Institut für Numerische und Angewandte Mathematiknum.math.uni-goettingen.de/preprints/files/2011-18.pdf · (MADM), namely simple additive rating (SAR), simple additive weighting

Institut für Numerische und Angewandte Mathematik

On the Similarities of Some Multi-Criteria Decision AnalysisMethods

J. Geldermann, A. Schöbel

Nr. 2011-18

Preprint-Serie des

Instituts für Numerische und Angewandte Mathematik

Lotzestr. 16-18

D - 37083 Göttingen

On the Similarities of Some Multi‐Criteria Decision AnalysisMethods

JUTTA GELDERMANNa,* and ANITA SCHÖBELbaUniversität Göttingen, Göttingen, GermanybInstitute for Numerical and Applied Mathematics, Georg‐August‐Universität Göttingen,Göttingen, Germany

ABSTRACT

Since multiple criteria characterise many practical decision problems, the research field of multicriteria decision analysis(MCDA) is flourishing. This paper compares the multi-attribute decision making (MADM) methods, simple additive rating(SAR), simple additive weighting (SAW) as well as the outranking approach PROMETHEE and explains their similarities.We present a case study on industrial paint application and provide theoretical evidence that PROMETHEE can mimic otherMADM algorithms. Copyright © 2011 John Wiley & Sons, Ltd.

KEY WORDS: outranking; MCDA; preference functions

1. INTRODUCTION

Thefield ofmulti‐criteria decision analysis (MCDA) hasdeveloped rapidly over the past quarter century,emerging from a number of divergent schools ofthought (Belton and Stewart, 2002; Figueira et al.,2005). These methods have been applied to manydecision situations, especially to technology assessment,for example, cleaner production (Geldermann andRentz, 2001; Hämäläinen, 2004; Zhou et al., 2004) orenergy selection (Hobbs andMeier, 2000; Greening andBernow, 2004; Pohekar and Ramachandran, 2004;Diakoulaki et al., 2005; Løken, 2007; Wang et al.,2009; Oberschmidt et al., 2010), during which eco-nomic, technical, and especially ecological criteria haveto be taken into account simultaneously.

In this paper, we recapitulate the most frequentlyused methods for multi attribute decision making(MADM), namely simple additive rating (SAR),simple additive weighting (SAW) and the outrank-ing‐approach preference ranking organization methodfor enrichment evaluations (PROMETHEE; Section2). On the basis of an industrial coating case study(Geldermann and Rentz, 2004, 2005), similaritiesbetween the different MADM approaches’ numericalresults are highlighted in Section 3. The mathematicalproof of the reported findings’ generality is provided.

2. APPROACHES TO MULTI‐CRITERIADECISION MAKING BASED ON THE RANKING

OF DISCRETE ALTERNATIVES

Various approaches toMADM have been developed tocompare various specific alternatives (Stewart, 1992;Belton and Stewart, 2002; Figueira et al., 2005).MADM’s basic notation considers the set A of Talternatives that have to be ranked, and K criteria thathave to be optimized,

A :¼ a1;…; aTf g : Set of discrete alternatives or techniquesF :¼ f1;…; fKf g : Set of criteria relevant for the decision:

(1)

The resulting multiple criteria decision problem canbe concisely expressed in a matrix format. The goalachievement matrix or decision matrix D : = (xtk) witht= 1,…, T and k = 1,…,K is a s(T×K ) matrix whoseelements xtk = fk(at) indicate the evaluation or value ofalternative at, with respect to criterion fk:

D ¼x11 ⋯ x1K⋮ xtk ⋮xT1 ⋯ xTK

24

35 :¼

f1 a1ð Þ ⋯ fK a1ð Þ⋮ fk atð Þ ⋮

f1 aTð Þ ⋯ fK aTð Þ

24

35

(2)

Among the most popular approaches are the multi‐attribute utility theory/multi‐attribute value theory

*Correspondence to: Chair of Production and Logistics,Universität Göttingen, Göttingen, Germany. E‐mail:[email protected]‐goettingen.de

Copyright © 2011 John Wiley & Sons, Ltd.Received 19 April 2010Accepted 5 May 2011

JOURNAL OF MULTI-CRITERIA DECISION ANALYSISJ. Multi-Crit. Decis. Anal. (2011)Published online in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/mcda.468

(MAUT/MAVT; Dyer, 2005; Siskos, 2005). Theanalytic hierarchy process (AHP) and analytic net-work processes (ANP; Saaty and Vargas, 2006) alsouse a kind of value function, which is derived bymeans of a nine‐point‐scale. One of the main under-lying assumptions of these approaches is that decisionmakers (DMs) are aware of the utility of differentcriteria values and are able to express the relativeimportance of different criteria in an unambiguousway. The goal of decision making is then to discloseand interpret the preferences of DMs in a transparentway.

The sometimes so‐called ‘European school ofthought’ (Roy and Vanderpooten, 1997) evolved asan attempt to overcome some shortcomings of MAUT/MAVT. Roy and Bouyssou (1993) and Brans et al.(1986) assumed that DMs are not fully aware oftheir preferences. Therefore, decision support is neededto help structure the decision situation and to demon-strate the influence of different criteria weightings. Thus,they developed the so‐called outranking approaches, andelimination and choice expressing reality (ELECTRE;Roy and Bouyssou, 1993) and PROMETHEE (Branset al., 1986; Brans and Mareschal, 2005) were their firstimplementations.

The emergence and existence of methods basedon value or utility theory or on outranking considera-tions are often the subject of discussion, mainlybecause they have no shared view of a psychologi-cally ‘correct’ modelling of human value judgement(Hämäläinen, 2004). This has given rise to con-troversial discussions in the scientific literature(Lootsma, 1996). Furthermore, numerous compar-isons have been published of the different ap-proaches (Hämäläinen, 2004).

A systematic investigation on the numerical differ-ences between the calculated results for one and thesame case study, however, is missing. Thus, in thefollowing, the algorithms for simple additive ranking(a very basic application of MAVT), simple additiveweighting (as a simplistic, but often used applicationof MAVT) and PROMETHEE (an Outrankingapproach) are briefly presented, before their applica-tion is shown for the investigated case study. The aimis not a comparison of the application, but anelucidation of the numerical differences or similaritiesbetween the approaches, as will be shown in Section4.1. It will be shown that PROMETHEE can mimicrythe simple additive ranking or the simple additiveweighting, if the preference functions are chosen in aspecific way. The mathematical proof, which demon-strates that the findings are general, can be found in theannex of this paper.

2.1. Simple additive ranking (SAR)As a very intuitive multi‐criteria approach, SAR isbased on the ranking of the alternatives of eachcriterion, the subsequent aggregation of the weightedranks and, finally, on the normalization of the obtainedvalues. The resulting value function indicates thegoodness of the alternatives: the higher the rank, thehigher the alternative’s value compared with the otherobserved alternatives (French, 1986):

v atð Þ ¼ 1T∑K

k¼1wk⋅Rk fk atð Þð Þ with wk ≥ 0 and ∑

K

k¼1wk¼1

(3)

where wk denotes a weighting factor for criterion k.Furthermore, Rk indicates the ordinal rank of analternative with regard to the considered criterion k.If the criterion is to be maximized, for example theprofit of an alternative, the value function Rk is definedas follows:

Rk max fk atð Þf gð Þ ¼ T if fk atð Þ→max1 if fk atð Þ→min

�

If the criterion is to be minimized, for example thematerial consumption or operational costs, the valuefunction Rk is defined in this way:

Rk min fk atð Þf gð Þ ¼ 1 if fk atð Þ→maxT if fk atð Þ→min

�

Normalization is not necessary to obtain a rankorder of the observed alternatives, but allows a moreintuitive graphical representation. Both descendingand ascending rank orderings can be defined and leadto the same resulting total rank order. The ordinalscore to assign has to be defined if two or morealternatives have an even score regarding a certaincriterion.

Simple additive rating is much criticized frommethodological points of view. But despite its roughdifferentiation between the alternatives, simple addi-tive ranking is a widely used approach, but prone torank reversals (cf. the so‐called Borda Count asdescribed by, for example, French (1986)). Never-theless, the rank ordering of each criterion withoutany further aggregation can help identify dominatingor dominated alternatives. Such a depiction also helpsto illustrate the multi‐criteria character of a decisionproblem in real world decision problems and groupdecisions with numerous participants (often fromvarious scientific backgrounds). The application ofSAR to the case study is illustrated in Section 3.

J. GELDERMANN AND A. SCHÖBEL

Copyright © 2011 John Wiley & Sons, Ltd. J. Multi-Crit. Decis. Anal. (2011)DOI: 10.1002/mcda

2.2. Simple additive weighting (SAW)TheMAVT is based on establishing a value function thatrepresents goal achievement according to each criterion,multiplied by the criterion’s specific weight (French,1986). The DM has to define a one‐dimensional value‐function vk( fk(at)). In the easiest case—also calledSAW—the value function is normalized to the interval[0,1], where each criterion’s best score on is assigned theutility value vk=1, and the worst is assigned vk=0.Formally, denoting

xmax;k :¼ maxt∈T

fk atð Þf gxmin;k :¼ min

t∈Tfk atð Þf g

the functions vk are given according to

vk fk at′ð Þð Þ ¼

fk at′ð Þ−xmin;k

xmax;k −xmin;kif fk atð Þ→max

xmax;k − fk at′ð Þxmax;k −xmin;k

if fk atð Þ→min

8>>><>>>:

(4)

Summing up the functions across all criteriak = 1,…,K, we obtain

v atð Þ ¼∑K

k¼1wk⋅vk fk atð Þð Þ with wk≥0 and ∑

K

k¼1wk ¼ 1

(5)

where wk are again the weighting factors of thecriteria k.

The higher the weighted sum of the utility values,the better the alternative. This concept is ratherintuitive for the decision‐maker.

Because MAUT is based on the concept of a trade‐off between different criteria’s scores, a complete‘compensation’ between attributes is possible. Con-sequently, a sufficiently large gain in a lesser attributewill eventually compensate for a small loss in a moreimportant attribute, no matter how important oneattribute is (Stewart, 1992). However, the effect ofcompensation is not always desired, as in environ-mental assessments: high emissions of one kind ofchemical substances (e.g. into the air) should not becounterbalanced by other environmental effects (e.g.little water effluents) (Geldermann and Rentz, 2005).

2.3. The outranking approach preference rankingorganization method for enrichment evaluation(PROMETHEE)Some mentioned disadvantages of MAVT, MAUT,AHP and ANP include, for example, the loss of

information because of the high aggregation of resultsand the fact that good and poor criteria values canfully compensate each other. However, because oftheir traceability and manageability in practice, thesemethods are often applied in real world decisionproblems. Outranking approaches have advantagesbecause they are not fully compensatory, and usuallyneed less information from the DM. They are able todeal with both quantitative and qualitative data on anopen scale and can integrate uncertain informationthrough probability distributions, fuzzy sets and thresh-old values (Geldermann et al., 2000; Haralambopoulosand Polatidis, 2003; Xu, 2005; Araz and Ozkarahan,2007; Løken, 2007; Ren et al., 2009). Furthermore, inaddition to strict preference and indifference, weakpreferences and incomparabilities can also be describedif there is not enough information to rank optionsunequivocally (Roy, 1996; Rogers and Bruen, 1998).ComparedwithAHP/ANP, fewer pairwise comparisonsare needed, and the valuations are not restricted toSaaty’s nine‐point scale (Albadvi et al., 2007; Anandand Kodali, 2008; Dagdeviren, 2008).

In order to overcome the assumptions of completecompensation and the existence of a ‘true’ ranking ofalternatives that only need to be discovered, outrankingmethods have been developed within the so‐calledEuropean or French MADM school (Stewart, 1992; Royand Bouyssou, 1993; Figueira et al., 2005).1 Outrankingtakes into account that preferences are not constant intime, not unambiguous and not independent of theprocess of analysis (Roy and Bouyssou, 1993).

The outranking method PROMETHEE (Branset al., 1986; Brans and Mareschal, 2005) derives thepreference values by generalized criteria, which thedecision‐maker can specifically define for eachconsidered criterion fk:

pk fk atð Þ− fk at′ð Þð Þ ¼ pk dð Þ∈ 0; 1½ � (6)

The degree of preference of an alternative atin comparison with at′ can vary from pk(d) = 0,which means indifference, across a zone of weakpreference to pk(d) = 1, indicating strict preference.Six types of generalized criteria that cover most ofthe decision problems have been suggested, but

1Following the work of (Ackoff, 1981), Outranking aims tobe more than a strict, ‘clinical’ problem resolution, orproblem solving in the classical, Operational Researchsense, but to foster problem dissolving approach, providinghints to change the constraints given with the decisionproblem.

ON THE SIMILARITIES OF SOME MCDA METHODS


the decision‐maker may model his preferences withthe help of individually shaped preference functionsas well. The six types of preference functions aredepicted in Figure 1.

In the case study in Section 3 and in Section 4.1,we will mainly focus on the type 3 preference function.Given a parameter p, the preference function of type 3is defined as

p dð Þ ¼0 if d < 0dp

if 0≤ d ≤ p

1 if d > p

8<:

Instead of a value function, an outranking relationπ :A×A→ [0, 1] is defined for every pair of alter-natives at; at′∈ A:

π : π at; at′ð Þ ¼ ∑K

k¼1wk⋅pk fk atð Þ−fk at′ð Þð Þ (7)

The preference index π at; at′ð Þ is a measure of theintensity of the DM’s preference for alternative at incomparison with alternative at′, if all criteria are consideredsimultaneously.

As a measure of the strength of the alternativesat ∈A, the outgoing flow is calculated:

Φþ atð Þ ¼ 1T⋅ ∑

T

t′¼1; t′≠tπ at; at′ð Þ (8)

As a measure of the weakness of alternative at ∈A,the incoming flow is calculated, measuring the ‘outrankedcharacter’ of at (analogously to the outgoing flow):

Φ− atð Þ ¼ 1T⋅ ∑

T

t′¼1;t′≠tπ at′ ; atð Þ (9)

Using the total number of considered alternativesT, normalization is not a necessary precondition, butthe same approach has to be chosen for both theoutgoing and incoming flow. As with the normal-ization of the weights, a comparison of differentevaluations is made easier. Mareschal (1998) suggestsa PROMETHEE algorithm without any normaliza-tion, whereas Brans and Mareschal (1994) proposes a

normalization factor of1

T−1.

PROMETHEE I derives the following partialpre‐order: at is preferred to at′ if Φþ atð Þ≥Φþ at′ð Þand Φ− atð Þ≤Φ− at′ð Þ , and at least one of the twoinequalities is strict. If a complete pre‐order isrequested, PROMETHEE II yields the so‐callednet‐flows as the difference between the outgoing andincoming flows, avoiding any incomparabilities:

Φnet atð Þ ¼ Φþ atð Þ−Φ− atð Þ (10)

However, the partial pre‐order derived by PRO-METHEE I may contain more realistic informationthrough the indication of incomparabilities.With the helpof a graphical representation, clusters of alternatives canbe derived, so that a group of best alternatives can beidentified. Moreover, the documentation of the incom-parabilities is helpful for the identification of furtherrequired information.

In the following section, the numerical results ofour case study are shown.

P(d)

d

P(d)

dq p

P(d)

dp

P(d)

dq

P(d)

d

P(d)

dq p

1

Type 1Usual criterion

1 1

1 1 1

Type 2Quasi-criterion

Type 3Criterion with linear preference

Type 4Level criterion

Type 5Criterion with linear preference and indifference area

Type 6Gaussian criterion

Figure 1. Six types of preference functions used for PROMETHEE.



3. CASE STUDY ON THE INDUSTRIAL PAINTAPPLICATION SECTOR

3.1. Description of case studyWe present a case study on the industrial paintapplication sector regarding polyvinyl chloride (PVC)parts for the automobile sector. Job coaters are affectedby the Solvent Directive of the European Union (1999/13/EC), which determines the emission limits for 20different categories of installations that use solvents andthus emit volatile organic compounds. In the prepara-tion of a job coater’s investment decision, the actualconsequences of emission reduction techniques’ im-plementation must be quantified for his or heroperations. In close cooperation with a job coater,several emission reduction potentials were analysed.The emissions and cost‐relevant process steps as wellas their interdependencies were investigated for thecoating processes of mobile phones and PVC partsdestined for the automobile industry. A computer‐aided mass and energy flow model (Geldermann andRentz, 2004) was the starting point of the investiga-tion into the current processes (Scenario 0) and toidentify the critical points of and hints for improve-ments of the process. Subsequently, the definition andmodelling of various scenarios formed the basis withwhich to support investment decisions. It should benoted that we use the term ‘scenario’ instead of‘alternative’ or ‘decision alternative’, because it refersto the strategic choice of the company depicted in ourcase study. Some scenarios are combinations ofvarious technical alternatives. In this way, the use ofa spray robot (Scenario 1), process modifications (e.g.the introduction of various off‐gas treatment measures(Scenario 2)), and substituting paints and organicsolvents (introducing water‐based paints and the useof recovered cleaning solvents (Scenario 3)) withsome modifications were investigated. For the fulldescription and discussion of the technical, economicand environmental implications of the case study, see(Geldermann and Rentz, 2005).

Table I summarizes the investigated scenarios forpaint application to PVC parts for the automobilesector. The data are only presented as a percentage ofchange in comparison with the status quo (Scenario 0:Manual Coating) to protect the cooperating company’sconfidential information. Negative percentages—asfor spray robots’ paint consumption—mean that lessmaterial is used in comparison with the status quo(Scenario 0). Only the decision‐relevant parameters areconsidered, which change from scenario to scenario.Although the calculated values in Table I are blindeddata (in order to keep trade secrets about exact values), T

able

I.Selectedresults

oftheinvestigated

scenariosforindustrial

paintapplication

Scenario

Resou

rces

Waste

Emission

ssolvent

emission

Primarymeasures

Waste

gas

cleaning

operating

costs(/a)

No.

Shortdescription

Paint

Dilu

ting

solvent

Con

sumption

slud

ges

Operating

costs

(/a)

Operating

costs

(/a)

Investment

(/a)

0(M

anualcoating)

00

00

00

00

1Robot

−45%

−45%

−60%

0%−4

2%−4

3%200

02A

Therm

alincineratio

non

drier

0%0%

0%0%

−3%

0%350

255

2BAdsorptionon

spraying

cabin

0%0%

0%0%

−60%

0%500

502C

Therm

alincineratio

non

spraying

cabin

0%0%

0%0%

−60%

0%750

20

2DBiofilteron

spraying

cabin

0%0%

0%0%

−72%

0%350

402E

Scenario2D

+1

−45%

−45%

−60%

0%−8

1%−4

3%550

403A

Water

basedcoat

6%6%

0%−7

5%−7

8%2%

250

3BScenario3A

+1

−41%

−41%

−60%

−75%

−87%

−44%

225

0



they allow the considered alternatives’ basic simulta-neous evaluation procedure to be illustrated with regardto the various, partly conflicting, criteria.

From a purely formal viewpoint, nine alternativesa1,…, a9, (namely, the scenarios 0, 1, 2A, 2B, 2C, 2D,2E, 3A and 3B) are compared with regard to eightcriteria. Table I shows the values of fk(at). All criteriaare to be minimized, and an equal weighting (of12.5%) is assumed. Changes in the weighting factorscan be systematically investigated with a subsequentsensitivity analysis (Mareschal, 1998; Geldermannand Rentz, 2001, 2005). For a brief overview of thegoal conflicts of a multi‐criteria decision problem,SAR can be applied. The graphical representation inFigure 2 clarifies that Scenario 3B shows the bestvalues with regard to the three criteria. None of thenine examined scenarios, however, dominates or isdominated.

The application of SAW yields a hierarchy arrangedin a more granular level. Figure 3 shows a graphicalrepresentation of the case study’s results if equalweighting of all criteria (12.5%) is assumed.

Finally, the outranking approach PROMETHEEwas applied. In this case study, the type 3 preferencefunction (criterion with linear preference, Figure 5)was deployed. The difference between the bestand the worst score on each criterion was taken asthe parameter p for the type 3 preference function,indicating the transition from weak to strict preferencebetween two considered alternatives with regard to therespective criterion. Figure 4 depicts the graphicalrepresentation of the obtained results. It should benoted that the calculated incoming flow is multipliedby (−1) in order to achieve an intuitively under-standable graphical representation. Also the partialpre‐order is given.

3.2. Comparison of the resultsOn closer examination, some similarities betweenthe simple additive weighting’s results and those ofthe PROMETHEE application become obvious.Firstly, according to the PROMETHEE net flowsΦnet (using type 3 preference functions with parameterp as the difference between the highest and lowestscore with regard to each criterion) and the aggregatedutility values from the simple additive weighting, theranking of the alternatives is the same. Moreover, thedifferences between the preference or utility values ofeach pair of alternatives are equal, as shown inTable II. This gives rise to the claim that the Outrankingpreference concept, which is based on pairwise compar-isons, and MAVT/MAUT utility functions concept arebasically the same.

We now explain this observation analytically. Allthree mentioned MADM approaches are additivemethods, because their definitions are based on theweighted sum of the respective utility, value orpreference functions (cf. equations (3), (5) and (7)).Therefore, we first further investigate the case of one

Figure 2. Graphical representation of the ordinal ranking of the investigated scenarios with regard to the considered criteriaaccording to simple additive ranking.

Figure 3. Graphical representation of the simple additiveweighting (SAW).



fixed criterion k. To illustrate the calculations, let usinvestigate four alternatives that have to be max-imized. The following notation is chosen to facilitatethe presentation:

x1 :¼ fk a1ð Þx2 :¼ fk a2ð Þx3 :¼ fk a3ð Þx4 :¼ fk a4ð Þ

We further (as before) use xmin= xmin, k and xmax =xmax, k for the minimum and maximum utility valuesover all the alternatives in criterion k. Within SAW, thedifferences between two alternatives’ utility values, saya1 and a2, are calculated as follows:

vk fk atð Þð Þ−vk fk at′ð Þð Þ ¼ xt−xmin

xmax−xmin−

xt′−xmin

xmax−xmin

¼ xt−xt′xmax−xmin

ð11Þ(11)

Within PROMETHEE, the calculation of the netflow is based on the preference values of the scores’differences

pk fk atð Þ−fk at′ð Þð Þ ¼ pk xt−xt′ð Þ

that can be presented in a matrix format:

a1 a2 a3 a4a1 0 pk x1−x2ð Þ pk x1−x3ð Þ pk x1−x4ð Þa2 pk x2−x1ð Þ 0 pk x2−x3ð Þ pk x2−x4ð Þa3 pk x3−x1ð Þ pk x3−x2ð Þ 0 pk x3−x4ð Þa4 pk x4−x1ð Þ pk x4−x2ð Þ pk x4−x3ð Þ 0

The net flow of each alternative with regard to eachcriterion is defined as

Φnetk atð Þ ¼ Φþ

k atð Þ−Φ−k atð Þ

where

Φþk atð Þ ¼ 1

T∑T

t′¼1;t′≠tpk vk xtð Þ−vk xt′ð Þð Þ

Φ−k atð Þ ¼ 1

T∑T

t′¼1;t′≠tpk vk xt′ð Þ−vk xtð Þð Þ

Note that these calculations can be easily doneusing the matrix above: Φþ

k atð Þ, the measure of thealternative’s strength is calculated as the sum of thepreference values in the rows for each alternative,whereas Φ−

k atð Þ, the measure for the weakness, is thesum of the preference values in the columns for eachalternative.

We now assume that x1 > x2 > x3 > x4 and exem-plarily justify for the case that t= 1 and t′= 2 that

Figure 4. Graphical representation of the PROMETHEEresults (Preference function of type 3, pk = xmax, k − xmin, k).

Table II. Comparison of the net fluxes Φnet calculated by PROMETHEE and the SAW utility values.

Scenario Φ+ Φ− Φnet Difference between previousalternatives

Utility value(SAW)

Difference between previousalternatives

3B 0.494 −0.013 0.481 0.9431 0.378 −0.065 0.313 0.169 (=0.481–0.313) 0.774 0.169 (=0.943–0.774)2E 0.371 −0.080 0.291 0.022 0.752 0.0223A 0.207 −0.186 0.021 0.269 0.483 0.2692D 0.061 −0.212 −0.151 0.173 0.310 0.1730 0.081 −0.257 −0.177 0.026 0.285 0.0262B 0.042 −0.240 −0.198 0.022 0.263 0.0222C 0.042 −0.267 −0.225 0.027 0.236 0.0272A 0.018 −0.374 −0.356 0.130 0.106 0.130

SAW, simple additive weighting; PROMETHEE, preference ranking organization method for enrichment evaluations.



Φnetk atð Þ−Φnet

k at′ð Þ ¼ vk fk atð Þð Þ−vk fk at′ð Þð Þ

as we observed in the case study. We use the type 3preference function with pk= xmax, k− xmin, k, that is,

pk dð Þ ¼0 if d<0d

xmax;k−xmin;kif d ≥0

8<:

This yields the following net flows for criterion kand the alternatives a1 and a2:

Φþ1 a1ð Þ ¼ 1

T

x1−x2xmax−xmin

þ x1−x3xmax−xmin

þ x1−x4xmax−xmin

� �Φ−

1 a1ð Þ ¼ 0

⇒Φnet a1ð Þ ¼ 14� 3x1−x2−x3−x4

xmax−xmin

Φ−1 a2ð Þ ¼ 1

T

x1−x2xmax−xmin

⇒Φnet a2ð Þ ¼ 14� 3x2−x1−x3−x4

xmax−xmin

The difference between the net flows of a1 and a2is therefore

Φnetk a1ð Þ−Φnet

k a2ð Þ¼14�3x1−x2−x3−x4

xmax−xmin−14�3x2−x1−x3−x4

xmax−xmin

¼ x1−x2xmax−xmin

¼ vk fk a1ð Þð Þ−vk fk a2ð Þð Þ:

The other cases could be considered analogously tothe example. However, in the following section, we willprovide a general proof and an extension of this result.

4. EQUIVALENCE OF SIMPLE ADDITIVEWEIGHTING AND PROMETHEE

In order to generalize the findings of the previous section,we prove the equivalence of SAW and PROMETHEE.Firstly, we consider the approaches regarding selectedparameters of the type 3 preference function. Secondly,we consider more general piecewise linear preferencefunctions.

4.1. Equivalence of SAW and PROMETHEEregarding a type 3 preference functionIn this section, we show that PROMETHEE and SAWare in fact equivalent (i.e. they yield the same order ofalternatives) if the following preference function

pk dð Þ¼0 if d≤0d

xmax;k−xmin;kif 0<d≤xmax;k−xmin;k:

8<: (12)

is considered. We first show the following observation.

Lemma 1Let pk be defined as in (12). Then, for t, t′ ∈ {1,…, T}with t≠ t′ we have

pk fk atð Þ− fk at′ð Þð Þ−pk fk at′ð Þ− fk atð Þð Þ ¼ fk atð Þ− fk at′ð Þxmax;k −xmin;k

ProofFirst, if fk atð Þ ¼ fk at′ð Þ, then both sides of the equationare zero and this case is true. We subsequentlyconsider two arbitrarily chosen t, t′ with fk atð Þ≠ fk at′ð ÞThen, for all k= 1,…,K, we have

• either fk atð Þ− fk at′ð Þ > 0 (hence fk at′ð Þ− fk atð Þ < 0)• or vice versa, that is, fk atð Þ− fk at′ð Þ < 0 (and hencefk at′ð Þ− fk atð Þ > 0).

Using (12), this means that

pk fk atð Þ−fk at′ð Þð Þ−pk fk at′ð Þ−fk atð Þð Þ

¼fk atð Þ−fk at′ð Þxmax;k−xmin; k

−0 if fk atð Þ−fk at′ð Þ > 0

0−fk at′ð Þ−fk atð Þxmax;k−xmin;k

if fk atð Þ−fk at′ð Þ < 0

;

8>>><>>>:

Hence, the result is the same in both cases.We can use this result to calculate the net flow and

derive the equivalence result as promised.

Theorem 1If the functions pk in PROMETHEE are definedaccording to (12), the order of at with regard to v(at) isthe same as the order with regard to Φnet(at).

ProofIn order to show that the order obtained is the same,we need to verify that

Φnet atð Þ−Φnet at′ð Þ > 0⇔v atð Þ−v at′ð Þ > 0:

Here, we prove the stronger statement that

Φnet atð Þ−Φnet at′ð Þ ¼ v atð Þ−v at′ð Þ:



To this end, we first calculate

Φnet atð Þ¼Φt atð Þ−Φ− atð Þ

¼ 1T

∑T

t′¼1;t′≠t

∑K

k¼1wk pk fk atð Þ−fk at′ð Þð Þ−pk fk at′ð Þ−fk atð Þð Þð Þ

¼ 1T

∑T

t′¼1;t′≠t∑K

k¼1wk

fk atð Þ−fk at′ð Þxmax;k−xmin;k

see Lemma 1

¼ 1T

∑T

t′¼1;t′≠t∑K

k¼1wk

fk atð Þ−xmin;k

xmax;k−xmin;k−fk at′ð Þ−xmin;k

xmax;k−xmin;k

� �

¼ 1T

∑T

t′¼1;t′≠t∑K

k¼1wk vk fk atð Þð Þ−vk fk at′ð Þð Þð Þ

¼ 1T

∑T

t′¼1;t′≠tv atð Þ−v at′ð Þ see 5ð Þ

¼ 1T

T−1ð Þv atð Þ− ∑T

t′¼1v at′ð Þ

!þ v atð Þ

!

¼ 1T

Tv atð Þ−∑T

t′¼1v at′ð Þ

!

¼ v atð Þ−︸

1T∑T

t′¼1v at′ð Þ

constant; independent of t

We conclude that

Φnet atð Þ−Φnet at′ð Þ ¼ v atð Þ− 1T∑T

s¼1v asð Þ

− v at′ð Þ− 1T∑T

s¼1v asð Þ

� �¼ v atð Þ−v at′ð Þ:

4.2. Extension to other preference functionsWe now consider an arbitrary piecewise linear functionpk : IR→ [0, 1] for each criterion k as depicted inFigure 5. Such functions are piecewise linear: Giventwo parameters p,q with p>q, the functions remainsconstant equal to zero for all values smaller than q, thenincrease linearly between the points (q, 0) and (p, 1), andfinally stay equal to one for all values greater than p.The equation of such a function is given below:

pk dð Þ ¼0 if d≤qk

1pk−qk

d−qkð Þ if qk < d < pk

1 if d≥pk

8><>:

Note that we do not require q≥ 0. If q = 0, wecount each pair of two alternatives at and at′ withdifferent utility values fk atð Þ≠ fk at′ð Þ exactly once,because exactly one of the two values

dtt′ ¼ fk atð Þ−fk at′ð Þ and dt′t ¼ fk at′ð Þ−fk atð Þ

is greater than zero. If q > 0, both values dtt′ and dt′tcan be less than q; hence, the contribution of this pairis not counted at all when the net flow is determined.In contrast to this, for q < 0, both values dtt′ > q anddt′t > q , that is, the pair, may contribute to the netflow of both alternatives, t and t′.

The case considered in Section 4.1 is a special onein which q= 0 and p is the maximal differencebetween the preference values (Case C1). For thiscase, we have shown that PROMETHEE and SAWare equivalent. The same result can also be obtainedwith regard to cases C2 and C3:

C1 Either q = 0 and p is the maximal differencebetween the preference values, or

C2 q is the minimal difference between the pre-ference values and p is the maximal differencebetween the preference values, or

C3 q is the minimal difference between the pre-ference values and p = 0.

Note that the proof of case C3 is analogous to theproof presented for C1 whereas the proof for C2 canbe found in the Appendix. Research is ongoing toidentify other classes of preference functions that canbe used to obtain other approaches’ results.

5. SUMMARY

Multi‐criteria decision aiding has been one of thefastest‐growing areas of operations research during thepast decades. Although the numerous approaches’distinctive features are the type of requested informa-tion that the DM and the weight elicitation procedures(e.g. PROMETHEE prescribes nomeans for the settingof weights) give, the formal aggregation of the obtainednumerical values reveals similarities. Because manysoftware packages offer various types of MCDAmodules, it is quite likely that a decision‐maker mayalso apply various methods and wonder why and whenshe or he yields exactly the same rank ordering.

q p

1

Figure 5. Preference function pk.



In this paper, we present the results of a comparisonof the rank orderings obtained in a case study in whichboth simple additive weighting—as a representative ofmulti attribute value theory approaches—and PRO-METHEE—as an Outranking approach—have beenapplied. In order to generalize the findings of theillustrative case study, we mathematically prove theequivalence of simple additive weighting and PRO-METHEE, firstly, for selected parameters of the type 3preference function and, secondly, for more generalpiecewise linear preference functions.

The findings of this paper give rise to the suppositionthat the classical MADM approach and also of theSAW, as an often applied simplifying heuristic, can bereproduced by PROMETHEE, if the preference func-tions are modelled accordingly. It should be noted that asuitable MCDAmethod has to be chosen carefully for aspecific decision problem (Guitouni and Martel, 1998).A sufficient awareness of the underlying philosophiesand theories, an understanding of the methods’ practicaldetails, and insight into practice are indispensable. Theproblem structuring’s impact on the decision processspecifically requires special attention (Belton andStewart, 2002). Experimental studies in psychologyand behavioural aspects of decision making reveal thathuman thinking should not only be modelled by logicalrules and calculations (Guitouni and Martel, 1998).Especially the findings of psychological research aboutbiases and heuristics have to be taken into account(Kahneman and Tversky, 1979; Kahneman et al., 1982;Hogarth, 1987; Hastie and Dawes, 2009) by the de-velopment of improved interactive software (Dyer et al.,1992) and more elaborated sensitivity analyses, forexample, examining the influence of the curvature of thevalue function on the obtained rank ordering (Bertsch,2008). Owing to the practical relevance of multi‐criteriadecision support, and given the challenges of modellinghuman thinking, clarity regarding the actual mathema-tical equivalence of various multi‐criteria aggregationprocedures contributes to transparent decision making.

APPENDIXIn the case of C2, we require that the maximumdifference xmax, k− xmin, k is mapped to 1 and theminimaldifference xmin, k− xmax, k is mapped to 0, and that thefunctions pk are linear between these points, that is, wehave

pk xmax;k−xmin;k� � ¼ 1

pk xmin;k−xmax;k� � ¼ 0:

and combining these two points by a linear functionyields

pk dð Þ ¼ 1

xmax;k−xmin;k� �

− xmin;k−xmax;k� � d− xmin;k−xmax;k

� ��

¼ d

2 xmax;k−xmin;k� �þ 1

2

(13)

Lemma 2For pk as defined as in (13), we have the followingrelation between pk and the functions vk of SAW:

pk fk atð Þ−fk at′ð Þð Þ ¼ 12vk fk atð Þð Þ−vk fk at′ð Þð Þ þ 1ð Þ

Proof

pk fk atð Þ−fk at′ð Þð Þ ¼ 12þ fk atð Þ−fk at′ð Þ2 xmax;k−xmin;k� � see 13ð Þ

¼ xmax;k−xmin;k þ fk atð Þ−fk at′ð Þ2 xmax;k−xmin;k� �

¼ xmax;k−xmin;k þ fk atð Þ−xmin;k þ xmin;k−fk at′ð Þ2 xmax;k−xmin;k� �

¼ 12þ 12vk fk atð Þð Þ− 1

2vk fk at′ð Þð Þ due to 4ð Þ:

Using these results, we can derive the followingformulas for the outgoing, the incoming and the netflow.

Lemma 3For pk as defined as in (13), we have

Φþ atð Þ ¼ 12

v atð Þ þ 1−1T

∑T

t′¼1;t′≠tv at′ð Þ

!

Φ− atð Þ ¼ 12

−v atð Þ þ 1þ 1T

∑T


!

Φnet atð Þ ¼ v atð Þ− 1T

∑T


Proof

Φþ atð Þ ¼ 1T

∑T

t′¼1; t′≠t∑K

k¼1wkpk fk atð Þ−fk at′ð Þð Þ

¼ 1T

∑T

t′¼1; t′≠t∑K

k¼1wk

12

vk fk atð Þð Þ−vk fk at′ð Þð Þ þ 1ð Þ see Lemma 2

¼ 1T

∑T

t′¼1; t′≠t

12

v atð Þ−v at′ð Þ þ 1ð Þ see 5ð Þ and use ∑Kk¼1wk ¼ 1

¼ 12

v atð Þ þ 1ð Þ− 12T

∑T

t′¼1; t′≠tv at′ð Þ



Analogously, we obtain the second part of thelemma. Subtracting the terms of the first and thesecond statement gives the third statement.

We finally can state the main result concerning pkas defined in (13).

Theorem 2If the functions pk in PROMETHEE are definedaccording to (13), we have: The order of at with regardto v(at) is the same as the order with regard to Φ+(at)and also the same as the order with regard to Φnet(at).

ProofWe use Lemma 3 and calculate

Φþ atð Þ ¼ 12

v atð Þ þ 1−1T

∑T


!

¼ 12

v atð Þ þ 1−1T∑T

t′¼1v at′ð Þ þ 1

Tv atð Þ

!

¼ 12

T þ 1T

v atð Þ þ 1−1T∑T

t′¼1v at′ð Þ

!

¼︸

T þ 12T

> 0

v atð Þ þ︸

12−

12 Tð Þ ∑

T

t′¼1v at′ð Þ:


Hence, the order of at with regard to Φ+ is thesame as the order of at with regard to v. Similarly, weobtain

Φnet atð Þ ¼ v atð Þ−

︸

1T þ 1

∑T

t′¼1v at′ð Þ

!;


hence all three orderings are the same.

REFERENCES

Ackoff RL. 1981. The art and science of mess management.Interfaces 110(1): 20 – 26.

Albadvi A, Chaharsooghi S, Esfahanipour A. 2007. Decisionmaking in stock trading: an application of PROMETHEE.European Journal of Operational Research 177(2):673– 683.

Anand G, Kodali R. 2008. Selection of lean manufacturingsystems using the PROMETHEE. Journal of Modellingin Management 3(1): 40 –70.

Araz A, Ozkarahan I. 2007. Supplier evaluation andmanagement system for strategic sourcing based on a

new multicriteria sorting procedure. International Jour-nal of Production Economics 106(2): 0585– 0606.

Belton V, Stewart T. 2002. Multiple Criteria DecisionAnalysis ‐ An Integrated Approach. Kluwer AcademicPress: Boston.

Bertsch V. 2008. Uncertainty Handling in Multi‐AttributeDecision Support for Industrial Risk Management.Universitätsverlag Karlsruhe.

Brans J‐P, Mareschal B. 1994. The PROMCALC and GAIAdecision support system for multi‐criteria decision aid.Decision Support System 12: 297–310.

Brans J‐P, Mareschal B. 2005. PROMETHEE methods. InMultiple Criteria Decision Analysis ‐ State of the ArtSurveys, Figueira J, Greco S, Ehrgott M (eds). Springer:New York; 163 –195.

Brans J‐P, Vincke P, Mareschal B. 1986. How to select andhow to rank projects: the PROMETHEE method. EuropeanJournal of Operational Research 24: 228–238.

Dagdeviren M. 2008. Decision making in equipmentselection: an integrated approach with AHP andPROMETHEE. Journal of Intelligent Manufacturing 19(4):397–406.

Diakoulaki D, Antunes CH, Martins AG. 2005. MCDA andenergy planning. In Multiple Criteria Decision Analysis ‐State of the Art Surveys, Figueira J, Greco S, Ehrgott M(eds). Springer: New York; 859– 897.

Dyer J. 2005. Maut ‐ multi attribute utility theory. InMultiple Criteria Decision Analysis: State of the ArtSurveys, Figueira J, Greco S, Ehrgott M (eds). Springer:New York; 265–295.

Dyer JS, Fishburn PC, Steuer RE, Wallenius J, Zionts S.1992. Multiple criteria decision making, multiattributeutility theory: the next ten years. Management Science38(5): 645 – 654.

Figueira J, Greco S, Ehrgott M. 2005. Multiple CriteriaDecision Analysis ‐ State of the Art Surveys. Springer:New York.

French S. 1986. Decision Theory ‐ An Introduction to theMathematics of Rationality. Ellis HorwoodLtd: Chichester.

Geldermann J, Rentz O. 2001. Integrated technique assess-ment with imprecise information as a support for theidentification of Best Available Techniques (BAT). ORSpectrum 23: 137–157.

Geldermann J, Rentz O. 2004. Decision support throughmass and energy flow management in the sector ofvehicle refinishing. Journal of Industrial Ecology 80(4):173 –187.

Geldermann J, Rentz O. 2005. Multi‐criteria analysis for theassessment of environmentally relevant installations. Journalof Industrial Ecology 90(3): 127–142.

Geldermann J, Spengler T, Rentz O. 2000. Fuzzy outrankingfor environmental assessment, case study: iron and steelmaking industry. Fuzzy Sets and Systems 115: 45–65.

Greening L, Bernow S. 2004. Design of coordinated energyand environmental policies. Use of multicriteria decisionmaking. Energy Policy 320(6): 721–735.

Guitouni A, Martel J‐M. 1998. Tentative guidelinesto help choosing an appropriate MCDA method.



European Journal of Operational Research 1090(2):501–521.

Haralambopoulos D, Polatidis H. 2003. Renewable energyprojects: structuring a multi‐criteria group decision‐makingframework. Renewable Energy 28(6): 961–973.

Hastie R, Dawes RM. 2009. Rational Choice in anUncertain World: The Psychology of Judgment andDecision Making (2nd edn). Sage Publications: London.

Hämäläinen RP. 2004. Reversing the perspective on theapplications of decision analysis.Decision Analysis 10(1).

Hobbs BF, Meier P. 2000. Energy Decisions and theEnvironment. A Guide to the Use of Multicriteria Methods.Kluwer Academic Publishers: Boston/Dordrecht/London.

Hogarth RM. 1987. Judgement and Choice (2nd edn).Wiley: New York.

Kahneman D, Tversky A. 1979. Prospect theory: an analysisof decisions under risk. Econometrica 47: 262–291.

Kahneman D, Slovic P, Tversky A. 1982. Judgement UnderUncertainty ‐ Heuristics and Biases. Cambridge Uni-versity Press: Cambridge.

Løken E. 2007. Use of multicriteria decision analysismethods for energy planning problems. Renewable andSustainable Energy Reviews 110(7): 1584 –1595.

Lootsma F. 1996. Comments on B. Roy and D. Vander-pooten: the European School of MCDA: emergence,basic features and current work. Journal of Multi CriteriaDecision Analysis 5: 37–38.

Mareschal B. 1998. Weight stability intervals in multi-criteria decision aid. European Journal of OperationalResearch 33: 54– 64.

Oberschmidt J, Geldermann J, Ludwig J, Schmehl M. 2010.Modified PROMETHEE approach for assessing energytechnologies. International Journal of Energy SectorManagement 4(2): 183–212.

Pohekar S, Ramachandran M. 2004. Application of multi‐criteria decision making to sustainable energy planning ‐ a

review. Renewable and Sustainable Energy Reviews80(4): 365–381.

Ren H, Gao W, Zhou W, Nakagami K. 2009. Multi‐criteriaevaluation for the optimal adoption of distributed residentialenergy systems in Japan. Energy Policy 37(12): 5484–5493.

Rogers M, Bruen M. 1998. Choosing realistic values ofindifference, preference and veto thresholds for usewith environmental criteria within ELECTRE. EuropeanJournal of Operational Research 107(3): 542–551.

Roy B. 1996. Multicriteria Methodology for DecisionAiding. Kluwer Academic Publishers: Dordrecht.

Roy B, Bouyssou D. 1993. Aide Multicritère á la Décision.Economica: Paris.

Roy B, Vanderpooten D. 1997. An overview on the Europeanschool of MCDA: Emergence, basic features and currentworks.European Journal ofOperationalResearch99: 26–27.

Saaty T, Vargas L. 2006. Decision Making with theAnalytic Network Process: Economic, Political, Socialand Technological Applications with Benefits, Opportu-nities, Costs and Risks. Springer: New York.

Siskos Y. 2005. UTA methods. In Multiple Criteria DecisionAnalysis ‐ State of the Art Surveys, Figueira J, Greco S,Ehrgott M (eds). Springer: New York; 297–343.

Stewart TJ. 1992. A critical survey on the status of multiplecriteria decision making theory and practice. OMEGA ‐International Journal of Management Science 200(5/6):569 –586.

Wang J, Jing Y, Zhang C, Zhao J. 2009. Review on multi‐criteria decision analysis aid in sustainable energydecision‐making. Renewable and Sustainable EnergyReviews 130(9): 2263–2278.

Xu Z. 2005. Deviation measures of linguistic preferencerelations in group decision making. Omega 33(3): 249–254.

Zhou P, Ang B, Poh K. 2004. Decision analysis in energyand environmental modeling: an update. Energy 310(14):2268–2286.



Institut für Numerische und Angewandte MathematikUniversität GöttingenLotzestr. 16-18D - 37083 Göttingen

Telefon: 0551/394512Telefax: 0551/393944

Email: [email protected] URL: http://www.num.math.uni-goettingen.de

Verzeichnis der erschienenen Preprints 2011

Number Authors Title

2011-1 M. Braack, G. Lube, L. Röhe Divergence preserving interpolation on anisotro-pic quadrilateral meshes

2011-2 M.-C. Körner, H. Martini, A.Schöbel

Minsum hyperspheres in normed spaces

2011-3 R. Bauer, A. Schöbel Rules of Thumb � Practical Online-Strategies forDelay Management

2011-4 S. Cicerone, G. Di Stefano, M.Schachtebeck, A. Schöbel

Multi-Stage Recovery Robustness for Optimiza-tion Problems: a new Concept for Planning un-der Disturbances

2011-5 E. Carrizosa, M. Goerigk, M.Körner, A. Schöbel

Recovery to feasibility in robust optimization

2011-6 M. Goerigk, M. Knoth, M.Müller-Hannemann, A. Schö-bel, M. Schmidt

The Price of Robustness in TimetableInformation

2011-7 L. Nannen, T. Hohage, A.Schädle, J. Schöberl

High order Curl-conforming Hardy space in�niteelements for exterior Maxwell problems

2011-8 D. Mirzaei, R. Schaback, M.Dehghan

On Generalized Moving Least Squares and Dif-fuse Derivatives


2011-9 M. Pazouki, R. Schaback Bases of Kernel-Based Spaces

2011-10 D. Mirzaei, R. Schaback Direct Meshless Local Petrov-Galerkin(DMLPG) Method: A Generalized MLSApproximation

2011-11 T. Hohage, F. Werner Iteratively regularized Newton methods with ge-neral data mis�t functionals and applications toPoisson data

2011-12 D. Rosca, G. Plonka Uniform spherical grids via equal area projectionfrom the cube to the sphere

2011-13 S. Hein, W. Koch, L. Nannen Trapped modes and Fano resonances in two-dimensional acoustical duct-cavity systems

2011-14 T. Peter, D. Rosca, G. Plonka-Hoch

Representation of sparse Legendre expansions

2011-15 F. Dunker, J.-P. Florens, T.Hohage, J. Johannes, E. Mam-men

Iterative Estimation of Solutions to Noisy Non-linear Operator Equations in Nonparametric In-strumental Regression

2011-16 R. Stück, M. Burger, T. Hohage The Iteratively Regularized Gauÿ-Newton Me-thod with Convex Constraints and Applicationsin 4Pi-Microscopy

2011-17 J. Brimberg, H. Juel, M. Kör-ner, A. Schöbel

Approximating a point set by a circle of an ar-bitrary norm

2011-18 J. Geldermann, A. Schöbel On the Similarities of Some Multi-Criteria Deci-sion Analysis Methods


Documents

Institut für Numerische und Angewandte Mathematiknum.math.uni-goettingen.de/preprints/files/2011-18.pdf · (MADM), namely simple additive rating (SAR), simple additive weighting