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This is a PROMETHEE Methods presentation.

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Thank you for your interest in the PROMETHEE methods.

Don’t forget to consult the following web sites:

www.promethee-gaia.com

www.decision-drive.com

www.sustainable-decisions.com

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http://www.promethee-gaia.com/



http://www.decision-drive.com/



http://www.sustainable-decisions.com/



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Unicriterion models have been developed from the beginning of OR.

The search for optimal solutions can be difficult but it is a well-stated

mathematical problem and a lot of work has been done to solve many

optimization problems.

On the other hand these models focus on a single objective that often is purely

monetary: maximize profit or minimize cost. This approach was acceptable in the

context of economic growth experienced after WW2. Starting with the oil crisis

in the 70’s, it began to appear short-sighted. Short term profit can be an objective

but it certainly is not the only one...

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Multicriteria models make it possible to take into account several criteria at the

same time. Either quantitative or qualitative.

It becomes possible to consider alternative criteria such as technical, social or

environmental factors besides monetary ones.

The price to pay for this advance is that no optimal solution generally exists to

multicriteria problems as the criteria are to some extent conflicting with each

others. The challenge of MCDA is to assist decision makers in finding best

compromise solutions.

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Here are a few examples of decision or evaluation problems where several

criteria are obviously important in order to compare and evaluate different

choices.

For instance, the location of a new plant should take into account financial

factors (investment and operation costs), technical factors (logistics, raw

materials availability, ...) as well as social and environmental impacts.

Similarly the purchase of new equipment should take into account various

qualitiy factors besides the price of the equipment. Finding the bes quality/price

ratio is a multicriteria problem.

Achieving sustainable development is also a multicriteria problem has it includes

at least three often conflicting dimensions: economy, men and environment.

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We consider problems including a set of actions (either potential decision or

items to evaluate) and a set of criteria that are used to model the preferences and

the objectives of the decision-maker.

The next slides show examples of the resulting two-way multicriteria evaluation

tables.

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Actions and criteria form the two dimensions of the multicriteria table.

As it can be seen, criteria are usually evaluated on specific and possibly quite

different scales.

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different scales.

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different scales.

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In the case of a plant location problem, the actions correspond to the potential

locations (sites) and criteria can possibly include investment cost, operations cost,

environmental impacts, ... And many others.

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In the case of the purchase of new equipment, the actions correspond to the

different available products and the criteria can include the price of the

equipment as well as quality factors such as the MTBF (mean time between

failures) or the quality of the maintenance for instance.

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We now consider a very simple yet didactic example to illustrate the

PROMETHEE and GAIA methods.

In particular we limit ourselves voluntarily to five criteria. In an actual problem

of course there could be more or different criteria.

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Six completely fictive cars are compared. They are given suggestive names that

indicate clearly their strengths and their weaknesses. In that way we will be able

to check the consistency of the results provided by PROMETHEE and GAIA.

In actual decision problems the decision maker rarely has such a large knowledge

and will progressively discover the conflicts between criteria and the specific

profiles of the actions using PROMETHEE and GAIA.

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Before we look at the PROMETHEE and GAIA methods it is important to

understand that four different problematics can arise in practice.

We focus on the two last one as PROMETHEE is a ranking method and GAIA

is a descriptive one.

PROMETHEE can also be used in a choice problem.

Extensions of PROMETHEE have also been proposed for sorting.

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A very common and simple approach to multicriteria problems is the weighted

sum.

In that case weights are allocated to the criteria to express the priorities of the

decision maker. Weights are positive numbers and represent the relative

importance of the criteria for the decision maker.

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For each action a, a global value V(a) is computed as the weighted sum of the

evaluations of that action over all the criteria.

It is then possible to rank all the actions according to their weighted sum values.

A typical example is the weighted average of exam scores for evaluating

students.

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This first example show that excessive compensations between criteria can

associate a high weighted sum value to an action that is not well balanced with

possibly important weaknesses on some criterion.

The weighted sum approach can propose unbalanced solutions that clearly are not

good compromises in a multicriteria context.

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This second example shows that actions with quite different profiles can have the

same weighted sum value.

Important information about the conflicts to arbitrate between criteria get lost in

the weighted sum approach.

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MCDA methods have developed in order to solve the shortcomings of the

weighted sum. Besides more specific methods such as interactive methods and

multi-objective programming, two main “schools” coexist.

The “American School” (MAUT) builds on the aggregation idea of the weighted

sum. A value or utility function is built from the different criteria and used to rank

the actions.

The “French School” (e.g. ELECTRE and PROMETHEE) relies on the notion

of outranking.

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This approach is common in the US and the UK.

It relies on the existence of an utility function integrating the different criteria

together with the preferences of the decision-maker in such a way that the best

action is the one with the largest utility value.

This raises a few questions.

1. Do we always make our decisions based on such a function? Are our

decisions always so rational?

2. How can we obtain the precise mathematical expression of our utility

function?

3. How can we introduce hesitation and knowledge acquisition in this approach?

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The construction of the utility function is a difficult problem. It can be done

either directly or indirectly.

The direct construction implies to obtain a lot of information from the decision

maker through series of questions. It can be tedious. And the quantity of

information gathered doesn’t necessarily guarantee its quality. Is it really

necessary to achieve a good decision?

Sensitivity analyses (what if analyses for instance) are made difficult as the

approach relies on the existence of a unique utility function.

Finally, an important hypothesis is that all the criteria can be aggregated and thus

compensation can arise. MAUT transforms a multicriteria problem in an

optimization, as if it were always possible to express all criteria (even social ones

or human safety) into monetary terms.

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Outranking methods appeared at the end of the 60’s with the first ELECTRE

methods. Compared to MAUT they require less information from the decision

maker and try to stay closer to the original multicriteria decision problem.

The basic idea was to build an outranking relation from pairwise comparisons of

the actions. This was done in a simplistic way in methods such as ELECTRE I

and II. The original idea was then refined during the 70’s with more sophisticated

methods like ELECTRE III and IV. These methods were ultimately less

successful because of their increasing complexity.

In 1983, the first PROMETHEE methods appeared. They proposed a simpler

alternative to ELECTRE III, with more emphasis on sensitivity analysis. They

were later completed with the GAIA descriptive approach.

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Any decision aid method requires additional information besides the multicriteria

table. Indeed the table contains no information about the preferences and

priorities of the decision-maker.

In the PROMETHEE and GAIA methods this is done in two steps:

1. Preference functions are used to model the perception of scales by the

decision maker.

2. Weights are allocated to the criteria to reflect the priorities of the decision

maker.

We are now going to introduce these elements and analyse our didactic example

with PROMETHEE and GAIA.

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PROMETHEE is based on pairwise comparisons of the actions.

When comparing two actions (like Action 1 and Action 3) it seems logical to look

at the differences on each criterion.

We need a way to translate these differences in function of the preferences of the

decision maker. That is the role of the preference function.

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A preference function is associated to each criterion separately.

It translates the difference between the evaluations of two actions on a given

criterion in terms of a preference degree measured between 0 and 1.

A value of 0 indicates no preference at all while a value of 1 means an

undisputable preference for the best evaluation.

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Let us compare two cars from our didactic example.

The leftmost column contains the differences that correspond to the advantages of

the Economic car. The rightmost column contains those of the Luxury 1 car.

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The preference functions associated to the criteria make it possible to translate all

the differences on the same preference degree scale (yellow columns).

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The problem is now to compare the two yellow columns and decide which action

(car) is the best.

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The problem is now to compare the two yellow columns and decide which action

(car) is the best.

At this stage we can introduce the weights of the criteria and compute the

weighted average of each yellow column.

With equal weights the Economic car is the best choice.

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When the weights are modified, the result can of course change!

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To summarize:

1. A preference function is associated to each criterion to reflect the perception

of the criterion scale.

2. Weights are allocated to the criteria to reflect the priorities of the decision

maker.

3. A multicriteria preference degree is computed and pairwise comparisons of

the actions are obtained.

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There are six types of preference functions that are implemented in the

PROMETHEE software (PROMCALC, Decision Lab and the forthcoming D-

Sight).

Usual, U-shape (less used) and Level are typically used for qualitative criteria

with discrete evaluation scales including a small number of levels.

V-shape, Linear and Gaussian (less used) are best suited for quantitative criteria.

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From the pairwise comparisons of all the actions, PROMETHEE is building

rankings of the actions according to preference flows.

Using the preference functions and the weights of the criteria, every action can

automatically be compared to each other.

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From this pairwise comparisons table we can extract information in order to rank

all the actions. This is done by computing three different preference flows.

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The positive (or leaving) flow measures the average degree to which an action is

preferred to the other ones. Actions with larger leaving positive flow values

should be ranked first.

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preferred to the other ones. Actions with larger leaving positive flow values


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preferred to the other ones. Actions with a larger leaving positive flow value


The negative (or entering) flow measures the average degree to which the other

actions are preferred to that action. Thus actions with a smaller negative flow

value should be ranked first.

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Usually both preference flows lead to somewhat different rankings as in a

multicriteria context there is usually no ranking completely consistent with all the

pairwise comparisons results.

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The net flow is the balance between the positive and the negative flows. It can be

used to rank all the actions from the largest positive values to the most negative

ones.

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ones.

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ones.

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To summarize:

•The best actions have a large positive flow value.

•The best actions have a small negative flow value.

•Positive and negative flows usually induce somewhat different rankings of the

actions.

•The net flow can be used to combine the information from the positive and

negative flows. The best actions have a large positive net flow value.

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The three PROMETHEE preference flows are the basis for the two

PROMETHEE rankings.

PROMETHEE I is prudent: its partial ranking of the actions includes only

preferences that are confirmed by both positive and entering flows.

Incomparabilities arise when both flows give opposite information. Usually, this

happens because the actions have quite different profiles and are thus difficult to

compare.

PROMETHEE II uses the net flow to rank completely all the actions from the

best to the worst one. In this case no incomparabilities are possible.

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The PROMETHEE I partial ranking for our didactic example includes many

incomparabilities. Yet it could be sufficient to solve a choice problem: the

Tourism B car seems to be the best choice.

The PROMETHEE II complete ranking is consistent with the partial ranking

but no incomparabilities are left. It could be more disputable as PROMETHEE

had to make decisions when comparing actions even in difficult cases.

The rankings are presented in the usual network representation with nodes and

arrows. This makes it difficult to appreciate the difference in flow values and to

evaluate the robustness of the rankings when preference parameters (preference

functions or weights) are modified.

An alternate representation of the PROMETHEE I partial ranking is the

“Diamond” implemented in the new D-SIGHT software.

Actions are drawn in the plane defined by the positive and negative flows.

Each action is represented by a point and a cone. Overlapping cones indicate

incomparabilities. It is also possible to see whether the flow values of two actions

are close to each other or not.

The two axes are angled in such a way the the vertical axis corresponds to the net

flow. Thus higher points are ranked first in the PROMETHEE II complete

ranking and a visual link exists between the two rankings.

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An alternate representation of the PROMETHEE I partial ranking is the

“Diamond” implemented in the new D-SIGHT software.

Actions are drawn in the plane defined by the positive and negative flows.

Each action is represented by a point and a cone. Overlapping cones indicate

incomparabilities. It is also possible to see whether the flow values of two actions

are close to each other or not.

The two axes are angled in such a way the the vertical axis corresponds to the net

flow. Thus higher points are ranked first in the PROMETHEE II complete

ranking and a visual link exists between the two rankings.

The « Thermometer » gives a more precise view of the complete ranking and of

its robustness.

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The bar chart representation is useful to visualize changes in the PROMETHEE

II ranking when the weights of the criteria are modified. It is central to the

Walking Weights procedure implemented in Decision Lab and D-SIGHT.

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Ranking methods are useful to finalize a decision. However they usually lack a

lot of information that can be helpful for the decision-maker. For instance:

-What can be expected when the weights of the criteria are modified?

-Which compromise solutions are possible and which are not?

-What are the origins of the incomparabilities? What are the conflicts to arbitrate

between criteria?

This information can be obtained from a descriptive approach. It can provide the

decision maker with a better understanding of the decision problem, help him/her

to better assess preference parameters and ultimately lead to better decisions.

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The descriptive counterpart of PROMETHEE is GAIA.

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A multi-dimensional graphical representation of multicriteria data is appealing as

it would clearly show important characteristics of the decision problem such as

the conflicts between the criteria and the distinctive profiles of the actions.

However such a representation is unpractical when the number of criteria

(dimensions) is larger than three.

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A multi-dimensional graphical representation of multicriteria data is appealing as

it would clearly show important characteristics of the decision problem such as

the conflicts between the criteria and the distinctive profiles of the actions.

However such a representation is unpractical when the number of criteria

(dimensions) is larger than three.

GAIA applies a Principal Components Analysis on the normalized

multidimensional space corresponding to the the unicriterion net flows. In that

way the preferences of the decision maker are taken into account in the

normalization. The two first principal components define the GAIA plane that

gives the best two-dimensional representation of the data.

Besides the actions (points), the criteria are also represented in the GAIA plane

by projecting their corresponding unit axes onto the plane.

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The GAIA plane contains a lot of information useful to:

-Learn about the actual conflicts between the criteria.

-Identify the profiles of the actions.

-Better assess the weighing of the criteria.

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Actions are represented by points:

-Similar profiles are located close to each other (such as the two Luxury cars).

-Quite different actions are located far away from each other (such as the Sport

and Economic cars).

Criteria are represented by axes:

-Axes pointing in similar directions indicate criteria that are in agreement with

each other.

-Opposite axes correspond to conflicting criteria.

-Longer axes correspond to more discriminating criteria.

-The direction of each criterion axis indicates in which direction the best values

are achieved on this criterion.

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For criterion Price the axis is oriented towards the right side of the GAIA plane.

Thus we can expect to see the cheapest cars (Economic) located to the right and

the most expensive ones (Luxury 1 and 2) to the left.

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For criterion Power the axis is oriented in the North-West direction.

Clearly the Sport car is the most powerful and the least powerful one is the

Economic car.

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One additional axis appears in the GAIA plane.

The decision axis corresponds to the PROMETHEE II ranking direction taking

into account the weights of the criteria.

It orientation can be interpreted similarly to a single criterion axis. However as

the weights are not taken into account in the Principal Components Analysis the

decision axis can be more or less well represented in the GAIA plane and

distortions can appear between the observed ranking and the actual

PROMETHEE II ranking.

One must be especially careful when the decision axis is shorter.

When the weights of the criteria are modified, the position of the decision axis

changes in the GAIA plane. It reflects the type of compromise that is proposed

by PROMETHEE.

The Walking Weights interactive procedure makes it easy to do such an analysis.

To better appreciate the robustness of the PROMETHEE rankings it is also

possible to represent the area in the GAIA plane where the tip of the decision

axis moves when the weights are changed within specified limits. That is GAIA-

Brain.

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In the leftmost example, the « brain » is located in the SW direction and it is

small: within these weight variations, the PROMETHEE ranking will not

change so much.

In the central example, the « brain » is large and overlaps the origin: the decision

axis can point in any direction and the PROMETHEE ranking could be quite

sensitive to weight variations.

The two rightmost examples correspond to smaller « brains » oriented in different

directions.

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The quality of the GAIA plane is an important factor to consider. It is measured

by D (the percentage of information retained in the GAIA plane).

The GAIA plane provides the decision maker with reliable information when D is

sufficiently large (for instance larger than 80%).

On the contrary, one must be very careful when D is small (smaller than 70% for

instance) as a lot of information was lost in the representation.

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Summary.

PROMETHEE is prescriptive. It constructs two different rankings on the set of

actions.

GAIA is descriptive. It provides a global synthetic representation of the decision

problem.

PROMETHEE and GAIA are linked to each other through the decision axis.

GAIA corresponds to the best possible 2-dimensional representation of the

multicriteria data. Its quality is however limited by the data themselves (D) and

the PROMETHEE II ranking can be poorly represented (especially when there

are strongly conflicting criteria and the decision axis is shorter).

Two alternative 2-dimensional representations are proposed in order to

circumvent these limitations: GAIA-Stick and GAIA-Criterion.

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In the GAIA-Stick plane, the horizontal axis is the decision stick. The horizontal

coordinates correspond to the PROMETHEE II net flow values and the ranking

is exactly represented from the right (Tourism B) to the left (Sport, very close to

Economic).

The vertical axis gives additional information related to the profiles of the

actions. For instance, Economic and Sport are very close to each other in the

PROMETHEE II ranking but they differ much in the vertical dimension:

Economic is clearly better on Price and Fuel (criteria pointing upward).

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In a GAIA-Criterion plane, the horizontal axis is determined by a single

criterion. In this example, the criterion Power is chosen. The horizontal

coordinates correspond to the unicriterion net flow values for Power with the best

(positive) values to the right (Sport is clearly the most powerful car) and the

worst (negative) ones to the left (Economic is the least powerful).

The vertical axis gives additional information related to the other criteria. For

instance, the Luxury 1 and 2 cars are very good on Comfort and Space (criteria

pointing downward).

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Please consult the www.promethee-gaia.com web site for more information

related to the PROMETHEE and GAIA methods.

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The PROMETHEE & GAIA methods have been implemented in different

software.

The pioneer was PROMCALC, a MS-DOS program developed at the beginning

of the 90’s at the ULB.

Ten years later Decision Lab appeared as a joint development between ULB and

Visual Decision, a Montreal-based company.

Today D-SIGHT is getting closer to its public release.

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Decision Lab is still available from Visual Decision ( www.visualdecision.com ).

http://www.visualdecision.com/

Currently (summer 2009) D-SIGHT is still in a beta stage. It is already available

to academic users for evaluation (check the D-SIGHT web site) and will be

publicly released before the end of the year.

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