fuzzy outranking for environmental assessment. case study: iron and steel making industry

Fuzzy Sets and Systems 115 (2000) 45–65www.elsevier.com/locate/fss

Fuzzy outranking for environmental assessment.Case study: iron and steel making industry

Jutta Geldermann a ;∗, Thomas Spengler b, Otto Rentz aa French–German Institute for Environmental Research (DFIU), University of Karlsruhe (TH), Hertzstra�e 16,

D-76187 Karlsruhe, Germanyb Department for Production Management, Technical University of Braunschweig, Katharinenstr. 3,

D-38106 Braunschweig, Germany

Received November 1998

Abstract

Recent years have seen a world-wide change in the environmental policy towards integrated pollution prevention, takinginto account all environmental media. Consequently, the environmental assessment of production techniques has to con-sider multiple criteria which cannot be aggregated to one single index. The concepts of Multi Criteria Decision Making(MCDM) seem a suitable means in order to implement Life Cycle Assessment (LCA) into integrated decision processes.The outranking methods as a special subgroup of MCDM methods are particularly suitable for integral decision makingthrough the notion of weak preference and incomparability, which better represent the real decision situation, as demon-strated in this paper. Especially the outranking method PROMETHEE brings together exibility and simplicity for theuser and is therefore chosen for the enhancement towards the evaluation of fuzzy data on preferences, scores and weights.c© 2000 Elsevier Science B.V. All rights reserved.

Keywords: Fuzzy numbers; Multicriteria analysis; Decision making; Outranking; Environmental assessment

1. Introduction

Recent years have seen a world-wide change in the environmental policy towards integrated pollutionprevention, taking into account all environmental media (air, water, land) and energy consumption. In theEuropean Union, this tendency is con�rmed in the Integrated Pollution Prevention and Control Directive(IPPC-Directive 96=61=EC), which obliges the Member States to take an integrated approach to the protectionof the environment in the licensing of environmentally relevant installations. The ‘best available techniques’BAT play an essential role in the material transformation of the IPPC-Directive, which will serve inter alia

∗ Corresponding author. Tel.: + 49-721-608-4583; fax: + 49-721-758909.E-mail address: [email protected] (J. Geldermann).

0165-0114/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(99)00021 -4

46 J. Geldermann et al. / Fuzzy Sets and Systems 115 (2000) 45–65

as a basis for the determination of reference values for emission limits and for the granting of permits forinstallations [24]. In environmental policy, the decision makers consequently have to de�ne the BAT fromall existing and emerging techniques as requested by the IPPC-Directive. The industry also has to choose themost appropriate technique for the particular installations.At the present time, no speci�c method for supporting these important decisions is being applied. Both

political and industrial decisions have to simultaneously take into account economic, technical, and ecolog-ical criteria. Especially the assessment of ecological criteria, however, is at an early stage of development.Therefore, characteristic �gures are not yet clearly de�ned. Moreover, the evaluation of cross-media aspectsremains a di�cult task, which considers transmedial problem shifting from one environmental medium toanother (like from air into water). An appropriate framework for ecological decisions is set by the Life CycleAssessment (LCA), which comprises goal and scope de�nition, inventory analysis, impact assessment andinterpretation [16]. Tools and methods for the interpretation of the impact assessment, however, have not beenyet developed.The collection of accurate data on the production techniques and the industrial installations poses further

di�culties: Due to varying measurements and the di�erences in the input parameters, comparable exact dataare rarely available. Therefore, a description of the techniques with the help of fuzzy numbers would seemto be more realistic than with crisp numbers. Only a exible comprehensive assessment approach can fosterthe discussion of the political and industrial decision makers on the most relevant aspects.As Zadeh [35] puts it, human goals matter in the case of decision processes, and therefore a wide gap exists

between theory and practice in decision analysis. The concept of fuzzy sets is a way to deal systematically withunsharp �gures, which better represent the reality. Therefore, in this paper, an approach for a fuzzy outrankingdecision support is presented. Based on the analysis of the decision situation in environmental assessments,a suitable concept for MCDM is selected. The outranking methods as a special subgroup of MCDM meetthe particular requirements of these soft decisions through the notion of weak preference and incomparability,which better represent the real decision situation [12,30]. Especially the outranking method PROMETHEE[3] brings together exibility and simplicity for the user and is therefore chosen for the evaluation of fuzzydata.Most of the outranking methods are per se based on a fuzzy notion since the comparisons do not hold with

the two-valued logic (true=false) [5]. This kind of fuzziness refers mainly to the objective function, which isnot always su�cient for soft decision analysis. More bene�ts can be gained by evaluating also fuzzy scoreson the criteria. Therefore, the integration of fuzzy algebra into the PROMETHEE algorithm is suggested inthis paper. Since it is particularly di�cult to realistically model the ‘true’ shape of the membership function,the so-called L-R-type fuzzy interval or trapezoidal fuzzy numbers are used [27]. The advantage can be seenin their good practicability and ease of understanding. The proposed approach is demonstrated with a casestudy presenting an environmental assessment of sinter techniques in the iron and steel making industry. Theexperiences gained in the practical application of the decision model are being discussed, before conclusionsare drawn.

2. Life cycle assessment for ecological assessment

The Life Cycle Assessment (LCA) has been developed as an instrument for the environmental assessmentof products [14,33]. Recently, the di�erent approaches for LCA have been harmonised to some extent in theISO 14040 document, which explains the four steps of LCA:1. Determination of objectives and scope

• Determination of the goal of the LCA, of the functional unit, of system boundaries and the requireddata quality.

J. Geldermann et al. / Fuzzy Sets and Systems 115 (2000) 45–65 47

2. Inventory analysis• Collection of data and calculation method for the quanti�cation of relevant inputs and outputs of theconsidered system as a basis for an Impact Assessment (consumption of resources as well as emissionsinto air, water and land).

• The standardisation of the data allows their comparison by means of conversion into the same quantitywith the functional unit as a uniform reference quantity.

3. Impact assessment• Assigning of inventory data to impact categories (classi�cation).• Modelling of the inventory data within impact categories (characterisation).• Possible aggregation of the results in very speci�c cases and only when meaningful (valuation).

4. Interpretation• Weighting by speci�c contribution as an indication for quantitative relevance of the concerned sub-stances.

• Weighting by environmental importance with respect to the signi�cance of the impact on the environmentas a criterion for judgement and prioritising.

• Verbal-argumentative �nal valuation of impact categories without using the results for further calcula-tions.

The methods and approaches to LCA, which have been developed so far, di�er mainly in the step of ImpactAssessment. The best developed methods are based on impact categories [14,33]. Currently, the followingimpact categories are mainly discussed:• Consumption of resources (including energy) [14,24,33]• Global warming [14,33]• Ozone depletion (stratosphere) [14,33]• Humantoxicity [14,24,33]• Ecotoxicity [14,24,33]• Acidi�cation aquatic and terrestrial [14,33]• Nutri�cation of water [14,33]• Photochemical oxidant formation [14,33]• Consumption of land [14,33]• Pollution (noise, odour) [14,33]• Health hazards at place of work (‘industrial safety’) [14]• Waste heat and radiation [14,33]• Hazardous waste [24]• Negative e�ects on environmental beauty and loss of biotopes (‘nature conservation’) [14]• Biodiversity [14]• Protection of the marine environment [24]Although these impact categories are still not complete and disputable in several aspects, this approach

has the best scienti�c research background for pointing out the relation between emissions and their potentialecological impacts. Therefore, at least hints for the environmentally most important aspects of production andconsequently for ecological improvement might be expected [24].In a classi�cation, the inputs and outputs identi�ed in the Inventory Analysis are assigned to the re-

spective impact categories. The ‘impact potential’ is then modelled by multiplying the amounts ofconsumption or emission with the respective impact assessment factors. It must be noted, that the lin-ear impact assessment factors are not designed for a realistic modelling of complex interdependencies andtoxicological issues, but only for a rough estimation of the potential environmental damage, without fur-ther consideration of local impacts or dose response. Due to the complexity of the interdependencies ofthe ecological e�ects caused by the di�erent substances, any environmental assessment can allow only asimpli�ed representation of the current situation by stressing individual problems. Due to a lack


of scienti�c work on cross-media aspects there is, however, no scienti�c basis available for aggregatingthe data gained from ecological assessments, so that a clear ranking of examined techniques based on trade-o� scores cannot be calculated. Therefore, a complete interconnection of the single problems will remainan unrealistic aim even in the future. The resulting ecological pro�le as a result of the impact assess-ment is therefore marked by the calculated impact potentials which are measured in di�erent units of mea-surements (e.g. kg CO2-equivalent, kg FCKW11-equivalent, m3 air), which is a typical decision situationfor MCDM.In order to di�erentiate the signi�cance of the impact potentials and of the decisive mass and energy ows

of the considered techniques, only an approximate weighting is possible. The weighting should be derivedfrom the state-of-the-art in science, but it will remain the most subjective part of any decision. Because ofthe given uncertainties, the decision situation can be characterised as a task for soft decision analysis. Inorder to meet the requirements set by the IPPC-Directive for policy and industry, a structured procedure forenvironmental assessment is needed [24]. Therefore, in the next section, a suitable approach for multicriteriadecision support is selected.

3. Selection of a suitable MADM method

The key philosophical departure point for Multicriteria Decision Making (MCDM) as a formal approachfor problem solving, as distinct from the classical approaches of Operations Research and ManagementScience, lies in the representation of several con icting criteria [31]. MCDM has been one of the fastestgrowing areas of Operational Research during the last two decades, as it is often realised that many con-crete problems contain several criteria [34]. The theory of MCDM can be divided into Multi ObjectiveDecision Making (MODM) and Multi Attribute Decision Making (MADM) [36]. MODM analyses a sub-set of a continuous vector space, usually restricted by constraints, by locating all e�cient solutions, beforedetermining the optimum dependent on the user’s preferences. Therefore, MODM o�ers feasible methodsfor operational planning, e.g. goal programming. For the comparison of several particular alternatives or instrategic planning, when a certain number of recycling techniques are to be pre-selected for further inves-tigation, the approaches of MADM should be used, because MADM investigates a �nite set of alternatives[13]. Basically, MADM comprises two steps [36]: (1) the aggregation of the judgements with regard toeach criterion and each alternative, and (2) the ranking of the alternatives according to the aggregationrules.To introduce the basic notation for MADM, consider the set A of T alternatives that has to be ranked, and

K criteria that have to be optimised,

A := {a1; : : : ; aT}: set of discrete alternatives or techniques at (t=1; : : : ; T );

F := {f1; : : : ; fk}: set of criteria relevant for the decision fk (k =1; : : : ; K);

then the resulting multiple criteria decision problem can be concisely expressed in a matrix format. The goalachievement matrix or decision matrix D := (xtk)t=1; :::; T; k=1;:::; K is a (T×K) matrix whose elements xtk =fk(at)indicate the evaluation or value of alternative at , with respect to aspect or criterion fk :

D =

x11 · · · x1K... xtk ...xT1 · · · xTK

:=

f1(a1) · · · fK (a1)... fk(at) ...

f1(at) · · · fK (aT )

: (1)


In the literature on decision theory, the MADM methods are classi�ed according to the given information[15,37]. In the environmental assessment, information on the criteria is mainly cardinal, but substitution ratescannot be given.From the methods suitable for cardinal information, the Analytical Hierarchy Process (AHP) is one of

the most frequently applied MADM methods [37], and a comfortable software package (Expert Choice) iscommercially available. AHP provides a logical, easy to use framework, where the decision maker has to assessthe weights of each criterion using a nominal nine-point scale, which is also available for fuzzy weights [36].In the context of environmental assessment, however, the nine-point-scale is not as suitable as for �nding acompromise between di�erent personal points of view. From an analytical point of view, the interpretation ofthe values on the nominal scale with verbal expression as a ratio scale using numbers is also doubted [31]. Thebasic principle of preference measurement is the establishment of a value function based on a simple additionof score representing goal achievement according to each criterion, multiplied by the particular weights.Therefore the concept of trade-o� between the scores on di�erent criteria is central to the interpretation ofthe value function, as it also underlies the Multi Attribute Value=Utility Theory (MAVT=MAUT) [31]. Thismeans that complete ‘compensation’ between attributes is possible, so that a su�ciently large gain in a lesserattribute will eventually compensate for a small loss in a more important attribute, no matter how importantone attribute is [1,31]. In environmental assessment, for example good results concerning releases into thewater might counterbalance worse emissions into the air, but this mathematical representation does not matchthe real e�ects caused by the emissions in the environment.In order to overcome the assumption of complete compensation and of the existence of a ‘true’ ranking of the

alternatives which only needs to be discovered, the outranking methods have been developed. Outranking rathertakes into account, that preferences are not constant in time, are not unambiguous, and are not independentof the process of analysis [28]. Therefore “outranking” could be thus de�ned: alternative at outranks at′ , ifthere is a “su�ciently strong argument in favour of the assertion that at is at least as good as at′ from thedecision maker’s point of view” [3]. Accordingly, the outranking-relation is the result of pairwise comparisonsbetween the alternatives with regard to each criterion [31,37].‘Classical’ decision making is based on strict preference (atPat′), i.e. alternative at is de�nitively preferred

to at′ , and indi�erence (atIat′), i.e. at is as good as at′ . But in reality, situations may also exist, in whichneither of a pair of alternatives outranks the other. This holds especially true for decision situations withnumerous criteria, as in environmental assessment. If the decision maker consequently cannot declare at betterthan at′ or vice versa, the outranking methods allow explicitly for incomparability (atRat′). Moreover, theconcept of weak preference (atQat′) is used, if for example the decision maker declares alternative at to bejust slightly better than at′ .The algorithms also take account of partial compensation: a good score for alternative a1 on criterion

fx might only o�set a poor score for criterion fy, when the di�erence between the scores exceeds a certainthreshold. Furthermore veto functions can be used, if it is essential for the decision that a particular score doesnot exceed or fall below a certain threshold. The result of the algorithm is a graph showing the partial pre-order of the alternatives, represented as nodes, with the outranking relations depicted as arcs. The outrankinggraph helps to gain insight into the decision maker’s preference structure and to focus attention on criticalissues, which might be regarded as the aim of soft decision analysis.Especially the principles of weak preference and incomparability are valuable in environmental decision

support, because they better represent the real situation. The ELECTRE approaches [28] are a popular mannerof rendering the outranking concept operational, but this method has a drawback through its complexity, stem-ming from the nuances in the comparisons: Through the underlying assumptions for the algorithm, the methodis rather di�cult to explain to decision makers in industry or politics, especially since the introduced thresholdsdo not have a realistic meaning [3,19]. To overcome these obstacles, the outranking method PROMETHEEhas been developed, bringing together exibility and simplicity for the user [3], which is being presented inthe next section.


Fig. 1. Generalised criteria for the use in PROMETHEE [3].

4. Algorithm of PROMETHEE

The outranking method PROMETHEE [3] o�ers a means of multicriteria analysis characterised by simplicityand clearness to the decision maker. All the used parameters have a real signi�cation, so that the decisionmaker can immediately interpret them. For the mathematical modelling of the preferences, the generalisedcriteria can be de�ned by the decision maker speci�cally for each considered criterion fk :

pk(fk(at)− fk(at′))=pk(d)∈ [0; 1]:

The degree of preference of an alternative at in comparison to at′ can vary from pk(d)= 0 which meansindi�erence over a zone of weak preference to pk(d)= 1 depicting strict preference. This concept of prefer-ences can be well explained to the political and industrial decision makers through the notion of the imaginaryquantity ‘preferability with respect to every considered criterion’. The pairwise comparisons can be illustratedby asking: ‘To what percentage is the statement true; that technique at is better than technique at′ as regardsthe considered criterion fk?’The six types of generalised criteria presented in Fig. 1 have been suggested which might cover most of

the decision problems [3], but the decision maker may also model his preferences with the help of speci�callyshaped preference functions.The algorithm for PROMETHEE can be outlined as follows [3]:

(1) Specify for each criterion fk a generalised preference function pk(d) (see Fig. 1).(2) De�ne a vector containing the weights, which are a measure for the relative importance of each criterion,

wT = [w1; : : : ; wk ]. If all the criteria are of the same importance in the opinion of the decision maker, allweights can be taken as being equal. The normalisation of the weights

∑Kk=1wk =1 is not necessarily

required.


Fig. 2. Outranking-graph. Fig. 3. Partial preorder as an example for a graphical represen-tation of the result on an outranking algorithm.

(3) De�ne for all the alternatives at; at′ ∈A the Outranking-Relation �:

� :

A×A→ [0; 1]

�(at ; at′)=K∑k=1

wk ·pk(fk(at)− fk(at′)):

The preference index �(at; at′) is a measure for the intensity of preference of the decision maker for analternative at in comparison with an alternative at′ for the simultaneous consideration of all criteria. Itis basically a weighted average of the preference functions pk(d) and can be represented as a valuedoutranking graph (cf. Fig. 2).

(4) As a measure for the strength of the alternatives at ∈A, the leaving ow is calculated:

�+(at)=1

T − 1 ·n∑t′=1t′ 6=t

�(at ; at′):

The leaving ow is the sum of the values of the arcs which leave node at and therefore yields a measureof the ‘outranking character’ of at .

(5) As a measure for the weakness of the alternatives at ∈A, the entering ow is calculated, measuring the‘outranked character’ of at (analogously to the leaving ow):

�−(at)=1

T − 1 ·n∑t′=1t′ 6=t

�(at′ ; at):

(6) A graphical evaluation of the outranking relation is derived: Basically, the higher the leaving ow and thelower the entering ow, the better the action. This result is graphically represented by a partial preorder(PROMETHEE I) or a complete order (PROMETHEE II).

The PROMETHEE I partial preorder is determined by a comparison of the leaving and entering ows bya set intersection in a manner that also allows the representation of weak preferences and incomparabilitiesof alternatives. In the valued outranking graph, an arc leads from alternative at to at′ , if at is preferred toat′ . In Fig. 3, alternative a1 dominates all other alternatives. If no arc exists between two alternatives, thenthese are incomparable with each another (e.g. a2 and a6). In the case of indi�erence, the arc is not pointed(a2 and a4).


In case a complete preorder is requested, PROMETHEE II yields the so-called net- ows as the di�erenceof the leaving and entering ows avoiding any incomparabilities:

�net(at)=�+(at)− �−(at): (2)

However, the partial preorder derived by PROMETHEE I may contain more realistic information through theindication of incomparabilities. This is especially important for decision making in the context of environmentalissues. With the help of the graphical representation, clusters of alternatives can be derived, so that the groupof best techniques can be identi�ed. The documentation of the incomparabilities is helpful for the identi�cationof any further information demand. Therefore, for environmental assessment, the partial preorder as derivedwith PROMETHEE I should be favoured [24,30].

5. Fuzzy outranking

Because of the underlying structure for multicriteria decision problems, the consideration of fuzzy logicwithin MCDM seems to be almost self-evident. Several surveys compile the fuzzy approaches to MCDM,cf. [4,5,25,36]. Most of the outranking methods are per se based on a fuzzy notion through the concepts of‘weak preference’ and ‘incomparability’ [5]. (For this reason, Ribeiro [25] might have excluded the outranking-approaches from her review on Fuzzy MADM.) This kind of fuzziness refers mainly to the objective functionand is applied to the comparison of the actions with respect to each criterion. Literature research reveals,that several attempts for the integration of fuzzy logic into multi-attribute decision support are also beingdiscussed for the evaluation of fuzzy information. These approaches show that fuzzy logic can be integratedinto outranking approaches, however, this particular modelling of imprecision has to be adjusted to the speci�cdecision situation [36]:• Tsoukias and Vincke [32] introduce a relational preference system which is based on a four-valued logic.The four statements for preference, which are true, false, none and both, allow for a �ner grading ofthe degree of preference than the two-valued logic, which allows only for yes=no decisions as regards thepreference. Nevertheless, the four-valued logic uses discrete, crisp values, which result in the evaluationof sixteen discrete logical values so that another kind of crisp thresholds is used (although the transitionsbetween the values are smoother). The increased e�ort for de�ning the four threshold values for eachcriterion can be seen as a disadvantage.

• Czyzak and Slowinski [6] propose for the evaluation of fuzzy numbers in ELECTRE III with fourcomparison indices using possibility and necessity concepts. The construction of the fuzzy outranking rela-tion takes into account the attitude of the decision maker towards compensation. The authors demonstratethe exploitation of fuzzy information for building an outranking relation. Some of the used parameters,however, “have rather technical character, so they should be speci�ed by an analyst acting in the nameof the decision maker” [6, p. 128]. In the given decision situation for environmental assessment, it mightbe more desirable to have parameters which are better understandable for the decision maker. Since en-vironmental data are rather hard to obtain, the gathering of the required data on possibility might be notadvisable.

• An interval version of PROMETHEE has been developed and applied for the comparison of buildingproducts’ design with ill-de�ned data for environmental use [17]. This approach seems to be a suitableenhancement of the PROMETHEE concept, however the increase in the intervals caused by the intervalarithmetic leads to problems. Moreover, crisp intervals are used, so that crisp numbers for the intervalboundaries are required.

• For multi-attribute decision making with cardinal numbers and additive aggregation, the outrankingmethods ELECTRE and PROMETHEE can be connected with the concept of linguistic variables [20].The derived fuzzy approach reduces the structure so that only linguistic variables remain which suit


Table 1Cases of uncertainty and their fuzzy modelling in the decision model

Sources of uncertainty [28] Fuzzy modelling of

Preferences Scores Weights

Imprecision caused by the di�culty of determining the scores of •actions on particular criteria

Indetermination, since the method of evaluation results from a • •relatively arbitrary choice between several possible de�nitions

Uncertainty, since the values involved vary over time and space • • •

many practical decision situations, while reducing the computational e�ort. For environmental assessment,the sole use of linguistic variables, however, might not be su�cient and convincing for the decisionmakers.

The integration of fuzzy logic into ELECTRE seems to be more advanced (e.g. [6,22]), but in general,PROMETHEE could also assimilate the fuzziness of performances in the evaluation table. Because of thelimitations of the proposed approaches in literature for environmental assessment, this paper suggests a con-cept of fuzzy outranking with the example of PROMETHEE. Table 1 shows the sources of uncertainty inthe decision situation as identi�ed by Roy and Bouyssou [28] and their corresponding modelling as fuzzynumbers. It can be seen that the fuzzy notion of the preferences may not be su�cient for an adequatecomprehensive decision support, but that also the consideration of fuzzy scores and fuzzy weights might beuseful in ill-structured decision situations. Therefore, this paper proposes the enhancement of PROMETHEEtowards fuzzy logic in order to consider not only fuzzy preferences, but simultaneously fuzzy scores and fuzzyweights.

6. PROMETHEE with trapezoidal fuzzy intervals

In this paper the use of fuzzy algebra throughout the PROMETHEE algorithm is suggested, as follows.Since it is particularly di�cult to realistically model the ‘true’ shape of the membership function, the useof the trapezoidal fuzzy numbers is proposed for the enhancement of PROMETHEE. This approximationwould seem to be su�cient and easily comprehensible for practitioners [26]. Delgado et al. [8] argue thatany fuzzy number can be represented as a trapezoidal or a ‘quasi-trapezoidal’ fuzzy number, having thesame basic attributes as the original fuzzy number. Although several formulae for fuzzy intervals are be-ing discussed and used [7], in this paper, the integration of fuzzy logic into outranking is proposed by amodi�ed PROMETHEE algorithm based on the arithmetic for trapezoidal fuzzy intervals [9,26,27] and putup for discussion. The membership function for the trapezoidal fuzzy intervals can be mathematically for-mulated as

�(x)=

0 for x6ml − � or mu + �6x;

1− ml − x�

for ml − � ¡ x ¡ ml;

1 for ml6 x6mu ;

1− x − mu�

for mu¡ x6mu + �;

(3)


where � and � are the left and right spread of the trapezoidal fuzzy interval, and with the interval [ml; mu]with ml and mu as the lower and upper boundaries of the numbers which belong with certainty to theset of available values. This trapezoidal fuzzy interval is represented by the notation M =(ml; mu; �; �)LR.Triangular fuzzy numbers are a speci�c case of the trapezoidal fuzzy interval with ml=mu, and crisp num-bers n can be formulated with ml=mu = n and �= �=0. The necessary algebraic operations are de�ned as[9,26,27]:Addition:

M ⊕ N = (ml; mu ; �; �)LR ⊕ (nl; nu ; ; �)LR = (ml + nl; mu + nu ; �+ �; + �)LR : (4)

Inverse:

− M = −(ml; mu ; �; �)LR = (−mu ;−ml; �; �)LR : (5)

Subtraction:

M N = (ml; mu ; �; �)LR (nl; nu ; ; �)LR = (ml − nu ; mu − nl; �+ �; � + )LR : (6)

Multiplication:

M ⊗ N = (ml; mu ; �; �)LR ⊗ (nl; nu ; ; �)LR ≈ (ml · nl; mu · nu ; ml + nl�− � ; mu�+ nu� + ��)LR : (7)

It should be noted, that the multiplication of fuzzy numbers of the same L-R-type do not result in generalin the same type of reference function. However, if the spreads of the trapezoidal fuzzy number are smallenough in comparison to the lower and upper boundary of the fuzzy interval, the approximation (7) can beused [26]. With these requisites, it is possible to redesign a fuzzy PROMETHEE algorithm (the single stepsare numbered as above, using the label F for “fuzzy PROMETHEE”):(F1) Specify for each criterion fk a generalised preference function pk(d).(F2) De�ne a vector containing the fuzzy weights (which do not need to be normalised to

∑Kk=1 wk =1):

wT = [w1; : : : ; wK ] with wk =(mwl ; mwu ; �

w; �w)LR :

(F3) De�ne for all the alternatives at; at′ ∈A the fuzzy outranking-relation �:

� :

A×A→ [0; 1]

�(at; at′)=K∑k=1

wk ⊗ pk(fk(at) fk(at′)):

With fk(at)= (ml;mu; �; �)LR and fk(at′)= (nl; nu; ; �)LR, the degree of preference for the comparisonof alternatives at and at′ with regard to criterion fk can be derived according to (for the extension ofthe preference from real numbers to fuzzy intervals, see Fig. 4):

pk(fk(at) fk(at′))= pk((ml;mu; �; �)LR (nl; nu; ; �))LR= pk(ml − nu;mu − nl; �+ �; � + )LR= (pk(ml − nu);p(mu − nl);pk(ml − nu)− pk(ml − nu − �+ �);pk(mu − nl + � + )− pk(mu − nl))LR

= (mpkl ;mpku ; �

pk ; �pk )LR :


Fig. 4. Preference function Type V applied to the di�erence of the trapezoid LR type fuzzy values fi(at) and fi(at′ ).

Next, the degrees of preference are multiplied by the respective weights for each criterion:

wk ⊗ pk(fk(at) fk(at′))= (mwkl ;m

wku ; �

wk ; �wk )LR ⊗ (mpkl ;mpku ; �

pk ; �pk )LR

≈ (mwkl · mpkl ;mwku · mpku ;mwkl · �pk + mpkl · �wk − �wk �pk ; mwku · �pk + mpku · �wk + �wk�pk )LR :

As the last step of de�ning the outranking relation �, the weighted preference degrees which have beencalculated for each criterion k are added:

�(at; at′) =K∑k=1

wk ⊗ pk(fk(at) fk(at′))

≈K∑k=1

(mwkl · mpkl ;mwku · mpku ;mwkl · �pk + mpkl · �wk − �wk �pk ;mwku · �pk + mpku · �wk + �wk�pk )LR

=

(K∑k=1

mwkl ·mpkl ;K∑k=1

mwku ·mpku ;K∑k=1

mwkl �pk + mpkl �

wk − �wk �pk ;K∑k=1

mwku �pk+mpku �

wk + �wk�pk)LR

= (m�l ;m�u; �

�; ��)LR :

(F4) As a measure for the strength of the alternatives at ∈A, the fuzzy leaving ow �+(at) is calculated:

�+(at) =

1T − 1 ·

T∑t′=1t′ 6=t

�(at; at′):


(F5) As a measure for the weakness of the alternatives, at ∈A, the fuzzy entering ow �−(at) is calculated:

�−(at)=

1T − 1 ·

T∑t′=1t′ 6=t

�(at′ ; at):

(F6) Graphical evaluation of the outranking relation.The rank ordering of the decision alternatives according to the aggregated fuzzy judgements is seen as “by

far not trivial” [36, p. 136]. Numerous models exist, but none of them can be considered as best under allcircumstances. Following the philosophy of outranking and especially of PROMETHEE, the applied approachshould be kept as plausible as possible for the decision maker. Therefore, the proposed evaluation is basedon the defuzzi�cation of the fuzzy leaving and entering ows. For the defuzzi�cation, the approach based onthe Centre of Area (COA) is selected:

xdefuzz =

∫x · �(x) dx∫�(x) dx

=

∫ ml

ml−�

(1− ml − x

�

)· x dx + ∫ muml

1 · x dx + ∫ mu+�mu

x − mu�

· x dx∫ ml

ml−�

(1− ml − x

�

)dx +

∫ muml1 dx+

∫ mu+�mu

(1− x − mu

�

)dx

=m2u − m2l + �ml + �mu + 1

3 (�2 − �2)

�+ � + 2mu − 2ml: (8)

The COA approach promises more reasonable results in the given decision situation than the Mean ofMaxima (MOM) or Maxima-Method (MAX) and allows a consistent evaluation of trapezoidal and triangularfuzzy data as well as of crisp data. Furthermore, no additional parameters are needed (e.g. as for �-cuts). Themain advantage of the defuzzi�cation, however, could be seen, that a basic sensitivity analysis of the chosenweights is possible with less e�ort, as explained in the following:The sensitivity intervals give the range of values that the weight of one criterion can take without altering the

results given in the initial set of weights, all other weights being kept constant [18]. The narrower the intervalboundaries, the more sensitive is the weighting of the respective criterion. This investigation is possible, sincethe PROMETHEE algorithms is basically an additive method.Where the weight for the investigated criterion equals 100%, the preference index can be de�ned as a

unicriterion preference index (with the only criterion as the investigated one). On the contrary, the preferenceindex can also be calculated for the preference index for all the criteria except for the investigated criterion.The straight line g(wfk ; at) in a diagram depicting both these preference indices goes through the preferenceindex calculated for the whole set of criteria. For the de�nition of the sensitivity interval, the points ofintersection of each pair of straight lines have to be calculated, from which those closest to the originalweight mark the sensitivity interval:

�net0 (at) + (�net1 (at)− �net0 (at)) · w∗ = �net0 (at′) + (�

net1 (at′)− �net0 (at′)) · w∗

⇔ w∗ �net0 (at)− �net0 (at′)�net1 (at′)− �net0 (at′)− �net1 (at) + �net0 (at)

(9)


Fig. 5. Sensitivity analysis.

with w∗=weight at the intersection,

�net0 (at) = preference index for alternative at with wfk =0

=

((�+(at)− �(at ; at′)− �−

(at) + �(at′ ; at))÷

((K∑k=1

wk

)− w∗

fk

))defuzz

with w∗fk as the originally chosen weight for that considered criterion

�net1 (at) = unicriterion preference index for alternative at with wfk =100%:

As an illustration, Fig. 5 sketches the sensitivity analysis for a criterion ‘eutrophication’. The originally cho-sen weight is 8.3%, for which the resulting complete order of the investigated alternatives is D→B→A→C.Beyond the lower boundary of the sensitivity interval at 2.4%, the rank order changes to B→D→A→C,while beyond the upper boundary of the sensitivity interval at 20.7% the rank order alters to D→A→B→C.It should be noted that the sensitivity analysis is performed by using the defuzzi�ed net preference ows

�net(at). Although it has been argued above that the net preference ow makes use of the concept of completecompensation and is therefore of less use in several decision situations, it is to be stressed that the netpreference ows are a suitable mean for an intuitive sensitivity analysis: Both the sensitivity interval and theresulting rank alterations can be shown in one graphical representation and can easily be explained to thedecision maker. Thus, the decision maker can easily grasp that for the example given in Fig. 5, alternative Bis largely dependent on the weighting of the investigated criterion.The good comprehensibility should be considered as an important advantage of this sensitivity analysis,

since it can focus the discussion of the decision makers on the most relevant, i.e. sensitive, criteria for theoverall decision. More elaborated approaches for sensitivity analysis exist, but it should be noted that theactual �gures of the sensitivity intervals should not be regarded as absolute values, but only for an estimationof the need for further discussion on the most appropriate weighting.

7. Case study: Environmental assessment for sinter plants

The procedure for the soft decision analysis based on the proposed algorithm for fuzzy PROMETHEE isdemonstrated with a case study from the iron and steel making industry. The sinter techniques are chosenas a case study, since the techniques and their emissions are relatively well documented, although di�erentmodes of data collection and varying measuring points limit the availability of comparable data. Therefore,the described techniques should be regarded as hypothetical [24].

58 J. Geldermann et al. / Fuzzy Sets and Systems 115 (2000) 45–65Table2

Decisiontableforthecasestudysinterproduction

Criteria

Totalimpactpotential

Unitpertsinter

Impactcategory

TechniqueA

TechniqueB

TechniqueC

TechniqueD

Photochemicaloxidantformation

(34;37;3;4) LR

(19;20;1;2) LR

(110;130;10;10) LR

(10;11;1;1) LR

10−3kgethene-equ.

Nutri�cation

(50;55;4;5) LR

(68;70;3;6) LR

(63;65;6;8) LR

(51;52;3;4) LR

10−3kgPO

3− 4-equ.

Acidi�cation

(1.1;1.2;0.1;0.2) LR

(1.7;1.8;0.2;0.2) LR

(1.7;1.8;0.2;0.4) LR

(1.1;1.2;0.1;0.4) LR

kgSO

2-equ.

Humantoxicity

(120;190;10;20) LR

(40;50;4;5) LR

(190;200;10;10) LR

(40;50;4;5) LR

106m3air

Ecotoxicity,air

(25;35;5;5) LR

(40;45;2;3) LR

(58;60;5;5) LR

(28;33;2;3) LR

106m3air

Ecotoxicity,water

(0;0;0;0) LR

(0;0;0;0) LR

(0;0;0;0) LR

(0.21;0.22;0.01;0.03) LR

lwater

Hazardouswaste

(0;0;0;0) LR

(0;0;0;0) LR

(0;0;0;0) LR

(0.15;0.15;0.02;0.04) LR

kgProtectionofthemaritimeenvironment

(2;3;0.5;0.5) LR

(0.04;0.06;0.01;0.02) LR

(7;8;1;1) LR

(0.1;0.2;0.05;0.05) LR

10−3kg

Datafrom

massandenergybalance

Consumptionofenergy

Fossilenergy

(1700;1700;50;50) LR

(1560;1560;50;50) LR

(1650;1650;50;50) LR

(1600;1600;50;50) LR

MJ

Electricenergy

(395;395;40;40) LR

(425;425;40;40) LR

(345;345;40;40) LR

(410;410;40;40) LR

MJ

AtmosphericemissionsPCDD=PCDF

(3.4;3.5;1;1) LR

(1.8;1.9;0.2;0.1) LR

(6;7;0.5;0.5) LR

(0.4;0.5;0.1;0.1) LR

10−9kg

Aquaticemissions

No

No

No

Yes

Table3

Exemplaryaggregationoftheecologicalandquantitativerelevanceoftheconsideredcriteria

Impactcategory

Speci�ccontribution

Quantitativerelevanceaccordingto

Ecologicalrelevanceofthe

Totalrelevance

speci�ccontribution

impactcategory

Formationofphotochemicaloxidants

5.5%

Low

Large

Medium

Eutrophication

25.4%

Moderate

Medium

Medium

Acidi�cation

47.0%

Medium

Medium

Medium

Humantoxicity

100%

Verylarge

Large

Verylarge

Ecotoxicity,air

45.7%

Medium

Medium

Medium

Ecotoxicity,water

–Low

Medium

Moderate

Hazardouswaste

–Low

Moderate

Moderate

Protectionofthemaritimeenvironment

–Low

Verylarge

Medium

Datafrom


Fossilenergy

–Low

Large

Medium

Electricenergy

–Low

Large

Medium

PCDD=PCDF

–Large

Large

Large

Aquaticemissions

–Low

Medium

Moderate


Fig. 6. Representation of the linguistic weights as triangular fuzzy numbers.

The sintering plant is the main important aggregate in the integrated iron and steel works for the preparationof iron ores. In a blast furnace, only iron ores of a relatively large size can be used. Small iron ore particlesas �nes and concentrates reduce the permeability and hamper the process in the blast furnace. The sinteringplant essentially consists of a large travelling grate of heat resistant cast iron. In the sintering process, thesmall particles are baked (sintered) into larger pieces (10–50mm), which can be fed into the blast furnace.The necessary heat for sintering is produced by coke as a fuel. Raw materials require blending prior to thesintering operation. Additives are mixed to the ore blend, like lime and olivine. In the sintering process, uedusts, mill scales etc. stemming from the production process of iron and steel making, can be recycled. Thesinter process is basically a source of various particulate matter and gaseous emissions, of which SO2, CO,PCDD=PCDF and to a less extent, NOx might be regarded as the most important ones.The investigated sinter techniques in this case study di�er mainly in their means of gas cleaning: an electric

precipitator, a fabric �lter in addition with an electric precipitator, a cyclone for the dust absorption, and amodi�ed wet scrubber. For a comprehensive explanation, including a detailed description of the investigatedtechniques and their Life Cycle Assessment, see [24].For the environmental assessment, �rstly the relevant mass and energy ows are compiled in the inventory

analysis for the reference quantity of 1 t sinter. Due to the dependency of the emissions on the varyingprocess inputs, as well as due to di�ering monitoring and reporting regulations, the given �gures are re-garded as trapezoidal fuzzy intervals. The classi�cation of the consumption and emissions takes place inthe impact assessment. The impact potentials of the techniques are calculated by multiplying the emissionswith the corresponding impact assessment factors. Several substances are classi�ed in multiple impact cat-egories, such as SO2 and NOx. The shortcomings of the used impact assessment factors are discussed in[24] (see also Section 2). Since the impact assessment factors only allow a rough approximation of thereal ecological impacts caused by the emissions, the impact potentials seem to be better represented byfuzzy intervals. Moreover, the data on emissions caused by the considered techniques should also be seen asfuzzy intervals because of the numerous interdependencies in the production process which cannot be exactlyquanti�ed.Table 2 presents the calculated impact potentials for the four investigated sinter techniques. It should be

noted that the energy demand is not converted into its contribution to the impact categories, but is taken intoaccount as total energy consumption. Emissions of de�nite relevance, which are not well represented withinthe impact categories, have also to be taken into account for a comprehensive environmental assessment.For the di�erentiation of the signi�cance of the considered criteria for the decision, the weighting in Table 3

is oriented towards the ecological relevance of the impact categories and the estimation of the quantitativerelevance of the impact potentials of the considered techniques by a comparison with the corresponding impactpotentials in the EU, if European emission data are available, called ‘speci�c contribution’. The aggregationto the total relevance takes place according to speci�ed decision rules [33]. It should be noted that thereis no de�ned substitution rate between the verbal predicates, which rather serve as a rough orientation fordi�erentiating the signi�cance of the criteria. In order to re ect the subjective character of the relevance,the derived weighting factors should be de�ned as fuzzy numbers and regarded as ‘holistic’. Fig. 6 depicts


Table4

Preferenceindicesforthecasestudysinterproduction

AB

CD

A(0;0;0;0) LR

(3.500;4.000;2.611;1.333) LR

(6.111;8.000;3.556;5.000) LR

(2.500;4.033;1.500;4.300) LR

B(4.653;4.653;2.030;3.772) LR

(0;0;0;0) LR

(7.000;7.333;3.000;4.333) LR

(2.820;3.213;1.820;3.804) LR

C(2.000;2.000;2.000;2.667) LR

(1.600;2.333;1.600;1.933) LR

(0;0;0;0) LR

(3.000;3.000;2.000;3.111) LR

D(5.431;7.508;2.764;5.225) LR

(3.978;4.750;2.744;4.074) LR

(9.000;9.000;4.533;3.333) LR

(0;0;0;0) LR

�−

(12.084;14.381;6.794;11.664)LR

(9.078;11.083;6.956;7.341)LR

(22.111;24.333;11.089;12.667) LR

(8.320;10.247;5.320;11.216) LR

�+

(12.111;16.033;7.667;10.633)LR

(14.473;15.419;6.850;11.910)LR

(6.600;7.333;5.600;7.711) LR

(18.408;21.258;10.042;12.633) LR

Table5

Preferencefunctions,parametersandweightingfactorsinthecasestudysinterproduction

Impactcategory

Preferencefunction

Preferenceparameters

Weightingfactors

Percentage

Formationofphotochemicaloxidants

3p=40

(3;3;1;1) LR

8.3%

Eutrophication

3p=5

(3;3;1;1) LR

8.3%

Acidi�cation

3p=0:3

(3;3;1;1) LR

8.3%

Humantoxicity

5p=50

(5;5;1;1) LR

13.9%

Ecotoxicity,air

3p=10

(3;3;1;1) LR

8.3%

Ecotoxicity,water

2(2;2;1;1) LR

5.6%

Hazardouswaste

2(2;2;1;1) LR

8.3%

Protectionofthemarineenvironment

3p=2

(3;3;1;1) LR

8.3%

Datafrom


Fossilenergy

3p=50

(3;3;1;1) LR

8.3%

Electricenergy

3p=30

(3;3;1;1) LR

8.3%

PCDD=PCDF

4p=3

(4;4;1;1) LR

11.1%

Aquaticemissions

2(2;2;1;1) LR

5.6%


the graphical representation of the linguistic total relevance as triangular fuzzy numbers, as they are used inthis exemplary case study.The algorithm for Fuzzy PROMETHEE described in the previous section is being applied to the evaluation

table (Table 4), using the indicated preference functions with the respective parameters given in Table 5.Aiming at a quick overview for the decision maker on the chosen weighting factors, the fuzzy weightingfactors are defuzzi�ed and expressed as percentages, in order to illustrate the spread of the weights in anappropriate graphical representation.If the weights as suggested in Table 3 are taken into account, fuzzy PROMETHEE reveals the results

given in Table 4. Fig. 7 displays the fuzzy leaving and entering ows for the considered criteria. The higherthe leaving ow and the lower the entering ow, the better the action. For the rank ordering of these fuzzynumbers, the defuzzi�cation following the Centre of Area (COA) approach is applied.To the political and industrial decision makers, the outranking graph can be explained as a graphical

representation of the ‘relative strength’ �+ and the ‘relative weakness’ �− of the considered techniques.Fig. 8 displays a suitable illustration of the outcome of fuzzy PROMETHEE for presentation to the decisionmakers: since the entering ows �− for the respective techniques are depicted with a negative sign, theirindication of the relative weakness of the considered techniques becomes more evident. To give the deci-sion makers a quick overview, also the net ows �net are depicted and o�ered as additional information.The partial preorder is derived from both the ranking according to the leaving ows �+ (D→B→A→C)

Fig. 7. Fuzzy leaving ows and fuzzy entering ows.

Fig. 8. Defuzzi�ed preference ows and partial preorder.


and according to the entering ows �− (B→D→A→C), which are aggregated as a set intersection, shownat the bottom of Fig. 8.This graph is to be interpreted by the decision makers. They may judge that Technique B and D come o�

well since for both techniques the leaving ows outweigh the entering ow signi�cantly, while Technique Ais in the medium range. This result would seem to be reasonable, since Technique C has been re�tted witha more modern emission reduction technique in the meantime. It could be argued that those techniques withleaving ows almost equal or larger than the entering ows should be regarded as ‘best consideredtechniques’.It has to be stressed, that the most important information of the fuzzy PROMETHEE result is the partial

pre-order, which also reveals incomparabilities and, even more, the sensitivity analysis which shows the mostsensitive weights and criteria. The diagrams obtained by the sensitivity analysis (cf. Fig. 9) communicatethe changing of the alternatives’ ranking at the respective weights. The more narrow a sensitivity interval is,the more sensitive is the respective criterion and its weighting. In this case study, the criteria nutri�cation,acidi�cation, ecotoxicity and aquatic emissions are more sensitive than the others. Also the slope of thealternative gives an indication: here, Technique D shows quite often di�erent results than the others.In the �nal discussion of the political and industrial decision makers, the BAT are to be determined. They

have to keep in mind the assumptions underlying the data preparation. Firstly, the limited data availability dueto the varying measurements restricts the reliability of the obtained results. Secondly, the subjective characterof the setting of weights and of the preference parameters should be considered. With this �nal interpretationby the decision makers it becomes evident that soft decision analysis only supports the decision, which itselfis left to the decision makers.

8. Discussion

Some practical experiences have already been gained from the application of PROMETHEE for environ-mental assessment in the context of environmental assessment and determining BAT [11,24]. Especially theconcept of generalised criteria as a basis for the preference functions is appreciated in practice, since it isbased on parameters which have a real meaning and are therefore operational and comprehensible. The pair-wise comparisons as a basis for the determination of the preferences are especially helpful in environmentalassessment where ‘ideal points’ are seldom available for decision support. Moreover, the decision makersshould be able to explain the results and their acquisition to others not participating in the decision process.For the success of a decision support approach, it should be taken into consideration, that an analyst is notalways with the decision makers.In the context of environmental assessment both with regard to LCA and to the determination of BAT, the

use of clear, exact statements is often impossible. Therefore, soft decision analysis would seem to be moree�ective, although at �rst sight, the results seem to be less exact. As opposed to an unguided discussion ofthe relevant issues, a structured course of decision support using fuzzy numbers and outranking techniques,should ensure adequate consideration of all relevant aspects and the disclosure of subjective assumptions.The latter is achieved especially by the calculation of the sensitivity intervals, which reveal those criteriawhose weighting in uences the overall result most. Hence, a thorough discussion of their weighting might benecessary for an acceptable decision. Moreover, a sensitivity analysis is particularly required by the ISO 14040on LCA.According to the IPPC-Directive, the economic and technical tenability of the BAT are also to be taken into

consideration, when determining the BAT. Since the considered techniques for BAT selection have alreadybeen tested on an industrial scale, their economic viability is assumed as a given fact at present. In future, theuse of fuzzy numbers for the description of economic and technical criteria is advisable, because statistical datalike the expected mean value and standard deviation are rarely obtainable. The reason is that investments for


Fig. 9. Sensitivity analysis for the chosen weights (ceteris paribus).


process-integrated emission reduction techniques and particularly for innovative techniques depend on hardlypredictable conditions, like the duration of the standstill of the complete production process or the costs foradjusting the infrastructure. These are general di�culties in systems engineering for special industrial branches,where unique projects do not form a basis for statistical interpretation. The use of linguistic variables couldbe recommended for the evaluation of the technical criterion, operation experience.It should be noted as a reservation, however, that any quanti�cation within the decision support fakes an

objectivity which does not exist. Nevertheless, this drawback has to be accepted in order to reduce the subjec-tivity of the decision or to improve transparency at least. Moreover, fuzzy logic can reduce the reservationsagainst the numerical results of the decision support, especially through the notion of fuzzy weights. In thiscontext, an extension of fuzzy PROMETHEE towards group decision support would seem to be desirable.

9. Conclusions

The outranking methods as a special subgroup of MCDM methods are suitable for soft decision makingthrough the notion of weak preference and incomparability, which better represent the real decision situation.None of the numerous approaches for fuzzy decision support discussed in the literature can be considered best,and the appropriateness depends on the decision situation and the attitude of the decision maker [36]. For en-vironmental assessment, the outranking method PROMETHEE brings together exibility and simplicity for theuser. Therefore the enhancement of the outranking approach PROMETHEE towards fuzzy logic using trape-zoidal fuzzy intervals might o�er a concept applicable to a broad variety of decision problems, as presented inthis paper. The ranking of the alternatives is based on the defuzzi�ed fuzzy preference ows (via the Centreof Area concept COA), which also serve as a basis for the newly introduced graphical sensitivity analysis.An example from the iron and steel making industry shows the applicability and usefulness of soft decisionanalysis.Since the outranking approaches use the concept of weak preferences, some preconditions of measurement

theory are not ful�lled [2]. Further research is required on the stability of the results, in order to betterexplain the rank reversals which sometimes occur. Some outranking algorithms also take account of partialcompensation: a good score for an alternative for a speci�c criterion might only o�set a poor score for anothercriterion, when the di�erence between the scores exceeds a certain threshold. Furthermore veto functions canbe used, if it is essential for the decision that a particular score does not exceed or fall below a certainthreshold. Both these concepts should also be integrated into PROMETHEE [11,12].The fuzzy PROMETHEE is integrated into the decision support system KOSIMEUS, which is a combi-

nation of process models simulated with a owsheeting program and a multicriteria decision support systemfor the evaluation of techniques taking into account ecological and so-called techno-economic criteria, thatdescribe the operational costs depending on the technical performance [11,30]. Several applications for the ironand steel making industry already exist. In summary, it can be stated that outranking via fuzzy PROMETHEEhelps to gain an insight into the decision maker’s preference structure and to focus attention on critical issues,which could be regarded as the aim of soft decision analysis. Especially the graphical representation of theoutranking graph and the sensitivity intervals are a suitable means for the �nal decision which must be leftto political and industrial decision makers.

Acknowledgements

The research project “Development of a multicriteria decision support system for the evaluation of process-integrated emission reduction techniques – Case study: Iron and steel making industry” is supported by theVolkswagen Stiftung, Hannover, Germany, which is greatly appreciated.


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fuzzy outranking for environmental assessment. case study: iron and steel making industry

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