Promethee

Description of the method

PROMETHEE (preference ranking organisation method for enrichment evaluations) is one of the so-called outranking methods (also called prevalence methods), along with ELECTRE (elimination et choix traduisant la realité) and ORESTE (organisation, rangement et synthèse de données relationelles). These procedures differ from the classic methods of multi-criteria decision-making in their basic assumptions about the decision-maker. In contrast to the classic methods, the outranking procedures' starting point is that a decision-maker does not have access to the information needed to form at least a weak order and make an optimum choice. The classic procedure assumptions that (a) a complete compensation or balancing between the attributes is possible and (b) an unambiguous estimate of the indifference or preference between two alternatives can be made are often unrealistic in multi-criteria problems. The assumptions underpinning outranking procedures differ on these points. Using PROMETHEE, graded preference estimations are permitted when assessing two alternatives, as well as strict preference and indifference judgements. Critical values, which indicate the difference in a criterion outcome at which a preference emerges, can also be included. Incomparability of alternatives caused by an inability to compensate may be considered as well, so often neither strong nor weak orders can be formed and no full ranking can be determined. However, the determination of an optimum alternative is not the purpose of the outranking procedures. Rather, they aim to support problem solving and contribute to identifying suitable alternatives.

To describe differentiated preference situations, the outranking procedures use a graduated relationship, the so-called outranking relationship (or prevalence relationship). This indicates the likelihood ftij, that the decision-maker estimates alternative i to be at least as good as alternative j. It must be formulated for every possible alternative pair. Pair comparisons between the alternatives are, as with the AHP, an essential feature of outranking procedures, so this approach is primarily suited to the assessment of relative profitability.

An evaluation of outranking relationships should help to solve any problem that is defined as the selection, arranging or ordering of alternatives. Since PROMETHEE has been primarily developed to determine rank orders, it aims to do so in the form of so-called pre-orders, for some or all alternatives. A pre-order is a specific order to which transitivity must not apply, and via which non-comparable factors can be incorporated.

Another fundamental characteristic of PROMETHEE is the use of generalised criteria. These consist of a typical series of so-called preference functions, which indicate the intensity of the preference for one alternative against another regarding a particular criterion. On the basis of the preferences determined using these functions, an outranking relationship and an outranking graph are produced.

This can be illustrated for a multi-criteria problem of the form:

A = (Ai,A2,...,Aj,...,Ai) represents the set of all alternatives and fc(Aj) represents A in real numbers in each case. Accordingly, fc(Ai) indicates the cardinally measured outcome of an alternative Ai with regard to the criterion c. This formulation of the multi-criteria problem assumes that all target measures are to be maximised. Therefore, criteria that are minimised must be transformed into a maximisation task (e.g. by multiplying by -1).

In PROMETHEE, a pair-wise comparison of all alternatives takes place for every criterion c. Thus, for an alternative Ai e A, a preference against the alternative Aj e A can be determined by calculating the difference dc between the outcomes fc(Aj) and fc(Aj) and converting this difference into a preference value using the preference function. For the preference function pc(Ai,Aj):

The preference value pc(Ai,Aj) indicates the level of dominance of alternative Ai over alternative Aj in regard to criterion c, and may have values between 0 and 1. For dc < 0, i.e. indifference or negative preference of Ai over Aj, a value of 0 is assigned to pc(Aj,Aj). For a strict preference for Ai over Aj, pc(Aj,Aj) amounts to 1.

Max (fj(Ai), f2(Ai),..., fc(Ai),..., f^)} with: Ai e A

With PROMETHEE it is possible to consider preference estimations (preference intensities) lying anywhere between indifference and strict preference. These are represented by preference values between 0 and 1. The higher the preference value, the more intense the preference: the increased intensity being the result of increasing differences d. The flexible means of assigning preference values pc to value differences using preference functions is another characteristic of PROMETHEE. Critical values can be included, as mentioned, for indifference and/or preference.

For most practical applications, six typical kinds of preference functions (the 'generalised criteria' mentioned above) are sufficient. Figure 6-13 shows these generalised criteria.

Usual criterion

Quasi-criterion

p(d) =

' 0, for d < 0 1, for d > 0

[ 1, for d > q

f p(d)

>

^p(d)

1 1 1 1 1 1

£

0 q -d

Criterion with linear preference

Step-criterion

p(d) =

1, for d > s

f 0, for d < q p(d) = i 0.5, for q < d < s

[1, for d > s

>

Vp(d)

/

p(d)

/ /

0,5 -

s d

0

q s d

Criterion with linear preference

GAUSS-criterion

and indifference area

0, for d < q

i 0, for d < 0

p(d) =.

d - q s - q 1, for

, for q < d < s d > s

1 1

I 2°2 J

%(d)

Vd)

!!

0

q s d

0

g d

Fig. 6-13: Generalised criteria with PROMETHEE

The usual criterion represents the classic case in decision theory, with a strict division between indifference (p(d) = 0, if d < 0 or f(Aj) < (Aj)) and strict preference (p(d) = 1, if d > 0 or f(Ai) > f(Aj)). The intensity, or degree, of preference is not considered.

The quasi-criterion differs from the usual criterion in that it includes a critical value (q) for indifference. This critical value equals the highest value of d at which indifference still exists. Small differences are then irrelevant. Strict preference, with p(d) = 1, applies to all values of d > q.

For a criterion with linear preference, a critical value for preference (s) is included, which represents the smallest value of the difference for which a strict preference exists. In the range between 0 and this critical value, preferences rise linearly, i.e. there is a proportionate relationship between differences and preference intensities.

For a step-criterion, critical values are considered for both indifference (q) and preference (s). Differences of d < q lead to indifference, differences above s indicate strict preference. In the range between q and s (including s), a weak preference with p(d) = 0.5 can be assumed. Alternatively, other preference values between 0 and 1, and more than two gradations, can be included as well.

A criterion with a linear preference and an indifference area also uses two critical values. This criterion represents a combination of the two previous criteria. It differs from the step-criterion in that a linear preference function is assumed to exist between the critical values.

Using the GAUSS-criterion, preference is strictly increasing with the difference d, beginning at d = 0. Even for high values of d, a strict preference (p(d) = 1) is not fully reached. With this criterion, a parameter o, which determines the turning point of the preference function, must be identified. The Gaussian distribution is included in the generalised criteria since the preference function based on it is quite stable, i.e. small changes in o result in only slight changes in preference.

PROMETHEE is carried out using the following steps:

1. Determination of the target criteria and data collection.

2. Selection of generalised criteria and definition of preference functions.

3. Determination of an outranking relationship.

4. Evaluation of the outranking relationship.

The first step, definition of the target criteria, requires a detailed analysis of the target system, as in all multi-criteria methods. After defining the targets, the possible outcomes for the available alternatives must be assigned cardinal numbers with respect to each criterion.

The second step, selection of generalised criteria and definition of preference functions is performed for every criterion and includes, if necessary, the specification of the generalised criteria by determining the associated parameters (s, q, o). This second step implies the assumption that the preference functions accurately reflect the preferences of the decision-maker in regard to the outcomes, or more precisely outcome differences, of each criterion.

To determine an outranking relationship (the third step), value differences must be calculated for all criteria and alternative pairs. Then, using the preference functions, the preference values are derived from the value differences. For every pair of alternatives (Ai,Aj) and every criterion two preference values are determined: a preference value indicating the preference for Ai against Aj; as well as one indicating the preference for Aj against Ai. One of the two values is always zero.

The relative importance of the criteria must also be fixed in this step. This is achieved using cardinally measured weighting factors wc for all criteria c. As with other methods, the weighting factors must fulfil the Condition (6.17):

Then, for the preference of every alternative Ai against another Aj, an outranking relationship can be determined using the weighted means of all criteria-specific preference values pc(Ai,Aj).

The values of the outranking relationship can be interpreted as preference indications that reflect the level of preference for Ai against Aj. After including all criteria, they can be interpreted similarly to the values pc(Aj,Aj) for a criterion c, that is n = 0 indicates indifference and n = 1 indicates strict preference. Between 0 and 1, the degree of preference rises with increasing values of n. For each alternative pair Ai,Aj, two values of the outranking relationship are determined (as for a single criterion).

The outranking relationships identified may be summarised as a square matrix. The elements of the main diagonal of this matrix represent the values n(Ai,Ai) at zero. Alternatively, the outranking relationship may be illustrated in the form of a graph. The nodes of the graph correspond to the alternatives, and the arrows correspond to the values of the outranking relationship between alternatives. Because, for two alternatives Ai and Aj, two outranking values are calculated, the graph contains two arrows between two nodes.

The fourth step of PROMETHEE is the evaluation of the outranking relationship. Based on the outranking graph, two flow measures can be determined for every node and every alternative. The outflow measure of a node (F+) is the sum of the assessments of all arrows (values of the outranking relationship) starting at the node:

It indicates the level of preference for one alternative against all others. The greater it is, the more preferable that alternative.

The inflow measure of a node (F-) is determined in the same way, as the sum of the estimates of all arrows flowing into the node: I

The inflow measure shows the extent to which an alternative is dominated by other alternatives. The higher it is, the greater the dominance by other alternatives.

Now, to set up a rank order of alternatives, each alternative is evaluated on the basis of the inflow and outflow measures. A suitable pre-order can be formulated to assess relative profitability. As a basis for this, an entire (pre)order is derived from both measures.

The pre-order resulting from the outflow measures, characterised by the symbols P+ (preference) and I+ (indifference), contains the following statements:

Ai is preferred to Aj (AiP+Aj), if F+(Ai) > F+(Aj) Ai is indifferent to Aj (AiI+Aj), if F+(Ai) = F+(Aj)

Accordingly, a pre-order based on the inflow measures (with the symbols P- and I-) may be formed:

Ai is preferred to Aj (AiP-Aj), if F-(Ai) < F-(Aj) Ai is indifferent to Aj (AiI-Aj), if F-(Ai) = F-(Aj)

After simultaneous inclusion of outflow and inflow measures, a pre-order of the following form can be produced to assess profitability (with the symbols P, I and U):

Ai is preferred to Aj (AjPAj), if (AiP+Aj and AiP-Aj) or (AiP+Aj and AiI-Aj) or (AiI+Aj and AiP-Aj) Ai is indifferent to Aj (AjIAj), if AiI+Aj and AiI-Aj Ai and Aj cannot be compared (AiUAj), if not AiPAj and not AilAj

If the relationship AiPAj is valid, the alternative Ai is clearly preferable to Aj - i.e. 'Ai outranks Aj'. For AilAj the decision-maker is indifferent between these options, and for AiUAj the alternatives are not comparable. A pre-order derived in this way is always a so-called partial pre-order when the alternatives (U) are not comparable. This is another difference between PROMETHEE and the methods discussed previously.

Example 6-4

Now the MAUT example is reconsidered. As in all outranking procedures, PROMETHEE is particularly suitable for decisions involving many alternatives. Therefore, the example is extended by a further two alternatives (A4, A5).

In the first step of PROMETHEE, the determination of target criteria and data collection, the following data are recorded for four target criteria (size of land (S), labour potential (LP), freight carrier (FC) and municipal factor of trade tax (MF)):

Target criteria Alternative

S

LP

FC

MF

A1

60,000

800

15

350

a2

42,500

1,100

12

250

a3

35,000

1,300

25

450

a4

35,000

900

14

300

a5

40,000

1,000

17

400

Tab. 6-4: Data for the five alternatives

Tab. 6-4: Data for the five alternatives

The second step involves selecting generalised criteria and defining preference functions for the four target criteria. Figure 6-14 contains the relevant generalised criteria and preference functions. It is assumed that they reflect the preferences of the decision-maker.

Criterion

Generalised criterion and preference function

Size of land

Quasi-criterion with parameter q = 5,000

(Criterion 1)

1, for d1 > 5,000

Labour potential

Step-criterion with parameters q = 50 and s = 200

(Criterion 2)

0, for d2 < 50

p2(d2) = ■

1, for d2 > 200

Freight carrier

Criterion with linear preference and indifference area;

(Criterion 3)

parameters q = 1 and s = 4

0, for d3 < 1

p3(d3) = ■

1, for d3 > 4

Municipal factor of

Criterion with linear preference, parameter s = 100

trade tax

(Criterion 4)

0, for d4 < 0

p4(d4) = ■

1, for d4 > 100

Fig. 6-14: Generalised criteria and preference functions in the example

In the third step, the outranking relationship is determined, with the weightings wc being assigned first. In this example they are:

Substituting into the Formula (6.35) for the value of the outranking relationship n(Aj,A2) for an alternative Aj compared to alternative A2:

The following is obtained:

n(A1,A2) = 0.3 • p1(A1,A2) + 0.35 • p2(AbA2) + 0.2 • p3(A1,A2) + 0.15 • p4(A1,A2).

By inserting the outcome differences between Ai and A2 in the preference functions and, subsequently, transforming the preference values, the following can be determined:

n(A1, A2) = 0.3 • p1(60,000 - 42,500) + 0.35 • p2(800 -1,100) + 0.2 • p3(15-12) + 0.15 • p4(-350- (-250))

n(A1 ,A2) = 0.3 • p1 (17,500) + 0.35 • p2(-300) + 0.2 • p3(3) + 0.15 • p4(-100)

n(A1, A2) = 0.3 -1 + 0.35 • 0 + 0.2 • 2 + 0.15 • 0 n(A1,A2) = 0.43

In the same way, the value n(A2,A1) can be calculated:

n(A2,A1) = 0.3 • p1(-17,500) + 0.35 • p2(300) + 0.2 • p3(-3) + 0.15 • p4(100) n(A2,A1) = 0.3 • 0 + 0.35-1 + 0.2 • 0 + 0.15-1 n(A2,A1) = 0.50

The other values of the outranking relationship can be determined in the same way. The matrix in Figure 6-15 shows the entire outranking relationship.

a1

a2

a3

a4

a5

f+

a1

0

0.43

0.45

0.30

0.38

1.56

a2

0.50

0

0.45

0.55

0.33

1.83

a3

0.55

0.38

0

0.55

0.55

2.03

a4

0.25

0.07

0.15

0

0.15

0.62

a5

0.24

0.20

0.08

0.31

0

0.83

f-

1.54

1.08

1.13

1.71

1.41

Fig. 6-15: The outranking relationship

The fourth step is the evaluation of the outranking relationship. Firstly, flow measures are determined. The outflow measure F+ results from adding the values of the columns for each alternative; the inflow measure F- results from summing up the values of the rows (see Figure 6-15). By simultaneously considering outflow and inflow measures, the partial pre-order shown in Figure 6-16 can be formulated.

a1

a2

a3

a4

a5

a1

x

-

-

A1PA4

A1UA5

a2

A2PA1

x

A2UA3

A2PA4

a2pa5

a3

A3PA1

A3UA2

x

A3PA4

A3PA5

a4

-

-

-

x

-

a5

A5UA1

-

-

A5PA4

x

Fig. 6-16: The partial pre-order

Fig. 6-16: The partial pre-order

In the matrix above it can be seen that the alternative Ai is preferable to A4 (A1PA4, there is: Ij+ > F4 and FT < F4 ); A2 is preferable to Ai (A2PA1, indicated by: F2 > Fi+ and F2 < Ff). The alternatives Ai and A5 are not comparable (A1UA5) because: Fi+ > F5 and F5" < FT .

This result can also be presented in the form of a directional graph. In this graph, the nodes represent the alternatives. An arrow from Ai towards Aj indicates that alternative i is preferable to alternative j. Indifference is expressed by lines without arrows drawn between the nodes. No connection between two nodes signifies a lack of comparability, i.e. no preference can be stated for either alternative.

From this analysis it is obvious that the alternatives A4, A1 and A5 are not relatively profitable (A4 is dominated by all the other alternatives; A1 and A5 are dominated by A2 and A3). Accordingly, either A2 or A3 should be selected; for these alternatives no preference can be stated, since the diagram shows no connection between the two (indicating a lack of comparability).

Assessment of the method

PROMETHEE (like the other outranking methods) can deal with a lack of comparability and incomplete information. In addition, critical values for preferences and preference intensities can be included in the profitability analysis.

The required computational effort is relatively low, and the data collection slightly simplified by the possibility of using generalised criteria. However, the preference functions, outcomes and weightings must be determined for each criterion. The measurements must be cardinal, which restricts the consideration of qualitative attributes.

The limitation to six generalised criteria, although not compulsory, might also be regarded as a problem. In general, there is doubt as to whether the preferences of the decision-maker can be encapsulated by generalised criteria, preference functions, and value differences (rather than absolute values). Again, the effects of uncertainty may be examined using sensitivity analysis.

In regard to the outranking relationship and the flow measures that form the basis of profitability assessments, it is assumed that target weightings can be assigned on a cardinal scale. The weighted means of all preference values (additive functions) as stated in the outranking relationship are purported to give an adequate comparison of alternatives. This also assumes - similar to the AHP and utility value analysis - that completely independent judgements are being made on each criterion. Using flow measures, it is assumed that preferences over other alternatives (outflow measures) as well as the 'domination' by other alternatives (inflow measures) will enable the formulation of a ranking. One weakness is that, as with the AHP, the pair comparisons depend upon the available alternatives and so the ranking obtained is unstable.

The inclusion of outflow and inflow measures is specific to the method. Due to the inclusion of inflow measures, PROMETHEE only allows limited compensation for unfavourable outcomes.

An order formed with PROMETHEE will reflect the preferences of the decision-maker only if the assumptions described above are fully met. Yet, such a preference statement is not the principal purpose of the procedure. Rather, and this is more important than with the other methods, decision support via preference and problem structuring is the main purpose of the PROMETHEE method.

To conclude the examination of multi-criteria methods, it should be pointed out that they share some similarities, in that they all operate by partitioning a problem. In each method the separate elements and target criteria must be determined and weighted, transformed into individual utility values or comparable values (partial utility values, local priorities, preference values) and, finally, summed taking the individual weightings into account.

Common features of utility value analysis and the AHP are primarily the step sequence and the additive total utility function. The AHP requires more effort, but has the advantage of examining the subjective estimates for consistency.

The MAUT differs from utility value analysis and the AHP in that it has a utility theory foundation and corresponding preconditions. Apart from that, the procedure is very similar to utility value analysis.

The PROMETHEE method has some similarity to the AHP, since it is based on the execution of pair comparisons. However, it offers decision support rather than a procedure for determining an optimum solution. In this regard, it differs from the other methods.

All procedures discussed in this chapter have specific advantages and disadvantages. It is therefore not possible to give a general recommendation for any one procedure; the choice of method depends on the problem being considered. A combination of methods, or elements of methods, may be useful - e.g. the target criteria weighting used with the AHP and MAUT may be applied within the framework of a utility value analysis.

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Crushing Your Goals and Achieving Success

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