a)L&V subse , (b)Tsubse . Appl. Sci. 2023,13, 12744 18 o 24 Appl. Sci. 2023, 13, x FOR PEER REVIEW 19 o 25 Figu e 6. Dis ibu ion o PE p e o s o GAV ens model. (a) L&V subse , (b) T subse . 6. Discussion As e iden om he li e a u e analysis, a emp s o de elop cos analysis models o sewe age p ojec s ha e been made [19–26]. In compa ison o he analyses p esen ed in his a icle, i can be no ed ha he selec ion o cos p edic o s, in e ms o hei na u e, is sim- ila . Howe e , i should no be o go en ha local cons uc ion ma ke condi ions and da a a ailabili y also in luence he esea ch. The inal se o cos p edic o s and hei alues s ongly depend on he possibili y o ob aining hem, which esul s in some diffe ences be ween he models p esen ed in he li e a u e. Among he men ioned wo ks, some a e based on he use o ANN [25,26]. Un o u- na ely, i is challenging o compa e he esul s o hese s udies wi h he indings o his esea ch. The ci ed wo ks used less da a o aining and es ing, and he models a e based on single ne wo ks. Mos impo an ly, he e is a lack o p ecise in o ma ion abou he ypes o neu al ne wo ks used and essen ial de ails ega ding he aining and es ing p o- cesses o ANN and he analysis o hei pe o mance. Thus, i was decided, o he pu pose o u he assessing esea ch esul s, o de elop a benchma k model based on mul iple eg ession using he classical leas squa e me hod. The benchma k model is he eina e e e ed o as MR. The gene al o mula o p edic- ions based on he MR model is p o ided below. 𝑦=𝛽+𝛽𝑥  (16) whe e: Figu e 6. Dis ibu ion o PEpe o s o GAVens model. (a)L&V subse , (b)Tsubse . An analysis o he PE p dis ibu ions allows us o selec he SAV ENS model as he one ha is sligh ly mo e s able when compa ing aining and es ing e o s. The sha es o PE p wi hin he ange <−30%; 30%> we e as ollows: •Fo SAVENS: 90.6% o he L&V subse and 94.7% o he Tsubse ; •Fo GAVENS: 84.0% o he L&V subse and 94.7% o he Tsubse . Table 9p esen s a summa y and pe o mance measu es o he wo de eloped models, speci ically RMSE and MAPE alues, as well as he maximum alues o APE p . The maxi- mum alues o APE p , which a e lowe o he SAV ENS model, con i m ha i s p edic i e pe o mance is sligh ly be e han ha o he GAVENS model. Table 9. Values o gene al p edic i e pe o mance measu es o he EoNN models. EoNN RMSEL&V RMSETMAPEL&V MAPETmax {APEPL&V} max {APEPT} SAVENS 436.8 344.5 15.1% 9.3% 57.6% 29.6% GAVENS 266.3 257.5 14.9% 9.9% 64.6% 31.3% Appl. Sci. 2023,13, 12744 19 o 24 In gene al, i can be concluded ha he ob ained esul s a e sa is ac o y. The de eloped p edic i e models p o ide cos es ima es wi hin he assumed and p e e ed ange o accu acy o he majo i y o bo h aining and, mos impo an ly, es ing cases. The model based on simple a e aging pe o ms sligh ly be e and o e s highe accu acy. 6. Discussion As e iden om he li e a u e analysis, a emp s o de elop cos analysis models o sewe age p ojec s ha e been made [ 19 – 26 ]. In compa ison o he analyses p esen ed in his a icle, i can be no ed ha he selec ion o cos p edic o s, in e ms o hei na u e, is simila . Howe e , i should no be o go en ha local cons uc ion ma ke condi ions and da a a ailabili y also in luence he esea ch. The inal se o cos p edic o s and hei alues s ongly depend on he possibili y o ob aining hem, which esul s in some di e ences be ween he models p esen ed in he li e a u e. Among he men ioned wo ks, some a e based on he use o ANN [ 25 , 26 ]. Un o u- na ely, i is challenging o compa e he esul s o hese s udies wi h he indings o his esea ch. The ci ed wo ks used less da a o aining and es ing, and he models a e based on single ne wo ks. Mos impo an ly, he e is a lack o p ecise in o ma ion abou he ypes o neu al ne wo ks used and essen ial de ails ega ding he aining and es ing p ocesses o ANN and he analysis o hei pe o mance. Thus, i was decided, o he pu pose o u he assessing esea ch esul s, o de elop a benchma k model based on mul iple eg ession using he classical leas squa e me hod. The benchma k model is he eina e e e ed o as MR. The gene al o mula o p edic ions based on he MR model is p o ided below. ˆ y=β0+∑ j βjxj(16) whe e: β0,βj— eg ession coe icien s. To ensu e he compa abili y o he MR benchma k model wi h he models based on he EoNN app oach de eloped du ing he esea ch, he compu a ion o coe icien s β0 and βj was pe o med wi h he use o subse C, equi alen o he aining subse o ANNs ha became membe s o SAV ENS and GAV ENS (including cases used o lea ning and alida ion p ocesses). Fo es ing he MR model, subse Twas used. Below a e he coe icien s ob ained om he eg ession analysis, along wi h he s anda d e o s o es ima ion p o ided in he b acke s. β0=−5588.5 (1420.85); β1= 1064.6 (376.12); β2=−39,347.1 (62,376.80); β3= 1414.0 (1025.67); β4= 2416.5 (727.67); β5= 2821.5 (548.23); β6= 30.8 (235.41); β7= 240.0 (358.94); β8= 3329.0 (763.03); β9= 1340.1 (635.74); β10 = 38,154.7 (53,700.36); β11 = 18,397.4 (20,636.69); β12 = 36.9 (449.73) Figu e 7displays he esul s o he MR model in he o m o a sca e plo o yand ˆ y alues (compa e wi h Figu es 3and 4). Co ela ion coe icien s Rwe e compu ed in a simila manne as in he case o he SAVENS and GAVENS models; •Fo MR: R= 0.926 o he Csubse and R= 0.940 o he Tsubse ; When compa ed o he co ela ions compu ed o he de eloped EoNN-based models, he di e ences ha occu a e ela i ely insigni ican . Howe e , an analysis o he sca e plo s and a compa ison wi h hose p esen ed o EoNN-based models e eal g ea e dispe - sion and de ia ions om he line o pe ec i in he case o he MR benchma k model; Appl. Sci. 2023,13, 12744 20 o 24 An analysis o he PE p dis ibu ions o he MR benchma k model e eals he supe i- o i y o EoNN-based models. Fo MR, he sha es o PE p wi hin he ange < − 30%; 30%> we e as ollows: •68.9% o he Csubse and 73.7% o he Tsubse ; Table 10 p o ides a summa y and pe o mance measu es o he MR benchma k model (compa e wi h Table 9). Appl. Sci. 2023, 13, x FOR PEER REVIEW 20 o 25 β 0 , β j — eg ession coefficien s. To ensu e he compa abili y o he MR benchma k model wi h he models based on he EoNN app oach de eloped du ing he esea ch, he compu a ion o coefficien s β 0 and β j was pe o med wi h he use o subse C, equi alen o he aining subse o ANNs ha became membe s o SAV ENS and GAV ENS (including cases used o lea ning and alida ion p ocesses). Fo es ing he MR model, subse T was used. Below a e he coefficien s ob- ained om he eg ession analysis, along wi h he s anda d e o s o es ima ion p o ided in he b acke s. β 0 = −5588.5 (1420.85); β 1 = 1064.6 (376.12); β 2 = −39 , 347.1 (62 , 376.80); β 3 = 1414.0 (1025.67); β 4 = 2416.5 (727.67); β 5 = 2821.5 (548.23); β 6 = 30.8 (235.41); β 7 = 240.0 (358.94); β 8 = 3329.0 (763.03); β 9 = 1340.1 (635.74); β 10 = 38 , 154.7 (53 , 700.36); β 11 = 18 , 397.4 (20 , 636.69); β 12 = 36.9 (449.73) Figu e 7 displays he esul s o he MR model in he o m o a sca e plo o y and ŷ alues (compa e wi h Figu es 3 and 4). Co ela ion coefficien s R we e compu ed in a simila manne as in he case o he SAV ENS and GAV ENS models; • Fo MR: R = 0.926 o he C subse and R = 0.940 o he T subse ; When compa ed o he co ela ions compu ed o he de eloped EoNN-based mod- els, he diffe ences ha occu a e ela i ely insigni ican . Howe e , an analysis o he sca - e plo s and a compa ison wi h hose p esen ed o EoNN-based models e eal g ea e dispe sion and de ia ions om he line o pe ec i in he case o he MR benchma k model; An analysis o he PE p dis ibu ions o he MR benchma k model e eals he supe i- o i y o EoNN-based models. Fo MR, he sha es o PE p wi hin he ange <−30%; 30%> we e as ollows: • 68.9% o he C subse and 73.7% o he T subse ; Table 10 p o ides a summa y and pe o mance measu es o he MR benchma k model (compa e wi h Table 9). Table 10. Values o gene al p edic i e pe o mance measu es o he MR model. RMSE C RMSE T MAP E C MAPE T max {APE PC } max {APE PT } MR 647.04 743.31 29.5% 19.3% 81.9% 111.5% Figu e 7. Sca e plo o eal-li e alues y and p edic ed alues ŷ o cos s o benchma k MR model. (a) C subse , (b) T subse . Figu e 7. Sca e plo o eal-li e alues yand p edic ed alues ˆ yo cos s o benchma k MR model. (a)Csubse , (b)Tsubse . Table 10. Values o gene al p edic i e pe o mance measu es o he MR model. RMSECRMSETMAPECMAPETmax {APEPC} max {APEPT} MR 647.04 743.31 29.5% 19.3% 81.9% 111.5% Figu e 8depic s he dis ibu ions o pe cen age e o s PE p , simila o hose p esen ed o he SAVENS and GAVENS models (compa e Figu es 5and 6). A compa ison o he alues in Tables 9and 10 e eals ha he o e all p edic ion pe o mance o he MR model based on linea eg ession is weake han he pe o mance o EoNN-based models (speci ically, he SAV ENS and he GAV ENS models) de eloped by his s udy o he pu poses o sewe age p ojec s cons uc ion cos s p edic ion. As desc ibed in he p eceding sec ion, he esea ch esul s a e sa is ac o y. The models de eloped using EoNN ha e demons a ed hei abili y o p edic cons uc ion cos s o sewe age p ojec s, o ei he s o mwa e uno s o was ewa e uno s, wi h accep able accu acy, mee ing he equi emen s o he expec ed ange o e o s in o e 90% o he es cases. The SAV ENS and GAV ENS models exhibi imp o ed p edic i e capabili ies when compa ed o indi idual ANNs, selec ed o be he ensemble membe s, used as s andalone models. This imp o emen can be a ibu ed o he e ec i e compensa ion o e o s inhe en in single ANNs when hey a e combined wi hin he ensemble. Mo eo e , combining se e al ANNs p o ides mo e objec i e cos p edic ions, as a ce ain bias o single ANNs ac ing in isola ion is ine i able due o he size o he aining se used in he cou se o esea ch. The wo applied app oaches, speci ically simple a e aging o he SAV ENS model and gene alized a e aging o he GAV ENS model, di e in e ms o he compu a ional e o equi ed o de e mine he weigh s o he membe ANNs. The i s app oach is s aigh o wa d—in ac , i in ol es aking he a i hme ical a e age o p edic ions as he EoNN ou pu . In con as , he second app oach demands mo e e o . Howe e , his can be Appl. Sci. 2023,13, 12744 21 o 24 e icien ly accomplished using a calcula ion shee o h ough p og amming and au oma ion o compu a ions. The gene alized a e aging app oach, wi h i s weigh op imiza ion, allows o a mo e nuanced di e en ia ion o he in luence o indi idual membe ANNs on he inal cos p edic ion. In summa y, bo h app oaches a e use - iendly and easily applicable. Appl. Sci. 2023, 13, x FOR PEER REVIEW 21 o 25 Figu e 8 depic s he dis ibu ions o pe cen age e o s PE p , simila o hose p esen ed o he SAV ENS and GAVE NS models (compa e Figu es 5 and 6). Figu e 8. Dis ibu ion o PE p e o s o benchma k MR model. (a) C subse , (b) T subse . A compa ison o he alues in Tables 9 and 10 e eals ha he o e all p edic ion pe - o mance o he MR model based on linea eg ession is weake han he pe o mance o EoNN-based models (speci ically, he SAV ENS and he GAV ENS models) de eloped by his s udy o he pu poses o sewe age p ojec s cons uc ion cos s p edic ion. As desc ibed in he p eceding sec ion, he esea ch esul s a e sa is ac o y. The mod- els de eloped using EoNN ha e demons a ed hei abili y o p edic cons uc ion cos s o sewe age p ojec s, o ei he s o mwa e unoffs o was ewa e unoffs, wi h accep a- ble accu acy, mee ing he equi emen s o he expec ed ange o e o s in o e 90% o he es cases. The SAV ENS and GAV ENS models exhibi imp o ed p edic i e capabili ies when compa ed o indi idual ANNs, selec ed o be he ensemble membe s, used as s andalone models. This imp o emen can be a ibu ed o he effec i e compensa ion o e o s inhe - en in single ANNs when hey a e combined wi hin he ensemble. Mo eo e , combining se e al ANNs p o ides mo e objec i e cos p edic ions, as a ce ain bias o single ANNs ac ing in isola ion is ine i able due o he size o he aining se used in he cou se o esea ch. The wo applied app oaches, speci ically simple a e aging o he SAV ENS model and gene alized a e aging o he GAV ENS model, diffe in e ms o he compu a ional effo equi ed o de e mine he weigh s o he membe ANNs. The i s app oach is s aigh o - wa d—in ac , i in ol es aking he a i hme ical a e age o p edic ions as he EoNN Figu e 8. Dis ibu ion o PEpe o s o benchma k MR model. (a)Csubse , (b)Tsubse . The key ad an age o an ensemble-based app oach, as a i med by he esea ch p e- sen ed, lies in he e icien u iliza ion o aining and es ing e o s ac oss ANNs a he han ocusing solely on a single ne wo k. This obse a ion ca ies pa icula signi icance in he con empo a y con ex , whe e apid p og ess in compu e echnology and he a ailabili y o e icien so wa e enables he explo a ion o nume ous ne wo ks wi hin a ela i ely sho ime ame. I is also essen ial o acknowledge he limi a ions o he de eloped models. Fi s ly, he models a e explici ly ailo ed o he speci ic local condi ions. This is a consequence o he e iden ac ha he da a used o aining and es ing we e collec ed in he Czech Republic. Howe e , i is wo h no ing ha he applica ion o he p oposed app oach in o he loca ions is easible, making he concep ual amewo k b oadly applicable. The second signi ican limi a ion s ems om he da a upda e scheduled o mid-2023. In his ega d, he p oposed model and app oach do no p o ide dynamic and au oma ic adjus men o cos a ia ions o e ime. This issue will be he subjec o u he esea ch. Appl. Sci. 2023,13, 12744 22 o 24 In summa y, he no el y o he p oposed model lies in i s use o AI ools, speci ically ANNs, and he combina ion o ained ANNs in he o m o an ensemble o achie e objec i e cos p edic ions o sewe age cons uc ion p ojec s. The li e a u e e iew indica es ha his app oach is o iginal, wi h no p io esea ch epo ing he de elopmen o simila models o such p ojec s. 7. Summa y The esea ch esul ed in he de elopmen o wo o iginal p edic i e models capable o o ecas ing he cons uc ion cos s o sewe age p ojec s based on ensembles o neu al ne wo ks. Based on he applied a i icial in elligence ool’s aining capaci y, in luenced by he collec ed da a, ea u es, and cha ac e is ics o he analyzed p ojec s, as well as he unde lying assump ions, he de eloped models demons a e he capabili y o o ecas cons uc ion cos s o ei he was ewa e o s o mwa e uno s, excluding combined uno s. The ensembles men ioned abo e consis o i e di e en MLP- ype ANNs. The ou pu s o hese ANNs a e combined using wo al e na i e app oaches: simple a e aging and gene alized a e aging. Al hough he p edic i e pe o mances o he wo EoNN-based models a e compa able, he one based on simple a e aging appea s o o e sligh ly be e esul s. The accu acy o cos p edic ions is sa is ac o y. Especially o he model based on simple a e aging, ha is, he SAV ENS model, mo e han 90% o cos p edic ion cases, bo h o aining and es ing, mee he accu acy equi emen s wi h pe cen age e o s alling wi hin he accep able ange o < − 30%; 30%>. While he de eloped model has i s limi a ions, i holds po en ial applica ions in es ima ing he cos s o cons uc ion o sewe age p ojec s in he Czech Republic. Fu he mo e, he p oposed gene al app oach may ind applicabili y in o he coun ies, al hough he models should be adap ed and ained using locally collec ed da a. Fu u e s udies will in ol e u he da a collec ion and he de elopmen o models based on AI ools. Addi ionally, u u e esea ch will ocus on inco po a ing cos a iabili y o e ime in o hese models in o de o o e come exis ing limi a ions. Au ho Con ibu ions: Concep ualiza ion, M.J., T.H. and M.V.; li e a u e e iew, M.J., T.H. and H.P.; me hodology, M.J.; sou ce documen s analysis and da a collec ion, T.H. and M.V.; da a cu a ion, M.J. and T.H.; o mal analysis and compu a ions, M.J., H.P. and M.S.; esul s analysis, M.J., H.P. and M.S.; discussion, M.J., T.H., M.V., H.P. and M.S.; inal conclusions, M.J., T.H., M.V., H.P. and M.S.; w i ing—o iginal d a p epa a ion, M.J.; e iew, M.J. and T.H.; edi ing, M.J.; unding acquisi ion, M.J., T.H. and M.V. All au ho s ha e ead and ag eed o he published e sion o he manusc ip . Funding: This esea ch was co- unded by p ojec no. FAST-S-23-8253 and p ojec no. FAST-J-23-8349 held by B no Uni e si y o Technology; s a u o y unds o he Facul y o Ci il Enginee ing, C acow Uni e si y o Technology; p og am o he Polish Minis y o Educa ion and Science “Implemen a ion doc o a e”, ag eemen numbe be ween he C acow Uni e si y o Technology and he Polish S a e T easu y/Minis e o Educa ion and Science—DWD/6/0520/2022. 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 : Some o all da a ha suppo he indings o his s udy is a ailable om he au ho s upon easonable eques . The da a a e no publicly a ailable o allow o ac ions aimed a comme cializing he esea ch indings. Con lic s o In e es : Au ho Hanna Pacyno was employed by he company Da acomp IT sp. z o.o. The emaining au ho s decla e ha he esea ch was conduc ed in he absence o any comme cial o inancial ela ionships ha could be cons ued as a po en ial con lic o in e es . Re e ences 1. Yang, M.-D.; Su, T.-C. 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Disclaime /Publishe ’s No e: The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). 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 .