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 .
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