Piecewise linear value functions for multi-criteria decision-making

Delft University of Technology

Piecewise linear value functions for multi-criteria decision-making

Rezaei, Jafar

DOI

10.1016/j.eswa.2018.01.004

Publication date

2018

Document Version

Final published version

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Expert Systems with Applications

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Rezaei, J. (2018). Piecewise linear value functions for multi-criteria decision-making. Expert Systems with

Applications, 98, 43-56. https://doi.org/10.1016/j.eswa.2018.01.004

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ContentslistsavailableatScienceDirect

Expert

Systems

With

Applications

journalhomepage:www.elsevier.com/locate/eswa

Piecewise

linear

value

functions

for

multi-criteria

decision-making

Jafar

Rezaei

Faculty of Technology Policy and Management, Delft University of Technology, Delft, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 24 August 2017 Revised 1 January 2018 Accepted 2 January 2018 Available online 3 January 2018 Keywords: Multi-criteria decision-making MCDM Decision criteria Value function Monotonicity

a

b

s

t

r

a

c

t

Multi-criteriadecision-making(MCDM)concernsselecting,rankingorsortingasetofalternativeswhich areevaluatedwithrespecttoanumber ofcriteria.ThereareseveralMCDMmethods,thetwocore el-ementsofwhichare(i)evaluatingtheperformance ofthealternativeswithrespecttothecriteria,(ii) findingtheimportance(weight)ofthecriteria.Thereareseveralmethodstofindtheweightsofthe cri-teria,however,whenitcomestothealternativemeasureswithrespecttothecriteria,usuallytheexisting MCDMmethodsusesimplemonotoniclinearvaluefunctions.Usuallyanincreasingordecreasinglinear functionisassumedbetweenacriterionlevel(overitsentirerange)anditsvalue.Thisassumption, how-ever,mightleadtoimproperresults.Thisstudyproposesafamilyofpiecewisevaluefunctionswhichcan beusedfordifferentdecisioncriteriafordifferentdecisionproblems.Severalreal-worldexamplesfrom existingliteratureareprovidedtoillustratetheapplicabilityoftheproposedvaluefunctions.Anumerical exampleofsupplierselection(includingacomparisonbetweensimplemonotoniclinearvaluefunctions, piecewiselinearvaluefunctions,andexponentialvaluefunctions)showshowconsideringpropervalue functionscouldaffectthefinalresultsofanMCDMproblem.

© 2018ElsevierLtd.Allrightsreserved.

1. Introduction

Decisiontheoryisprimarilyconcernedwithidentifyingthebest

decision. In many real-world situations the decision is to select

the bestalternative(s) fromamong aset ofalternatives

consider-ing a set of criteria. This subdivision of decision-making, which

has gained enormousattention, due to its practical value, in the

pastrecentiscalledmulti-criteriadecision-making(MCDM).More

precisely, MCDMconcerns problemsin whichthe decision-maker

faces m alternatives (a1,a2,…, am), which should be evaluated

withrespect toncriteria(c1,c2,…, cn), inorderto findthe best

alternative(s), rankorsortthem.Inmostcases,an additive value

functionisusedtofindtheoverallvalueofalternativei,Ui,as

fol-lows: Ui= n  j=1 wjui j, (1)

where uij is the value of alternative i withrespect to criterion j,

andwjshowstheimportance(weight)ofcriterionj.Insome

prob-lems,thedecision-makerisabletofinduijfromexternalsourcesas

objectivemeasures,insome otherproblems,uij reflectsa

qualita-tiveevaluationprovidedbythedecision-maker(s),expertsorusers

assubjectivemeasures.Priceofacarisanobjectivecriterionwhile

comfortofacarisasubjectiveone.Forobjectivecriteria,we

usu-E-mail address: j.rezaei@tudelft.nl

allyuse physicalquantities, for instance,‘InternationalSystem of

Units’(SI),whileforsubjectivecriteria,wedonothavesuch

stan-dards,which iswhywemostlyusepairwise comparison,

linguis-ticvariables,or Likertscales inorder toevaluate thealternatives

withregard tosuch criteria. Inorder tofindthe weights, wj,the

decision-makermight use different tools and methods, from the

simplestway,whichisassigningweightstothecriteriaintuitively,

touse simple methodslike SMART (simple multi-attribute rating

technique)(Edwards,1977),tomorestructuredmethodslike

mul-tipleattributeutility theory(MAUT)(Keeney&Raiffa, 1976),

ana-lytichierarchyprocess(AHP)(Saaty,1977),andbestworstmethod

(BWM) (Rezaei, 2015, 2016). While these methods are usually

called‘multi attribute utility and value theories’ (Carrico, Hogan,

Dyson,& Athanassopoulos, 1997),there isanother classof

meth-ods, called outranking methods, like ELECTRE (ELimination and

ChoiceExpressingREality)family (Roy, 1968), PROMETHEE

meth-ods (Brans, Mareschal, & Vincke, 1984) which do not necessarily

need the weights to select, rank or sort the alternatives. What,

however,isincommoninthesemethodsisthewaytheyconsider

thenatureofthe criteria.Thatisto say,inthe currentliterature,

one of the commonassumptions aboutthe criteria (most of the

timeitisnotexplicitlymentionedintheliterature),is

monotonic-ity.

Definition1 (Keeney&Raiffa,1976).Leturepresentsavalue

func-tionforcriterionX,thenuismonotonicallyincreasingif:

[x1>x2]⇔[u

(

x1

)

>u

(

x2

)

]. (2)

https://doi.org/10.1016/j.eswa.2018.01.004

Definition2 (Keeney&Raiffa,1976).Leturepresentsavalue

func-tionforcriterionX,thenuismonotonicallydecreasingif:

[x1>x2]⇔[u

(

x1

)

<u

(

x2

)

]. (3)

Afunctionwhichisnotmonotoniciscallednon-monotonicand

mayhavedifferentshapes.Forinstance,avaluefunctionwiththe

first part increasing and the second part decreasing called

non-monotonic, by splitting of which, we have two monotonic

func-tions.

This assumption – monotonicity – however,is an

oversimpli-fication in some real-world decision-making problems. Another

simplification is the use of simple linear functions over the

en-tirerangeofacriterion.Consideringthetwoassumptions

(mono-tonicity, linearity),we usually seesimple increasing and

decreas-inglinearvalue functionsforthedecisioncriteriainMCDM

prob-lems.Theliteratureisfullofsuchapplications.Forinstance,many

of the studies reviewed in the following review papers

implic-itly adoptsuch assumptions: the MCDM applications in supplier

selection (Ho, Xu, & Dey, 2010), in infrastructure management

(Kabir, Sadiq, & Tesfamariam, 2014), in sustainable energy plan-ning(Pohekar&Ramachandran,2004),andinforestmanagement

andplanning(Ananda&Herath,2009).Whileinsomestudiesthe

use of monotonic and/or linear value function might be logical,

their use in some other applications might be unfitting. For

in-stance, Alanne, Salo, Saari, andGustafsson (2007), for evaluation

of residential energy supply systems use monotonic-linear value

functionsforall the selected evaluationcriteriaincluding “global

warmingpotential (kgCO2 m−2 a−1)”, and“acidificationpotential

(kgSO2 m−2 a−1)”.Consideringamonotonic-linearvaluefunction

forsuchcriteriaimpliesthatthedecision-makeracceptsanylevel

ofsuch harmful environmentalcriteriafor an energysupply

sys-tem. However, if the decision-maker does not accept some high

levels of such criteria (which seems logical), a piecewise linear

function might better represent the preferences of the

decision-maker(seethedecrease-levelvaluefunctioninthenextsection).

Some authors have discussed nonlinear monotonic value

functions (e.g., exponential value functions by Kirkwood, 1997;

Pratt, 1964). Others use qualitative scoring to address the

non-monotonicity(Brugha,2000;Kakeneno&Brugha,2017; O’Brien&

Brugha,2010).Wecanalsofindsomeformsofelicitingpiecewise

linear value function in Jacquet-Lagreze and Siskos (2001), and

Stewartand Janssen (2013). Some other value function

construc-tion orelicitation frameworks can be found inHerrera,

Herrera-Viedma, and Verdegay (1996), Lahdelma and Salminen (2012),

MustajokiandHämäläinen(2000),StewartandJanssen(2013),and

Yager (1988). Although in PROMETHEE we usedifferent types of

piecewisefunctionsforpairwisecomparisons(Brans,Mareschal,&

Vincke,1984),thefunctionsarenotusedto evaluatethedecision

criteria.So,despitesome efforts inliterature,thereisnoalibrary

ofsome standard piecewise linear value functions which can be

usedindifferent methodslike AHPorBWM. It isalso important

tonotethat whileinmanystudiesvaluefunctionsareelicited

ac-cordingtothepreferencedatawehavefromthedecision-maker(s),

inMCDM,usuallyweusethevaluefunctionasasubjectiveinput.

Thisimpliesthat,inMCDMmethods(exceptafewmethods,such

asUTA),thevaluefunctionisnotelicited,butanapproximationis

used.Thisalsosuggeststhattherichliteratureondeterminingand

elicitingvaluefunctionsisnotactuallyhelpingMCDMmethodsin

thisarea. In thispaper, first,a number ofpiecewise linearvalue

functionswithdifferentshapesareproposed tobeconsidered for

thedecision criteria. It is then shown, with some real-world

ex-amples,howsuchconsiderationmightchangethefinalresultsofa

decisionproblem.Acomparisonbetweensimplemonotonic linear

valuefunctions, piecewise linearvalue functions,andexponential

valuefunctionsisconducted,whichshowstheeffectivenessofthe

proposedpricewisevalue functions. Thisisasignificant

contribu-Fig. 1. Increasing value function.

tion tothis field anditis expectedto be widelyused by MCDM

applications.

Inthenextsection,somepiecewiselinearvaluefunctionsalong

withsome real-worldexamples are presented, whichis followed

bysomeremarksinSection3.InSection4,somenumerical

analy-sesareusedtoshowtheapplicabilityofconsideringtheproposed

valuefunctionsinadecisionproblem.InSection5,the

determina-tionofthevalue functionsisdiscussed.InSection 6,thepaperis

concluded,somelimitationsofthestudyarediscussed,andsome

futureresearchdirectionsareproposed.

2. Piecewiselinearvaluefunctions

Inthissection, anumberof piecewisevalue functionsare

de-fined for decisioncriteria. We providesome example casesfrom

the existing literature or practical decision-making problems to

support1 _{each value function.In all thefollowing valuefunctions}

weconsider[dl

j, duj]asthedefineddomainforthecriterionbythe

decision-maker;xijshowstheperformanceofalternativeiwith

re-specttocriterionj;anduijshowsthevalueofalternativeiwith

re-specttocriterionj.Forinstance,ifadecision-makerwantstobuy

a carconsideringpriceasone criterion,ifall thealternatives the

decision-makerconsiders are between€17,000 and€25,000, then

thecriterionmightbedefinedforthisrange[17,000, 25,000].

2.1. Increasing

Increasing value function isperhaps the mostcommonly used

functioninMCDMapplications.Itbasicallyshowsthatasthe

crite-rionlevel,xij,increases,itsvalue,uij,increasesaswell.Itisshown

inFig.1andformulatedasfollows:

ui j=

⎧

⎨

⎩

xi j− dlj du j− dlj , dl j≤ xi j≤ d u j, 0, otherwise. (4)

Forthisfunctionwecanthinkof:

• Product quality in supplier selection (Xia & Wu, 2007).

Con-sideringa set of suppliers, a buyermay always prefer a

sup-plierwithahigherproductqualitycomparedtoasupplierwith

lowerproductquality.

• Energyefficiencyinalternative-fuelbusselection(Tzeng,Lin,&

Opricovic,2005). Consideringa set ofbuses,a bus withmore

efficientfuel energymight always be preferred toa bus with

lessefficientfuelenergy.

1 It is worth-mentioning that the studies we discuss to support each value func-

tion have some theoretical or practical support for the proposed value functions. It does not, however, mean that those studies have used these value functions in their analysis.

Fig. 2. Decreasing value function.

Fig. 3. V-shape value function.

2.2. Decreasing

Decreasing value function showsthatasthe criterionlevel, xij,

increases,its value,uij,decreases.ItisshowninFig.2and

formu-latedasfollows: ui j=

⎧

⎨

⎩

du j− xi j du j− dlj , dl j≤ xi j≤ d u j, 0, otherwise. (5)

Forthisfunctionwecanthinkof:

• Product pricein supplier selection (Xia & Wu, 2007).

Consid-eringasetof suppliers,a supplierwithalower product price

might always be preferred to a supplier with higher product

price.So,ahigherproductpricehasalowervalue.

• Maintenancecostinalternative-fuelbusselection(Tzengetal.,

2005).Consideringasetofbuses,abuswithlessmaintenance

costmightbepreferredtoabuswithhighermaintenancecost.

So,ahighermaintenancecostisassociatedwithalowervalue.

2.3. V-shape

V-shapevaluefunction showsthatasthecriterion level,xij,

in-creasesuptoacertainlevel,dm

j,itsvalue,uij,decreasesgradually,

andafterthatcertainlevel,dm

j,itsvalue,uij,increasesgradually.It

isshowninFig.3andformulatedasfollows:

ui j=

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

dm j − xi j dm j − d l j , dl j≤ xi j≤ dmj, xi j− dmj du j− dmj , dm j ≤ xi j≤ duj, 0, otherwise. (6)

Forthisfunction,wecouldnotfindmanyexamples,anditmay

representa smallnumber ofvery particulardecision criteria. For

thisfunctionwecanthinkof:

• Relativemarketshareinselectingafirmforinvestment(Wilson

&Anell, 1999). WilsonandAnell(1999)found that for

invest-ment decision-making,firmswith low andhighmarket share

aremoredesirabletotheinvestors.Thisimpliesthatthevalue

Fig. 4. Inverted V-shape value function.

ofafirmdecreaseswhileitsmarketshareincreasesuptoa

cer-tainlevel,dm_{,andafterthatitsvalueincreasesagain.}

• Firm size in R&D productivity(Tsai & Wang, 2005). Tsai and

Wang(2005)foundthatbothsmallandlargefirmshavehigher

R&Dproductivitycomparedtomedium-sizedfirms.Thisistrue

for both high-tech andtraditional industries. Thismeans that

therelationshipbetweensizeandvalue(measuredbyR&D

pro-ductivity)isV-shapewithaminimumlevelofvalueassignedto

acertainsizeofdm

j.

2.4.InvertedV-shape

InvertedV-shapevaluefunctionshowsthatasthecriterionlevel,

xij,increasesuptoacertain level,dmj,itsvalue,uij,increases,and

afterthatcertain level,dm

j,itsvalue, uij,decreases.Itisshownin

Fig.4andformulatedasfollows:

ui j=

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

xi j− dlj dm j − d l j , dl j≤ xi j≤ dmj, du j− xi j du j− dmj , dm j ≤ xi j≤ duj, 0, otherwise. (7)

Forthisfunctionwecanthinkof:

• Commute time in selecting a job (Redmond &

Mokhtar-ian, 2001).For manypeople,the idealcommute, dm

j,is larger

than zero.Thisimpliesthatcommute timesbetweenzeroand

theoptimalcommutetime,andbetweentheoptimalcommute

timeandlargertimes,havelowervaluethantheoptimal

com-mute time for such individuals. This suggests an inverted

V-shapevaluefunction.

• Cognitive proximity in innovation partner selection

(Nooteboom, 2000). For a company there is an optimal

cognitive distanceto thepartner they areworkingon

innova-tion(dm

j).Thisimpliesthat anydistancelessthandmj orlarger

thandm

j haslessvalue.

2.5.Increase-level

Increase-levelvaluefunctionshowsthatasthecriterionlevel,xij,

increasesuptoacertainlevel,dm

j,itsvalue,uij,increases,andafter

that certain level,dm

j,its value, uij,will remain atthe maximum

level.ItisshowninFig.5andformulatedasfollows:

ui j=

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

xi j− dlj dm j − dlj , dl j≤ xi j≤ dmj, 1, dm j ≤ xi j≤ duj, 0, otherwise. (8)

Fig. 5. Increase-level value function.

Fig. 6. Level-decrease value function.

• Fill rate in supplier selection (Chae, 2009). Although a buyer

preferssupplierswithhigherfillrate,whichimpliesthatasthe

fillrateincreasesitsvalueincreases,thebuyermightbe

indif-ferenttoanyincreaseafteracertainlevel,dm

j,asusually

buy-erspre-identifyadesirableservicelevelwhichissatisfiedbya

certainminimumlevelofsupplier’sfillrate.

• Diversity ofrestaurants inhotellocation selection (Chou,Hsu,

& Chen, 2008). In order to find the best location for an

international hotel, a decision-maker prefers locations with

more divers restaurants.However, reaching a level,dm

j, might

fullysatisfyadecision-makerimplyingthatthedecision-maker

mightnotbesensitivetoanyincreaseafterthatcertainlevel.

2.6.Level-decrease

Level-decrease value function showsthat asthe criterion level,

x_ij,increasesuptoacertainlevel,dm

j,itsvalue,uij,remainsat

max-imumlevel,andafterthatcertainlevel,dm

j,itsvalue,uij,decreases

gradually.ItisshowninFig.6andformulatedasfollows:

ui j=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

1, dl j≤ xi j≤ dmj, du j− xi j du j− dmj , dm j ≤ xi j≤ duj, 0, otherwise. (9)

Forthisfunctionwecanthinkof:

• Distanceinselectingauniversity(Carricoetal.,1997).Whilea

student prefersa closeruniversity toa farther university, this

preference might start after a certain distance, dm

j, implying

thatanydistancebetween[dl

j,dmj]isoptimalandindifferentfor

thestudent.

• Leadtime insupplierselection(Çebi&Otay,2016).Althougha

supplierwithashorterleadtime ispreferred,iftheleadtime

is in a limit such that it doesnot negatively affect the

com-pany’s production, the company might then be indifferent to

thatrange.

Fig. 7. Level-increase value function.

Fig. 8. Decrease-level value function.

2.7. Level-increase

Level-increase value function shows that asthe criterion level,

xij,increasesuptoacertainlevel,dmj,itsvalue,uij,remainsat

min-imumlevel,andafterthatcertainlevel,dm

j,itsvalue,uij,increases.

ItisshowninFig.7andformulatedasfollows:

ui j=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

0, dl j≤ xi j≤ dmj, xi j− dmj du j− dmj , dm j ≤ xi j≤ duj, 0, otherwise. (10)

Forthisfunctionwecanthinkof:

• Leveloftrustinmakingabuyer–supplierrelationship(Ploetner

&Ehret,2006).Trustincreasesthelevelofpartnershipbetween

abuyerandasupplier,howeveritisonlyeffectiveaftera

cer-tainthreshold,dm

j.

• Level of relational satisfaction in evaluating quality

commu-nication in marriage (Montgomery, 1981). Below a minimum

levelofrelationalsatisfaction,dm

j,qualitycommunication

can-nottakeplacethusresultsinminimumvalue.

2.8. Decrease-level

Decrease-level value function showsthat asthe criterion level,

xij,increasesuptoacertainlevel,dm_j,its value,uij,decreases,and

afterthatcertain leveldm

j,itsvalue,uij,willremain atminimum.

ItisshowninFig.8andformulatedasfollows:

ui j=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

dm j − xi j dm j − dlj , dl j≤ xi j≤ dmj, 0, dm j ≤ xi j≤ duj, 0, otherwise. (11)

Forthisfunctionwecanthinkof:

• Carbon emission intransportation mode selection (Hoen, Tan,

Fransoo, & van Houtum, 2014). In selecting a transportation

mode,themorethecarbonemissionbythemode,thelessthe

Fig. 9. Increasing stepwise value function.

zerovaluetoa modewithcarbonemissionhigherthana

cer-tainlevel,dm

j.

• Distancewhen selectinga school(Frenette,2004). Ithasbeen

shownthat the longerthe distanceto the school thelessthe

preferencetoattendthatschool.It isalsoclearthat,forsome

people,thereisnovalueafteracertaindistance,dm

j.

2.9. Increasingstepwise

Increasing stepwise value function shows that asthe criterion

level, xij, increasesup to a certain level,dmj,its value remains at

a certain level, u0, andafter that certain level, dmj, its value, uij,

jumpstoahigherlevel(maximum)andremainsatthemaximum.

ItisshowninFig.9andformulatedasfollows:

ui j=

⎧

⎨

⎩

u0, dlj≤ xi j≤ dmj, 1, dm j ≤ xi j≤ duj, 0, otherwise. (12) where0<u0<1.

Forthisfunctionwecanthinkof:

• Suppliers capabilities in supplier segmentation (Rezaei &

Ortt, 2012). Suppliers of a company are evaluated based on

theircapabilities,andthensegmentedbasedontwolevels(low

andhigh)withrespecttotheir capabilities.Assuch asupplier

scored betweendl

j anddmj isconsidered asa low-level

capa-bilitiessupplier whileasupplier scoredbetweendm

j andduj is

consideredasahigh-levelcapabilitiessupplier.

• Symmetry in selecting a close type of partnership

(Lambert, Emmelhainz, & Gardner, 1996). In order to have

asuccessful relationshipbetweensupplychainpartners, there

should be some demographical similarities (for instance, in

terms of brand image, productivity) between them. So, more

symmetry means closer relationship. However, if we consider

two levels of closeness, it is clear that for some level of

symmetrythevalueofclosenessremainsthesame.

For the increasing stepwise value function, a criterion might

havemorethanonejump.Forinstance,ifadecision-makerwants

toconsider threelevelslow,medium,andhighwhensegmenting

thesupplierswithrespecttotheir capabilities,thenan increasing

stepwisefunctionwithtwojumpsshouldbedefinedforthis

crite-rion.Thefollowingvaluefunctionisageneralincreasing stepwise

valuefunctionwithkjumps.

ui j=

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

u0, dl_j≤ xi j≤ dm_j1, u1, dmj1≤ xi j≤ dmj2, . . . 1, dmk j ≤ xi j≤ d u j, 0, otherwise. (13) where0<u0<u1<…<1.

Fig. 10. Decreasing stepwise value function.

2.10.Decreasingstepwise

Decreasing stepwise value function shows that asthe criterion

level,xij,increasesuptoa certainlevel,dmj,its value,uij,remains

atathemaximumlevel,andafterthatcertainlevel,dm

j,itsvalue,

u_ij,jumpsdowntoalowerlevel,u0,andremainsatthatlevel.Itis

showninFig.10andformulatedasfollows:

ui j=

⎧

⎨

⎩

1, dl j≤ xi j≤ dmj, u0, dmj ≤ xi j≤ duj, 0, otherwise. (14) where0<u0 <1.

Forthisfunctionwecanthinkof:

• Considering supply risk in portfolio modeling (Kraljic, 1983).

For a company, an item witha higherlevel of risk results in

lessvalue,however,duetoportfoliomodeling,thereisno

dif-ferencebetweenalllevelsofriskinthedomain[dl

j, dmj].

Simi-larly,alllevelsofriskinthedomain[dm

j, duj]resultinthesame

value.

• Delay in logistics service provider selection (Qi, 2015). Some

companies consider stepwise value function for delay in

de-liveringtheitemsbya logisticsserviceprovider,which means

thatthevalueofthatproviderdecreaseswhendelayincreases,

howeveritisconstantwithincertainintervals.

For the decreasing stepwise function, a criterion might have

morethanonejump.Thefollowingvaluefunctionisageneral

de-creasingstepwisefunctionwithkjumps.

ui j=

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

1, dl j≤ xi j≤ dmj1, u1, dmj1≤ xi j≤ dmj2, . . . uk, dmkj ≤ xi j≤ d u j, 0, otherwise. (15) where0<uk<…<u1 <1.

3. Someremarksonthevaluefunctions

Here, a number of remarks are discussed, shedding light on

some aspects of the proposed value functions, which might be

usedinreal-worldapplications.

Remark1. Shape andparameters ofa valuefunction is

decision-maker-dependent,implyingthat(i)whileadecision-maker

consid-ers, for instance, a level-increasing function for the size of

gar-den when buying a house, another decision-makerconsiders an

increasing stepwise function, and (ii) while two decision-makers

considerincreasingstepwisefunctionforthesizeofgardenwhen

buyinga house, the parameters they consider fortheir functions

(

dl

Fig. 11. Increasing-level-decreasing value function.

Table 1

Suppliers performance with respect to different decision criteria ( x ij ).

Criteria

Supplier Quality Price ( €/item) Trust CO 2 (g/item) Delivery (day)

1 85 27 4 10 0 0 3 2 90 28 2 1500 4 3 80 26 5 20 0 0 3 4 75 25 5 10 0 0 2 5 95 29 7 1700 3 6 99 30 6 20 0 0 1

Remark2. Adecision-makermightconsidera hybridvalue

func-tion fora criterion. For instance, a criterion might be

character-ized withan increasing-level-decreasing, which is a combination

ofincreasing-levelandlevel-decreasing.Thisfunctioncan alsobe

considered as a special form of inverted V-shape function. It is

showninFig.11andformulatedasfollows:

ui j=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

xi j− dlj dm j − dlj , dl j≤ xi j≤ dmj1, 1, dm1_{≤ x} i j≤ dm2, du j− xi j du j− dmj , dm2_{≤ x} i j≤ du_j, 0, otherwise. (16)

Forinstance,adecision-makerhastoselectthebestR&D

part-neramong10partners.Oneofthecriteriaisdistanceandthe

com-panygivesless preferenceto very closeorvery distant partners,

which are distributed in the range [10km, 2000km]. The

com-panyconsidersanoptimaldistanceof[200km, 500km].This

im-pliesthatdistancefollowsan‘increasing-level-decreasing’function

forthisdecision-maker:[dl

j, d m1 j ,d m2 j ,d u j]=[10, 200, 500, 2000].

4. Numericalandcomparisonanalyses

In this section, we show how to incorporate the proposed

piecewise value functionsinto account when applyingan MCDM

method,andweshowthattheresultsmightbedifferentwhenwe

considertheproposedpiecewisevaluefunctions.

WeconsideranMCDMproblem,whereabuyershouldselecta

supplierfromamongsixqualifiedsuppliersconsideringfive

crite-ria:qualitywhichismeasuredby1−

α

,where

α

showsthelot-size

averageimperfectrate;price(euro)peritem;trust,whichis

mea-suredbya Likertscale (1:verylow to 7:very high);CO2 (gram)

peritem;delivery(day),theamountoftimewhichtakestodeliver

itemsfromthesuppliertothebuyer(allthecriteriaarecontinuous

excepttrust). Table1showsthe performanceof thesixsuppliers

withrespecttothefivecriteria.

Thebuyerhasusedanelicitation method2 _{tofindtheweights}

whichareasfollows:

w∗_quality =0.20,w∗_price=0.30, w∗_trust=0.27, w∗CO2 =0.08, w

∗

delivery=0.15.

And we assume that the decision-maker considers piecewise

valuefunctionsforthesecriteria(see,Table2).

So, as can be seen from Table 2, the decision-maker

consid-ersalevel-increaselinearfunctionforqualitywiththelowestand

highestvalues of0 and100, respectively.For thedecision-maker

anynumberbelow85hasno valueatall. Forcriterionprice, the

decision-makergivesthehighestvalue toanypricebelow15

(al-though in the existing set of suppliersthere is no supplier with

a price within thisrange), after which thevalue decreases till it

reachesto themaximum priceof30.Forcriterion trust whichis

measured using a Likert scale (1:very low to 7: very high),any

number less than 3 has no value for the decision-maker, while

thevalue gradually increasesbetween3 and7.ForCO2 emission,

thereisadecreasingvaluefunctionfrom0to1500gperitem,

af-terwhichtill2000g,allthenumbershavezerovalue.Finally,for

deliverythere isa simpledecreasingfunctionwithminimumand

maximumvaluesof0and5days.

Byusingthefollowingequation, wecan findtheoverall value

ofeachsupplierandthenrankthemtofindthebestsupplier.

Ui= n



j=1

wjui j (17)

where,uij is the value of the performance of supplier i with

re-specttocriterionj (usingtheequationsinTable2forthedatain

Table1),andwjistheweightofcriterionjasfollows:

w∗_quality=0.20,w∗_price=0.30, w∗trust=0.27,

w∗_CO2=0.08,w∗_delivery=0.15.

The value scores and the aggregated values are presented in

Table3(seealsoFig.13forthefinalresults).

AscanbeseenfromTable3,supplier6withthegreatestoverall

valueof0.51isrankedasthefirstsupplier.Suppliers5,4,3,1,and

2arerankedinthenextplaces.

4.1.Comparingwiththesimplelinearvaluefunctions

Inexistingliterature,consideringthenatureofthecriteria,the

valuesarecalculated,forinstance,usingthe followingsimple

lin-earvaluefunction:

ui j=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

xi j− dlj du j− d l j

, ifmorexi j ismoredesirable

(

suchasquality

)

,

du j− xi j du j− d l j

, ifmorexi jislessdesirable

(

suchasprice

)

.

(18)

Eq.(18)is usedtofindthevaluesofthecriteriaforeach

sup-plier usingthe data inTable 1. Considering the criteria weights,

andu_ij (Eq.(18)forthe datainTable 1) usingEq.(17)the value

scoresandalsotheaggregatedoverallscoreofeachalternativecan

becalculatedwhichareshowninTable4(seealsoFig.13forthe

finalresults).

InTable 4 itis assumedthat quality andtrust arecriteriafor

whichthehigherthebetter,whilefortheothercriteria(price,CO2,

anddelivery),thelowerthebetter.Infact,weconsidersimple

lin-ear functions(increasinganddecreasingrespectively)forthe two

groupsofcriteria.

2 Please note that we report some weights for the criteria as the aim of the study

Table 2

Piecewise value functions.

Shape Value function

u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0 , d l j ≤ x i j ≤ d mj , x i j − d mj d u j − d mj d m j ≤ x i j ≤ d uj , 0 , otherwise . = ⎧ ⎪ ⎨ ⎪ ⎩ 0 , 0 ≤ x i j ≤ 85 , x i j − 85 100 − 85 , 85 ≤ x i j ≤ 100 , 0 , otherwise . u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 , d l j ≤ x ≤ d mj , d u j − x i j d u j − d mj , d m j ≤ x ≤ d uj , 0 , otherwise . = ⎧ ⎪ ⎨ ⎪ ⎩ 1 , 13 ≤ x i j ≤ 15 , 30 − x i j 30 − 15 , 15 ≤ x i j ≤ 30 , 0 , otherwise . u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0 , d l j ≤ x i j ≤ d mj , x i j − d mj d u j − d mj , d m j ≤ x i j ≤ d uj , 0 , otherwise . = ⎧ ⎪ ⎨ ⎪ ⎩ 0 , 1 ≤ x i j ≤ 3 , x i j − 3 7 − 3 , 3 ≤ x i j ≤ 7 , 0 , otherwise . u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ d m j − x i j d m j − d lj , d l j ≤ x i j ≤ d mj , 0 , d m j ≤ x i j ≤ d uj , 0 , otherwise . = ⎧ ⎪ ⎨ ⎪ ⎩ 1500 − x i j 1500 − 0 , 0 ≤ x i j ≤ 1500 , 0 , 1500 ≤ x i j ≤ 2000 , 0 , otherwise . u i j = ⎧ ⎨ ⎩ d u j − x i j d u j − d lj , d l j ≤ x i j ≤ d uj , 0 , otherwise . = _{5 − x} i j 5 − 0 , 0 ≤ x i j ≤ 5 , 0 , otherwise .

According toTable 4,the best supplieris supplier 4,whichis

ranked as the3rd one considering the piecewise value functions

(Table3). The rankingof theother suppliersis alsodifferent.So,

ascanbeseen,suchdifferencesareassociatedtothewaywe

cal-culatethevalueofthecriteria.Ifwelookatthecriteriontrust,for

instance(Table3),wecanseethatonlythenumbersgreaterthan

3canbeusedforcompensatingtheothercriteria.Inotherwords,

thevalues1,2and3forthiscriterionhavenoselectionpower.No

suppliercancompensateitsweaknessinothercriteriabyhavinga

valuebetween1and3fortrust.However,suchimportantissueis

entirelyignoredinthesimplewayofdeterminingthevalue

Table 3

Value scores, u ij , and the aggregated overall score considering the proposed piecewise value

functions.

Supplier Quality Price Trust CO 2 Delivery Aggregated value Rank

1 0.00 0.20 0.25 0.50 0.40 0.23 5 2 0.33 0.13 0.00 0.00 0.20 0.14 6 3 0.00 0.27 0.50 0.00 0.40 0.28 4 4 0.00 0.33 0.50 0.50 0.60 0.37 3 5 0.67 0.07 1.00 0.00 0.40 0.48 2 6 0.93 0.00 0.75 0.00 0.80 0.51 1 Table 4

Value scores, u ij , and the aggregated overall scores considering the simple linear value func-

tions.

Supplier Quality Price Trust CO 2 Delivery Aggregated value Rank

1 0.42 0.60 0.40 1.00 0.33 0.50 4 2 0.63 0.40 0.00 0.50 0.00 0.29 6 3 0.21 0.80 0.60 0.00 0.33 0.49 5 4 0.00 1.00 0.60 1.00 0.67 0.64 1 5 0.83 0.20 1.00 0.30 0.33 0.57 2 6 1.00 0.00 0.80 0.00 1.00 0.57 3

Fig. 12. Exponential value functions.

isevenofahigherimportanceforcompensatorymethodssuchas

AHPandBWM.

4.2.Comparingwiththeexponentialvaluefunctions

Another importantwayto approximate thevalue functions in

practiceistheuseofexponentialvaluefunctions(Kirkwood, 1997;

Pratt, 1964). The exponential value functions can specifically be

used when the preferences are monotonically increasing or

de-creasing. Although this approach is not popular in MCDM

do-main,andwewere notabletofindanyapplicationofthesevalue

functionsparticularly in MCDM field, we would like to compare

ourresults to the results ofapplying thesefunctions, whichare,

to some extent, close to some of our proposed piecewise value

functions(suchaslevel-increase,level-decrease,increase-level,and

decrease-level).Usingthe samenotationsasbeforeand

consider-ingashape parameter

ρ

whichiscalled‘risktolerance’,a

mono-tonically increasing exponential value function can be shown as

follows: ui j=

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

1− exp



−

xi j− dlj

/

ρ



1− exp



−

du j− dlj

/

ρ



,

ρ

=Infinity xi j− dlj du j− dlj , otherwise. (19)

Amonotonically decreasing exponential value function can be

shownasfollows: ui j=

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

1− exp



−

du j− xi j

/

ρ



1− exp



−

du j− dlj

/

ρ



,

ρ

=Infinity du j− xi j du j− dlj , otherwise. (20)

Fig. 12 showsthe monotonically increasing exponential value

functions(fordifferentvaluesof

ρ

)(a),andthemonotonically

de-creasingexponentialvaluefunctions(fordifferentvaluesof

ρ

)(b).

Risk-averse decision-makershave

ρ

> 0 (hill-like functionsin

Fig.12),whilerisk-seekingdecision-makershave

ρ

<0(bowl-like

functionsinFig.12).

ρ

=Infinity(straight-lineinFig.12)showsthe

valuefortheriskneutraldecision-makers.Infact,

ρ

=Infinity

pro-ducesthesimplelinearvaluefunctionswhichareverypopularin

MCDMfield.

In order to do the comparison analysis, we use exponential

value functions for the criteria of the aforementioned example

(Table 1) to check the similarities and differences. To make a

faircomparison,we tryto generate3 _{the corresponding}

exponen-tial value functionsof thepiecewise value functions(Table2) as

3 To see how these value functions are elicited considering the decision-maker

closeaspossible.Forquality,amonotonicallyincreasing

exponen-tialvaluefunctionwithnegative

ρ

wouldbeappropriate.Forprice

amonotonicallydecreasingexponentialvaluefunctionwitha

posi-tive

ρ

,fortrust,amonotonicallyincreasingexponentialvalue

func-tion with a negative

ρ

, for CO2, a monotonically decreasing

ex-ponential value function with a negative

ρ

, and, finally, for

de-liverya monotonicallydecreasingexponentialvaluefunction with

ρ

=Infinitywouldbesuitable.Table5showsthefunctions,where

functionswithdifferent

ρ

sareshownandamoresuitable oneis

showninbold.

Using theexponential value functionsof Table5, forthe data

ofTable 1,we get thevalue scores andthe aggregatedvaluesas

presentedinTable6(seealsoFig.13forthefinalresults).

Table 5

Exponential value functions.

Shape Function u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 − exp [ −( x i j − d lj) /ρ] 1 − exp[ −( d u j − d lj) /ρ] , ρ = Infinity x i j − d lj d u j − d lj , otherwise . = ⎧ ⎨ ⎩ 1 − exp [ −( x i j − 0 ) /ρ] 1 − exp[ −( 100 − 0 ) /ρ] , ρ = Infinity x i j − 0 100 − 0 , otherwise . u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 − exp [ −( d u j − x i j) /ρ] 1 − exp[ −( d u j − d lj) /ρ] , ρ = Infinity d u j − x i j d u j − d lj , otherwise . = ⎧ ⎨ ⎩ 1 − exp [ −( 30 − x i j) /ρ] 1 − exp[ −( 30 − 0 ) /ρ] , ρ = Infinity 30 − x i j 30 − 0 , otherwise . u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 − exp [ −( x i j − d lj) /ρ] 1 − exp[ −( d u j − d lj) /ρ] , ρ = Infinity x i j − d lj d u j − d lj , otherwise . = ⎧ ⎨ ⎩ 1 − exp [ −( x i j − 1 ) /ρ] 1 − exp[ −( 7 − 1 ) /ρ] , ρ = Infinity x i j − 1 7 − 1 , otherwise . u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 − exp [ −( d u j − x i j) /ρ] 1 − exp[ −( d u j − d lj) /ρ] , ρ = Infinity d u j − x i j d u j − d lj , otherwise . = ⎧ ⎨ ⎩ 1 − exp [ −( 20 0 0 − x i j) /ρ] 1 − exp[ −( 20 0 0 − 1500 ) /ρ] , ρ = Infinity 20 0 0 − x i j 20 0 0 − 0 , otherwise .

Table 5 ( continued ) Shape Function u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 − exp [ −( d u j − x i j) /ρ] 1 − exp[ −( d u j − d lj) /ρ] , Infinity d u j − x i j d u j − d lj , otherwise . = ⎧ ⎨ ⎩ 1 − exp [ −( 5 − x i j) /ρ] 1 − exp[ −( 5 − 0 ) /ρ] , ρ = Infinity 5 − x i j 5 − 0 , otherwise . Table 6

Value scores, u ij , and the aggregated overall scores considering the exponential value func-

tions.

Supplier Quality Price Trust CO 2 Delivery Aggregated value Rank

1 0.85 0.45 0.05 0.08 0.40 0.22 5 2 0.90 0.33 0.00 0.02 0.20 0.16 6 3 0.80 0.55 0.13 0.00 0.40 0.26 4 4 0.75 0.63 0.13 0.08 0.60 0.32 3 5 0.95 0.18 1.00 0.00 0.40 0.46 1 6 0.99 0.00 0.37 0.00 0.80 0.38 2

Fig. 13. Final results using three types of value functions.

AscanbeseenfromTable6,supplier5withthegreatest

over-allvalue of0.46isrankedasthefirst supplier,which isdifferent

fromwhat we get from the proposed piecewise value functions

(Table3).Whilesupplier5wasrankedthe2ndbasedonour

pro-posedvaluefunctions, usingthe exponentialvalue functions, this

supplierbecomesnumber2. Other suppliers(1,2,3,4) have the

samerankingbasedonthetwoapproaches.Thedifferencesare

ob-viouslyassociatedtothewaywegetthevaluesofthecriteria.We

alsocheckedsomeother close

ρ

valuesforthe exponentialvalue

functions. There are some changes in the aggregatedvalues, yet,

therankingisthesame.

Asitcanbeseen,theresultsofthetwoapproaches(piecewise

valuefunctionsandexponentialfunctionsaremuchclosertoeach

other than tothe resultsof theregular simple linearvalue

func-tions).

Our observation is that the exponential value functions can

play a role close to a number of proposed value functions in

thisstudysuchasincrease-level,decrease-level,level-increase,and

level-decrease. In order to make an exponential value functions

closetooneofthementionedproposedvaluefunctionsweshould

choose

ρ

valuesclose to zero.If we try to make the

ρ

asclose

Fig. 14. Fitting an exponential value function to a level-decrease value function.

otherpartbecomesverysteepandnotrepresentative.Ontheother

hand,ifwewanttochoosea

ρ

value whichbetterrepresentsthe

slopeofthefunctionforthe“increase” orthe“decrease” part,then

itisimpossibletocoverthe“level” partofthevaluefunction

prop-erly.

For example, let usconsider the criterion price again. In Fig.

14,itcanbeseenthat,ifweconsider

ρ

=1,thelevelpartisfully

covered, but the decrease part ofthe exponential value function

is verymuch differentfromthe decreasepartofthe linear

func-tion.Evenifwechoose

ρ

=3, whichdoesnotfullycoverthelevel

partoftheproposedfunction,thedecreasepartisreallydifferent.

On the other hand,we can find

ρ

=8 as a close one to the

de-creasepartof thelinearfunction, butthistime itis notpossible

tocoverthelevelpartproperly.So,althoughaverygood

approx-imation,theexponentialvaluefunctionsmightnotbesuitable for

casesinwhichadecision-makerhasaclearvalue-indifference

in-terval (alevel part) for a criterion. However, thesefunctions are

indeedsuitablewhenthedecision-makerhasdifferentpreferences

onthelowerandontheupperpartsofthecriterionmeasure.

Fromthefigurewecanalsoseethatwhilewecouldmakethe

two piecewise and exponential value functions, to some degree,

closeto each other,they are toodifferentfromthe simplelinear

valuefunction.Itisalsoclearthatnoneofthesimplelinearvalue

functionsortheexponential valuefunctionscanrepresentthe

V-orinvertedV-shapevaluefunctions.

Asa generalconclusion,we thinkthattheproposed piecewise

linear functions havetwo salient features: (i) simplicity; and(ii)

representativeness.Thatis,itiseasytoworkwithlinearfunctions,

anditiseasyforapractitionertofindamorerepresentative

func-tionfromtheproposedlibraryofthepricewisevaluefunctionsfor

a particularcriterion.The cut-off pointscanalsobe estimatedby

the decision-maker.The simple monotonic-linear value functions,

which are dominant inexisting literature, are very simple.

How-ever, theymight notbe representative insome cases.Finally,the

exponentialvaluefunctionsmighthaveabetterrepresentativeness

(comparedtothe simplemonotonic-linearvalue functions).

How-ever,theyarenotsimple.Workingwithnon-linerfunctionsisnot

easyforpractitioner,and,moreimportantly,itisverydifficultfora

practitionertoestimateavalue for

ρ

(the shapeparameterofthe

exponential valuefunctions),asitcannotbe easilyinterpreted by

apractitioner(pleasenotethatweconsideravaluefunctionasan

inputforanMCDMprobleminthisstudy).

5. Determiningthevaluefunctions

One ofthe big challenges inreal-world decision-making is to

find a proper value function for a decision criterion. This,

per-haps, has been one of the main reasons why the use of simple

linear value functions in multi-criteria decision-making is

domi-nant. The linear value functions are easy for modeling purposes

andcan, tosome extent,representthe reality.More complicated

valuefunctions,althoughmightbeclosertorealityofthe

decision-maker’spreferences, are moredifficultto be elicitedandare

dif-ficult for modeling purposes. We refer the interested readers to

someexistingproceduresforidentifyingvaluefunctions(Fishburn,

1967;Keeney & Nair, 1976; Kirkwood, 1997;Pratt, 1964; Stewart & Janssen, 2013). We think that the proposed value functions in

thispaperdonothavethedisadvantageofnonlinearityandatthe

same time have the advantagesof being closer to the real

pref-erencesofthedecision-makerasthey providesomediversityand

flexibility in modeling the functions. As we do not consider the

nonlinearityofthe value functionswedo not usetheconcept of

risk tolerance in determining the value functions as it has been

used by others.We rather propose a simple procedure, which is

morepractical.

A decision analyst, could first show the value functions in

Table 2 to the decision-maker to see which one most suits the

preference structure of the decision-maker. Once the

decision-makerselectsa particularvalue function, theother details ofthe

function, such as the lower bound, the upper bound and the

thresholds canbe determined. We should highlightagain that in

mostMCDMmethods,thevaluefunctionisnotelicited.Itisrather

simply assumed to have a particular shape, and this is why we

thinkhavingapre-specifiedsetofstandardvaluefunctionswhich

canbeusedassubjectiveapproximationoftherealpreferencesof

thedecision-makercanmakeasignificantimpactontheresults.

6. Conclusion,limitationsandfutureresearch

This study proposes a set of piecewise value functions for

multi-criteriadecision-making(MCDM)problems.Whilethe

exist-ingapplicationsofMCDMmethodsusuallyusetwogeneralsimple

increasing and decreasing linear value functions, this study

pro-vides severalreal-world examples to support the applicability of

someotherformsofvaluefunctionsforthecriteriausedinMCDM.

Itis alsoexplicated how, insome decisionproblems,a

combina-tion of two or more value functions can be used for a

dif-ferentMCDM methods indifferent decisionproblems. A

numeri-calexampleofsupplierselectionproblem(includingacomparison

betweensimple monotonic linear value functions, piecewise

lin-earvaluefunctions,andexponentialvaluefunctions)showedhow

theuseofthe proposed valuefunctionscould affectthe final

re-sults.Consideringthesevaluefunctionscouldbetterrepresentthe

realpreferencesofthedecision-maker.Itcanalsohelpreducethe

inappropriatecompensations ofthedecisioncriteria, forinstance,

throughusingalevel-increasingfunctionwhichassignszerovalue

toany value ofthe criterion below a certain threshold.The

pro-posedvalue functionsare presented in a generalform such that

they can be tailor-made for a specific decision-maker. Thatis to

say,notonlyitispossiblefortwodifferentdecision-makerstouse

twodifferentvalue functionsfora singlecriterion.It isalso

pos-sibleto use different domain (e.g.min and max) values forthat

particularvaluefunction.

Despite the advantages of the proposed value functions, they

havesomelimitations.Althoughtheproposedvaluefunctions

con-sidersomereal-worldfeaturesofthedecisioncriteria,theyare

lin-ear which might be, to some degree, a simplification. We think

thatthe biggerproblemin existingliterature of MCDMis

mono-tonicity assumption and not linearity assumption. Nevertheless,

moreresearchneedstobeconductedtoempiricallyfindtheshare

ofeach.Furthermore,toformulatethedecisioncriteriaoneshould

payenoughattentiontochecktherealcontributionofthedecision

criterion into the ultimate goal of the decision-making problem.

Forinstance,if acriterion contributesto anothercriterion which

hasarealroleinmakingthedecision,oneshouldexcludethe

ini-tialone. Foradetaileddiscussiononthismatter, interested

read-ersarereferred toBrugha(1998).Oneinterestingfuturedirection

wouldbetoapplytheproposedvaluefunctionsinsomereal-world

MCDMproblemsandcomparetheirfitnesstotheothervalue

func-tions.Inthisregard,findingamoresystematicapproachto

deter-minethe valuefunctionsin practicewouldbe alsovery

interest-ing.Itwouldbealsointerestingtostudythecasesinwhichthere

are more than one decision-maker. As different decision-makers

maychoose differentvalue functions, different domains, and

dif-ferent thresholds for a single criterion, proposing a way to find

the final output of the MCDM problem for the group would be

an interesting future research. Finally, finding a sensitivity

anal-ysisforthe proposed value functions isrecommended.

Consider-ingthestudiesofBertschandFichtner(2016),Bertsch,Treitz,

Gel-dermann, and Rentz (2007), Insua and French (1991), Wulf and Bertsch(2017)couldgiveinterestingideastomakesuchsensitivity

analysisframework.

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Jafar Rezaei is an associate professor of operations and supply chain management at Delft University of Technology, the Netherlands, where he also obtained his Ph.D. One of his main research interests is in the area of multi-criteria decision-making (MCDM) analysis. He has published in various academic journals, including International Journal of Production Economics, International Journal of Production Research, Industrial Marketing Management, Applied Soft Computing, Applied Mathematical Modelling, IEEE Transactions on Engineering Management, Journal of Cleaner Production, European Journal of Operational Research, Information Science, Omega, and Expert Systems with Applications.