PROMETHEE-MD-2T method for project selection

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European Journal of Operational Research 195 (2009) 841–849

PROMETHEE-MD-2T method for project selection

N. Halouani a,*, H. Chabchoub a, J.-M. Martel b

a University of Economic Sciences and Management, Sfax, Tunisiab Laval University, Quebec, Canada

Available online 12 November 2007

Abstract

Project selection is a real problem of multicriteria group decision making (MCGDM) where each decision maker expresses his/herpreferences depending on the nature of the alternatives and on his/her own knowledge over them. Thus, information, as much quanti-tative as qualitative, coexists. The traditional methods of MCGDM developed for project selection usually discriminates in favour ofquantitative information at the expense of qualitative information, and this is due to the capability to integrate this first type of infor-mation inside their procedure. In this article, two new multicriteria 2-tuple group decision methods called ‘‘Preference Ranking Orga-nisation Method for Enrichment Evaluation Multi Decision maker 2-Tuple-I and II” (PROMETHEE-MD-2T-I and II) arepresented. They are able to integrate inside their procedure both quantitative and qualitative information in an uncertain context. Thishas been performed by integrating a 2-tuple linguistic representation model dealing with non-homogeneous and imprecise informationdata made up by valued intervals, numerical and linguistic values into the aggregation operators of Promethee methods. Although theyhave been developed for project selection problems, these proposed methods can be applied to all kinds of decision-making problemswith heterogeneous and multigranular information.

Therefore, the application of these methods to real problems will lead to better results in MCGDM. It offers to the decision makerssimpler and wider application of the aggregation operators of the Promethee multicriteria method without sacrificing any of its advan-tageous properties and by taking into account the imprecision of data. A numerical simulation of the proposed method is carried out toshow its possibilities.� 2007 Elsevier B.V. All rights reserved.

Keywords: Project selection; Promethee method; 2-Tuple model; Interval valued; Numerical values; Linguistic terms; MCGDM

1. Introduction

Multicriteria decision making (MCDM) analysis hasbeen widely used to deal with decision-making problemsinvolving multiple criteria selection of alternatives. Differ-ent methods to cope with MCDM problems have beendeveloped such as Promethee (Brans et al., 1984). Thesemethods can only handle quantitative information andtherefore cannot be well applied when there is qualitativeinformation (as occurs in real situations of project selec-

0377-2217/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2007.11.016

* Corresponding author.E-mail addresses: [email protected], halouani.nesrin@gmail.

com (N. Halouani).

tion). To manage this problem, the Promethee methodhas been used in MCDM on project selection problems,in which qualitative information is traditionally trans-formed to numerical one using an ordinal scale (Al-Rash-dan et al., 1999). However, the drawbacks of thistransformation are the possible major problem occurrencein projects of considerable scope in which a bad choicecould lead to unforeseeable repercussions. Prometheemethod was extended by Goumas and Lygerou (2000) todeal with fuzzy input data, but they have limited the trans-formation of qualitative data to only the performances ofthe alternatives as fuzzy numbers while the preferences ofthe decision makers remained as crisp values. In fact, thechoice of the parameters expressing the preference or opin-ion of the decision maker, as crisp values, greatly influence

842 N. Halouani et al. / European Journal of Operational Research 195 (2009) 841–849

the outcome. Since the opinion of different persons rarelycoincide, in most cases there is not a single generallyaccepted solution and the results should be viewed in thisperspective.

Many other authors extended the Promethee method todifferent areas in a stochastic context (Marinoni, 2005), in acontext where some informations are uncertain (Wang andLin, 2003), through the mathematical programming (Fer-nandez-Castro and Jimenez, 2005), by using fuzzy numbers(Goumas and Lygerou, 2000; Radojevic and Petrovic,1997; Srinivasa Raju, 1995) and with intervals (Gelder-mann et al., 2000; Al-Rashdan et al., 1999; Le Teno andMareschal, 1998). All these extensions do not treat at thesame time the qualitative aspect and the imprecision ofboth alternatives performances, weighting factors, decisionmakers’ preferences and parameters for the generalized cri-teria. Also, they do not deal with heterogeneous and mul-tigranulary data, which are numerical values, linguisticterms and interval valued in a group decision-making pro-cess (GDM).

Thus, and due to the imprecision and subjectivity of theinformation associated with the selection process, the crispvalues are completely unsuitable when solving GDM selec-tion problems. A much more realistic approach would beto use linguistic estimators instead of numerical values, inother words, to use linguistic variables in the processes ofthe different Promethee approach. Therefore, a fuzzymodel of linguistic representation with 2-tuples (Herreraet al., 2005) will be used. The application of the 2-tuplemodel to Promethee methods improves considerably theobtained results.

The Promethee methodology seems to be completelyadequate to project selection problems because it modelspreferences within its procedures in a simple and flexiblemanner. Also, it is perfectly intelligible for decision makerssince it represents one of the most intuitive multicriteriadecision methods. Therefore, it is chosen for the enhance-ment towards the evaluation of fuzzy data on preferences,scores, parameters of generalized criteria and weights.

Hence, within the framework of this paper, new meth-ods of project selection to deal with non-homogeneousdata made up by valued intervals, numerical and linguisticvalues in a MCGDM context are presented. New aggrega-tion operators were defined in order to allow developingthe computing processes on these types of information.These new operators are considered as an extension/improvement of those initially used in Promethee method,in terms of the precision, flexibility and facilitated use andof implementation.

This article is divided into four important sections. Inthe first, the 2-tuple model of linguistic representation isdetailed. The second presents a description of Prometheeapproach. The third is interested on the development ofthe proposed methods of project selection. An applicationof the developed methods is carried out through an illustra-tive example which is the subject of the last section of thispaper.

2. The linguistic 2-tuple representation model

The 2-tuple representation model (Herrera and Marti-nez, 2000a,b) is used to represent linguistic and numericalinformation. However, all suitable information for a deci-sion problem is represented by this 2-tuple model, in theform of a simple internal representation which includesall original information, generally of multigranulary nature(a linguistic variable with different granularity and/orsemantics, for each criterion leading to fuzzy evaluations).Then, this information representation is introduced intothe selected multicriterion procedure of aggregation, Prom-ethee, and the results obtained are presented in the initialfield of information.

The 2-tuple representation model is based on a conceptof symbolic translation which it uses to represent linguisticinformation by means of a linguistic pair of values named2-tuple (s,a), where s is a linguistic term and a a numericalvalue representing the symbolic translation.

Definition 1. Let b be the result of an aggregation of theindexes of a set of labels assessed in a linguistic term setS = {s0, . . . , sg}, i.e., the result of a symbolic aggregationoperation. b 2 [0,g], being g + 1 the cardinality of S. Leti = round(b) and a = b � i two values, such that, i 2 [0,g]and a 2 [�0.5,0.5) is called a Symbolic Translation (Her-rera et al., 2005).

The symbolic translation of a linguistic term si, is an esti-mated numerical value in [�0.5,0.5) which support the”information difference” between a quantity of informa-tion b 2 [0,g] obtained after the operation of symbolicaggregation and the value nearest in {0, . . . ,g} which indi-cates the index of the linguistic term nearest in S.

From this concept, we must develop a linguisticrepresentation model which represents linguistic informa-tion by means of a 2-tuple (si,ai), si 2 S and ai 2[ � 0.5,0.5):

– si represents the linguistic label of information and– ai is a numerical value expressing the translation starting

from the original result b with the index nearest to thelabel, i, in the set of linguistic term (si), i.e. the symbolictranslation.

This model defines a set of transformation functionsbetween the numerical values and the 2-tuples.

Definition 2. Let S = {s0, . . . , sg} a linguistic term set andb 2 [0,g] a value supporting the result of a symbolicaggregation operation, then the 2-tuple that expresses theequivalent information to b is obtained with the followingfunction (Herrera and Martinez, 2000b):

D : ½0; g� ! S � ½�0:5; 0:5Þ;

DðbÞ ¼ ðsi; aÞ; withsi i ¼ roundðbÞa ¼ b� i a 2 ½�0:5; 0:5Þ;

�

N. Halouani et al. / European Journal of Operational Research 195 (2009) 841–849 843

where round(�) is the usual operation round, si is the indexof the label nearest with b and a is the value of the symbolictranslation.

From this definition, a function D�1 can be defined, suchas, from a 2-tuple (si,a) it turns over its equivalent numer-ical value b 2 [0,g] � R, in the following way:

D�1 : S � ½�0:5; 0:5Þ ! ½0; g�D�1ðsi; aÞ ¼ aþ i ¼ b

From Definitions 1 and 2, it is obvious that the conver-sion of a linguistic term into linguistic 2-tuple consists inadding a value 0 as symbolic translation: si 2 S) (si,0).

3. The Promethee method

As in the majority of the multicriteria decision aid meth-ods, Promethee is built on the basis of a set A = {a1, . . . ,am}of m alternatives which must be ordered, and a wholeF = {f1, . . . , fn} of n criteria which must be optimized.

The decision multicriterion problem resulting can thenbe expressed in the form of a decision matrix (m � n) whoseelements indicate the evaluation or the value of the alterna-tive ai according to the criterion fj. Thus, the Prometheealgorithm can be summarized as follows (Brans et al.,1986; Geldermann et al., 2000):

1. To indicate for each criterion fj a generalized preferencefunction Pj(d) with d = fj(ai) � fj(ak).

2. To define a weighting vector, which is a measurementsof the relative importance of each criterion, W ={w1, . . . ,wj}.

3. To define for all the alternatives ai,ak 2 A the preferencerelation p:

p :

A� A! ½0; 1�;

pðai; akÞ ¼Pnj¼1

wjP jðfjðaiÞ � fjðakÞÞ:

8<:The preference index p(ai,ak) is an intensity measurementof the total preference of the decision maker for an alter-native ai compared to an alternative ak and that by takinginto account all the criteria simultaneously. It is, then, aweighted average of the preference functions Pj(d).

4. To calculate outgoing flow which is a measure of alter-native force ai 2 A like

/þðaiÞ ¼1

n

Xn

i¼1i–k

pðai; akÞ:

5. To calculate entering flow which is a measure of the out-classed character of an alternative ai 2 A, as

/�ðaiÞ ¼1

n

Xn

i¼1i–k

pðak; aiÞ:

6. Preference relation evaluation.

Basically, more the outgoing flow is large and more theentering flow is weak, better is the alternative. Promethee-Imethod lead to a partial pre-order which is obtained bycomparing the outgoing–entering flows and by carryingout the intersection between the two total pre-orders(obtained by leaving and entering flows) what makes it pos-sible to emphasize incomparable alternatives.

If a complete pre-order is necessary, Promethee-IImethod calculates net flow like the difference betweenentering and outgoing flows; thus, we must avoids allincomparability between two alternatives: /(ai) = /+(ai) �/�(ai).

4. MCGDM methods for project selection:

Promethee-MD-2T

We present in this section a short description of the var-ious stages used for PROMETHEE-MD-2T. In fact, ourunderlying approach with this decision-making model isprimarily conceived to use two different approachesnamely: Promethee and 2-tuple. This integrated modelexploits the capacities of Promethee and 2-tuple methodsby dealing with qualitative criteria and divergent decisionmaker’s preferences, in a MCGDM context. This modelis composed of the seven following stages.

4.1. Identification of decision criteria

The first phase of this stage consists in drawing up a‘‘rough” list of potential criteria which could take intoaccount all aspects of the multidimensional evaluation.This list will be derived from the general policy of the orga-nization, and will be prepared by the various decision mak-ers and experts taking part in the evaluation. The secondphase consists in refining this preliminary list of criteriato an ‘‘essential minimum” number. This refinement mustbe done with the intention to satisfy the requirements ofa coherent family of criterion.

Our methods do not provide specific directives for theweighting of the criteria, but they suppose that the decisionmakers are suited to do it suitably, at least when the num-ber of the criteria is not too high.

4.2. Identification of decision maker’s preferences

Let A = {a1,a2, . . . ,am} be the set of alternatives (or pro-jects) that the decision makers want to evaluate with a fam-ily of criteria F = {f1, f2, . . . , fn}. This family of criteria isthe same one for all the decision makers.

We associate with each criterion fj a preference functionP d

j : eA � eA ! ½0; 1� (d 2 {1, . . . ,D). This function makes itpossible to model the preferences of each of the D decisionmakers according to the criterion fj. When the decisionmakers compare two alternatives ai and ak, P d

j ðai; akÞ repre-sents the level of their preferences for ~ai compared to ~ak byconsidering only the criterion fj. The more P d

j ðai; akÞ tends

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towards 1, more the preference of the decision makers forthe alternative ai grows.

In order to facilitate the selection of a specific preferencefunction, six types of functions were proposed by Bransand Vincke (1985). In what follows, we presents these sixtypes by adapting them to the context of this work.

Type I: True criterionThis criterion is defined by

P dj ð~ai; ~akÞ ¼

0 if ~ddj 6 0;

1 if ~ddj > 0;

8<:– P d

j ð~ai; ~akÞ ¼ 0; 8~ddj 6 0 indicate the indifference of

the decision maker between the alternatives ~ai and~ak when ~dd

j ¼ fjð~aiÞ � fjð~akÞ ¼ 0) fjðeAiÞ ¼ fjðeAkÞ.– P d

j ð~ai; ~akÞ ¼ 1; 8~ddj > 0 indicate the strict preference

of the decision maker of the alternative ~ai comparedto ~ak when fjð~aiÞ � fjð~akÞ > 1.

– If the decision maker identifies that the criterion fj isof type I, he does not have any particular parameterto determine.

This type does not include extension, it gives simply theopportunity to the decision maker of using the criterionaccording to a true criterion when it is necessary.

Type II: Quasi-criterionThis criterion is defined by

P dj ð~ai; ~akÞ ¼

0 if ~ddj 6 ~qd

j ;

1 if ~ddj > ~qd

j :

8<:– P d

j ð~ai; ~akÞ ¼ 0; 8~ddj 6 ~qd

j indicate the indifference ofthe decision maker between the alternatives ~ai and~ak if the difference ~dd

j ¼ fjð~aiÞ � fjð~akÞ 6 ~qdj .

– P dj ð~ai; ~akÞ ¼ 1; 8~dd

j > ~qdj indicate the strict preference

of the decision maker of the alternative ~ai comparedto ~ak when ~dd

j ¼ fjð~aiÞ � fjð~akÞ > ~qdj .

– If the decision maker identifies that the criterion fj isof type II, the only parameter which he must deter-mine is ~qd

j .

P dj ð~ai; ~akÞ ¼

~ddjepdj

if 0 6 ~ddj 6 epd

j ;

1 if ~ddj > epd

j :

8><>:The preference of the decision maker between the alter-natives ~ai and ~ak is carried out in a progressive way. Theintensity of the preference grows linearly withthe growth of ~dd

j ¼ fjð~aiÞ � fjð~akÞ up to the value epdj .

After this value the preference becomes strict and willbe equal to 1.If the decision maker identifies that this criterion is oftype III, he must define the value of the threshold epd

j

from which, and his preference becomes strict. Ifepdj ¼ 0, we find the true criterion case.

Type IV: Criterion with stagesThis criterion is defined by

P dj ð~ai; ~akÞ ¼

0 if ~ddj 6 ~qd

j ;

1=2 if ~qdj 6

~ddj 6 epd

j ;

1 if ~ddj > epd

j :

8>>><>>>:The indifference of the decision maker between the alter-natives ~ai and ~ak takes place when the value of the differ-ence ~dd

j ¼ fjð~aiÞ � fjð~akÞ does not exceed ~qdj .

Between the values ~qdj and epd

j , the preference of the deci-sion maker for the alternative ~ai compared to the alter-native ~ak is weak and it is equal to 1/2. For a differencelarger than epd

j , his preference for the alternative ~ai

becomes strict and it is equal to 1.If the decision maker identifies that the criterion is oftype IV, he must fix the values of the two thresholdsepd

j and ~qdj .

Type V: Criterion with linear preference and field ofindifferenceThis criterion is defined by

P dj ð~ai; ~akÞ ¼

0 if ~ddj 6 ~qd

j ;

~ddj�~qd

jepdj�~qd

j

if ~qdj 6

~ddj 6 epd

j ;

1 if ~ddj > epd

j :

8>>>><>>>>:When the difference ~dd

j ¼ fjð~aiÞ � fjð~akÞ does not exceedthe value ~qd

j , the decision maker is indifferent betweenthe two alternatives ~ai and ~ak. Beyond ~qd

j , the preferenceof the decision maker follows an increasing linear func-tion (with a maximum of 1) until the difference~dd

j ¼ fjð~aiÞ � fjð~akÞ is equal to epdj . After this value, the

preference becomes strict and equal to 1.The decision maker identifies the criterion of the type Vand fixes the values of the thresholds epd

j and ~qdj .

Type VI: Gaussian criterion

P dj ð~ai; ~akÞ ¼

0 if ~ddj 6 0;

1� expð�ð~ddj Þ

2=2r2Þ if ~dd

j > 0:

(According to this criterion, the preference of the deci-

sion maker grows with the increase in ~ddj ¼ fjð~aiÞ � fjð~akÞ.

The value r is the standard deviation of a normal distribu-tion and corresponds to the distance between the originand the inflection point of the preference function curve.

The decision maker determines only the value of r.

4.3. Standardisation of information

The third stage of the project evaluation model proce-dure is the standardization of the input information. Thisstage requires the choice of ST = {s0, . . . , sg} like BLTS

N. Halouani et al. / European Journal of Operational Research 195 (2009) 841–849 845

(Basic Linguistic Term Set). Then, a set of terms with anumber of terms more wide than the number than a personis able to discriminate must be chosen (normally, 11 or 13).Normally, we choose a BLTS with 15 terms symmetricallydistributed as follows.

4.4. Transformation of the input information into F(ST)

The fourth stage of the evaluation procedure consists intransforming information of the first two stages into fuzzysets in the BLTS, F(ST). This stage is carried out accordingto the following order:

– transformation of numerical values in [0,1] intoF(ST),

– transformation of linguistic terms into F(ST),– transformation of valued intervals into F(ST).

4.4.1. Transformation of numerical values in [0, 1] into

F(ST)

Let F(ST) be the set of the fuzzy sets in ST = {s0, . . . , sg},we must transform a numerical value # 2 [0,1] into a fuzzyset in F(ST) calculating the value of membership of # in thefuzzy number associated with the linguistic terms of ST.

Definition 3. The function sNST transforms a numericalvalue into a fuzzy set in ST (Herrera et al., 2005):

sNST : ½0; 1� ! F ðST Þ;

sNST ð#Þ ¼ fðs0; c0Þ; . . . ; ðsg; cgÞg; si 2 ST and ci 2 ½0; 1�;

ci ¼ lsið#Þ ¼

0 if # R supportðlsiðxÞÞ;

#�aibi�ai

if ai 6 # 6 bi;

1 if bi 6 # 6 di;

ci�#ci�di

if di 6 # 6 ci:

8>>>>><>>>>>:Herrera et al. (2005) consider the membership functions

lSið�Þ, for the linguistic labels si 2 ST, represented by a

parametric function (ai,bi,di,ci). A particular case is the lin-guistic evaluations whose functions of membership are atriangle: bi = di.

4.4.2. Transformation of linguistic terms into F(ST)

Definition 4. Let ST = {l0, . . . , lp} and ST = {s0, . . . , sg} twolinguistic term sets of, such that, g P p. Then, a linguistictransformation function, sSST , is defined as (Herrera et al.,2005)

sSST : S ! F ðST Þ;sSST ðliÞ ¼ ðsk; c

ikÞ=k 2 f0; . . . ; gg

� �8li 2 S;

ci ¼ maxy minflliðyÞ; lsk

ðyÞg;

where F(ST) is the set of fuzzy sets defined in ST, llið�Þ and

lSkð�Þ are related to membership of the fuzzy sets associated

with the terms li and sk, respectively.

Therefore, the result of sSST for any linguistic value of S

is a fuzzy set defined in the BLTS, ST.

4.4.3. Transformation of valued intervals into F(ST)

Let I = [i,�i] be a valued interval in [0,1], to make thistransformation we suppose that the valued interval has arepresentation inspired of the membership function of thefuzzy sets as follows (Kuchta, 2000):

lIð#Þ ¼0 if # < i;

1 if i 6 # 6 �i;

0 if �i < #;

8><>:where # is a value in [0,1].

Definition 5. Let ST = {l0, . . . , lp} a BLTS. Then, thefunction sIST transforms an interval valued I in [0,1] intoa fuzzy set in ST, is defined as follows (Herrera et al., 2005)

sIST : I ! F ðST Þ;sIST ðIÞ ¼ ðsk; c

ikÞ=k 2 f0; . . . ; gg

� �;

ci ¼ maxy minflliðyÞ; lsk

ðyÞg;

where F(ST) is the set of fuzzy sets defined in ST, llið�Þ and

lSkð�Þ are the membership functions of the fuzzy sets asso-

ciated with the valued interval I and sk, respectively.

Once the input information was transformed into fuzzyset F(ST) in ST = {s0, . . . , sg}, the resulting values from thisstage must be transformed into 2-tuple, it is the goal of thenext stage.

4.5. Transformation of the linguistic values into 2-tuple

In this phase, we transform the fuzzy sets in the linguis-tic BLTS into 2-tuples in the BLTS to facilitate the rankingprocess involved in the exploitation phase of the evaluationprocess. Herrera and Martinez (2000b) present a function vwhich transforms a fuzzy set into a numerical value in aninterval of granularity of ST, [0,g]:

vðF ðST ÞÞ ¼ vðfðsi; cjÞ; j ¼ 0; . . . ; ggÞ ¼Pg

j¼0jcjPgj¼0cj

¼ b;

846 N. Halouani et al. / European Journal of Operational Research 195 (2009) 841–849

where the fuzzy set F(ST), can be obtained from sNST , sIST ,sSST .

Therefore, by applying the function D to b (definition 2)we obtain a collective preference relation whose values areexpressed by means of linguistic 2-tuples: D(v(s(#))) =D(b) = (si,a).

This obtained preference relation must be aggregate,which is the subject of the following stage.

4.6. The 2-tuples aggregation process

In the previous stage, the decision problem informationis represented by 2-tuples, and here it is incorporated, byusing the concept of symbolic translation D which makesit possible to transform the aggregation results to a 2-tupleand vice versa without any loss of information. This aggre-gation process will depend on the manner of using the mul-ticriteria method with fuzzy evaluations. In our case, theaggregation operators used are those of Promethee multi-criteria methods.

The implementation of the suggested method can bebrought back to the execution of the three following stages:choice of the parameters of the preference functions, pref-erences evaluation and determination of the preferencerelations.

4.6.1. Choice of the parameters of the preference functions

The values of the discrimination thresholds are includedin the evaluation process like ~pd

j ¼ DðvðsRST ðpdj ÞÞÞ and

~qdj ¼ DðvðsRST ðqd

j ÞÞÞ with pdj , qd

j 2 R; these thresholds arethe linguistic variables where the decision makers take lin-guistic terms to express their personal preferences.

4.6.2. The preference relation

The preference relation ~p is the function which trans-lates how an alternative (project) outclasses another. Thepreference index gives in fact a quantification of thisrelation of outclassing. It approximately represents thecredibility of the assertion ‘‘the alternative (project) ~ai out-

classes the alternative (project) ~ak”. This preference relationis characterized like an application of eA � eA ! ST�½�0:5; 0:5Þ. This degree of credibility of outclassing is cal-culated by means of the individual and total preferenceindices.

4.6.2.1. Notation. Let

– ST = {s0, . . . , sg} a set of linguistic terms with a suffi-ciently large cardinality,

– F ¼ fef 1; . . . ; eF ng a set of n fuzzy criteria,– A ¼ f~a1; . . . ; ~amg a set of m projects (alternatives) with

fuzzy evaluations,– d ¼ f~d1; . . . ; ~dDg a set of D decision makers,– P d

j ð~ai; ~akÞ 2 ½0; 1� is a preference function selected byeach decision maker, such as

– P dj : eA � eA ! ½0; 1�,

– ~wdj ¼ ðwd

j ; adj Þ 2 ST � ½�0:5; 0:5Þ: individual weight (for

each decision maker) which represents the relativeimportance of the fuzzy criterion j in the evaluationprocess,

– eW d ¼Pn

j¼1ðwdj ;a

dj Þ 2 ST � ½�0:5;0:5Þ ) eW d ¼

Pnj¼1 ~wd

j ¼f~wd

1 ; . . . ; ~wdng,

– b~wd

j ¼ D�1ðwdj ; a

dj Þ 2 ½0; g�,

– j 2 {1, . . . ,n}: n criteria,– i,k 2 {1, . . . ,m}: m alternatives.

4.6.2.2. Individual preference index. For each alternative(project) ~ai, belonging to the set of the alternatives (pro-jects) A ~pdð~ai; ~akÞ is a preference index of ~ai compared with~ak, taking in account all the criteria.

The mathematical formulation of the individual prefer-ence index, made by analogy with the preference index ofPromethee method, is as follows:

~pd : eA � eA ! ST � ½�0:5; 0:5Þ;

~pdð~ai; ~akÞ ¼1eW d

Xn

j¼1

~wdj � P d

j ð~ai; ~akÞ

¼ 1Pnj¼1ðwd

j ; adj ÞXn

j¼1

ðwdj ; a

dj Þ � P d

j ð~ai; ~akÞ:

However, to have a 2-tuple result, which expresses theequivalent information, we must use the function D whichtransforms the numerical values into 2-tuple without lossof information.

This preference index express the intensity ~ai comparedwith ~ak according to the point of view of each decisionmaker.

4.6.2.3. Total preference index. In a MCGDM approach,each member of the group introduces his own sets of valuesof the parameters into the individual multicriterion table.Thus, we have the individual preference indexes. We applies,then, a synthesis method to incorporate these individualpreferences in order to establish a total result. Our choicefor the synthesis method is fixed at the arithmetic mean.

We obtains, then, the following formula of the totalpreference index:

epð~ai; ~akÞ ¼ DXD

d¼1

1

m:D�1ðepdð~ai; ~akÞÞ

!

¼ D1

m

XD

d¼1

bdi

!:

The total preference index makes it possible to calculatethe average of a set of linguistic values without any loss ofinformation.

4.6.3. Preferences evaluation

The alternative (project) ~a1 must be compared with thealternative ~a2 but still with the other alternatives (projects)~a3; . . . ; ~am. In order to evaluate all the possible alternatives(projects), we define these following flows:

N. Halouani et al. / European Journal of Operational Research 195 (2009) 841–849 847

4.6.3.1. Entering flows. Entering flow ~/�ð~aiÞ is a measure-ment of which ~ai is dominated by the other alternatives(projects). This flow is calculated in the following way:

~/� : eA ! ST � ½�0:5; 0:5Þ is defined like

~/�ð~aiÞ ¼1

m� 1

Xk

~pð~ak; ~aiÞ ¼ D1

m� 1� D�1

Xm

k¼1;k–i

~pk;i

! !

¼ D1

m� 1� D�1

Xm

k¼1;k–i

ðpk;i; ak;iÞ ! !

D1

m� 1� D�1

Xm

k¼1;k–i

b~pk;i

! !8eAi 2 eA

with, b~pk;i ¼ D�1ðpk;i; ak;iÞ and m is the cardinality of eA.

4.6.3.2. Outgoing flows. Outgoing flow ~/�ð~aiÞ is a measure-ment of which ~ai is dominated by the other alternatives(projects). This flow is calculated in the following way:

~/þ : eA ! ST � ½�0:5; 0:5Þ is defined like

~/þð~aiÞ ¼1

m� 1

Xk

~pð~ai; ~akÞ

¼ D1

m� 1� D�1

Xm

k¼1;k–i

~pi;k

! !

¼ D1

m� 1� D�1

Xm

k¼1;k–i

ðpi;k; ai;kÞ ! !

¼ D1

m� 1� D�1

Xm

k¼1;k–i

b~pi;k

! !8eAi 2 eA

with, b~pi;k ¼ D�1ðpi;k; ai;kÞ and m is the cardinality of eA.

Fig. 1. Variables ef 1, ef 2 and ef 3.

4.6.3.3. Net flows. ~/ : eA ! ½�g; g� is defined like

~/ð~aiÞ ¼ D�1ð~/þð~aiÞÞ � D�1ð~/�ð~aiÞÞ

¼ 1

m� 1

Xm

k¼1;k–i

~b~pi;k � ~b~p

k;i 8Ai 2 eAwith, b~p

i;k ¼ D�1ðpi;k; ai;kÞ, b~pk;i ¼ D�1ðpk;i; ak;iÞ and m is the

cardinality of eA.

4.7. Exploitation process

Finally, from generated flows of the aggregation pro-cess, a new relation of outclassing of the set of alternatives(projects) is obtained, which is ~ai outclasses ~ak; Sð~ai; ~akÞ.Thus, we have, two types of arrangement.

4.7.1. Arrangement of Promethee-MD-2T-I

This method is based on the following considerations:‘‘an alternative will be as much better as its outgoing flowis large and/or its entering flow is weak”. In other term, if~/þð~aiÞP ~/þð~akÞ and ~/�ð~aiÞ 6 ~/�ð~akÞ.

4.7.2. Arrangement of Promethee-MD-2T-II

Promethee-MD-2T-II method determines a total pre-order (all the alternatives (projects) are ordered even if theyare incomparable). This total pre-order will be induced onthe basis of notion of the flow net ~/ð~aiÞ which we definedabove.

A priori, it could seem more useful for the decisionmaker to have the total pre-order rising from Promethee-MD-2T-II. However, we should not lose sight of the factthat the partial pre-order rising from Promethee-MD-2T-I is more realistic because it takes into account outclassingsrising from the input–output flow and, moreover, it makesit possible to highlight the incomparabilities.

Finally, using the relation of outclassing S, we obtain aclassification of the various alternatives (projects) of theproblem.

5. Numerical example

Let us suppose that an organization has an amount ofmoney to invest. There are three possible projects to investthis amount of money a1, a2 and a3. In this organization, allthe decisions are made according to opinions’ provided bythe persons in charge for three departments, R1, R2 and R3:department of marketing, department of risk analysis anddepartment of growth analysis, through four criteria: netpresent value ef 1, index of risk ef 2, environmental impactef 3 and social development ef 4. Being given that eachresponsible belongs to a different field of knowledge somecan have more facility to express their opinions with num-bers, while others prefer to express their opinions by meansof the linguistic expression, and even of the valued inter-vals. The three responsible try to reach a participativedecision.

Let us suppose that the first responsible expresses hispreferences by means of linguistic values in a linguistic termset, S. The second expresses his preference relation usingthe numerical values in [0,1]. And the third responsiblecan express his preferences using the values of preferenceinterval valued in [0,1]. We use for the fuzzy criteria ef 1,

Fig. 2. Variable ef 4.

848 N. Halouani et al. / European Journal of Operational Research 195 (2009) 841–849

ef 2 and ef 3 the linguistic variables given in Fig. 1, and forthe criterion ef 4 the linguistic variables of Fig. 2.

The preferences of each of the three decision makers arein the following Table 1.

These heterogeneous and multigranulary data must betransformed into 2-tuples through the functions describedin the preceding section. Then, we will have the 2-tuplesvalues present in the following Table 2.

Table 1Input data of each decision maker

d1 d2

F1 F2 F3 F4 F1 F2 F3

a1 H L M H 0.51 0.12 0.2a2 M VL M L 0.2 0.02 0.31a3 VH M H H 0.76 0.31 0.68wj VH H VH M 0.31 0.2 0.37Type 2 2 3 3 2 2 3pj L VL 0.1qj H VL 0.51 0.12

Table 22-Tuples input data of each decision maker

d1 d2eF 1eF 2

eF 3eF 4

eF 1eF 2

~a1 (S10,�0.36) (S4,0.5) (S7,0.29) (S10,0.44) (S7,0.12) (S2,�0.29~a2 (S7,0.29) (S2,0.45) (S7,0.29) (S4,�0.45) (S3,�0.14) (S0,0.27)~a3 (S12,0.3) (S7,0.29) (S10,�0.36) (S10,0.44) (S11,�0.43) (S4,0.43)ewj (S12,0.3) (S10,�0.36) (S12, 0.3) (S7,0.29) (S4,0.43) (S3,�0.14Type 2 2 3 3 2 2epj (S4,�0.45) (S2,0.45)~qj (S10,�0.36) (S2,0.45) (S7, 0.12) (S2,�0.29

Table 3Individual preference indexes of each decision makerepd d1 d2

~a1 ~a2 ~a3 ~a1

~a1 – (S3,�0.37) (S0,0) –~a2 (S0,0) – (S0,0) (S6,�0.3)~a3 (S6,�0.19) (S8,0.44) – (S12,0.3)

From these transformed data and by calculating theindividual preference indexes relating to each of the threedecision makers, we find the results in Table 3.

These individual preference indexes are aggregate inthe following way in order to establish a total result inTable 4.

We arrive now at the stage of flows exploitation alreadyarisen. The exploitation process is carried out through twotypes of arrangement:

5.1. Arrangement of Promethee-MD-2T-I

To be able to lead to this arrangement, we must firstlycalculate the outclassing relation between the variousprojects.

According to Table 5, we obtain the followingarrangement:

d3

F4 F1 F2 F3 F4

0.6 [0.55,0.7] [0.05,0.15] [0.4,0.6] [0.6,0.8]0.152 [0.2,0.4] [0,0.06] [0.4,0.6] [0.05,0.15]0.77 [0.75,0.9] [0.4,0.6] [0.55,0.7] [0.6,0.8]0.12 [0.75,0.9] [0.2,0.3] [0.35,0.6] [0.25,0.35]3 2 2 3 30.05 [0.1,0.15] [0.01,0.06]

[0.55,0.7] [0.16,0.22]

d3eF 3eF 4

eF 1eF 2

eF 3eF 4

) (S3,�0.14) (S8,0.29) (S9,�0.33) (S2,�0.23) (S7,0) (S10,�0.28)(S4,0.43) (S2,0.17) (S4,0.3) (S0,0.46) (S7,0) (S2, �0.23)(S9,0.43) (S12,�0.29) (S11,0.5) (S7,0) (S9,�0.33) (S10,�0.28)

) (S5, 0.29) (S2,�0.29) (S7,�0.33) (S4,�0.41) (S7, �0.33) (S4,0.23)3 3 2 2 3 3(S2, 0.17) (S5,�0.43) (S2,�0.25) (S0,0.14)

) (S9,�0.33) (S2,0.29)

d3

~a2 ~a3 ~a1 ~a2 ~a3

(S2,0.1) (S0,0) – (S3,0) (S7,0)– (S0,0) (S7,0) – (S7,0)(S10,�0.17) – (S7,0) (S3,�0.46) –

Table 4Total preference indexes

~p ~a1 ~a2 ~a3~/þð~aiÞ

~a1 – (S3,�0.42) (S0,0) (S1,0.29)~a2 (S2,�0.1) – (S0,0) (S1,�0.05)~a3 (S6,0.4) (S7,�0.06) – (S7,�0.33)~/�ð~aiÞ (S4,0.15) (S7,�0.24) (S0,0) –

Table 5Outclassing relation of Promethee-MD-2T

N. Halouani et al. / European Journal of Operational Research 195 (2009) 841–849 849

5.2. Arrangement of Promethee-MD-2T-II

We was already found that ~/ð~a1Þ ¼ �1:143, ~/ð~a2Þ ¼�2:9 and ~/ð~a3Þ ¼ 2:56. From where, we obtain the follow-ing arrangement:

6. Conclusion

Project selection is a complex decision-making problem.It handles a large amount of data, which can come fromquantitative and qualitative sources alike and so it wouldbe useful to develop suitable decision-making methods tofacilitate the project selection task. Currently, a new inte-gration of linguistic 2-tuples and MCDM Promethee isproposed. The 2-tuple representation model is used to rep-resent linguistic, numerical and interval valued informa-tion. However, all suitable information for a decisionproblem is represented by this 2-tuple model, in the formof a simple internal representation which includes all origi-nal information, generally, of multigranulary nature (a lin-guistic variable with different granularity and/or semantics,for each fuzzy criterion). Then, this information represen-tation is introduced into the selected multicriteria aggrega-tion process of Promethee method, and the obtained resultsare presented in the initial field of information. Thus, theapproach proposed here offers to the decision makers sim-pler and wider application of the aggregation operators ofthe Promethee multicriteria method without sacrificing any

of its advantageous properties. Also, it introduces newinformation types into the decision-making process result-ing in a more realistic selection process where the impreci-sion of the data is taken into account, and helps to gain aninsight into the decision maker’s preference structure.Although, it is illustrated by a numerical selection problem,however, it can also be applied to any other areas of man-agement decision problems.

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