Analysing the Compensatory Properties of the Outranking Approach PROMETHEE

Schä , Sebas ian; Pohl, E ik; Gelde mann, Ju a
A icle — Published Ve sion
Analysing he Compensa o y P ope ies o he Ou anking
App oach PROMETHEE
Jou nal o Mul i‐C i e ia Decision Analysis
P o ided in Coope a ion wi h:
John Wiley & Sons
Sugges ed Ci a ion: Schä , Sebas ian; Pohl, E ik; Gelde mann, Ju a (2025) : Analysing he
Compensa o y P ope ies o he Ou anking App oach PROMETHEE, Jou nal o Mul i‐C i e ia
Decision Analysis, ISSN 1099-1360, Wiley, Hoboken, NJ, Vol. 32, Iss. 2,
h ps://doi.o g/10.1002/mcda.70013
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Jou nal o Mul i- C i e ia Decision Analysis, 2025; 32:e70013
h ps://doi.o g/10.1002/mcda.70013
Jou nal o Mul i- C i e ia Decision Analysis
RESEARCH ARTICLE OPEN ACCESS
Analysing he Compensa o y P ope ies o he Ou anking
App oach PROMETHEE
Sebas ianSchä 1,2 | E ikPohl1 | Ju aGelde mann1
1Chai o Business Adminis a ion and P oduc ion Managemen , Uni e si y o Duisbu g- Essen, Duisbu g, Ge many | 2Depa men o En i onmen al
Social Sciences, Eawag: Swiss Fede al Ins i u e o Aqua ic Science and Technology, Dübendo ,Swi ze land
Co espondence: Sebas ian Schä (sebas ian.schae @eawag.ch)
Recei ed: 5 May 2025 | Re ised: 5 May 2025 | Accep ed: 12 May 2025
Funding: The au ho s ecei ed no speci ic unding o his wo k.
Keywo ds: compensa ion| mul iple c i e ia decision analysis| ou anking| PROMETHEE
ABSTRACT
The PROMETHEE me hods a e inc easingly applied in en i onmen al and public policy decision- making due o hei comp e-
hensi eness and explainabili y. Howe e , he li e a u e con ains di e ing s a emen s ega ding hei compensa o y p ope ies.
Compensa ion in mul iple c i e ia decision agg ega ion p ocedu es is commonly unde s ood as allowing a gain in one c i e ion o
o se a loss in ano he one. In ce ain domains, such as en i onmen al o public policy decision- making, i may be undesi able,
as some impac s may esul in losses oo se e e o be coun e balanced by good pe o mance on o he c i e ia. The e o e, i may
be necessa y o limi he ex en o which an agg ega ion p ocedu e pe mi s compensa ion o o explici ly con ol i as needed.
Guidelines and de ailed analy ical ools, howe e , ha help use s and analys s o con ol compensa ion in he PROMETHEE
me hods emain sca ce and o en lack anspa ency. In his s udy, we analyse he compensa o y beha iou o he PROMETHEE
I and II me hods and iden i y he key de e minan s o compensa ion in hese me hods. Based on hese insigh s, we de elop low
insensi i i y in e als o assess he sensi i i y o a gi en decision model owa ds compensa o y e ec s and p o ide a se o gene al
guidelines o con olling compensa ion in he PROMETHEE I and II me hods o any gi en pai o c i e ia. The indings a e
illus a ed a hand o an en i onmen al managemen case s udy. By combining he guidelines wi h low insensi i i y in e als,
use s and analys s gain access o measu es o a ying g anula i y o e alua e and con ol compensa ion in a PROMETHEE de-
cision model.
1 | In oduc ion
An essen ial pa o mul iple c i e ia decision analysis
(MCDA) is he o malised p ocedu e o mo e om a decision
model o a syn hesis o he in o ma ion ha has been ob-
ained abou he di e en op ions as well as he objec i es and
p e e ences o e e yone in ol ed (Bel on and S ewa 2002).
Fo his ask, a ich se o di e en mul iple c i e ia agg e-
ga ion p ocedu es (MCAP) has eme ged om he scien i ic
discou se. They a e o en di e en ia ed in o wo schools o
hough (Vansnick1990; Roy and Vande poo en1996), anal-
ogous o he undamen al decision mechanisms de eloped
by de Bo da (1781) and Condo ce  (1785). F equen ly used
app oaches om he so- called Ame ican school o hough
(Von Win e eld and Edwa ds 1986; Keeney 1992) a e he
mul iple a ibu e alue and u ili y heo y (MAVT/MAUT)
(Keeney1992; Keeney and Rai a1993), while he P e e ence
Ranking O ganiza ion METhod o En ichmen E alua ions
(PROMETHEE) (B ans e al.1986) and he Elimina ion and
Choice T ansla ing Reali y (ELECTRE) app oach (Roy1991)
a e o en applied ou anking app oaches om he F ench o
Eu opean school o hough (Roy and Bouyssou1993). These
wo dis inc schools a e essen ially based on di e en assump-
ions and axioms when es ablishing he se o syn hesised
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion and ep oduc ion in any medium,
p o ided he o iginal wo k is p ope ly ci ed.
© 2025 The Au ho (s). Jou nal o Mul i-C i e ia Decision Analysis published by John Wiley & Sons L d.
2 o 21 Jou nal o Mul i- C i e ia Decision Analysis, 2025
in o ma ion in he o m o p e e ence s uc u es (see e.g.,
Vansnick 1986; Moulin 1988; Munda 2016; Roy 2016). This
leads o a di e ging in e p e a ion o c i e ia weigh s and di e -
en agg ega ion p ope ies (Munda2008; Figuei a e al.2016;
B ans and De Sme 2016; Ma el and Ma a azzo2016).
One agg ega ion p ope y ha dis inguishes he a i-
ous MCAPs is he possible occu ence o compensa ion.
Compensa ion is commonly unde s ood as allowing a pe -
o mance gain in one objec i e o c i e ion o o se a pe o -
mance loss on ano he one (Roy and Słowiński 2013). Due
o hei di e en axioma ic ounda ions, i is o en a gued
ha MCAP om he Ame ican school o hough a e based
on a compensa o y agg ega ion logic, while ou anking ap-
p oaches gene ally conside his p ope y as unwan ed in he
p ocess o es ablishing p e e ence ela ions (Vansnick1990;
Roy and Słowiński 2013). Depending on he decision p ob-
lem, compensa o y e ec s could ei he be desi ed (e.g., o
employee pe o mance e alua ions) o should be delibe a ely
a oided (e.g., o ensu e he concep o s ong sus ainabili y)
(Cinelli e  al. 2014). Especially in con ex s whe e nume -
ous ace s o sus ainabili y need conside a ion and coun-
e balancing e ec s be ween ce ain dimensions need o be
a oided, o limi ed, he issue o compensa ion is o pa icula
in e es . A ew selec ed examples comp ise ene gy echnol-
ogy assessmen (Diakoulaki e al.2005; Tsou sos e al.2009;
Obe schmid e al.2010; S an zali and A a ossis2016), p o-
duc ion managemen (Hämäläinen 2004; Gelde mann and
Ren z 2001; Tong e  al. 2022), en i onmen al managemen
(Kike e al.2005; Huang e al.2011; Liene e al.2015), and
policy making (Fe e i2016; Salo and Hämäläinen2010).
While he e is schola ly wo k a ailable ha o e s guidance
on he choice o agg ega ion unc ions o s ee compensa-
o y e ec s in alue heo y based app oaches (e.g., Langhans
e al.2014; Cinelli e al.2014) p o ide a comp ehensi e body o
esea ch on he compensa ion deg ee o common MCDA me h-
ods, esea ch ha o e s a de ailed analysis o he compensa-
ion mechanisms o he PROMETHEE me hods seems limi ed.
Much mo e, we ind equi ocal s a emen s ega ding he com-
pensa o y beha iou o he PROMETHEE me hods in he sci-
en i ic discou se. In Cinelli e al.(2020, 2022), PROMETHEE
I, II, TRI and V (B ans and De Sme 2016; Figuei a e al.2004;
B ans and Ma eschal1992) a e classi ied as bo h no and pa -
ially compensa o y. Fu he ele an wo ks cha ac e ise he
PROMETHEE me hods as non- compensa o y ou anking me h-
ods (Pi lo 1997; G eco e al.2021; Cos a and Al es2021). In
o he li e a u e i is s a ed ha PROMETHEE me hods ‘a oid
ullcompensa ion’ (P ado e al.2012) o a e ei he ully, pa ial
o non- compensa o y, depending on he selec ed me hod and i s
con igu a ion (Beze a e al.2021; Benoi and Rousseaux2003;
Moghaddam e  al. 2011; Gui ouni and Ma el 1998; Ishizaka
and Resce2021). Only ecen ly, Dejaege e and De Sme (2023)
p oposed he new PROMETHEE
𝛾
me hod, which can delib-
e a ely be used in o ally compensa o y o non- compensa o y
manne o ci cum en unwan ed compensa o y beha iou o
PROMETHEE I in ce ain si ua ions. Addi ionally, hey ema k
ha ‘[…] i is no clea whe he PROMETHEE is conside ed as
non- compensa o y o pa ially compensa o y’ (Dejaege e and De
Sme 2023, 149). Howe e , we a e no awa e o any wo k ha
discloses he p econdi ions o compensa o y beha iou wi hin
he PROMETHEE me hods and p o ides a comp ehensi e se
o guidelines ha allows applican s o s ee i in he desi ed
manne .
To complemen exis ing esea ch, his a icle will p o ide he
ollowing main con ibu ions:
• The compensa o y p ope ies o he PROMETHEE me h-
ods a e highligh ed. In pa icula , he de e minan s o
compensa ion in he PROMETHEE I and II me hods a e
disclosed.
• A measu e o in es iga e he sensi i i y o a gi en decision
model ha is agg ega ed acco ding o PROMETHEE I o
II owa ds compensa o y e ec s is de eloped. In his way,
i is possible o iden i y he c i e ia o which compensa-
ion can occu and o speci y he ex en o which compen-
sa o y e ec s a e esponsible o p ese ing a p e e ence
s uc u e.
• A se o guidelines o he design and pa ame e isa ion o
a PROMETHEE model o con ol compensa ion in a ine-
g ained esolu ion, anging om ull compensa ion o no
compensa ion a all, is p oposed.
The emainde o he a icle is s uc u ed as ollows. We
i s de ine he undamen als o a mul iple c i e ia decision
p oblem and a i e a de ini ions o he no ion o compen-
sa ion (Sec ion 2). In Sec ion3, he PROMETHEE me hods
a e in oduced, wi h pa icula emphasis on PROMETHEE
I and II. Following his, he compensa o y beha iou o he
PROMETHEE me hods is cha ac e ised by using a di e en-
ial calculus app oach and showcased by a se ies o nume ical
examples (Sec ion4). The esul s a e discussed in Sec ion5
and used o elabo a e guidelines o applican s and analys s.
Conclusions a e d awn in Sec ion6.
2 | Compensa ion in Mul iple C i e ia
Agg ega ion P ocedu es
Gene ally speaking, MCDA me hods apply dis inc agg ega ion
p ocedu es o syn hesise p e e ence in o ma ion and e alua e
a se o po en ial cou ses o ac ion. In his sec ion, he o mal
no a ion o hese MCAPs is in oduced and he no ion o com-
pensa ion is de ined.
2.1 | Mul iple C i e ia Agg ega ion P ocedu es
In a mul iple c i e ia decision p oblem, a se o po en ial ac ions
A
≔
{
a
1
,a
2
,…,a
i
,…,a
m}
is o be e alua ed on a se o a i-
bu es o c i e ia
G
≔
{
g
1
,g
2
,…,g
j
,…,g
n}
acco ding o he
p e e ences o one o mo e decision- make (s) (DM). A decision
able hen summa ises he a ailable al e na i es, c i e ia and
he pe o mance o each al e na i e on each c i e ion, deno ed
as
gj(
a
i)
.
An MCAP applies a speci ied ma hema ical p ocedu e on he
decision able o e alua e he o malised decision p oblem in
o de o p oduce a desi ed esul which allows o he e al-
ua ion o al e na i es in a comp ehensi e way (Roy 2016).
3 o 21
Depending on he desi ed ype o esul , decision p oblems a e
ypically dis inguished in o ou di e en ypes: Choice, so -
ing, anking and desc ip ion p oblema ics (Roy2016). To each
he desi ed ou come, MCAPs es ablish p e e ence s uc u es
which o malise he compa ison o any pai o al e na i es
(ai,ax)
∈
A
×
A
in clea o uzzy language (K an z e al.1971;
Fishbu n1999; Roubens and Vincke1985; Mo e i e al.2016).
A p e e ence s uc u e is a se o bina y p e e ence ela ions
ha desc ibe a DM's a i udes owa ds a subse o o de ed
pai s om
A×A
, ha is, al e na i es, such ha o each pai
o al e na i es exac ly one ela ion holds (Mo e i e al.2016).
Acco dingly, bina y p e e ence ela ions
R
a e p e e ence
s a emen s on pai s o
A
. They allow o exp ess si ua ions o
p e e ence o indi e ence, ypically deno ed
⟨P,I⟩
. In some
cases, weak p e e ence can also be exp essed (Vincke1988).
Fo a mo e ex ensi e o e iew on he di e en ypes o p e -
e ence s uc u es ha can be es ablished, see o example
Mo e i e al.(2016).
Each MCAP ollows a di e en logic in he c ea ion o p e e -
ence s uc u es. App oaches ha maximise a alue, u ili y o
sco ing unc ion a e o en e e ed o as agg ega ing MCAPs
(Vincke1992). They exploi an agg ega ed sco e o es ablish bi-
na y p e e ence ela ions.
In ou anking me hods, a p e e ence s uc u e is cha ac e -
ised by bina y ou anking ela ions, deno ed
S
, ha can o can
no be ansi i e and comple e, depending on he p e alence
o incompa abili y be ween al e na i es (Bouyssou 1996).
The no ion o incompa abili y goes back o he conco dance-
disco dance p inciple, an essen ial axioma ic ounda ion o
ou anking app oaches (Bouyssou and Pi lo 2009; Figuei a
e al.2013). The conco dance- disco dance p inciple acknowl-
edges ha he exis ence o p e e ence o indi e ence on a pai
o al e na i es is no always possible no desi ed (Vincke1992).
The e o e, an addi ional bina y ela ion desc ibing incompa-
abili y be ween al e na i es (
J
) is in oduced (Tsoukiàs and
Vincke1995). In
⟨P,I,J⟩
p e e ence s uc u es, he ela ion
P
desc ibes si ua ions in which one al e na i e is clea ly p e-
e ed o e ano he , while he ela ions
I
and
J
bo h deno e
ha nei he al e na i e is p e e ed (Mo e i e al.2016). Fo
he e lexi e and symme ic indi e ence ela ion
I
, he lack
o p e e ence is due o hei equi alen alua ion. The sym-
me ic and i e lexi e incompa abili y ela ion
J
, howe e ,
e e s o si ua ions in which a lack o in o ma ion o con lic -
ing e alua ions does no allow o he exp ession o p e e ence
and
¬(
a
i
Pa
x)
,¬
(
a
x
Pa
i)
, and
¬(
a
i
Ia
x)
simul aneously holds
(Mo e i e al.2016).
The esul ing p e e ence s uc u e can hen be de ined as ol-
lows Dejaege e and De Sme (2023):
The ou anking ela ion
aiSax
deno es he asse ion ha ‘ai is
a leas as good as ax’ o ‘ai ou anks ax’. Tha means
ai
can be
conside ed a leas as good as
ax
since he e is su icien e idence
suppo ing such a s a emen and no con adic ing e idence,
gi en he a ailable in o ma ion (Bouyssou and Vansnick1986).
2.2 | Compensa o y and Non- Compensa o y
Mul iple C i e ia Agg ega ion P ocedu es
The no ion o compensa ion in MCAPs has been a subjec o ex-
ensi e s udy and he academic li e a u e p esen s a se o de ini-
ions. Mos commonly, he compensa o y cha ac e o an MCAP
is connec ed o i s axioma ic ounda ions.
Agg ega ing MCAPs in e p e weigh s as ade- o s be ween
a ibu es and ypically agg ega e he pe o mance o al e na-
i es on hese a ibu es in an addi i e manne , o example, ia
simple addi i e weigh ing (SAW) in he o m
∑n
j=1
wj⋅gj
�
ai
�
.
Such MCAPs a e conside ed inhe en ly compensa o y o wo
analogous easons. Fi s , he addi i e agg ega ion logic implies
ha a low pe o mance on one a ibu e can be o se by a el-
a i ely good pe o mance on ano he a ibu e wi h ega ds
o he agg ega ed pe o mance o an al e na i e (Langhans
e al.2014). Fu he mo e, he weigh s ha a e a ached o he
di e en objec i es a e in e p e ed as ade- o s be ween objec-
i es (Moulin1988). Thus, hey a e equi alen o he objec i es'
subs i u ion a es and a ully compensa o y agg ega ion logic
is inhe en ly necessa y (Munda2016). An illus a i e example
which de i es his equi alence o he case o SAW is p o ided
by (Munda2008, chap e 4). The no ion o compensa ion in his
case can be de ined as he p ope y ha he p e e ence ela ions
emain una ec ed when a loss on one objec i e is accompanied
by compa able gain on ano he one, adjus ed o he weigh s (i.e.,
subs i u ion a e) (Roy and Mousseau1996; Haag e al.2019).
In ou anking me hods, he MCAP is no conce ned wi h max-
imising an unde lying alue o u ili y unc ion. Ins ead, pai -
wise pe o mance assessmen s o al e na i es a e conduc ed on
he se o c i e ia o es ablish p e e ence s uc u es, whe e he
weigh s only ep esen he ela i e impo ance o each c i e ion
(Figuei a e al.2016). This leads o wo undamen al di e ences
compa ed o agg ega ing MCAPs:
1. The conco dance- disco dance p inciple con ines compen-
sa o y e ec s in he cons uc ion o p e e ence s uc u es
o he in a- c i e ial compa isons o al e na i es and only
up o a speci ied h eshold. Wi hin he pai wise compa -
isons, small pe o mance disad an ages o an al e na i e
ha do no su pass his h eshold a e comple ely dis e-
ga ded. The h eshold alue de ines he ex en o wha can
be conside ed a ‘small’ di e ence in pe o mance o dis-
inguish be ween ‘su icien ’ and ‘insu icien ’ e idence o
p e e ence (Pi lo 1997).
2. The conco dance- disco dance p inciple is gene ally no
conce ned wi h pe o mance subs i u ion be ween he di -
e en c i e ia. The pe o mance di e ences ha exceed he
speci ied h esholds in in a- c i e ial compa isons a e no
able o a ec he e alua ion on ano he c i e ion wi hou
u ning ela i e disad an ages in o ad an ages and ice
e sa. I is hus i ele an o he cons uc ion o p e e ence
ela ions by how much he pe o mances o al e na i es
a
iPax⇔aiSax∧¬
(
axSai
)
aiIax⇔aiSax∧axSai
ai
Ja
x
⇔¬
(
a
i
Sa
x)
∧¬
(
a
x
Sa
i)
4 o 21 Jou nal o Mul i- C i e ia Decision Analysis, 2025
wi h ega d o a gi en c i e ion di e i he h eshold is
exceeded, and any changes in pe o mance ac oss c i e ia
will no be able o a ec he p e e ence s uc u e (Figuei a
e al.2010).
Since he p e e ence s uc u es in ou anking me hods a e only
a ec ed by whe he an al e na i e ou anks ano he one and
ice e sa, hey a e gene ally classi ied as non- compensa o y
MCAPs (Fishbu n1976; Bouyssou and Vansnick1986). A he
same ime, he conco dance- disco dance p inciple does no
s ic ly imply ha ou anking ela ions a e non- compensa o y,
gi en ha small enough disad an ages a e no conside ed
in he cons uc ion o hem (Dejaege e and De Sme  2023).
Fu he mo e, no all ou anking me hods ully adop he
conco dance- disco dance p inciple. The PROMETHEE I and II
me hods a e examples o ou anking me hods ha a e no based
on he conco dance- disco dance p inciple, which leads o di -
e en compensa o y p ope ies. We cha ac e ise hei compen-
sa o y beha iou a e a i ing a a se o wo king de ini ions.
2.3 | De ini ions
So a , no widely accep ed and p ecise de ini ion o compen-
sa ion in MCAP could be iden i ied in he li e a u e. Fo he
pu pose o his wo k, compensa ion in MCAP is de ined as he
possibili y o in e - c i e ial pe o mance subs i u ion (Roy and
Mousseau1996). In e - c i e ial pe o mance subs i u ion e e s
o he balancing o o se ing o a disad an age on a c i e ion,
in e ms o he esul ing p e e ence s uc u es, by a su icien
ad an age on ano he c i e ion.
The e o e, an MCAP is conside ed compensa o y i a p e e ence
ela ion be ween wo al e na i es is al e ed by a change in pe -
o mance on one c i e ion and can be eins a ed by adjus ing
he al e na i es' pe o mance on ano he c i e ion (Roy and
Słowiński2013).
De ini ion 1. (Compensa o y MCAP). Conside a decision
p oblem wi h mul iple c i e ia
G
≔
{g1,
…
,gn}
and al e na i es
A
≔
{
a
1
,…,a
m}
, whe e
gj(
a
i)
deno es he pe o mance o al-
e na i e
ai
on c i e ion
gj
. Le
ai
and
ax
hen be wo dis inc al-
e na i es om
A
wi h
aiRax
de i ed om he MCAP, whe e
R
is
he bina y p e e ence ela ion ob ained om he MCAP.
We say ha he MCAP is compensa o y, i he e a e wo c i e-
ia
gj
and
gk
, such ha o a change o
gj(
a
i)
which al e s he
p e e ence ela ion
aiRax
, we can ind a co esponding change in
gk(
a
i)
ha eins a es he ini ial p e e ence ela ion
aiRax
.
De ini ion 2. (Non- compensa o y MCAP). Conside a deci-
sion p oblem wi h mul iple c i e ia
G
≔
{g1,
…
,gn}
and al e -
na i es
A
≔
{
a
1
,…,a
m}
, whe e
gj(
a
i)
deno es he pe o mance
o al e na i e
ai
on c i e ion
gj
. Le
ai
and
ax
hen be wo dis inc
al e na i es om
A
wi h
aiRax
de i ed om he MCAP, whe e
R
is he bina y p e e ence ela ion ob ained om he MCAP.
We say ha he MCAP is non- compensa o y, o any wo c i e ia
gj
and
gk
, and any change o
gj(
a
i)
which al e s he p e e ence
ela ion
aiRax
, i he e is no co esponding change in
gk(ai)
ha
eins a es he ini ial p e e ence ela ion
aiRax
.
A change in he pe o mance o an al e na i e
ai
on a c i e ion
gj
is hence o h deno ed
Δ
g
j(
a
i)
. I is conside ed a ‘gain’ i i in-
c eases he pe o mance di e ence o he nex bes al e na i e
on he same c i e ion o educes he gap o an al e na i e ha
pe o ms be e on his c i e ion. In u n, a change in pe o -
mance ha dec eases he pe o mance di e ence be ween
al e na i e
ai
and a lowe pe o ming al e na i e on he same
c i e ion is conside ed a ‘loss’. These de ini ions also apply o
o he ini ial p e e ence s uc u es, o example, i
ai
and
ax
a e
conside ed indi e en . Compensa ion can occu i any such pe -
o mance changes al e s a gi en p e e ence ela ion be ween
ai
and any o he al e na i e o a su icien ly la ge alue o
Δ
g
j(
a
i)
.
In addi ion, we de ine a o al ‘non- compensa o y MCAP’ since
he gi en de ini ion o non- compensa o y MCAPs does no
co e cases in which a p e e ence ela ion can no be des oyed
by he s a ed pe o mance subs i u ions due o he na u e o he
agg ega ion p ocess (Fishbu n1976; Roy and Mousseau1996).
De ini ion 3. (To ally non- compensa o y MCAP). An
MCAP is conside ed o ally non- compensa o y i he p e e ence
si ua ion be ween wo dis inc pai s o al e na i es
(ai,ax)
and
(
a
i
′,a
x
′
)
is conside ed on an o dinal scale on each c i e ion
gj
(Roy and Mousseau1996). Tha is, he p e e ence ela ion is in-
dependen om he di e ence in pe o mance and he e is no
possibili y o compensa ion ela i e o any c i e ion, so ha
holds o all al e na i es
a′
𝚤
and
a′
x
ha a e deduced om (
ai
,
ax
)
by changing hei pe o mance on any c i e ion wi hou al e ing
he o dinal o de o al e na i es on a c i e ion. Thus, in a o ally
non- compensa o y MCAP he e is no possibili y o compensa-
o y e ec s be ween di e en c i e ia as long as he p e e en ial
p o ile o al e na i es is kep (Bouyssou1986).
3 | The PROMETHEE I and II Me hods
Gi en hese de ini ions, ou anking app oaches ollowing
he conco dance- disco dance p inciple gene ally can be con-
side ed non- compensa o y. Howe e , PROMETHEE I and
II a e ou anking me hods ha do no s ic ly ollow he
conco dance- disco dance p inciple. Ins ead, hey es ablish
a anking o al e na i es based on a se o alued ou ank-
ing ela ions (B ans e al.1986). This means ha a nume ic
alue, desc ibing he in ensi y o a p e e ence ela ion, is a -
ached o a pai o al e na i es (Bouyssou and Pi lo 2009). In
PROMETHEE, he alued ou anking ela ions a e used o
es ablish sco es o ou anking lows, which ep esen he pe -
o mance o an al e na i e compa ed o he o he al e na i es
(B ans and De Sme 2016).
To de e mine he ou anking lows, he di e ences be ween
pe o mances o each pai o al e na i es om
A
a e compu ed
o all c i e ia in he i s s ep:
(1)
[
a
i
Ra
x
=a
�
𝚤
Ra
�
x
∧a
x
Ra
i
=a
�
x
Ra
�
𝚤 ]
(2)
d
j
(a
i
,a
x
)
=
g
j
(a
i
)
−
g
j
(a
x
),
∀
a
i
,a
x∈
A
j=1, …,n

5 o 21
By means o a p e e ence unc ion

i is hen possible o calcu-
la e he indi idual p e e ences o he DM:
A p e e ence unc ion is a mono onic unc ion o
dj(
a
i
,a
x)
and
maps he in a- c i e ial p e e ences o he DM, no malised o
he in e al [0, 1], while Equa ion(4) holds, so ha he p e e -
ence unc ion o c i e ia o be minimised is as in Equa ion(5).
In gene al, he shape and de ini ion o a p e e ence unc ion can
be selec ed by he DM. To educe he cogni i e load o modelling,
a se o six non- dec easing p e e ence unc ions has been es ab-
lished which a e conside ed sui able o mos con ex s (B ans and
De Sme 2016). These unc ions and hei pa ame e s a e depic ed
in TableA1 in he AppendixA. Depending on he shape o he
p e e ence unc ion, addi ional h eshold pa ame e alues may be
elici ed om he DM. The indi e ence h eshold
qj
delimi s si u-
a ions in which he di e ence in pe o mance be ween wo al e -
na i es is oo small o allow any s a emen o p e e ence o ei he .
Pa ame e
pj
models he h eshold o a si ua ion o s ic p e e -
ence. The in lec ion poin
𝜎j
o he Gaussian c i e ion ( ype VI)
allows o model he DM's sensi i i y owa ds pe o mance di e -
ences in a non- linea manne , as depic ed in Table1. A lowe alue
esul s in g ea e sensi i i y o small di e ences in pe o mance
(single do - dash line
⋅−
), while a ela i ely high alue means ha
p e e ence sensi i i y o la ge di e ences in pe o mance be-
ween wo al e na i es is highe (double do - dash line
⋅⋅−
).
The DMs in a- c i e ial p e e ences a e hen agg ega ed o
a global p e e ence index o all pai s o al e na i es as in
Equa ion(6):
A weigh ing coe icien
wj≥0
deno es he ela i e impo ance
o each c i e ion
gj
.
The p e e ence indexes can hen be used o compu e he ou -
anking low sco es o an al e na i e:
The posi i e low sco e deno ed
𝜙+
agg ega es he e idence
om all pai wise compa isons ha ein o ce a si ua ion o
p e e ence o an al e na i e o e all he o he al e na i es
unde conside a ion; he nega i e low sco e
𝜙−
in u n sums
up all e idence speaking agains such a s a emen (Linko
e al.2021, 9).
In PROMETHEE I,
⟨P,I,J⟩
p e e ence s uc u es a e es ablished
based on he posi i e and nega i e low sco es.
Thus, he PROMETHEE I anking o al e na i es o a gi en
decision p oblem is he in e sec ion o he ankings ob ained
om he posi i e and nega i e low sco es and o ms a pa -
ially o de ed se (B ans and De Sme 2016; Dejaege e and De
Sme 2023).
The PROMETHEE II me hod yields a o al anking o al-
e na i es and
⟨P,I⟩
p e e ence s uc u es by means o he
PROMETHEE ne low:
whe e
is he single c i e ion low o an al e na i e wi h espec o c i-
e ion
gj∈G
.
(3)

j
(a
i
,a
x
)
=j
(d
j
(a
i
,a
x
)),
∀
a
i
,a
x∈
A
j
=1, …,
n
(4)
j(
a
i
,a
x)
>0⇒
j(
a
x
,a
i)
=
0
(5)
j(
a
i
,a
x)
=F
j
−d
j(
a
i
,a
x)
(6)
𝜋(
ai,ax
)
=
n
∑
j
=1
wj⋅j
(
ai,ax
)
∀ai,ax∈
A
(7)
𝜙
+
(
ai
)
=
1
m−1
∑
ax∈A
𝜋
(
ai,ax
)
∀ai∈
A
𝜙
−
(
ai
)
=1
m−1
∑
ax∈A
𝜋
(
ax,ai
)
∀ai∈
A
(8)
a
iPIax⇔
⎧
⎪
⎨
⎪
⎩
𝜙
+�
ai
�
≥𝜙
+�
ax
�
∧𝜙
−�
ai
�
<𝜙
−�
ax
�
o
𝜙+�ai�>𝜙+�ax�∧𝜙−�ai�≤𝜙−�ax
�
aiIIax⇔𝜙+�ai�=𝜙+�ax�∧𝜙−�ai�=𝜙−�ax�
a
iJIax⇔⎧
⎪
⎨
⎪
⎩
𝜙+�ai�<𝜙+�ax�∧𝜙−�ai�<𝜙−�ax
�
o
𝜙+
�
a
i�
>𝜙+
�
a
x�
∧𝜙−
�
a
i�
>𝜙−
�
a
x�
(9)
𝜙
ne
(
ai
)
=𝜙+
(
ai
)
−𝜙−
(
ai
)
=
n
∑
j
=
1
𝜙j
(
ai
)
⋅w
j
(10)
𝜙
j
(
ai
)
=
1
m−1
∑
a
x
∈A
[
j
(
ai,ax
)
−j(ax,ai)
]
TABLE 1 | Decision able wi h h ee al e na i es and ou c i e ia used in he s e eo ypical cases.
C i e ion Pe o mance P e e ence in o ma ion
Name
A1
A2
A3
wj
Pola i y Func ion
g1
g1(
A
1)
g1(
A
2)
g1(
A
3)
w1
Max
1
g2
g2(A1)
g2(A2)
g2(A3)
w2
Max
2
g3
g3(A1)
g3(A2)
g3(A3)
w3
Max
3
g4
g4(
A
1)
g4(
A
2)
g4(
A
3)
w4
Max
4
6 o 21 Jou nal o Mul i- C i e ia Decision Analysis, 2025
The o al anking o all al e na i es is hen gi en by o de ing hem
acco ding o hei ne low sco es, while indi e ence may occu :
The PROMETHEE me hod amily also p o ides me hods o
so ing (Figuei a e al.2005; De Sme 2019) and choice p ob-
lems (B ans and Ma eschal1992). Fo a comple e o e iew o
PROMETHEE me hods (B ans and De Sme 2016; Bel on and
S ewa 2002), o he encompassing su ey pape on hei de-
elopmen , ex ensions and u u e di ec ions (B ans 2015), we
kindly e e he eade o he espec i e li e a u e.
4 | Compensa ion in he PROMETHEE Me hods
Since he PROMETHEE me hods do no s ic ly ollow he
conco dance- disco dance p inciple, hey exhibi a a ying com-
pensa ion beha iou ha can be ei he non- compensa o y o explic-
i ly allow o compensa o y e ec s be ween c i e ia. Speci ically,
he alued ou anking ela ions ha he PROMETHEE me hods
p oduce allow o compensa ion based on he elici a ion o selec-
ion o p e e ence unc ions, p e e ence pa ame e s and c i e ia
weigh s. The compensa ion beha iou o hese me hods hus de-
pends on he choices made du ing he elici a ion and modelling
o he decision p oblem. The in o ma ion ha is equi ed o con-
s uc hese alued ou anking ela ions, namely he ou anking
lows, also allows one o cap u e compensa o y e ec s o he
PROMETHEE me hods and cha ac e ise de e minan s o com-
pensa ion. This will be he ocus o his sec ion.
4.1 | Cha ac e ising he Compensa ion Beha iou
o he PROMETHEE II Me hod
The compensa o y beha iou o he PROMETHEE me hods is
connec ed o he alued ou anking ela ions ha a e p oduced
o es ablish a anking o al e na i es. In o de o cha ac e ise
he compensa ion beha iou o he PROMETHEE me hods,
selec ed s e eo ypical cases a e analysed. We limi he analysis
o he compensa o y beha iou o PROMETHEE I and II since
hey a e he main me hods o his amily and a e in ended o be
used o anking p oblema ics.
The s e eo ypical cases comp ise a se o h ee decision al e -
na i es ha a e o be e alua ed agains a se o ou ca dinal
c i e ia ha a e o be maximised, so ha :
•
A
=
{
A
1
,A
2
,A
3}
,
•
G
=
{
g
1
,g
2
,g
3
,g
4}
, and
•
gj∈[0,100]
.
The al e na i es' pe o mances, he chosen p e e ence unc ion
o each c i e ion and he co esponding c i e ia weigh s a e o be
a ied acco ding o he s udied case, leading o he gene ic deci-
sion able p esen ed in Table1, which summa ises he s uc u e
o he s udied decision p oblems and he nomencla u e adop ed
in his s udy. This o ms he s a ing poin o he analysis o he
PROMETHEE me hods' compensa o y p ope ies. Subsequen ly, a
change in he pe o mance o a selec ed al e na i e wi h ega ds o
a speci ic c i e ion is in oduced ( o mally deno ed
Δgj(Ai)
). In he
cases p esen ed he e, his change ep esen s a pe o mance loss (o
gain) whose e ec on he ou anking lows is o be compensa ed
o by means o a pe o mance inc ease (dec ease) o he same al-
e na i e on ano he c i e ion (deno ed
Δgk(Ai)
). I is also analysed
how he selec ion o p e e ence unc ions, h eshold pa ame e s
and c i e ia weigh s in luence he compensa o y beha iou .
4.1.1 | The Equi alence o PROMETHEE II o
Addi i ely Agg ega ing MCAPs
Ce ain ins ances o addi i ely agg ega ing MCAPs (e.g.,
SAW o MAVT/MAUT wi h linea alue unc ions and an
addi i e model) a e equi alen o PROMETHEE II wi h ype
III p e e ence unc ions and su icien ly la ge h eshold pa-
ame e s, as shown by Gelde mann and Schöbel (2011) o
Ma eschal (2015). Fo hese ins ances, bo h me hods also
exhibi compa able compensa ion beha iou , whe e a loss in
pe o mance on one c i e ion can be ully o se by a co e-
sponding gain on ano he c i e ion. I may he e o e be a gued
ha he PROMETHEE II me hod is undamen ally compen-
sa o y acco ding o he de ini ion adop ed in his wo k (see
Sec ion2.3).
An exempla y model o PROMETHEE ha co esponds o
an MAVT model wi h linea alue unc ions and an addi i e
agg ega ion unc ion is s a ed in Table 2. The c i e ia in he
PROMETHEE model a e modelled ia he widely used linea
p e e ence unc ion ( ype III) and he p e e ence h eshold is
se o he pe o mance di e ence be ween he bes and wo s
pe o ming al e na i e o each c i e ion (
p
j=g
max
j
−g
min
j
).
Using he piecewise linea p e e ence unc ions o ype V wi h
pe inen pa ame e isa ion is likewise possible in his example
(Gelde mann and Schöbel2011).
(11)
a
iP
II
ax⇔𝜙
ne (
ai
)
>𝜙
ne (
ax
)
a
i
IIIa
x
⇔𝜙ne
(
a
i)
=𝜙ne
(
a
x)
TABLE 2 | Decision able wi h h ee al e na i es, ou c i e ia and p e e ence in o ma ion o be used o he applica ion o he PROMETHEE
me hods.
C i e ion Pe o mance P e e ence in o ma ion
Name
A1
A2
A3
wj
Pola i y Func ion
pj
g1
100 10 15 0.25 Max Type III 90
g2
090 70 0.25 Max Type III 90
g3
13 21 66 0.25 Max Type III 53
g4
78 100 42 0.25 Max Type III 58
7 o 21
The al e na i es' global alues and
𝜙ne
sco es a e agg ega-
ion acco ding o MAVT and PROMETHEE II a e shown in
Figu e1. They a e indica ed by he squa e ma ks connec ed
by a solid line. Acco ding o bo h me hods, he o al anking
o al e na i es is
A2≻A3≻A1
, while
≻
deno es a si ua ion o
p e e ence. Fu he mo e, Figu e 1 displays he equi alen
compensa ion beha iou o bo h me hods. One can iden i y a
pe o mance gain o
A3
on he i s c i e ion ha dissol es he
p e e ence ela ion be ween
A2
and
A3
and yields indi e ence
be ween hese al e na i es o bo h me hods. In his case, such
a
Δg1(A3)
akes he alue o 28.6. The global alues, as well as
he ne lows o
A2
and
A3
, deno ed
V′
and
𝜙′
ne
a e equal a e he
pe o mance inc ease o
A3
on c i e ion
g1
. This is isualised by
he wo ho izon ally aligned do ed lines. Gi en equal c i e ia
weigh s, a co esponding pe o mance loss on he second c i-
e ion,
Δ
g
2(
A
3)
=−
28.6
, ees ablishes he o iginal p e e ence
ela ions as well as he exac sco es o
V(A3)
and
𝜙ne (A3)
.
This beha iou can be gene alised, so ha he ex en o equi ed
pe o mance gain on c i e ion
g2
o compensa e o a loss on
g1
is
I depends on he a io o he c i e ia weigh s, as well as he p e e -
ence h esholds o bo h o he in ol ed c i e ia o accoun o di -
e en uni s o measu emen . Fu he mo e, he condi ion ha he
linea p e e ence unc ion ( ype III) is used and pa ame e ised so
ha i is equally shaped o a linea alue unc ion mus hold:
We do no ecommend o design a ype III p e e ence unc-
ion acco ding o Equa ion (13) in p ac ical applica ion and
u he discussing i in Sec ion5.3. The gene alised p oo o
Equa ion(12) can be ound in he AppendixA.
Despi e he adjus ed pe o mances, he
𝜙ne
sco es o all o he
al e na i es also emain cons an a e applying pe o mance
changes o
Δ
g
1(
A
3)
=
28.6
and
Δ
g
2(
A
3)
=−
28.6
, as he o ange
ba in he h ee panels on he igh hand side o Figu e2 shows.
Gi en he me hods pai wise- compa ison logic o p oduce a se o
alued ou anking ela ions his is no necessa ily expec ed. The
compensa ing e ec ha is esponsible o his ci cums ance
becomes isible upon in es iga ion o co esponding single c i-
e ion lows. The bo om igh panel shows he single c i e ion
lows o Al e na i e 3. Compa ed o he single c i e ion lows
be o e he pe o mance adjus men s, which is indica ed by he
shaded a ea, he ela i e weakness o
A3
ega ding c i e ion
g1
is educed while he con a y happens o he pe o mance
on c i e ion
g2
. In a non- compensa o y se ing, his change in
single c i e ion lows would no happen. Logically, he sin-
gle c i e ion lows o
A1
and
A2
a e also changing on he wo
pe o mance- adjus ed c i e ia due o he PROMETHEE calcu-
la ion p ocedu e. This means ha ac oss all h ee al e na i es,
only he single c i e ion lows ega ding he i s wo c i e ia a e
changing. All o he single c i e ion lows emain cons an . The
co esponding e ec o pe o mance subs i u ion o he MAVT
model wi h addi i e agg ega ion unc ion and o he alue agg e-
ga ion me hods is ex ensi ely s udied by Langhans e al.(2014).
4.1.2 | Non- Compensa o y Modelling in
PROMETHEE II
Howe e , he e a e also modelling a ian s o PROMETHEE II
whe e compensa o y e ec s a e la gely o e en comple ely a oided.
I compensa ion needs o be a oided, his can be done qui e in-
ui i ely by changing he p e e ence unc ion o he a ec ed c i-
e ia. The esul s o swi ching o a ype I p e e ence unc ion o
c i e ion
g1
and
g2
o he s a ed decision p oblem a e gi en in
Table3. Following he de ini ion o his p e e ence unc ion, he
pe o mance al e a ions
Δg1(A3)
=
28.6
and
Δg2(A3)
=−
28.6
do nei he a ec he PROMETHEE ne lows no he single c i-
e ion lows. Thus, compensa o y e ec s can no occu when
con ining o his p e e ence unc ion ype.
Due o he calcula ion logic o he ou anking lows and he
piecewise de ini ion o he emaining p e e ence unc ion ypes,
(12)
Δ
g2
(
Ai
)
=
p
2⋅
w
1
p1
⋅
w2
⋅Δg1
(
Ai
)
(13)

j
�
dj
�
=
⎧
⎪
⎨
⎪
⎩
0dj≤0
dj
gmax
j
−gmin
j
0≤gmax
j−g
min
j
FIGURE 1 | Global alues and PROMETHEE ne lows o decision p oblem in Table2 and e ec s o changing he pe o mance o
A3
on
g1
.
8 o 21 Jou nal o Mul i- C i e ia Decision Analysis, 2025
FIGURE 2 | PROMETHEE single c i e ion lows and ne lows o he decision p oblem in Table2 (panels on he le hal ). The h ee panels on
he igh side highligh he compensa o y e ec s when dec easing he pe o mance o
A3
on
g1
and compensa ing o his loss by a gain on c i e ion
g2
. Al hough he PROMETHEE ne lows o
A3
emain simila since he loss is compensa ed, he compensa ion becomes obse able in he changed
single c i e ion lows o he wo a ec ed c i e ia (
𝜙1
and
𝜙2
in his ins ance).
TABLE 3 | PROMETHEE single c i e ion lows and ne lows o he decision p oblem in Table2 be o e and a e al e ing he pe o mance o
A3
on c i e ion
g1
and
g2
by −28.6 and 28.6 poin s.
C i e ia pe o mance Al e na i e
PROMETHEE lows
𝝓1(ai)
𝝓2(ai)
𝝓3(ai)
𝝓4(ai)
𝝓ne (ai)
g1(
A
3)
=
70;
g
2(
A
3)
=
15
A1
0.25 −0.25 −0.144 0.03 −0.114
A2
−0.25 0.25 −0.087 0.172 0.085
A3
0 0 0.231 −0.203 0.029
g1(
A
3)
=
43.6;
g2(
A
3)
=
41.4
A1
0.25 −0.25 −0.144 0.03 −0.114
A2
−0.25 0.25 −0.087 0.172 0.085
A3
0 0 0.231 −0.203 0.029
No e: C i e ia
g1
and
g2
a e modelled ia he usual p e e ence unc ion o ype I. The linea p e e ence unc ion and pa ame e isa ion is kep o
g3
and
g4
. The ne
lows (bold) o an al e na i e a e he sum o he single c i e ion lows.
15 o 21
possible compensa o y e ec s and may no adequa ely map he
p e e ences o he DM. In addi ion, he e is e en a isk o allow
o ull compensa ion on c i e ia whe e he al e na i es pe o m
ela i ely simila . The ollowing example will highligh he ben-
e i s and d awbacks o his app oach o con ol compensa ion.
5.2.2 | Compensa o y P ope ies o he Type I
P e e ence Func ion
As no ed abo e, changing o a ype I p e e ence unc ion
does no necessa ily elimina e he issue o compensa ion in a
PROMETHEE model. I does, howe e , subs an ially educe
he sensi i i y o i owa ds compensa o y e ec s. Conside he
same decision p oblem as in Table8 wi h he only modi ica ions
being (i) c i e ia
g1
and
g9
a e bo h o ype I and (ii) all c i e ia
weigh s a e equal. These changes lead o la ge low insensi i -
i y in e als compa ed o he ini ial model, as Table9 shows. An
analogous pa e n could be obse ed i o he c i e ia han
g1
o
g9
we e changed o ype I.
A he same ime, changing o a ype I p e e ence unc ion can
also c ea e unwan ed ins ances o ull compensa ion. This hap-
pens when he al e na i es' pe o mances ma kedly change be-
yond he low insensi i i y in e als. Figu e6 displays he single
c i e ion and ne lows when he pe o mance o
A4
on c i e ia
g1
and
g9
changes beyond he low insensi i i y in e als displayed
in Table9. In a i s s ep,
g9(
A
4)
is inc eased by 0.2 uni s.
G9
is
an en i onmen al c i e ion and o be minimised, so ha his
change e lec s a pe o mance loss. The s iped ba s indica e he
new lows a e his change is applied. The pe o mance loss o
A4
on he en i onmen al c i e ion
g9
leads o a ank change so
ha
A1
becomes he p e e ed al e na i e. In a second s ep, he
pe o mance sco e o
A4
on he echnical c i e ion
g1
is educed
by 31.4 uni s. Since
g1
is also o be minimised, i e lec s a pe o -
mance gain. The e ec s ha his change has a e isualised by
he do ed ba s in he igu e. The in oduced pe o mance gain
on
g1
ully compensa es he loss in lows on
g9
and eins a es he
ini ial ne lows and ank o de . The single c i e ion lows on all
o he c i e ia a e no a ec ed by he pe o mance changes and
hus no displayed.
This example highligh s ha he p e iously in oduced guide-
lines o e s a ing poin s o conside ing he issue o compen-
sa ion in he PROMETHEE modelling p ocess. In addi ion,
he low insensi i i y in e als o e a mo e de ailed means o
analyse he occu ence and ex en o compensa ion in a a ge ed
manne . They a e pa icula ly po en i addi ional in o ma ion
on he unce ain y in he pe o mance sco es is known. In his
case, p e e ence h esholds can be modi ied conside ing his
TABLE 9 | Flow insensi i i y in e als o an en i onmen al managemen decision p oblem wi h adap ed p e e ence unc ions.
C i e ion
g1
g2
g3
g4
g5
g6
g7
g8
g9
g10
g11
Func ion ype IIV IV III III IV IV V I IV IV
Δ
g
j(
A
1)
[−937.4,9.8)
{}
[−1, 0]
{} {} {} {}
[−70,4.2)
(−0.02,∞)
{} {}
Δgj(A2)
(−9.8,49.6)
{}
[−1, 0]
(−0.12, ∞)
(−280.36, ∞)
{} {}
(−2.2,5)
(−0.22,0.02)
{} {}
Δ
g
j(
A3
)
(−49.6,31.4)
{}
[0, 2]
{} {} {} {}
(−1.2,2.2)
(−0.2,0.22)
{} {}
Δ
g
j(
A4
)
(−31.4,∞)
{}
[−1, 0]
[−1.34,0.06)
{} {} {}
(−4.2,1.2)
[−3.47,0.2)
{} {}
No e: Bold- aced in e als indica e changes.
FIGURE 6 | PROMETHEE single c i e ion lows and ne lows o he decision p oblem in Table9. C i e ia
g1
and
g9
modelled ia ype I p e e -
ence unc ion. Ini ial ne lows a e indica ed by solid ba s. S iped ba s (pe o mance o
A4
on
g9
changed beyond insensi i i y bounda ies) indica e
he change in ne lows and anking be ween
A1
and
A4
. Do ed ba s (pe o mance o
A4
on
g1
changed in opposi e pola i y) showcase ull compen-
sa ion be ween bo h c i e ia.

16 o 21 Jou nal o Mul i- C i e ia Decision Analysis, 2025
unce ain y o ci cum en unwan ed compensa o y e ec s in a
a ge ed and g anula manne .
5.3 | Discussion
The p e ious analysis is based on a numbe o delibe a e as-
sump ions ega ding he de ini ion o compensa ion and he nu-
ances in which i can occu . We now discuss hese assump ions
and he p ac ical implica ions ha can be d awn om hem.
Fo he pu pose o his wo k, he gene al no ion o compensa ion
in an MCAP has been de ined as he p ope y ha p e e ence
ela ions, which a e dissol ed by changes in an al e na i e's pe -
o mance, can be eins a ed by a pe o mance change on any
o he c i e ion o he same al e na i e (see Sec ion2.3).
A de ini ion analogous o o he anking app oaches, such as
MAUT/MAVT, could e e o he ne lows ins ead o he ank-
ing o al e na i es de i ed om compa ing hei ne lows.
Adap ed o he PROMETHEE me hods, compensa ion would
hen be de ined as he possibili y o ne low subs i u ion be-
ween c i e ia. Acco ding o his e ised de ini ion, an ins ance
o PROMETHEE is ully compensa o y i he e exis s a pai o
pe o mance sco e changes o a gi en al e na i e ha main ains
i s ne low. Howe e , such a de ini ion may no be a easonable
choice in ligh o he me hod's di e en axioma ic ounda ions
compa ed o he MAUT/MAVT (see Sec ion2.1). Addi ionally,
he ac ual ne lows a e ypically no used when in e p e ing a
anking ha has been conduc ed acco ding o PROMETHEE
II, as o en done o alue heo y- based app oaches (Coquele
e al.2024). We he e o e a gue ha he b oad de ini ion o a
compensa ing MCAP adop ed in his wo k, based on he un-
damen al abili y o al e ing o p ese ing p e e ence s uc u es
(Sec ion2.3) is mo e easonable.1 This undamen al de ini ion o
compensa ion can hen be ope a ionalised o he PROMETHEE
me hods by means o he single c i e ion lows. I allows o nu-
ance be ween pa ial and ull compensa ion based on he abili y
o single c i e ion low subs i u ion and is o highe use o p ac-
ical applica ion o PROMETHEE I and II.
In his ega d, we p esen ed a ully compensa o y case o
PROMETHEE II in Sec ion4.1.1. I is based on a se o a i icial
assump ions ha appea no e y ealis ic o p ac ical appli-
ca ion. We especially do no ecommend se ing he p e e ence
h eshold o a ype III p e e ence unc ions in such a way since
we belie e ha i con adic s he concep ual ounda ions o he
PROMETHEE me hods. We only use i he e o showcase ha
he e can be ins ances o compensa ion in he PROMETHEE
me hods. Mo e p ecisely, his case shows ha PROMETHEE
II can be ully compensa o y, acco ding o how we de ine he
no ion o compensa ion in MCAP. I p o ides con adic ing e i-
dence o some s a emen s in he li e a u e on he compensa o y
cha ac e is ics o he PROMETHEE me hods.
The heo e ical insigh s gained om his analysis lead o se e al
p ac ical conside a ions, especially in managing compensa o y
e ec s wi hin PROMETHEE models and acili a ing communi-
ca ion wi h DMs. As highligh ed in he case s udy (Sec ion5.2),
he ype I unc ion can be an app oach o ci cum en compen-
sa ion be ween c i e ia. Howe e , i equi es ca e ul a en ion
since he e may be pe o mance changes ha in oduce ull com-
pensa ion in o a ela ionship o wo c i e ia (see Sec ion5.2.2).
Reso ing o a ype I p e e ence unc ion also comes a he ex-
pense o being unable o model he DMs p e e ences in ine es-
olu ion, which is a dis inc ea u e o he PROMETHEE me hods
compa ed o o he ou anking me hods. The ype I c i e ion is
o en c i icised because i does no ake in o accoun impe ec
in o ma ion, ambigui y o o he ac o s ha migh cause a DM
o hesi a e in exp essing a clea p e e ence o one ac ion o e
ano he (B ans e al.1986; Roy e al.2014). In u n, when he DM
eques s ano he p e e ence modelling, o example, ia he ype
III p e e ence unc ion, bu simul aneously desi es no compen-
sa ion o his c i e ion, ca e ul communica ion and adjus men s
a e equi ed. To econcile hei p e e ence o no compensa ion,
i would be necessa y o swi ch o o he unc ions and in oduce
p e e ence o indi e ence h esholds ha model his pu pose.
This equi es a dialogue wi h he DM and a case- speci ic and de-
libe a e ade- o be ween modelling choices.
6 | Conclusion
In his wo k, we conduc ed an in- dep h analysis o he compen-
sa o y p ope ies o he PROMETHEE ou anking me hods. Ou
ocus has been on cha ac e ising he de e minan s o compensa-
ion wi hin PROMETHEE I and II. Compensa o y e ec s can
p ese e a p e e ence s uc u e when pe o mance on mul iple
objec i es changes a he same ime. Since his beha iou may
be undesi ed o ce ain objec i es, o example, o en o ce he
p inciple o s ong sus ainabili y, a p ese a ion o he anking
due o compensa o y mechanisms may no e lec he DMs ac-
ual p e e ences. In pa icula , we ha e e ealed how he choice
o p e e ence unc ions and elici a ion o hei pa ame e s can
allow o ins ances o ull o pa ial compensa ion ega ding
he de i ed p e e ence s uc u e. Ul ima ely, he e a e ins ances
whe e PROMETHEE II and also PROMETHEE I can be com-
pensa o y o any ype o p e e ence unc ion. The indings
con ibu e o he exis ing esea ch by disclosing de e minan s
o compensa ion in PROMETHEE I and II and o e ing mecha-
nisms o di e en g anula i y o cap u e and con ol compensa-
o y e ec s in hese me hods.
We ha e in oduced a compensa ion sensi i i y analysis as a
no el ool o in es iga e he sensi i i y o a PROMETHEE deci-
sion model owa ds compensa o y e ec s and highligh ed di -
e en app oaches o con ol compensa ion. Flow insensi i i y
in e als help decision- make s (DMs) and analys s o iden i y
componen s o a decision model ha may igge compensa o y
e ec s and conduc app op ia e changes. Speci ically, hey allow
de e mining he changes in pe o mance sco es ha will a ec
he ou anking lows and hus can compensa e o changes on
o he c i e ia. This in o ma ion can be u ilised o con ol com-
pensa o y e ec s in a ine- g ained esolu ion by means o ad-
jus ing he p e e ence unc ions o he h eshold pa ame e s.
An en i onmen al managemen case s udy demons a es ha
con olling compensa ion in a PROMETHEE model can ha e
a subs an ial impac on he inal anking. By delibe a ely con-
olling o compensa ion, i is possible o a oid unin ended shi s
in he anking o al e na i es ha migh happen due o compen-
sa o y e ec s be ween a pa icula pai o c i e ia. Con olling
17 o 21
o compensa ion is enabled by e e ing o he se o gene al
guidelines and by using he esul s o he compensa ion sensi-
i i y analysis de eloped in his s udy. I allows one o ake he
aspi a ion o a non- compensa o y decision model in o accoun
in he design and pa ame e isa ion o a PROMETHEE model.
The ques ion o how o deal wi h compensa o y p ope ies in
MCAP and he ecogni ion o incompa abili y has con ibu ed
ma kedly o he eme gence o some ou anking me hods. As
highligh ed by Dejaege e and De Sme  (2023), compensa o y
p ope ies could be linked o he exis ence o incompa abili y. In
u he esea ch, a de ailed analysis o he links be ween com-
pensa ion and (in)compa abili y in PROMETHEE I, II could
p o ide in e es ing insigh s. In pa icula , he in es iga ion o
whe he and when compensa ion in PROMETHEE II leads o
incompa abili y in PROMETHEE I and compa isons wi h he
newly p oposed PROMETHEE
𝛾
me hod. Ano he cen al cha -
ac e is ic o ou anking p ocedu es is he logic o pai wise com-
pa isons and he much- discussed ank e e sal phenomenon,
which is s ongly ela ed o i . The ela ions be ween ank e e -
sal and compensa ion, howe e , a e no ye disclosed.
Fu u e esea ch could also explo e u he applica ions o he
compensa ion sensi i i y analysis ac oss di e en p oblem con-
ex s and se ings. An analy ical s udy o explo e he cha ac e is-
ics o a decision p oblem, in e ms o he numbe o al e na i es
and c i e ia, could be an in e es ing poin o depa u e. In addi-
ion, an in es iga ion in o wha ex en and how he p esen ed
indings on he compensa ion beha iou o he PROMETHEE
me hods can sys ema ically in o m he p e e ence elici a ion
p ocess also poses an in iguing a enue o esea ch.
Acknowledgemen s
The au ho s a e g a e ul o Da id S zalka o his Mas e 's hesis a he
Uni e si y o Duisbu g- Essen, which p o ided addi ional insigh s in o
he ully compensa o y case p esen ed in his wo k. We also hank ou
s uden assis an , Ma hias Ge ula , who helped us wi h he o ma -
ing o he manusc ip and he design o igu es. We kindly acknowl-
edge suppo by he Open Access Publica ion Fund o he Uni e si y
o Duisbu g- Essen. Open Access unding enabled and o ganized by
P ojek DEAL.
Con lic s o In e es
The au ho s decla e no con lic s o in e es .
Da a A ailabili y S a emen
The da a ha suppo he indings o his s udy a e a ailable in he
Suppo ing In o ma ion o his a icle.
Endno es
1 E en hough, pu ing inc eased a en ion on he ac ual low sco es ha
PROMETHEE yields could also yield impo an insigh s in he analy-
sis o esul s (Dejaege e e al.2022; Dejaege e and De Sme 2023).
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Suppo ing In o ma ion
Addi ional suppo ing in o ma ion can be ound online in he
Suppo ing In o ma ion sec ion.
Appendix A
Gene alisa ion o he Compensa o y Case o PROMETHEE II
Theo em 1. I he p e e ence unc ions o wo c i e ia in PROMETHEE
a e de ined as in Equa ion(13), he equi ed gain in c i e ion
gk
o achie e
compensa ion w. . . o he PROMETHEE II ne low o Al e na i e
ai
o
a loss in c i e ion
gj
is gi en by
P oo . The PROMETHEE ne low o an al e na i e is de e mined ac-
co ding o Equa ions(9) and (10). Any alue o
Δgj(ai)
≠
0
hen al e s
he PROMETHEE ne low o an Al e na i e by
Δ𝜙ne (ai)
, while
The PROMETHEE ne low o an al e na i e a e he pe o mance on
wo c i e ia, c i e ion
gj
and any o he c i e ion
gk
, has been al e ed is
deno ed
𝜙ne (
a
i)′
. The pe o mance change ha is equi ed on any o he
c i e ion
gk
o eins a e he o iginal PROMETHEE ne low and hus
e i y ha
(A1)
Δ
gk
(
ai
)
=−
p
k⋅
w
j
p
j
⋅w
k
⋅Δgj
(
ai
)
���
Δ𝜙ne �ai��
��
=
�
���
������
𝜙ne �ai�−
⎡
⎢
⎢⎢⎢⎣
1
m−1
n
�
j=1�
ax∈A
⎡
⎢
⎢⎢⎢⎣
Δj�ai,ax�−Δj�ax,ai�
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
g
j(
a
i)
=g
j(
a
i)
+Δg
j(
a
i)
⎤
⎥
⎥⎥⎥⎦
⋅wj
⎤
⎥
⎥⎥⎥⎦
����������
𝜙
ne
(
ai
)
�
!
=𝜙ne
(
ai
)
+Δ𝜙ne
(
ai
)
⏟⏞⏞⏟⏞⏞⏟
=0
20 o 21 Jou nal o Mul i- C i e ia Decision Analysis, 2025
is hen gi en by
Unde he condi ion ha
Δ
𝜙
ne (
a
i)
=
0
, we hen ha e ha
𝜙
ne
(ai)
�
=𝜙+(ai)
�
−𝜙−(ai)
�
=1
m−1∑
ax∈A,
i≠x
𝜋(ai,ax)−1
m−1∑
ax∈A,
i≠x
𝜋(ax,ai)
=1
m−1∑
ax∈A,
i≠x
n
∑
j=1
j(dj(ai,ax))⋅wj−1
m−1∑
ax∈A,
i≠x
n
∑
j=1
j(dj(ax,ai))⋅wj
=1
m−1
⋅∑
ax∈A[gj(ai)+Δgj(ai)−gj(ax)
pj
⋅wj+gk(ai)+Δgk(ai)−gk(ax)
pk
⋅wk+…gn(ai)−gn(ax)
pn
⋅wn
]
−
1
m−1
⋅
∑
ax
∈
A[
gj
(
ax
)
−gj
(
ai
)
+Δgj
(
ai
)
pj
⋅wj+gk
(
ax
)
−gk
(
ai
)
+Δgk
(
ai
)
pk
⋅wk+…gn
(
ax
)
−gn
(
ai
)
pn
⋅wn
].
Δ𝜙
ne (
ai
)
=𝜙
ne (
ai
)
−𝜙
ne (
ai
)�
=
1
m−1
⋅[wj⋅(gj(ai)−gj(ax))
pj
+wk⋅(gk(ai)−gk(ax))
pk
−wj⋅(gj(ai)+Δgj(ai)−gj(ax))
pj
+wk⋅(gk(ai)+Δgk(ai)−gk(ax))
pk
]
=pk⋅wj(gj(ai)−gj(ax)−gj(ai)+gj(ax)+Δgj(ai))
−pj⋅wk(gk(ax)−gk(ai)−gk(ax)+gk(ai)+Δgk(ai))
=pk⋅wi⋅Δgj(ai)−pj⋅wk⋅Δgk(ai)
=Δgk
(
ai
)
=−pk⋅wj
p
j
⋅w
k
⋅Δgj
(
ai
)

21 o 21
TABLE A1 | O e iew on he six gene alised p e e ence unc ions in he PROMETHEE me hods (B ans and De Sme 2016).
C i e ion G aph De ini ion Pa ame e
Type I: Usual c i e ion

j
(
dj
)
=
{
0dj≤
0
1d
j
>
0
—
Type II: Usual c i e ion wi h indi e ence a ea

j
(
dj
)
=
{
0dj≤q
j
1d
j
>q
j
qj
Type III: C i e ion wi h linea p e e ence

j�dj�=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
0dj≤0
dj
pj
0≤dj≤p
j
1dj>pj
pj
Type IV: Le el c i e ion

j�dj�=
⎧
⎪
⎨
⎪
⎩
0d
j≤
q
j
1
2qj<dj≤p
j
1d
j
>p
j
pj,qj
Type V: C i e ion wi h linea p e e ence and
indi e ence a ea

j�dj�=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
0dj≤qj
dj−qj
pj−qj
qj<dj≤p
j
1dj>pj
pj,qj
Type VI: Gaussian c i e ion

j
�
dj
�
=
⎧
⎪
⎨
⎪
⎩
0dj≤
0
1−e−
d2
j
2𝜎2
jdj>
0
𝜎j