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Chapter 5
PROMETHEE METHODS
Jean-Pierre Brans
Centrum voor Statistiek en Operationeel Onderzoek
Vrije Universiteit Brussel
Pleinlaan 2, B-1050 Brussels
Belgium
jpbrans@vub.ac.be
Bertrand Mareschal
Service de Mathématiques de la Gestion
Université Libre de Bruxelles
Boulevard du Triomphe CP 210-01, B-1050 Brussels
Belgium
bmaresc@ulb.ac.be
Abstract
Keywords:
This paper gives an overview of the PROMETHEE-GAIA methodology for
MCDA. It starts with general comments on multicriteria problems, stressing that
a multicriteria problem cannot be treated without additional information related
to the preferences and the priorities of the decision-makers. The information re-
quested by PROMETHEE and GAIA is particularly clear and easy to define for
both decision-makers and analysts. It consists in a preference function associ-
ated to each criterion as well as weights describing their relative importance. The
PROMETHEE I, the PROMETHEE II complete ranking, as well as the GAIA
visual interactive module are then described and commented. The two next sec-
tions are devoted to the PROMETHEE VI sensitivity analysis procedure (human
brain) and to the PROMETHEE V procedure for multiple selection of alternatives
under constraints. An overview of the PROMETHEE GDSS procedure for group
decision making is then given. Finally the DECISION LAB software implemen-
tation of the PROMETHEE-GAIA methodology is described using a numerical
example.
MCDA, outranking methods, PROMETHEE-GAIA, DECISION LAB.
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MULTIPLE CRITERIA DECISION ANALYSIS
1.
History
2.
Multicriteria Problems
The PROMETHEE I (partial ranking) and PROMETHEE II (complete ranking)
were developed by J.P. Brans and presented for the first time in 1982 at a
conference organised by R. Nadeau and M. Landry at the Université Laval,
Québec, Canada (L’Ingéniérie de la Décision. Elaboration d’instruments d’Aide
à la Décision). The same year several applications using this methodology were
already treated by G. Davignon in the field of Heath care.
A few years later J.P. Brans and B. Mareschal developed PROMETHEE
III (ranking based on intervals) and PROMETHEE IV (continuous case). The
same authors proposed in 1988 the visual interactive module GAIA which is
providing a marvellous graphical representation supporting the PROMETHEE
methodology.
In 1992 and 1994, J.P. Brans and B. Mareschal further suggested two nice
extensions: PROMETHEE V (MCDA including segmentation constraints) and
PROMETHEE VI (representation of the human brain).
A considerable number of successful applications has been treated by the
PROMETHEE methodology in various fields such as Banking, Industrial Loca-
tion, Manpower planning, Water resources, Investments, Medicine, Chemistry,
Health care, Tourism, Ethics in OR, Dynamic management, ... The success
of the methodology is basically due to its mathematical properties and to its
particular friendliness of use.
Let us consider the following multicriteria problem:
where
A
is a finite set of possible alternatives and
a set of evaluation criteria. There is no objec-
tion to consider some criteria to be maximised and the others to be minimised.
The expectation of the decision-maker is to identify an alternative optimising
all the criteria.
Usually this is a ill-posed mathematical problem as there exists no alternative
optimising all the criteria at the same time. However most (nearly all) human
problems have a multicriteria nature. According to our various human aspira-
tions, it makes no sense, and it is often not fair, to select a decision based on one
evaluation criterion only. In most of cases at least technological, economical,
environmental and social criteria should always be taken into account. Multi-
criteria problems are therefore extremely important and request an appropriate
treatment.
The basic data of a multicriteria problem (5.1) consist of an evaluation table
(Table 5.1).
PROMETHEE Methods
165
Let us consider as an example the problem of an individual purchasing a car.
Of course the price is important and it should be minimised. However it is clear
that in general individuals are not considering only the price. Not everybody
is driving the cheapest car! Most people would like to drive a luxury or sports
car at the price of an economy car. Indeed they consider many criteria such as
price, reputation, comfort, speed, reliability, consumption, … As there is no
car optimising all the criteria at the same time, a compromise solution should
be selected. Most decision problems have such a multicriteria nature.
The solution of a multicriteria problem depends not only on the basic data
included in the evaluation table but also on the decision-maker himself. All
individuals do not purchase the same car. There is no absolute best solution!
The best compromise solution also depends on the individual preferences of
each decision-maker, on the “brain” of each decision-maker.
Consequently, additional information representing these preferences is re-
quired to provide the decision maker with useful decision aid.
The natural dominance relation associated to a multicriteria problem of type
(5.1) is defined as follows:
For each
where P, I, and R respectively stand for preference, indifference and incompa-
rability. This definition is quite obvious. An alternative is better than another if
it is at least as good as the other on all criteria. If an alternative is better on a cri-
terion and the other one better on criterion it is impossible to decide which
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MULTIPLE CRITERIA DECISION ANALYSIS
is the best one without additional information. Both alternatives are therefore
incomparable!
Alternatives which are not dominated by any other are called efficient solu-
tions. Given an evaluation table for a particular multicriteria problem, most of
the alternatives (often all of them) are usually efficient. The dominance relation
is very poor on P and I. When an alternative is better on one criterion, the other
is often better on another criterion. Consequently incomparability holds for
most pairwise comparisons, so that it is impossible to decide without additional
information. This information can for example include:
Trade-offs between the criteria;
A value function aggregating all the criteria in a single function in order
to obtain a mono-criterion problem for which an optimal solution exists;
Weights giving the relative importance of the criteria;
Preferences associated to each pairwise comparison within each criterion;
Thresholds fixing preference limits;
Many multicriteria decision aid methods have been proposed. All these meth-
ods start from the same evaluation table, but they vary according to the addi-
tional information they request. The PROMETHEE methods require very clear
additional information, that is easily obtained and understood by both decision-
makers and analysts.
The purpose of all multicriteria methods is to enrich the dominance graph, i.e.
to reduce the number of incomparabilities (R). When a utility function is built,
the multicriteria problem is reduced to a single criterion problem for which an
optimal solution exists. This seems exaggerated because it relies on quite strong
assumptions (do we really make all our decisions based on a utility function
defined somewhere in our brains?) and it completely transforms the structure
of the decision problem. For this reason B. Roy proposed to build outranking
relations including only realistic enrichments of the dominance relation (see
[86] and [87]). In that case, not all the incomparabilities are withdrawn but
the information is reliable. The PROMETHEE methods belong to the class of
outranking methods.
In order to build an appropriate multicriteria method some requisites could
be considered:
Requisite 1: The amplitude of the deviations between the evaluations of the
alternatives within each criterion should be taken into account:
PROMETHEE Methods
167
This information can easily be calculated, but is not used in th e efficiencytheory.
When these deviations are negligible the dominance relation can possibly be
enriched.
Requisite 2: As the evaluations of each criterion are expressed in their
own units, the scaling effects should be completely eliminated. It is not accept-
able to obtain conclusions depending on the scales in which the evaluations
are expressed. Unfortunately not all multicriteria procedures are respecting this
requisite!
Requisite 3: In the case of pairwise comparisons, an appropriate multicriteria
method should provide the following information:
a is preferred to b;
a and b are indifferent;
a and b are incomparable.
The purpose is of course to reduce as much as possible the number of incompa-
rabilities, but not when it is not realistic. Then the procedure may be considered
as fair. When, for a particular procedure, all the incomparabilities are system-
atically withdrawn the provided information can be more disputable.
Requisite 4: Different multicriteria methods request different additional in-
formation and operate different calculation procedures so that the solutions
they propose can be different. It is therefore important to develop methods be-
ing understandable by the decision-makers. “Black box” procedures should be
avoided.
Requisite 5: An appropriate procedure should not include technical param-
eters having no significance for the decision-maker. Such parameters would
again induce “Black box” effects.
Requisite 6: An appropriate method should provide information on the con-
flicting nature of the criteria.
Requisite 7: Most of the multicriteria methods are allocating weights of
relative importance to the criteria. These weights reflects a major part of the
“brain” of the decision-maker. It is not easy to fix them. Usually the decision-
makers strongly hesitate. An appropriate method should offer sensitivity tools
to test easily different sets of weights.
The PROMETHEE methods and the associated GAIA visual interactive
module are taking all these requisites into account. On the other hand some
mathematical properties that multicriteria problems possibly enjoy can also be
considered. See for instance [95]. Such properties related to the PROMETHEE
methods have been analysed by [7] in a particularly interesting paper.
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MULTIPLE CRITERIA DECISION ANALYSIS
The next sections describe the PROMETHEE I and II rankings, the GAIA
methods, as well as the PROMETHEE V and VI extensions of the methodology.
The PROMETHEE III and IV extensions are not discussed here. Additional in-
formation can be found in [17]. Several actual applications of the PROMETHEE
methodology are also mentioned in the list of references.
3.
The PROMETHEE Preference Modelling Information
3.1
Information between the Criteria
The PROMETHEE methods were designed to treat multicriteria problems of
type (5.1) and their associated evaluation table.
The additional information requested to run PROMETHEE is particularly
clear and understandable by both the analysts and the decision-makers. It con-
sists of:
Information between the criteria;
Information within each criterion.
Table 5.2 should be completed, with the understanding that the set
represents weights of relative importance of the different criteria.
These weights are non-negative numbers, independent from the measurement
units of the criteria. The higher the weight, the more important the criterion.
There is no objection to consider normed weights, so that:
In the PROMETHEE software PROMCALC and DECISION LAB, the user
is allowed to introduce arbitrary numbers for the weights, making it easier to
express the relative importance of the criteria. These numbers are then divided
by their sum so that the weights are normed automatically.
Assessing weights to the criteria is not straightforward. It involves the prior-
ities and perceptions of the decision-maker. The selection of the weights is his
space of freedom. PROMCALC and DECISION LAB include several sensitiv-
ity tools to experience different set of weights in order to help to fix them.
PROMETHEE Methods
169
3.2
Information within the Criteria
PROMETHEE is not allocating an intrinsic absolute utility to each alternative,
neither globally, nor on each criterion. We strongly believe that the decision-
makers are not proceeding that way. The preference structure of PROMETHEE
is based on pairwise comparisons. In this case the deviation between the eval-
uations of two alternatives on a particular criterion is considered. For small
deviations, the decision-maker will allocate a small preference to the best al-
ternative and even possibly no preference if he considers that this deviation
is negligible. The larger the deviation, the larger the preference. There is no
objection to consider that these preferences are real numbers varying between
0 and 1. This means that for each criterion the decision-maker has in mind a
function
where:
and for which:
In case of a criterion to be maximised, this function is giving the preference
of over for observed deviations between their evaluations on criterion
It should have the following shape (see Figure 5.1). The preferences equals 0
when the deviations are negative.
The following property holds:
Figure 5.1. Preference function.
For criteria to be minimised, the preference function should be reversed or
alternatively given by:
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MULTIPLE CRITERIA DECISION ANALYSIS
We have called the pair the generalised criterion associated
to criterion Such a generalised criterion has to be defined for each criterion.
In order to facilitate the identification six types of particular preference functions
have been proposed (see table 5.3).
PROMETHEE Methods
171
In each case 0, 1 or 2 parameters have to be defined, their significance is
clear: q is a threshold or indifference;
p is a threshold of strict preference;
s is an intermediate value between
and
The indifference threshold is the largest deviation which is considered as
negligible by the decision maker, while the preference threshold is the smallest
deviation which is considered as sufficient to generate a full preference.
The identification of a generalised criterion is then limited to the selection
of the appropriate parameters. It is an easy task.
The PROMCALC and DECISION LAB software are proposing these six
shapes only. As far as we know they have been satisfactory in most real-world
applications. However there is no objection to consider additional generalised
criteria.
In case of type 5 a threshold of indifference and a threshold of strict pref-
erence have to be selected.
In case of a Gaussian criterion (type 6) the preference function remains
increasing for all deviations and has no discontinuities, neither in its shape, nor
in its derivatives. A parameter has to be selected, it defines the inflection point
of the preference function. We then recommend to determine first a and a
and to fix in between. If is close to the preferences will be reinforced for
small deviations, while close to they will be softened.
As soon as the evaluation table is given, and the weights and
the generalised criteria are defined for
the PROMETHEE procedure can be applied.
4.
The PROMETHEE I and II Rankings
4.1
Aggregated Preference Indices
The PROMETHEE procedure is based on pairwise comparisons (cfr. [8]–[16],
[59], [60]). Let us first define aggregated preference indices and outranking
flows.
Let
and
let:
is expressing with which degree is preferred to over all the criteria
and how is preferred to In most of the cases there are criteria for
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MULTIPLE CRITERIA DECISION ANALYSIS
which is better than and criteria for which is better than consequently
and are usually positive. The following properties hold for all
It is clear that:
As soon as and are computed for each pair of alternatives of
A, a complete valued outranking graph, including two arcs between each pair
of nodes, is obtained (see Figure 5.2).
Figure 5.2. Valued outranking graph.
4.2
Outranking Flows
Each alternative is facing other alternatives in A. Let us define the
two following outranking flows:
the positive outranking flow:
the negative outranking flow:
PROMETHEE Methods
173
Figure 5.3. The PROMETHEE outranking flows.
The positive outranking flow expresses how an alternative is outranking
all the others. It is its power, its outranking character. The higher the
better the alternative (see Figure 5.3(a)).
The negative outranking flow expresses how an alternative is outranked by
all the others. It is its weakness, its outranked character. The lower the
better the alternative (see Figure 5.3(b)).
4.3
The PROMETHEE I Partial Ranking
The PROMETHEE I partial ranking is obtained from the positive
and the negative outranking flows. Both flows do not usually induce the same
rankings. PROMETHEE I is their intersection.
where respectively stand for preference, indifference and incompa-
rability.
When a higher power of is associated to a lower weakness of with
regard to The information of both outranking flows is consistent and may
therefore be considered as sure.
When both positive and negative flows are equal.
When a higher power of one alternative is associated to a lower weak-
ness of the other. This often happens when is good on a set of criteria on which
is weak and reversely is good on some other criteria on which is weak. In
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MULTIPLE CRITERIA DECISION ANALYSIS
such a case the information provided by both flows is not consistent. It seems
then reasonable to be careful and to consider both alternatives as incomparable.
The PROMETHEE I ranking is prudent: it will not decide which action is best
in such cases. It is up to the decision-maker to take his responsibility.
4.4
The PROMETHEE II Complete Ranking
4.5
The Profiles of the Alternatives
PROMETHEE II consists of the complete ranking. It is often the
case that the decision-maker requests a complete ranking. The net outranking
flow can then be considered.
It is the balance between the positive and the negative outranking flows. The
higher the net flow, the better the alternative, so that:
When PROMETHEE II is considered, all the alternatives are comparable. No
incomparabilities remain, but the resulting information can be more disputable
because more information gets lost by considering the difference (5.16).
The following properties hold:
When is more outranking all the alternatives on all the criteria,
when it is more outranked.
In real-world applications, we recommend to both the analysts and the
decision-makers to consider both PROMETHEE I and PROMETHEE II. The
complete ranking is easy to use, but the analysis of the incomparabilities often
helps to finalise a proper decision.
As the net flow provides a complete ranking, it may be compared with
a utility function. One advantage of is that it is built on clear and simple
preference information (weights and preferences functions) and that it does rely
on comparative statements rather than absolute statements.
According to the definition of the positive and the negative outranking flows
(5.13) and (5.14) and of the aggregated indices (5.10), we have:
PROMETHEE Methods
175
Consequently,
if
is the single criterion net flow obtained when only criterion is
considered (100% of the total weight is allocated to that criterion). It expresses
how an alternative is outranking or outranked by
all the other alternatives on criterion
The profile of an alternative consists of the set of all the single criterion net
flows:
Figure 5.4. Profile of an alternative.
The profiles of the alternatives are particularly useful to appreciate their
“quality” on the different criteria. It is extensively used by decision-makers to
finalise their appreciation.
According to (5.20), we observe that the global net flow of an alternative is
the scalar product between the vector of the weights and the profile vector of
this alternative. This property will be extensively used when building up the
GAIA plane.
5.
The GAIA Visual Interactive Module
Let us first consider the matrix of the single criterion net flows of all
the alternatives as defined in (5.21).
5.1
The GAIA Plane
The information included in matrix M is more extensive than the one in the
evaluation table 5.1, because the degrees of preference given by the generalised
criteria are taken into account in M. Moreover the are expressed on their
own scale, while the are dimensionless. In addition, let us observe, that
M is not depending on the weights of the criteria.
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MULTIPLE CRITERIA DECISION ANALYSIS
Consequently the set of the alternatives can be represented as a cloud of
points in a space. According to (5.18) this cloud is centered at
the origin. As the number of criteria is usually larger than two, it is impossible
to obtain a clear view of the relative position of the points with regard to the
criteria. We therefore project the information included in the
space on a plane. Let us project not only the points representing the alternatives
but also the unit vectors of the coordinate-axes representing the criteria. We
then obtain:
Figure 5.5. Projection on the GAIA plane.
The GAIA plane is the plane for which as much information as possible
is preserved after projection. According to the principal components analysis
technique it is defined by the two eigenvectors corresponding to the two largest
eigenvalues of the covariance matrix of the single criterion net flows.
Of course some information get lost after projection. The GAIA plane is a
meta model (a model of a model). Let be the quantity of information preserved.
PROMETHEE Methods
177
In most applications we have treated so far was larger than 60% and in many
cases larger than 80%. This means that the information provided by the GAIA
plane is rather reliable. This information is quite rich, it helps to understand the
structure of a multicriteria problem.
5.2
Graphical Display of the Alternatives and of the
Criteria
Let be the projections of the points representing
the alternatives and let
be the projections of the
unit
vectors of the coordinates axes of representing the criteria. We then obtain
a GAIA plane of the following type:
Figure 5.6. Alternatives and criteria in the GAIA plane.
Then the following properties hold (see [59] and [16]) provided that is
sufficiently high:
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MULTIPLE CRITERIA DECISION ANALYSIS
The longer a criterion axis in the GAIA plane, the more discrim-
inating this criterion.
Criteria expressing similar preferences are represented by axes
oriented in approximatively the same direction.
Criteria expressing conflicting preferences are oriented in oppo-
site directions.
Criteria that are not related to each others in terms of preferences
are represented by orthogonal axes.
Similar alternatives are represented by points located close to
each other.
Alternatives being good on a particular criterion are represented
by points located in the direction of the corresponding criterion
axis.
P1:
P2:
P3:
P4:
P5:
P6:
On the example of Figure 5.6, we observe:
That the criteria and are expressing similar preferences and
that the alternatives and are rather good on these criteria.
That the criteria and are also expressing similar preferences
and that the alternatives and are rather good on them.
That the criteria and are rather independent
That the criteria and are strongly conflicting with the criteria
and
That the alternatives and are rather good on the criteria
and
That the alternatives and are rather good on the criteria
and
That the alternatives and are never good, never bad on all the criteria,
Although the GAIA plane includes only a percentage of the total infor-
mation, it provides a powerful graphical visualisation tool for the analysis of a
multicriteria problem. The discriminating power of the criteria, the conflicting
aspects, as well as the “quality” of each alternative on the different criteria are
becoming particularly clear.
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179
5.3
The PROMETHEE Decision Stick. The
PROMETHEE Decision Axis
Let us now introduce the impact of the weights in the GAIA plane. The vector
of the weights is obviously also a vector of According to (5.20), the PRO-
METHEE net flow of an alternative is the scalar product between the vector
of its single criterion net flows and the vector of the weights:
This also means that the PROMETHEE net flow of is the projection of the
vector of its single criterion net flows on Consequently, the relative positions
of the projections of all the alternatives on provides the PROMETHEE II
ranking.
Figure 5.7. PROMETHEE II ranking. PROMETHEE decision axis and stick.
Clearly the vector plays a crucial role. It can be represented in the GAIA
plane by the projection of the unit vector of the weights. Let be this projection,
and let us call the PROMETHEE decision axis.
On the example of Figure 5.7, the PROMETHEE ranking is:
A realistic view of this ranking is given in the GAIA plane although
some inconsistencies due to the projection can possibly occur.
If all the weights are concentrated on one criterion, it is clear that the PRO-
METHEE decision axis will coincide with the axis of this criterion in the
GAIA plane. Both axes are then the projection of a coordinate unit vector
of When the weights are distributed over all the criteria, the PROME-
THEE decision axis appears as a weighted resultant of all the criterion axes
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MULTIPLE CRITERIA DECISION ANALYSIS
If is long, the PROMETHEE decision axis has a strong decision power
and the decision-maker is invited to select alternatives as far as possible in its
direction.
If is short, the PROMETHEE decision axis has no strong decision power.
It means, according to the weights, that the criteria are strongly conflicting and
that the selection of a good compromise is a hard problem.
When the weights are modified, the positions of the alternatives and of the
criteria remain unchanged in the GAIA plane. The weight vector appears as a
decision stick that the decision-maker can move according to his preferences in
favour of particular criteria. When a sensitivity analysis is applied by modify-
ing the weights, the PROMETHEE decision stick and the PROMETHEE
decision axis are moving in such a way that the consequences for decision-
making are easily observed in the GAIA plane (see Figure 5.8).
Decision-making for multicriteria problems appears, thanks to this method-
ology, as a piloting problem. Piloting the decision stick over the GAIA plane.
Figure 5.8. Piloting the PROMETHEE decision stick.
The PROMETHEE decision stick and the PROMETHEE decision axis pro-
vide a strong sensitivity analysis tool. Before finalising a decision we recom-
mend to the decision-maker to simulate different weight distributions. In each
case the situation can easily be appreciated in the GAIA plane, the recom-
mended alternatives are located in the direction of the decision axis. As the
alternatives and the criteria remain unchanged when the PROMETHEE deci-
sion stick is moving, the sensitivity analysis is particularly easy to manage.
Piloting the decision stick is instantaneously operated by the PROMCALC and
the DECISION LAB softwares. The process is displayed graphically so that
the results are easy to appreciate.
PROMETHEE Methods
181
6.
The PROMETHEE VI Sensitivity Tool (The “Human
Brain”)
The PROMETHEE VI module provides the decision-maker with additional
information on his own personal view of his multicriteria problem. It allows
to appreciate whether the problem is hard or soft according to his personal
opinion.
It is obvious that the distribution of the weights plays an important role in
all multicriteria problems. As soon as the weights are fixed, a final ranking
is proposed by PROMETHEE II. In most of the cases the decision-maker is
hesitating to allocate immediately precise values of the weights. His hesitation
is due to several factors such as indetermination, imprecision, uncertainty, lack
o
f
control, … on the real-world situation.
However the decision-maker has usually in mind some order of magnitude
on the weights, so that, despite his hesitations, he is able to give some intervals
including their correct values. Let these intervals be:
Let us then consider the set of all the extreme points of the unit vectors
associated to all allowable weights. This set is limiting an area on the unit
hypersphere in Let us project this area on the GAIA plane and let us call
(HB) (“Human Brain”) the obtained projection. Obviously (HB) is the area
including all the extreme points of the PROMETHEE decision axis for all
allowable weights.
Figure 5.9. “Human Brain”.
Two particular situations can occur:
(HB) does not include the origin of the GAIA plane. In this case, when
the weights are modified, the PROMETHEE decision axis remains
globally oriented in the same direction and all alternatives located in this
direction are good. The multicriteria problem is rather easy to solve, it is
a soft problem.
S1:
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MULTIPLE CRITERIA DECISION ANALYSIS
Figure 5.10. Two types of decision problems.
Reversely if (HB) is including the origin, the PROMETHEE decision
axis can take any orientation. In this case compromise solutions can
be possibly obtained in all directions. It is then actually difficult to make
a final decision. According to his preferences and his hesitations, the
decision-maker is facing a hard problem.
S2:
In most of the practical applications treated so far, the problems appeared
to be rather soft and not too hard. This means that most multicriteria problems
offer at the same time good compromises and bad solutions. PROMETHEE
allows to select the good ones.
7.
PROMETHEE V: MCDA under Constraints
PROMETHEE I and II are appropriate to select one alternative. However in
some applications a subset of alternatives must be identified, given a set of
constraints. PROMETHEE V is extending the PROMETHEE methods to that
particular case. (see [13]).
Let be the set of possible alternatives and let us associate
the following boolean variables to them:
The PROMETHEE V procedure consists of the two following steps:
STEP 1: The multicriteria problem is first considered without constraints.
The PROMETHEE II ranking is obtained for which the net flows
have been computed.
PROMETHEE Methods
183
STEP 2: The following {0,1} linear program is then considered in order to
take into account the additional constraints.
where ~ holds for =, or The coefficients of the objective function (5.25)
are the net outranking flows. The higher the net flow, the better the alternative.
The purpose of the {0,1} linear program is to select alternatives collecting as
much net flow as possible and taking the constraints into account.
The constraints (5.26) can include cardinality, budget, return, investment,
marketing,... constraints. They can be related to all the alternatives or possibly
to some clusters.
After having solved the {0,1} linear program, a subset of alternatives sat-
isfying the constraints and providing as much net flow as possible is obtained.
Classical 0-1 linear programming procedures may be used.
The PROMCALC software includes this PROMETHEE V procedure.
8.
The PROMETHEE GDSS Procedure
The PROMETHEE Group Decision Support System has been developed to pro-
vide decision aid to a group of decision-makers
(see [54]). It has been designed to be used in a GDSS room in-
cluding a PC, a printer and a video projector for the facilitator, and R working
stations for the DM’s. Each working station includes room for a DM (and pos-
sibly a collaborator), a PC and Tel/Fax so that the DM’s can possibly consult
their business base. All the PC’s are connected to the facilitator through a local
network.
There is no objection to use the procedure in the framework of teleconference
or video conference systems. It this case the DM’s are not gathering in a GDSS
room, they directly talk together through the computer network.
One iteration of the PROMETHEE GDSS procedure consists in 11 steps
grouped in three phases:
PHASE I: Generation of alternatives and criteria
PHASE II: Individual evaluation by each DM
PHASE III: Global evaluation by the group
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MULTIPLE CRITERIA DECISION ANALYSIS
Feedback is possible after each iteration for conflict resolution until a final
consensus is reached.
8.1
PHASE I: Generation of Alternatives and Criteria
STEP 1: First contact Facilitator — DM’s
The facilitator meets the DM’s together or individually in order to enrich his
knowledge of the problem. Usually this step takes place in the business base of
each DM prior to the GDSS room session.
STEP 2: Problem description in the GDSS room
The facilitator describes the computer infrastructure, the PROMETHEE meth-
odology, and introduces the problem.
STEP 3: Generation of alternatives
It is a computer step. Each DM implements possible alternatives including their
extended description. For instance strategies, investments, locations,production
schemes, marketing actions, … depending on the problem.
STEP 4: Stable set of alternatives
All the proposed alternatives are collected and displayed by the facilitator one
by one on the video-screen, anonymously or not. An open discussion takes
place, alternatives are canceled, new ones are proposed, combined ones are
merged, until a stable set of alternatives is reached.
This brainstorming procedure is extremely useful, it often generates alternatives
that were unforeseen at the beginning.
STEP 5: Comments on the alternatives
It is again a computer step. Each DM implements his comments on all the
alternatives. All these comments are collected and displayed by the facilitator.
Nothing gets lost. Complete minutes can be printed at any time.
STEP 6: Stable set of evaluation criteria
The same procedure as for the alternatives is applied to define a stable set of
evaluation criteria Computer and open dis-
cussion activities are alternating. At the end the frame of an evaluation table
(Type Table 5.1) is obtained. This frame consists in a matrix. This
ends the first phase. Feedbacks are already possible to be sure a stable set of
alternatives and criteria is reached.
8.2
PHASE II: Individual Evaluation by each DM
Let us suppose that each DM has a decision power given by a non-negative
weight so that:
PROMETHEE Methods
185
STEP 7: Individual evaluation tables
The evaluation table has to be completed by each DM. Some evalua-
tion values are introduced in advance by the facilitator if there is an objective
agreement on them (prices, volumes, budgets, …). If not each DM is allowed
to introduce his own values.
All the DM’s implement the same matrix, if some of them are not
interested in particular criteria, they can simply allocate a zero weight to these
criteria.
STEP 8: Additional PROMETHEE information
Each DM develops his own PROMETHEE-GAIA analysis. Assistance is given
by the facilitator to provide the PROMETHEE additional information on the
weights and the generalised criteria.
STEP 9: Individual PROMETHEE-GAIA analysis
The PROMETHEE I and II rankings, the profiles of the alternatives and the
GAIA plane as well as the net flow vector are instantaneously obtained,
so that each DM gets his own clear view of the problem.
8.3
PHASE III: Global Evaluation by the Group
STEP 10: Display of the individual investigations
The rankings and the GAIA plane of each DM are collected and displayed by
the facilitator so that the group of all DM’S is informed of the potential conflicts.
STEP 11: Global evaluation
The net flow vectors of all the DM’s are collected by
the facilitator and put in a matrix. It is a rather small matrix which is
easy to analyse. Each criterion of this matrix expresses the point of view of a
particular DM.
Each of these criteria has a weight and an associated generalised criterion
of Type so that the preferences allocated to the deviations between
the values will be proportional to these deviations.
A global PROMETHEE II ranking and the associated GAIA plane are then
computed. As each criterion is representing a DM, the conflicts between them
are clearly visualised in the GAIA plane. See for example Figure 5.11 where
is strongly in conflict with and
The associated PROMETHEE decision axis gives the direction in which
to decide according to the weights allocated to the DM’s.
If the conflicts are too sensitive the following feedbacks could be considered:
Back to the weighting of the DM’s. Back to the individual evaluations. Back
to the set of criteria. Back to the set of alternatives. Back to the starting phase
and to include an additional stakeholder (“DM”) such as a social negotiator or
a government mediator.
The whole procedure is summarised in the following scheme (Figure 5.12):
186
MULTIPLE CRITERIA DECISION ANALYSIS
Figure 5.11. Conflict between DM’s.
Figure 5.12. Overview PROMETHEE GDSS procedure.
9.
The DECISION LAB Software
DECISION LAB is the current software implementation of the PROMETHEE
and GAIA methods. It has been developed by the Canadian company Visual
Decision, in cooperation with the authors. It replaces the PROMCALC software
that the authors had previously developed.
DECISION LAB is a Windows application that uses a typical spreadsheet
interface to manage the data of a multicriteria problem (Figure 5.13).
PROMETHEE Methods
187
Figure 5.13. Main window.
All the data related to the PROMETHEE methods (evaluations, preference
functions, weights, ...) can be easily defined and input by the user. Besides,
DECISION LAB provides the user with additional features like the definition
of qualitative criteria, the treatment of missing values in the multicriteria table
or the definition of percentage (variable) thresholds in the preference functions.
Categories of alternatives or criteria can also be defined to better identify sub-
groups of related items and to facilitate the analysis of the decision problem.
All the PROMETHEE and GAIA computations take place in real-time and
any data modification is immediately reflected in the output windows. The PRO-
METHEE rankings, action profiles and GAIA plane are displayed in separate
windows and can easily be compared (Figure 5.14).
Several interactive tools and displays are available for facilitating extensive
sensitivity and robustness analyses. It is possible to compute weight stability
intervals for individual criteria or categories of criteria. The walking weights
display (Figure 5.15) can be used to interactively modify the weights of the cri-
teria and immediately see the impact of the modification on the PROMETHEE
II complete ranking and on the position of the decision axis in the GAIA plane.
This can particularly useful when the decision-maker has no clear idea of the
appropriate weighting of the criteria and wants to explore his space of freedom.
The PROMETHEE GDSS procedure is also integrated in DECISION LAB
through the definition of several scenarios for a same decision problem. Sce-
narios share the same lists of alternatives and criteria but can include different
preference functions, different sets of weights and even different evaluations for
some criteria. Each scenario can be analysed separately using PROMETHEE
and GAIA. But it is also possible to aggregate all the scenarios and to generate
188
MULTIPLE CRITERIA DECISION ANALYSIS
Figure 5.14. PROMETHEE rankings, action profiles, GAIA plane.
Figure 5.15. Walking weights.
the PROMETHEE group rankings as well as the group GAIA plane. Conflicts
between decision-makers can easily be detected and analysed.
At the end of an analysis, the DECISION LAB report generator can produce
tailor-made reports including the tables and graphics required by the user. The
PROMETHEE Methods
189
reports are in the html format so that they can easily be edited in a word processor
or published on paper or on the web.
DECISION LAB can easily be interfaced with other programs like for in-
stance databases. Its own interface can also be adapted to specific needs (special
menus or displays, additional analysis modules, ...).
The next step in PROMETHEE software is a web-based implementation
which is being developed under the Q-E-D name (Quantify-Evaluate-Decide).
The Q-E-D demo web site will be launched during the spring 2003 at http: //
www.q-e-d.be.
Additional information on DECISION LAB can also be obtained on the
following web sites: http: //www. idm-belgium. com and http: //www.
visualdecision.com.
Acknowledgments
The authors wish to address their most sincere acknowledgements to their col-
league and friend Professor Johan Springael for his valuable assistance and
comments.
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