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Compa ing e e ence poin based in e ac i e mul iobjec i e op imiza ion me hods
wi hou a human decision make
© The Au ho (s) 2022
Published e sion
Chen, Lu; Mie inen, Kaisa; Xin, Bin; Ojaleh o, Vesa
Chen, L., Mie inen, K., Xin, B., & Ojaleh o, V. (2023). Compa ing e e ence poin based
in e ac i e mul iobjec i e op imiza ion me hods wi hou a human decision make . Jou nal o
Global Op imiza ion, 85(3), 757-788. h ps://doi.o g/10.1007/s10898-022-01230-3
2023
Jou nal o Global Op imiza ion
h ps://doi.o g/10.1007/s10898-022-01230-3
Compa ing e e ence poin based in e ac i e mul iobjec i e
op imiza ion me hods wi hou a human decision make
Lu Chen1,2,3 ·Kaisa Mie inen4·Bin Xin1,2,3 ·Vesa Ojaleh o4
Recei ed: 22 Ma ch 2020 / Accep ed: 14 Augus 2022
© The Au ho (s) 2022
Abs ac
In e ac i e mul iobjec i e op imiza ion me hods ha e p o en p omising in sol ing op i-
miza ion p oblems wi h con lic ing objec i es since hey i e a i ely inco po a e p e e ence
in o ma ion o a decision make in he sea ch o he mos p e e ed solu ion. To ind he
app op ia e in e ac i e me hod o a ious needs in ol es analysis o he s eng hs and weak-
nesses. Howe e , ex ensi e analysis wi h human decision make s may be oo cos ly and o
ha eason, we p opose an a i icial decision make o compa e a class o popula in e ac i e
mul iobjec i e op imiza ion me hods, i.e., e e ence poin based me hods. Wi hou in ol ing
any human decision make s, he a i icial decision make wo ks au oma ically o in e ac
wi h di e en me hods o be compa ed and e alua e he inal esul s. I makes a di e ence
be ween a lea ning phase and a decision phase, ha is, lea ns abou he p oblem based on
in o ma ion acqui ed o iden i y a egion o in e es and e ines solu ions in ha egion o ind
a inal solu ion, espec i ely. We adop di e en ypes o u ili y unc ions o e alua ion solu-
ions, p esen co esponding pe o mance indica o s and p opose wo examples o a i icial
decision make s. A se ies o expe imen s on benchma k es p oblems and a wa e esou ces
planning p oblem is conduc ed o demons a e how he p oposed a i icial decision make s
can be used o compa e e e ence poin based me hods.
BBin Xin
[email p o ec ed]
Lu Chen
[email p o ec ed]
Kaisa Mie inen
[email p o ec ed]
Vesa Ojaleh o
[email p o ec ed]
1School o Au oma ion, Beijing Ins i u e o Technology, Beijing 100081, China
2S a e Key Labo a o y o In elligen Con ol and Decision o Complex Sys ems, Beijing 100081,
China
3Beijing Ad anced Inno a ion Cen e o In elligen Robo s and Sys ems, Beijing 100081, China
4Uni e si y o Jy askyla, Facul y o In o ma ion Technology, P.O. Box 35 (Ago a), FI-40014 Uni e si y
o Jy askyla, Finland
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Jou nal o Global Op imiza ion
Keywo ds Mul ic i e ia op imiza ion ·In e ac i e mul iobjec i e op imiza ion ·Lea ning
phase ·Decision phase ·Pe o mance compa ison ·Re e ence poin
1 In oduc ion
Many eal-wo ld op imiza ion p oblems in ol e con lic ing objec i es which a e o be op i-
mized simul aneously. These p oblems a e known as mul iobjec i e op imiza ion p oblems.
The ul ima e aim o mul iobjec i e op imiza ion is o suppo a decision make (DM) o
ind his/he mos p e e ed solu ion. Up o now, a ious mul iobjec i e op imiza ion me h-
ods ha e been p oposed by bo h he Mul iple C i e ia Decision Making (MCDM) and he
E olu iona y Mul iobjec i e Op imiza ion (EMO) communi ies [7,20,30,56]. They can be
di ided in o no-p e e ence me hods, a p io i me hods, a pos e io i me hods, and in e ac i e
me hods acco ding o he ole o he decision make in he solu ion p ocess [21,29]. Among
hem, in e ac i e me hods ha e been widely de eloped due o se e al ad an ages [9,29,30,
33,45]. Fo example, he DM can lea n p og essi ely om he solu ion p ocess and adjus
his/he p e e ences. In his case, only solu ions ha he DM is in e es ed in will be ob ained.
Thus, he compu a ional complexi y can be educed in compa ison wi h app oxima ing he
whole se o Pa e o op imal solu ions.
In he solu ion p ocess wi h an in e ac ion me hod, we can o en iden i y wo phases:
a lea ning phase and a decision phase [33]. The lea ning phase is impo an o a DM o
lea n abou he possibili ies and limi a ions o he p oblem. In he decision phase, he DM
can ocus on he egion iden i ied in he lea ning phase o ind be e solu ions. Di e en
in e ac i e me hods equi e di e en ypes o p e e ence in o ma ion om he DM, such as
e e ence poin s, desi abili y o ade-o s, o he classi ica ion o objec i e unc ions [33].
Among hem, e e ence poin based me hods a e e y popula owing o se e al easons. Fo
example, i is con enien and in ui i e o he DM o speci y a e e ence poin consis ing o
desi able objec i e unc ion alues since he me hod p o ides objec i e unc ion alues o
he DM, and, hus, no cogni i e mapping is necessa y [30]. Besides, e e ence poin s can
be speci ied wi hou conside ing he ade-o s among objec i es. Thus, he DM’s bu den is
ela i ely low [24].
The p oblem o how o compa e in e ac i e me hods a ises na u ally wi h he eme gence o
di e se in e ac i e me hods. A pos e io i me hods like mul iobjec i e e olu iona y algo i hms
can be compa ed wi h some pe o mance me ics (see, e.g., [25,54]). Howe e , compa ing
in e ac i e me hods is no e y s aigh o wa d. Only limi ed s udies (see, e.g., [3,4,8,10–12,
26,28,35,37,39,44,58]) ha e ocused on his opic and ew o hem a e alid o e e ence
poin based in e ac i e me hods. In his pape , we concen a e on he compa ison o e e ence
poin based me hods.
The DM is a key elemen o applying and also compa ing in e ac i e me hods. Using
human DMs o compa e in e ac i e me hods has some di icul ies. Fi s ly, i may be a he
expensi e because human DMs in ol ed in he expe imen s should ha e app op ia e domain
expe ise. Fu he mo e, he o de o using in e ac i e me hods will in luence he compa ison
esul s because a human DM lea ns and wha he/she has lea n when in e ac ing wi h a
me hod will a ec his/he decisions on he subsequen me hods. To elimina e such e ec s, a
su icien ly la ge numbe o human DMs is equi ed so ha di e en g oups o DMs would
use he me hods in di e en o de s. Some esea ch in ol ing human DMs in he expe imen s
has been conduc ed, and he DMs we e asked o exp ess hei eelings abou in e ac i e
me hods such as ease o use and he deg ee o sa is ac ion on he inal solu ion [3,8,11,
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Jou nal o Global Op imiza ion
12]. Howe e , hese expe imen s we e limi ed in e.g., he ange o p oblems s udied and he
numbe o human DMs in ol ed, see [6,55].
Conside ing he abo e di icul ies, a good al e na i e is o de elop a i icial DMs (ADMs)
which can eplace human DMs o compa ing in e ac i e me hods quan i a i ely whene e
human DMs a e no a ailable. In such compa isons, he e a e no eal p e e ences o ollow.
Ins ead, ADMs in e ac wi h each in e ac i e me hod and measu e he esul s o he me hod.
Opposed o using human DMs, expe imen ing wi h ADMs is cheape and less ime consum-
ing. Besides, a la ge numbe o expe imen s can be conduc ed wi hou conside ing human
a igue [6], and quan i a i e esul s can be ob ained. Mos exis ing ADMs ake o ms o u ili y
unc ions (also e e ed o as alue unc ions). Fo example, Malakoo i and Ra ind an [28]
used linea u ili y unc ions as ADMs o compa e mul iobjec i e linea p og amming me h-
ods. Mo e e al. [35] assumed a nonlinea u ili y unc ion. Ree es and Gonzalez [39] adop ed
linea and mul ilinea u ili y unc ions. Fou ypes o u ili y unc ions including quad a ic,
squa e- oo , exponen ial and L4-no m we e conside ed in [44]. L´opez-Ib´añez and Knowles
[26] u ilized linea scala izing unc ions o assess an in e ac i e EMO me hod.
Ne e heless, u ili y unc ions as ADMs canno compa e e e ence poin based me hods
because hey canno p o ide e e ence poin in o ma ion equi ed by hese me hods. U il-
i y unc ions a e only sui able o compa ing non ad hoc in e ac i e me hods [45]suchas
ade-o based in e ac i e me hods since u ili y unc ions can p o ide he ype o p e e -
ence in o ma ion equi ed by hese me hods. To ou bes knowledge, only a ew ADMs ha e
been de eloped o compa ing e e ence poin based me hods [1,2,4,37,38]. The ADMs
de eloped in [4,37,38] all consis s o a s eady pa which includes a p e-de ined aspi a ion
poin o a alue unc ion, and a cu en con ex which includes he knowledge abou he
p oblem gained du ing he solu ions p ocess. In [37], e e ence poin s a e gene a ed by a
decision ee-based app oach. The ADM p oposed in [4] e ol es he e e ence poin h ough
pa icle swa m op imiza ion. Re e ence poin s a e gene a ed in [38] by upda ing he po en ial
egion acco ding o he p ede ined alue unc ion. In hese ADMs, he sea ch is main ained
owa ds he s eady pa , which may hinde he explo a ion o he whole Pa e o on . The
ADMs p oposed in [1,2] dis inguish he lea ning phase and decision phase. Di e en ways
o gene a ing e e ence poin s a e de eloped o he wo phases. Howe e , hese wo ADMs
a e only sui able o compa ing e e ence poin based in e ac i e EMO me hods.
In his pape , we p opose a new ADM o compa ing e e ence poin based me hods
quan i a i ely. Simila o he ADMs de eloped in [1,2], ou ADM also has a lea ning phase
and a decision phase. I should be poin ed ou ha his is he only simila i y be ween ou
ADM and hose wo ADMs. Ou ADM is designed wi h a modula s uc u e con aining
h ee modules: lea ning, e alua ion, and decision. The lea ning module and he decision
module a e used o gene a e e e ence poin s in he wo phases, espec i ely. Speci ically,
a s uc u ed lea ning app oach is designed in he lea ning module o gene a e e e ence
poin s sys ema ically based on Pa e o op imal solu ions de i ed so a o explo e di e en
egions o he sea ch space, which acili a es iden i ying a egion o in e es (ROI). Besides, a
polyhed al cone-based me hod is p oposed o upda e e e ence poin s in he decision module
so ha he solu ions in he ROI can be adjus ed acco ding o he cu en p e e ed solu ion.
The e alua ion module has wo oles. One is e alua ing solu ions ound so a and iden i ying
a p e e ed solu ion a each i e a ion o he decision phase. The o he is o measu e he inal
esul s o in e ac i e me hods a he las i e a ion.
Compa ed o he ADMs de eloped in [4,37,38], ou ADM does no need o con e ge
o a ixed pa a he e y beginning. On he con a y, i beha es di e en ly in wo phases,
e lec ing he DM’s di e en needs. In compa ison wi h he wo ADMs p oposed in [1,2],
ou ADM gene a es e e ence poin s in qui e di e en ways. Those wo ADMs assume ha
123
Jou nal o Global Op imiza ion
solu ion p ocesses o all e e ence poin based EMO me hods being compa ed a e conduc ed
simul aneously. The e e ence poin is upda ed based on he solu ions gene a ed by all me h-
ods. I a new me hod is o be compa ed, he whole compa ison will s a om he beginning,
and all me hods ha e o be un again. Ou ADM ea s all me hods o be compa ed indi idu-
ally and does no ha e his limi a ion. Mo eo e , ou ADM is applicable wi h any e e ence
poin based me hods, no only EMO me hods. I should be emphasized ha ou ADM does
no enhance e e ence poin based me hods, i in e ac s wi h di e en me hods and measu es
he inal esul s o hem o he sake o compa ing hem au oma ically.
By adop ing u ili y unc ions wi h and wi hou noise o e alua e solu ions and p o iding
wo associa ed pe o mance indica o s o e alua e he inal esul s in he e alua ion module,
wo examples o he p oposed ADM a e p esen ed in his pape . They a e used o show how
he p oposed ADM can be u ilized o compa e e e ence poin based me hods and cap u e
he di e ences among hose me hods. The main con ibu ions a e summa ized as ollows:
•A new modula ADM is p oposed o compa ing e e ence poin based in e ac i e mul i-
objec i e op imiza ion me hods au oma ically and quan i a i ely. I is no limi ed o some
speci ic ype o me hods, only he p e e ence in o ma ion mus be a e e ence poin .
•The p oposed ADM akes in o accoun di e en needs ha ypically can be seen in he
lea ning and he decision phases, and wo di e en mechanisms o gene a ing e e ence
poin s a e de eloped acco ding o di e en needs.
•The p oposed ADM’s modula s uc u e o e s a lexible way o c ea ing di e en ype
o ADMs which mimic di e en in e ac ion p ocesses.
•Two pe o mance indica o s a e p oposed o be used wi h he new ADM.
The es o he pape is o ganized as ollows. In Sec . 2, some concep s and no a ions a e
gi en. Sec ion 3is de o ed o p esen ing he p oposed ADM in e ms o i s ac ions in each
phase and p oposing wo pe o mance indica o s o be used wi h he ADM. Two examples o
he ADM a e p o ided, and how o u ilize one o hem wi h a e e ence poin based me hod is
also illus a ed. Sec ions 4and 5demons a e how he ADM can be used o compa e di e en
e e ence poin based me hods on benchma k es p oblems and on a eal-wo ld p oblem,
espec i ely. Finally, conclusions a e d awn and some u u e wo k is discussed b ie ly in
Sec . 6.
2 Concep s and no a ions
Gene ally, a mul iobjec i e op imiza ion p oblem can be de ined as ollows:
minimize (x)={ 1(x), 2(x),..., k(x)}
subjec o x∈S⊂Rn,(1)
whe e k(k≥2)objec i e unc ions i:S→R o i∈{1,2,...,k}a e o be minimized
simul aneously. The decision ec o x=(x1,x2,...,xn)belongs o he easible egion
S⊂Rn. Fo each x, he objec i e ec o z= (x)belongs o he objec i e space Rk.The
image Z= (S)o he easible egion is called he easible objec i e se .
Gi en wo easible decision ec o s x1and x2,x1is said o domina e x2i x1is no wo se
han x2in all objec i es and s ic ly be e in a leas one objec i e. A easible decision ec o x
is said o be Pa e o op imal i and only i he e is no easible decision ec o which domina es
i . The co esponding objec i e ec o is called a Pa e o op imal objec i e ec o . We deno e
he se o all Pa e o op imal decision ec o s as E.Theimage (E)o Eis called Pa e o
on . No e ha we use he e m Pa e o op imal solu ion o e e o a Pa e o op imal objec i e
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Jou nal o Global Op imiza ion
ec o in his pape . The mos p e e ed solu ion (MPS) is he Pa e o op imal solu ion ha
he DM is mos sa is ied wi h [33].
As s a ed in he in oduc ion, we concen a e on compa ing e e ence poin based me h-
ods, a he han p oposing o enhancing such kind o me hods. In he ollowing, we gi e a
b ie in oduc ion on e e ence poin based me hods. A e e ence poin g=(g1,g2,...,gk)
is composed o aspi a ion le els o he DM o all objec i es e lec ing desi able objec i e
alues. I is said o be achie able i i s aspi a ion le els can be achie ed o imp o ed simul a-
neously by a easible solu ion; o he wise, i is said o be unachie able [41]. When p o iding
e e ence poin s, i is o en use ul o he DM o know he anges o he objec i e alues in
he Pa e o on . We deno e z∗=(z∗
1,z∗
2,...,z∗
k)wi h z∗
i=minx∈S i(x) o i=1,2,...,k
o be he ideal objec i e ec o which gi es he minimum alue o each objec i e in he
easible egion. The nadi objec i e ec o znad =(znad
1,znad
2,...,znad
k)is composed o
he maximum alue o each objec i e in he Pa e o on , i.e., znad
i=maxx∈E i(x) o
i=1,2,...,k. I is usually di icul o ob ain znad and one may ha e o se le o app oxi-
ma ions, o example, a payo able [29] o o he special ways [15,47]. An es ima e o znad
can also be de e mined by using he in o ma ion p o ided by he DM, i a ailable.
A each i e a ion, a e e ence poin based me hod uses he e e ence poin (s) p o ided by
he DM o gene a e one o mo e Pa e o op imal solu ions. I he DM is no sa is ied wi h any
solu ion, he/she is expec ed o speci y a new e e ence poin o he me hod o gene a e new
solu ions. Figu e 1illus a es how an example o a e e ence poin based me hod wo ks in
wo i e a ions. This me hod p oduces k+1 Pa e o op imal solu ions based on one e e ence
poin . The solu ion p ocess will con inue un il he DM inds a sa is ac o y solu ion.
Up o now, many e e ence poin based me hods ha e been p oposed. Some me hods, such
as he e e ence poin me hod [51], he e e ence di ec ion (RD) me hod [23], he ligh beam
sea ch (LBS) [22], and he sa is icing ade-o me hod [36], ely on sol ing ans o med
single-objec i e subp oblems o gene a e Pa e o op imal solu ions. This means ha a each
i e a ion, hey o mula e a scala izing unc ion like an achie emen scala izing unc ion (ASF)
based on he DM’s e e ence poin and minimize i by using an app op ia e single-objec i e
op imize . Va ious o ms o ASFs ha e been de eloped (see, e.g., [27,31,42,51]), and a
common ASF is he ollowing augmen ed ASF [52]:
s=max
i=1,...,k{wi( i(x)−gi)}+ρ
k

i=1
wi( i(x)−gi), (2)
whe e w=[w1,...,w
k]is a weigh ing ec o and ρis a su icien ly small posi i e numbe .
One can p o e ha i gene a es Pa e o op imal solu ions o bo h achie able and unachie able
e e ence poin s and any Pa e o op imal solu ion wi h ade-o s be ween ρand 1/ρ [29].
Fig. 1 An example o how a
e e ence poin based me hod
wo ks
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Jou nal o Global Op imiza ion
Fig. 2 In e ac ion be ween ADM
and a e e ence poin based
in e ac i e me hod
Fig. 3 F amewo k o he
p oposed ADM
Some e e ence poin based me hods sol e (1) di ec ly by using a mul iobjec i e e olu-
iona y algo i hm a each i e a ion. They use he DM’s e e ence poin o guide he e olu ion
o he popula ion owa ds he DM’s p e e ed egion on he Pa e o on . I he DM does
no ind any sa is ac o y solu ion, he/she can supply a new e e ence poin . A new un o
he mul iobjec i e e olu iona y algo i hm will be implemen ed o ind new solu ions. E en
hough e olu iona y algo i hms canno usually gua an ee Pa e o op imali y, o simplici y we
e e o hei solu ions as Pa e o op imal ones he e. Examples o his ype o me hods include
R-NSGA-II [17], RD-NSGA-II [13], LBS-NSGA-II [14], g-NSGA-II [34], -NSGA-II [43],
p e e ence based e olu iona y algo i hm (PBEA) [49], in e ac i e weigh ing achie emen
scala izing unc ion gene ic algo i hm (in e ac i e WASF-GA) [40], and o he s, o su eys,
see, e.g., [5,53].
In he nex sec ion, he ADM we build o compa ing e e ence poin based me hods is
in oduced.
3 P oposed ADM
To compa e e e ence poin based me hods, he ADM needs o pa icipa e in he solu ion
p ocess. The in e ac ion be ween he ADM and a me hod o be conside ed is shown in Fig. 2.
The ADM has solu ions p o ided by he me hod as inpu and a new e e ence poin as ou pu .
The s uc u e o he p oposed ADM is gi en in Fig. 3. We assume ha he ADM will i s
ha e a lea ning phase and hen s a a decision phase. The lea ning and decision modules
a e esponsible o gene a ing e e ence poin s in he lea ning phase and he decision phase,
espec i ely. The e alua ion module wo ks in he decision phase. We deno e he i e a ion
numbe in he whole solu ion p ocess by , and he i e a ion numbe in he decision phase
by dwhich coun s om 0 when his phase s a s. In wha ollows, he ADM is in oduced
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Jou nal o Global Op imiza ion
Fig. 4 Neighbo ing solu ions in a
bi-objec i e case
in e ms o i s ac ions in each phase. The s eps o u ilizing i wi h a e e ence poin based
me hod a e also gi en.
3.1 ADM’s ac ions in he lea ning phase
Owing o he modula s uc u e o he p oposed ADM, di e en ypes o lea ning modes o
ADMs can be de eloped o simula e di e en ways o speci ying e e ence poin s. As s a ed
in he in oduc ion, we ocus on a s uc u ed lea ning app oach o acili a e a sys ema ic
lea ning o easible solu ions. A each i e a ion, he solu ions which ha e been gene a ed by
he conside ed in e ac i e me hod a e used o iden i y a ela i ely la ge unexplo ed egion
o he objec i e space. Then, a e e ence poin is de e mined acco dingly wi h he desi e o
ob aining new solu ions in ha egion. He e an unexplo ed egion e e s o a egion inside
which no solu ions ha e been ound by he conside ed me hod ye . The de ails a e as ollows.
Fi s ly, we de e mine a se ies o unexplo ed egions by inding neighbo ing solu ions. We
deno e an ex eme poin as he objec i e ec o which has he minimum alue o one o he
objec i e unc ions on he Pa e o on [48]. Conside ing all he ex eme poin s and all he
non-domina ed solu ions gene a ed by he in e ac i e me hod and passed o he ADM, we
gi e a gene al de ini ion o neighbo ing solu ions o any numbe o objec i es: Fo any wo
solu ions zaand zb, deno e zab =[zab
1,...,zab
k]wi h zab
i=min(za
i,zb
i), i=1,2,...,k.
Then, zaand zba e de ined as neighbo s i no o he solu ions a e domina ed by zab. Unde
his de ini ion, he egion domina ed by zab is an unexplo ed egion whe e he in e ac i e
me hod has no gene a ed solu ions inside i ye .
Figu e 4gi es an example o a bi-objec i e case. Conside ing he i e poin s A, B, C,
D, and E, ou pai s o neighbo ing solu ions can be ound: (A, B), (B, C), (C, D), and (D,
E). All o he pai s o solu ions a e no neighbo s. Fo ins ance, B and D a e no neighbo s
because zBD domina es C. Figu e 5shows a h ee-objec i e case. Any solu ion in he Pa e o
on sa is ies 1+ 2+ 3=1. I is easy o e i y ha (A, D), (B, E), (C, F), (D, F), and
(E, F) a e neighbo s. The g ay egion, i.e., he egion domina ed by (0,0,0.4), ep esen s he
unexplo ed egion de e mined by (A, D). Fo p oblems wi h mo e han h ee objec i es, i is
also easy o ind all neighbo ing solu ions because we only need o de e mine he dominance
ela ions o solu ions. By inding all pai s o neighbo ing solu ions, a se ies o unexplo ed
egions o he objec i e space can be ob ained.
Secondly, as he lea ning phase aims o lea n abou he possibili ies o he p oblem, ou
ADM is assumed o ega d he “la ges ” unexplo ed egion as p omising and wan o explo e
i o inding new solu ions. Tha is, among mul iple unexplo ed egions, he “la ges ” one
is iden i ied as he nex egion o be explo ed. Conside ing ha di e en unexplo ed egions
may o e lap when hey sp ead upwa ds while wha we eally wan o sea ch is he egion nea
he Pa e o on , we use he dis ance be ween each pai o neighbo ing solu ions o measu e
he size o each unexplo ed egion. An unexplo ed egion is ega ded as he “la ges ” i
123
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Fig. 5 Neighbo ing solu ions in a h ee-objec i e case. The g ay egion is he unexplo ed egion de e mined
by A and D
he co esponding pai o neighbo ing solu ions has he la ges dis ance among all pai s’
dis ances. Di e en o ms o dis ance can be used and he no malized Euclidean dis ance is
used in his pape :
d(za,zb)=


k

i=1za
i−zb
i
znad
i−z∗∗
i2
,(3)
whe e z∗∗ =[z∗∗
1,z∗∗
2,...,z∗∗
k]is a u opian objec i e ec o which is sligh ly be e han z∗.
The componen s o z∗∗ a e gi en by z∗∗
i=z∗
i−εi o all i=1,2,...,kwhe e εiis a small
posi i e numbe [45]. The u opian poin is used ins ead o z∗in case ha he denomina o in
(3) is ze o o e y small.
No e ha when he Pa e o on is disconnec ed o an in e ac i e me hod poo ly esponds
o he e e ence poin , he e may be he case ha none o he newly gene a ed solu ions loca e
wi hin he desi ed la ges unexplo ed egion. In his case, he la ges unexplo ed egion may
always be selec ed as he la ges in he ollowing i e a ions. To a oid his si ua ion, he same
egion is allowed o be chosen only once.
Finally, suppose he wo neighbo ing solu ions co esponding o he la ges unexplo ed
egion a e za∗and zb∗, he poin za∗b∗wi h za∗b∗
i=min(za∗
i,zb∗
i) o all i=1,...,kis aken
as he new e e ence poin . Wi h his poin , he ADM wan s o explo e he la ges unexplo ed
egion o inding solu ions in i .
To sum up, he s eps o he s uc u ed lea ning app oach o gene a ing a new e e ence
poin is as ollows:
1. Find all pai s o neighbo ing solu ions among all ex eme poin s and all he solu ions
gene a ed by he in e ac i e me hod and passed o he ADM.
2. Selec he pai wi h he la ges no malized Euclidean dis ance, deno ed by za∗and zb∗.
123
Jou nal o Global Op imiza ion
Fig. 11 The e e ence poin s and
co esponding Pa e o op imal
solu ions in he lea ning phase
(a) (b)
Fig. 12 The e e ence poin s and co esponding Pa e o op imal solu ions in he decision phase: a he ou h
i e a ion; b he i h i e a ion
4 Nume ical expe imen s on benchma k es p oblems
In his sec ion, we use ADM1 and ADM2 as examples o demons a e how he p oposed ADM
can be u ilized o compa e e e ence poin based me hods on benchma k es p oblems. Fou
popula me hods, i.e., he e e ence poin me hod [51], R-NSGA-II [17], g-NSGA-II [34],
and -NSGA-II [43] a e used as examples o e e ence poin based me hods. As in oduced in
Sec . 3.3.2, he e e ence poin me hod gene a es k+1 Pa e o op imal solu ions by minimizing
k+1 ASFs a each i e a ion. The o he h ee me hods use he DM’s e e ence poin o modi y
NSGA-II [16]. Ne e heless, hei ways o modi ica ion a e di e en . R-NSGA-II modi ies
he c owding dis ance mechanism o NSGA-II o p e e solu ions close o he e e ence
poin . g-NSGA-II uses g-dominance ela ion o eplace he Pa e o dominance in NSGA-II
o emphasize solu ions which sa is y all aspi a ion le els o achie e none o he aspi a ion
le els. -NSGA-II subs i u es -dominance o he Pa e o dominance in NSGA-II o make
solu ions close o he e e ence poin mo e p e e ed. In he li e a u e, he h ee me hods
a e usually used as a p io i me hods. We use hem in e ac i ely in his pape as ollows.
A each i e a ion, he modi ied NSGA-II is pe o med o a ce ain numbe o gene a ions
123

Jou nal o Global Op imiza ion
o ind Pa e o op imal solu ions co esponding o he ADM’s cu en e e ence poin . The
p ocedu es o he modi ied NSGA-II a di e en i e a ions a e independen . Tha is o say, he
modi ied NSGA-II is e un by s a ing om a newly andomly gene a ed ini ial popula ion
a each i e a ion. In ou expe imen s, he codes o he la e wo me hods come om he
MATLAB pla o m Pla EMO [50].
4.1 Expe imen al se ings
4.1.1 Tes p oblems
Th ee- and i e-objec i e DTLZ1, DTLZ2, DTLZ3, DTLZ4 and DTLZ7 [18] a e used as es
p oblems. The Pa e o on o DTLZ1 is linea . DTLZ2, DTLZ3, and DTLZ7 ha e conca e
Pa e o on s. DTLZ7 has 2k−1disconnec ed Pa e o op imal egions.
4.1.2 Pa ame e se ings o in e ac i e me hods
Fo he e e ence poin me hod, we adop he same implemen a ion as used in Sec . 3.3.The
pa ame e alues used in DE/ and/1/bin a e: (1) popula ion size NP =5n; (2) he scaling
ac o F=0.5; (3) he c osso e p obabili y CR =0.5; and (4) he maximum numbe o
gene a ions Gmax =400. No e ha DE/ and/1/bin is called k+1 imes a each i e a ion
o he e e ence poin me hod. Fo he o he h ee me hods, he popula ion size is 100 and
200 o h ee-objec i e and i e-objec i e p oblems, espec i ely. The clea ing pa ame e 
in R-NSGA-II is se as 0.01 [17] and he non- -dominance h eshold δin -NSGA-II is se
as 0.3 [43]. Since hese h ee me hods gene a e many solu ions a each i e a ion, we selec
k+1 solu ions o he ADM by clus e ing. This implies ha he ou me hods will p esen he
same numbe o solu ions o he ADM a each i e a ion. To ge compa able esul s wi h he
ou me hods, hei o al numbe s o unc ion e alua ions a e kep he same a each i e a ion.
4.1.3 Pa ame e se ings o ADMs
I has been epo ed in he li e a u e ha he median numbe o i e a ions pe o med wi h
in e ac i e me hods is o en be ween h ee and eigh [19]. Too many i e a ions will b ing
a hea y bu den o he DM. On he o he hand, a e e ence poin based me hod is likely o
ha e be e esul s when mo e i e a ions a e adop ed since mo e solu ions can be ob ained.
In o de o allow he ou me hods o ha e be e pe o mance wi hin an accep able numbe
o i e a ions, we se he o e all numbe o i e a ions Tas eigh . In addi ion, we se Tl=
5,Td=3. In ac , one can se T,Td,andTl he way one wishes.
The o mulas o he disu ili y unc ions in (4)and(5) a e used. Table 1gi es he weigh ing
ec o s used in hem. Fo each es p oblem, he di e ence be ween znad and z∗in each
dimension is la ge enough, so we use z∗di ec ly ins ead o z∗∗ (i.e., εi=0, o all i∈
{1,...,k})in(4), (5), and (7). Fo ADM2, he s anda d de ia ions o he noise o 
U−in he
decision phase a e se as σ1=0.2×(U−max −U−∗), σ2=σ1/2,σ
3=σ2/2. A e he
solu ions gene a ed a he inal i e a ion a e gi en o he ADM, σ dwill be educed o ze o,
which means ha he noise o ADM2’s disu ili y unc ion disappea s inally. The MPS on
each es p oblem and co esponding op imal disu ili y alue a e lis ed in Table 1.
123
Jou nal o Global Op imiza ion
Table 1 The weigh ing ec o s, MPSs, and he op imal disu ili y alues
P oblem wz
MPS U−∗
3-obj DTLZ1 [1,1.2,1.5][0.2,0.1667,0.1333]0.4
3-obj DTLZ2/3/4 [1,1.2,1.5][0.6838,0.5698,0.4558]0.6838
3-obj DTLZ7 [1,1.2,3][0.8094,0.6745,3.6771]0.9419
5-obj DTLZ1 [1,1.2,1.2,1.2,1.5][0.12,0.1,0.1,0.1,0.08]0.24
5-obj DTLZ2/3/4 [1,1.2,1.2,1.2,1.5][0.5324,0.4437,0.4437,0.4437,0.3549]0.5324
5-obj DTLZ7 [4,4,1,1,2][0.1951,0.1951,0.7804,0.7804,6.3026]0.9080
4.1.4 Ini ial e e ence poin
We use ou di e en ini ial e e ence poin s o each es p oblem, as lis ed in Table 2.They
a e deno ed by i p1, i p2, i p3, and i p4 whe e i p1 and i p2 a e unachie able while i p3
and i p4 a e achie able. Besides, i p1 and i p3 a e close o he Pa e o on while i p2 and
i p4 a e a away om he Pa e o on . On each es p oblem, he wo ADMs will use he
same ou ini ial e e ence poin s when applying each in e ac i e me hod. The app oach o
gene a ing a ious ini ial e e ence poin s is gi en in he supplemen a y ma e ial.
4.2 Expe imen al esul s and analysis
4.2.1 Compa ison o he ou me hods by using he p oposed ADM
In his subsec ion, he p oposed ADM1 and ADM2 a e u ilized o compa e he ou e e ence
poin based me hods. Since andomness is in ol ed in he solu ion p ocess, he solu ions
gene a ed by each me hod wi h he same e e ence poin may a y in di e en uns. Hence, o
e e y ADM, each me hod is un 21 imes independen ly on each es p oblem unde each ini ial
e e ence poin . This means ha each me hod is applied o a o al o 2×4×10×21 =1680
imes. In each un, he alues o he wo indica o s di e ence and dis ance a e calcula ed by
using (6)and(7), espec i ely.
•Compa ison acco ding o he mean alues. The mean alues and he s anda d de ia ions
o each me hod o e 21 independen uns a e lis ed in Tables 3and 4.In hese wo
ables, he i s h ee columns show he es p oblem, he numbe o objec i es, he ini ial
e e ence poin , espec i ely. The ou h o se en h columns p esen esul s o he ou
me hods in e ms o di e ence. No e ha he e e ence poin me hod is abb e ia ed as
RPM. The alues in pa en heses a e he s anda d de ia ions. Simila ly, he eigh h o
ele en h columns gi e esul s in e ms o dis ance. In o de o acili a e compa ison, we
anked he ou me hods on each ins ance by using 1, 2, 3, and 4 acco ding o hei mean
alues. The me hod wi h ank 1 has he smalles mean alue. All he anks a e lis ed in
he ou h o ele en h columns o each able. The a e age ank o each me hod o e 40
ins ances is shown in he las ow o each able.
Acco ding o Tables 3and 4, no me hod is absolu ely supe io o he o he me hods on
all ins ances. The e e ence poin me hod anks he i s on mo e han hal o he ins ances
unde each indica o , no ma e which ADM is u ilized. Mos o i s mean di e ence alues
a e less han 10% and he co esponding s anda d de ia ions a e less han 5%, which means
ha i can gene ally ind solu ions close o he eal MPS and i has a ela i ely good s abili y.
123
Jou nal o Global Op imiza ion
Table 2 The ini ial e e ence poin s o each es p oblem
P oblem i p1i p2i p3i p4
3-obj DTLZ1 [0.22,0.20,0.01][0.1,0.08,0.07][0.42,0.12,0.11][0.35,0.33,0.32]
3-obj DTLZ2/3/4 [0.72,0.61,0.08][0.34,0.28,0.23][0.96,0.45,0.36][0.84,0.78,0.73]
3-obj DTLZ7 [0.15,0.69,4.19][0.4,0.34,3.15][0.80,0.81,3.41][0.53,0.46,5.58]
5-obj DTLZ1 [0.19,0.1,0.05,0,0.14][0.06,0.05,0.05,0.05,0.04][0.09,0.09,0.08,0.4,0.04][0.31,0.3,0.3,0.3,0.29]
5-obj DTLZ2/3/4 [0.56,0.5,0.54,0.06,0][0.27,0.22,0.22,0.22,0.18][0.48,0.93,0.31,0.36,0.25][0.77,0.72,0.72,0.72,0.68]
5-obj DTLZ7 [0.23,0.37,0.2,0.82,7.2][0.1,0.1,0.39,0.39,4.77][0.82,0.76,0.77,0.74,6.21][0.64,0.76,0.76,0.55,9.01]
123
Jou nal o Global Op imiza ion
Table 3 Resul s o he ou me hods when using ADM1 o compa e hem
P oblem k i p Di e ence Dis ance
RPM R-NSGA-II g-NSGA-II -NSGA-II RPM R-NSGA-II g-NSGA-II -NSGA-II
DTLZ1 3 i p13.906(2.809)17.632(3.521)23296(2514)414.79(8.204)30.052(0.040)10.152(0.079)232.67(31.51)40.180(0.090)3
i p2 0.429(0.388)13.693(3.574)24316(3342)417.63(9.682)30.005(0.006)10.043(0.035)247.96(38.01)40.228(0.118)3
i p3 3.349(1.875)16.756(4.409)24580(3941)416.75(8.619)30.047(0.027)10.091(0.067)245.98(38.73)40.210(0.108)3
i p4 1.422(0.773)11.788(1.881)25362(4280)419.22(8.854)30.022(0.014)10.030(0.034)254.27(43.50)40.238(0.123)3
5i p1 2.914(1.175)115.17(3.395)226257(7450)476.46(41.03)30.045(0.013)10.286(0.026)2377.1(61.07)41.000(0.523)3
i p2 0.956(0.368)113.97(4.783)225853(6222)492.55(47.04)30.020(0.005)10.241(0.094)2357.3(46.57)41.204(0.616)3
i p3 5.587(1.756)114.74(4.773)226430(6424)480.62(51.01)30.104(0.031)10.274(0.068)2378.0(70.19)40.973(0.581)3
i p4 1.802(0.561)110.54(4.449)226552(8138)4102.4(60.35)30.025(0.006)10.218(0.077)2369.2(89.57)41.292(0.768)3
DTLZ2 3 i p1 6.941(0.000)29.955(2.459)45.134(1.710)19.739(2.486)30.180(0.000)20.344(0.106)40.083(0.038)10.339(0.113)3
i p2 0.411(0.000)110.15(7.317)45.400(5.961)33.894(3.975)20.006(0.000)10.227(0.226)40.052(0.051)20.065(0.120)3
i p3 2.666(0.000)110.25(7.718)47.401(3.510)34.491(5.152)20.022(0.000)10.130(0.090)40.090(0.046)30.055(0.073)2
i p4 1.590(0.000)17.758(11.35)36.066(3.758)219.61(3.567)40.019(0.000)10.129(0.208)30.091(0.071)20.514(0.125)4
5i p1 11.72(0.000)120.97(6.935)3149.8(26.52)413.19(1.892)20.444(0.000)10.636(0.082)31.759(0.226)40.515(0.069)2
i p2 5.308(0.648)125.69(5.980)3153.8(23.68)411.72(9.191)20.074(0.014)10.646(0.146)31.774(0.169)40.300(0.243)2
i p3 3.317(0.040)132.51(7.910)3152.5(22.70)418.07(5.770)20.054(0.000)10.789(0.090)31.843(0.206)40.538(0.123)2
i p4 3.878(0.000)112.11(5.415)2157.3(30.93)421.85(5.707)30.043(0.000)10.445(0.164)21.745(0.194)40.493(0.209)3
123
Jou nal o Global Op imiza ion
Table 3 con inued
P oblem k i p Di e ence Dis ance
RPM R-NSGA-II g-NSGA-II -NSGA-II RPM R-NSGA-II g-NSGA-II -NSGA-II
DTLZ3 3 i p1 6.225(2.786)18.431(3.884)25497(3598)420.52(7.009)30.127(0.086)10.244(0.144)240.95(25.41)40.280(0.081)3
i p2 4.687(3.733)16.114(7.005)26373(5582)424.17(5.204)30.053(0.047)10.093(0.138)244.18(30.19)40.334(0.089)3
i p3 2.878(0.885)18.956(6.379)25761(3978)421.42(6.080)30.042(0.028)10.123(0.117)241.92(27.46)40.304(0.105)3
i p4 3.840(3.441)15.347(4.752)24755(3621)423.85(4.047)30.036(0.025)10.065(0.064)235.15(25.51)40.330(0.089)3
5i p1 10.26(4.590)117.69(5.127)291875(32154)443.73(43.91)30.255(0.124)10.558(0.063)31012(329.9)40.538(0.460)2
i p2 7.792(1.938)127.27(9.837)2100039(22665)449.77(70.71)30.139(0.052)10.608(0.193)21051(226.3)40.680(1.044)3
i p3 10.25(3.005)135.46(7.867)2100121(17770)459.73(46.90)30.218(0.098)10.780(0.114)31067(185.0)40.678(0.517)2
i p4 6.935(2.092)113.41(7.478)2102039(18981)435.24(36.73)30.130(0.094)10.440(0.138)21098(194.6)40.493(0.432)3
DTLZ4 3 i p1 7.433(2.155)217.17(8.346)44.709(1.574)19.660(2.855)30.191(0.097)20.488(0.116)40.087(0.051)10.196(0.100)3
i p2 5.597(3.541)113.14(7.981)45.738(3.600)28.574(3.223)30.070(0.048)20.405(0.210)40.066(0.047)10.111(0.064)3
i p3 2.276(2.440)18.411(6.360)37.260(4.905)29.091(3.194)40.033(0.035)10.126(0.118)40.113(0.137)20.116(0.049)3
i p4 3.686(2.809)17.257(4.904)37.127(3.615)28.967(3.323)40.045(0.034)10.160(0.152)40.113(0.075)20.141(0.100)3
5i p1 12.06(1.248)122.22(6.094)329.53(15.45)413.56(3.784)20.473(0.073)30.636(0.075)40.457(0.161)10.292(0.093)2
i p2 10.60(5.805)124.66(7.845)332.88(18.01)413.22(4.034)20.290(0.174)20.694(0.159)40.430(0.220)30.228(0.063)1
i p3 14.93(4.936)218.41(5.926)330.02(12.77)414.07(4.692)10.484(0.080)30.512(0.160)40.436(0.150)20.247(0.092)1
i p4 11.13(5.915)210.05(6.992)127.25(13.13)417.76(5.382)30.225(0.176)10.332(0.195)30.339(0.137)40.313(0.129)2
DTLZ7 3 i p1 8.653(2.336)210.71(2.684)37.787(1.566)134.58(7.567)40.294(0.053)20.557(0.220)30.203(0.052)10.711(0.076)4
i p2 1.124(0.510)11.625(1.742)31.543(1.116)239.09(13.94)40.050(0.027)30.049(0.049)20.045(0.029)10.739(0.083)4
i p3 2.882(0.005)28.312(2.937)31.832(0.955)138.16(11.31)40.065(0.007)20.448(0.188)30.047(0.026)10.715(0.090)4
i p4 6.184(3.809)35.480(4.598)22.417(1.522)139.04(10.96)40.218(0.101)20.238(0.251)30.075(0.051)10.713(0.078)4
5i p1 6.255(2.704)112.50(3.639)338.63(12.49)49.730(2.277)20.387(0.247)10.756(0.102)40.694(0.217)30.670(0.115)2
i p2 4.118(2.366)32.039(0.606)249.37(12.29)41.224(1.004)10.246(0.144)30.089(0.019)20.830(0.229)40.061(0.043)1
i p3 2.525(0.594)178.34(24.63)446.84(20.87)276.59(24.07)30.156(0.032)10.751(0.224)30.812(0.277)40.724(0.233)2
i p4 8.263(2.989)132.80(18.07)246.18(24.30)436.12(23.24)30.532(0.204)10.836(0.289)30.856(0.347)40.727(0.305)2
A e age ank 1.25 2.6 3.275 2.875 1.375 2.85 3.075 2.7
123

Jou nal o Global Op imiza ion
Table 4 Resul s o he ou me hods when using ADM2 o compa e hem
P oblem k i p Di e ence Dis ance
RPM R-NSGA-II g-NSGA-II -NSGA-II RPM R-NSGA-II g-NSGA-II -NSGA-II
DTLZ1 3 i p1 7.448(3.965)110.14(3.975)24364(4158)415.08(7.673)30.135(0.086)10.206(0.098)341.18(38.76)40.204(0.107)2
i p2 2.292(2.079)19.780(6.291)26360(5281)414.92(8.040)30.031(0.034)10.125(0.092)266.91(65.26)40.194(0.107)3
i p3 5.864(4.218)114.33(5.915)23054(2330)417.10(6.936)30.094(0.081)10.178(0.072)228.01(20.14)40.215(0.086)3
i p4 5.462(3.816)22.849(2.022)13475(3492)417.01(6.650)30.084(0.059)20.039(0.026)133.86(31.01)40.220(0.081)3
5i p1 6.647(3.566)120.00(6.093)225960(7819)468.24(39.00)30.126(0.076)10.335(0.048)2364.8(56.89)40.890(0.470)3
i p2 3.344(3.579)117.84(5.221)228451(7976)4119.0(61.82)30.049(0.044)10.270(0.089)2389.8(67.88)41.558(0.899)3
i p3 8.391(2.258)119.99(7.230)225372(6442)493.16(40.69)30.144(0.042)10.331(0.085)2363.5(51.77)41.218(0.532)3
i p4 3.396(2.621)112.88(4.980)228505(7419)4106.7(47.97)30.056(0.044)10.227(0.069)2384.7(57.93)41.370(0.677)3
DTLZ2 3 i p1 11.02(2.209)311.72(2.967)48.398(2.182)110.61(1.936)20.299(0.102)20.365(0.095)40.181(0.079)10.351(0.102)3
i p2 7.367(6.026)215.21(7.737)46.685(3.701)17.858(6.193)30.108(0.103)30.309(0.226)40.073(0.049)10.099(0.121)2
i p3 4.252(2.468)120.98(9.284)411.45(7.515)37.383(7.678)20.063(0.045)10.318(0.151)40.141(0.086)30.100(0.129)2
i p4 4.360(4.996)15.626(6.465)27.824(4.101)322.72(8.860)40.044(0.044)10.073(0.135)20.122(0.091)30.437(0.204)4
5i p1 14.87(2.660)121.05(5.596)3149.6(31.45)415.60(3.284)20.413(0.089)10.604(0.076)31.672(0.255)40.536(0.056)2
i p2 6.539(3.265)132.09(9.009)3149.2(21.04)412.99(8.607)20.100(0.069)10.691(0.128)31.724(0.241)40.201(0.182)2
i p3 4.177(1.110)135.86(7.674)3152.7(25.34)420.61(5.650)20.071(0.012)10.808(0.095)31.807(0.223)40.569(0.128)2
i p4 6.716(2.247)111.88(5.468)2149.4(20.79)425.89(6.056)30.093(0.042)10.360(0.195)21.762(0.193)40.478(0.190)3
DTLZ3 3 i p1 10.35(5.313)110.36(2.826)27123(4675)421.06(6.440)30.237(0.130)10.294(0.134)349.94(28.90)40.274(0.101)2
i p2 8.769(3.583)111.08(7.427)25880(2451)424.18(6.470)30.099(0.037)10.156(0.134)241.02(13.51)40.347(0.108)3
i p3 4.742(2.349)112.17(8.390)26236(4704)421.47(7.421)30.079(0.051)10.154(0.128)246.87(35.13)40.318(0.126)3
i p4 8.520(5.052)24.671(2.823)16544(9890)422.51(5.238)30.098(0.066)20.053(0.038)143.95(53.84)40.308(0.090)3
5i p1 12.82(4.487)117.93(5.621)299566(22136)436.22(27.31)30.323(0.132)10.559(0.087)31064(243.9)40.475(0.330)2
i p2 10.16(3.808)129.14(6.047)289091(17358)468.08(61.60)30.157(0.071)10.654(0.194)2978.9(157.1)40.784(0.689)3
i p3 13.19(4.363)137.63(9.944)388988(20326)433.94(32.36)20.325(0.159)10.781(0.142)31001(194.3)40.477(0.438)2
i p4 11.01(3.314)29.095(3.472)199941(19039)443.67(35.16)30.272(0.140)10.301(0.159)21131(220.3)40.551(0.344)3
123
Jou nal o Global Op imiza ion
Table 4 con inued
P oblem k i p Di e ence Dis ance
RPM R-NSGA-II g-NSGA-II -NSGA-II RPM R-NSGA-II g-NSGA-II -NSGA-II
DTLZ4 3 i p1 11.46(2.682)215.84(7.303)410.25(6.227)111.56(2.783)30.293(0.126)30.475(0.104)40.239(0.161)10.290(0.120)2
i p2 9.462(5.975)220.72(11.03)46.553(4.204)110.94(6.328)30.121(0.097)20.442(0.167)40.091(0.095)10.169(0.126)3
i p3 5.500(3.969)117.26(7.572)49.466(7.111)210.17(5.487)30.076(0.057)10.300(0.170)40.112(0.089)20.179(0.123)3
i p4 6.842(3.559)19.285(5.935)210.10(4.993)310.77(6.271)40.081(0.051)10.213(0.180)40.172(0.095)30.161(0.129)2
5i p1 14.74(1.643)119.92(6.157)334.21(20.55)414.88(4.338)20.520(0.047)30.600(0.078)40.518(0.183)20.322(0.091)1
i p2 12.07(6.294)133.80(14.96)428.32(15.14)312.82(3.744)20.314(0.193)20.715(0.156)40.357(0.189)30.225(0.076)1
i p3 17.79(6.135)225.32(11.16)339.56(20.36)414.47(4.808)10.499(0.108)20.525(0.131)40.508(0.268)30.287(0.133)1
i p4 10.61(3.242)111.99(5.213)231.13(16.76)417.99(5.432)30.263(0.174)10.406(0.165)30.412(0.160)40.333(0.138)2
DTLZ7 3 i p1 8.653(2.336)211.99(2.066)37.787(1.566)134.58(7.567)40.294(0.053)20.537(0.214)30.203(0.052)10.711(0.076)4
i p2 1.124(0.510)14.923(3.009)31.543(1.116)239.09(13.94)40.050(0.027)20.186(0.149)30.045(0.029)10.739(0.083)4
i p3 2.882(0.005)28.071(3.302)31.832(0.955)138.16(11.31)40.065(0.007)20.282(0.157)30.047(0.026)10.715(0.090)4
i p4 6.184(3.809)28.587(5.035)32.417(1.522)139.04(10.96)40.218(0.101)20.368(0.248)30.075(0.051)10.713(0.078)4
5i p1 8.329(0.967)114.22(4.030)338.63(12.49)413.32(3.572)20.597(0.178)10.747(0.078)40.694(0.217)20.726(0.055)3
i p2 6.165(2.691)33.527(1.884)249.37(12.29)43.116(2.870)10.237(0.176)30.111(0.045)20.830(0.229)40.096(0.055)1
i p3 45.33(40.40)191.00(2.716)446.84(20.87)288.80(3.046)30.548(0.373)11.028(0.196)40.812(0.277)20.904(0.156)3
i p4 9.317(3.379)163.76(15.03)346.18(24.30)272.52(15.66)40.579(0.171)10.899(0.172)40.856(0.347)20.889(0.197)3
A e age ank 1.46 2.54 3.04 2.96 1.38 2.63 3.17 2.83
123
Jou nal o Global Op imiza ion
R-NSGA-II usually anks he las on h ee-objec i e DTLZ2 and DTLZ4, and anks he
second o he hi d on he o he es p oblems. Simila ly, -NSGA-II anks he second o he
hi d in mos cases. F om he a e age ank, R-NSGA-II and -NSGA-II ank he second o
he hi d among he ou me hods. When ADM2 is u ilized, R-NSGA-II is sligh ly be e
han -NSGA-II in e ms o each indica o . When ADM1 is used, R-NSGA-II has a be e
di e ence alue while -NSGA-II is be e unde he dis ance indica o . This shows ha a
be e disu ili y alue and a close dis ance a e no always consis en . I is meaning ul o use
di e en indica o s o measu e he me hods’ pe o mance.
Al hough anking he i s o second on h ee-objec i e DTLZ7, g-NSGA-II anks he
las on mos o he o he ins ances, which makes i he wo s one among he ou me hods.
Meanwhile, i s mean di e ence alues on DTLZ1 and DTLZ3 a e e y la ge. When checking
he solu ions ob ained by g-NSGA-II, we ound ha he inal popula ion o g-NSGA-II a
each i e a ion is usually a away om he Pa e o on , especially on DTLZ1 and DTLZ3.
This shows ha g-NSGA-II has a ela i ely weak capabili y o con e ge o he Pa e o on .
In ac , g-NSGA-II has a d awback ha i may p e e a solu ion a o ano he solu ion which
domina es ain he Pa e o sense, which can hinde i s con e gence.
F om he di e ences be ween he mean alues on h ee- and i e-objec i e p oblems, i
can be obse ed ha he e e ence poin me hod has no signi ican di e ences while he o he
h ee me hods gene ally ge be e mean alues on h ee-objec i e p oblems. This e lec s he
deg ada ion o he h ee me hods’ pe o mance wi h he inc ease o he numbe o objec i es.
•Compa ison acco ding o he boxplo s. Fo he sake o an in ui i e compa ison o he
ou me hods, he boxplo s o he di e ence/dis ance alues o e 21 independen uns o each
me hod a e d awn. He e we only p esen he boxplo s o he di e ence alues when ADM1
is used, as shown in Fig. 13. All he boxplo s can be ound in he supplemen a y ma e ial
and hey gi e simila esul s o Fig. 13 does. In Fig. 13, M1, M2, M3, and M4 ep esen
he e e ence poin me hod, R-NSGA-II, g-NSGA-II, and -NSGA-II, espec i ely. As he
di e ence alues o g-NSGA-II a e much la ge han hose o he o he me hods on se e al
p oblems, in o de o see he di e ences among he o he me hods, he maximum di e ence
alue o he boxes is es ic ed o be no mo e han 100%. Values la ge han 100% a e no
shown.
F om Fig. 13,wecansee ha hedi e ence alues o he e e ence poin me hod a e usually
smalle and he leng hs o he boxes a e usually sho e han he o he me hods on mos es
p oblems. This means ha he e e ence poin me hod gene ally inds be e solu ions and is
mo e s able han he o he s. g-NSGA-II pe o ms a he badly on h ee- and i e-objec i e
DTLZ1, DTLZ3. Howe e , i has ela i ely smalle boxes which a e close o ze o han he
o he h ee me hods on h ee-objec i e DTLZ7. R-NSGA-II and -NSGA-II usually ank in
he middle among he ou me hods. These obse a ions a e simila o wha we ha e de i ed
acco ding o he mean alues and s anda d de ia ions o he ou me hods.
•Compa ison acco ding o s a is ical es s. To compa e he ou me hods pai wise, he
Wilcoxon ank sum es a a signi icance le el o 0.05 is conduc ed on each pai o me hods.
We use h ee symbols +,≈,and− o ep esen ha me hod ipe o ms s a is ically be e
han, equal o, and wo se han me hod j, espec i ely. The numbe s o hese symbols o e
40 ins ances wi h espec o each ADM and each indica o a e gi en in Tables 5,6,7and 8.
In Tables 5,6,7and 8, bo h he indica o s gi e a he simila esul s. The e e ence poin
me hod pe o ms s a is ically be e han o equal o he o he h ee me hods in mos cases,
no ma e which ADM is used. On he con a y, g-NSGA-II is in e io o he o he me hods in
mo e han a hal o he ins ances. These obse a ions a e consis en wi h wha we ound om
he mean alues in Tables 3and 4. R-NSGA-II is a li le be e han -NSGA-II when ADM1
123
Jou nal o Global Op imiza ion
(a)
(b)
Fig. 13 Boxplo s o di e ence alues (%) o e 21 independen uns o each me hod when using ADM1 o
compa e me hods: aboxplo s on 3-objec i e es p oblems; bboxplo s on 5-objec i e es p oblems
123
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