Multiple Attribute Decision Making: Methods and ...

7

PROMETHEE Method

Brans et al. (1984, 1985) consider a new family of outranking methods, called PROMETHEE (Preference Ranking Organization METHods for Enrichment Evaluations) for solving MADM problems. These methods are based on a generalization of the notion of criterion. In this period, a basic concept of fuzzy outranking relation is first considered and built into each criterion by pairwise comparison measures for alternatives to different relation-degrees in each other. These different relation-degrees are then used to set up a partial preorder (PROMETHEE I), a complete preorder PROMETHEE II), or an interval order (PROMETHEE III) on a finite set of feasible solutions. Another method, called PROMETHEE IV, is introduced for the case where the set of feasible solutions is continuous. These results can easily be apprehended by the decision maker, as illustrated in a numerical application.

7.1 THE NOTION OF THE PROMETHEE METHOD Let a multiattribute decision-making problem be represented as:

{

}

Max g1 ( ai ) , g2 ( ai ) ,… , g j ( ai ) ,… , gn ( ai )|ai ∈ A ,

(7.1)

where A = {ai | i = 1,2,…, m} is a set of possible actions (or alternatives) and g = {g j | j = 1,2,…, n} is a set of considered criteria; g j(ai) represents performance of action ai with respect to the jth criterion. If, for a given pair of alternatives, a and b have g j(a) ≥ g j (b) for j = 1,2,…, n and at least one inequality is strict, then a dominates b. According to Brans et al. (1984), the PROMETHEE methods belong to the outranking methods consisting in enriching the dominance order. They include three phases: 1. Construction of generalized criteria 2. Determination of an outranking relation on A 3. Evaluation of this relation in order to give an answer (7.1) In the first phase, a generalized criterion is associated to each criterion g j by considering a preference function. In the second phase, a multicriteria preference index is defined in order to obtain a valued outranking relation representing the preference of decision makers. The evaluation of outranking relations are obtained by considering for each action a leaving and entering flow.

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7.2 PROMETHEE I, II, III, IV Brans et al. (1984) first suppose that A is a finite set of possible alternatives. A partial preorder (PROMETHEE I) or a complete preorder (PROMETHEE II) on A can first be proposed to the decision maker. PROMETHEE III provides an interval order emphasizing indifference; PROMETHEE IV deals with continuous sets of possible alternatives. The PROMETHEE methods request additional information but only a few parameters are to be fixed and they all have a real economic significance. Six possible types of generalized criteria can be considered in PROMETHEE methods and, as shown as Table 7.1, each of them can be very easily defined because only one or two parameters are to be fixed: 1. q is a difference threshold. It is the largest value of d below which the decision maker considers there is indifference; 2. p is a strict preference threshold. It is the lowest value of d above which the decision maker considers there is strict preference; 3. σ is a well-known parameter directly connected with the standard deviation of a normal distribution. Let A be a finite set of alternatives for MCDM problems, and suppose a preference function fj has been defined for each g j, for each couple of alternatives a,b ∈ A; i.e., when a  b in j criterion, fj(a,b) = f j(dab| j) indicates that the degree of alternative a prefers to alternative b (a over b) with different distance of performance value dab| j = gj(a) − gj (b) in j criterion; and π(a,b) is a preference index over all the criteria defined by: π ( a, b ) =

n

∑ f (a, b)

(7.2a)

∑ w f (a, b)

(7.2b)

1 n

j

j =1

or π ( a, b ) =

n

j j

j =1

where Equation 7.2a shows the criteria are all equal and Equation 7.2b shows the criterion weight is wj in criterion j and j = 1,2,…,n. wj can be obtained by the analytic hierarchy process (AHP) or the analytic network process (ANP) based on a network relationship map (NRM) from DEMATEL or interpretive structural modeling (ISM) techniques. The preference index π(a,b) gives the intensity of preference of the decision maker for a over b, all criteria being considered. We have 0 ≤ π(a,b) ≤ 1. Moreover, in order to evaluate the alternatives of A by using the outranking relation, they define the following flows:

PROMETHEE Method

97

TABLE 7.1 Generalized Criteria Types of Criteria Type I: Usual criterion

Analytical Definition

Shape

Parameter NA

⎧0, d = 0; H (d ) = ⎨ ⎩1, | d | > 0.

1 d

Type II: Quasi-criterion

1 –q

Type III: V-sharp criterion

Type IV: Level-criterion

Type V: Linear criterion

Type VI: Gaussian criterion

p

1 –p

d

p

q, p

H

| d | ≤ q; q < | d | ≤ p; otherwise.

⎧0, ⎪ ⎪ |d | − q H (d ) = ⎨ p − q , ⎪ ⎪⎩1,

d

q H

⎧ |d | ⎪ , | d | ≤ p; H (d ) = ⎨ p ⎪ | d | > 0. ⎩1, ⎧0, ⎪ H (d ) = ⎨1/2, ⎪1, ⎩

q

H

⎧⎪0, | d | ≤ q; H (d ) = ⎨ ⎪⎩1, otherwise.

1 1/2

–p –q | d | ≤ q;

H

q < | d | ≤ p;

1

otherwise.

–p –q

d

qp

q, p

d

qp

σ

H

2 ⎧ ⎫ H (d ) = 1 − exp ⎨− d 2 ⎬ ⎩ 2σ ⎭

−σ

σ

d

Source: From Brans, J.P., B. Mareschal, and Ph. Vincke, Operational Research, Elsevier Science Publishers B.V., North-Holland, 1984b.

1. The leaving flow: φ+ ( a ) =

∑ π (a, b), b ∈A

(7.3)

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Multiple Attribute Decision Making: Methods and Applications

2. The entering flow: φ− (a) =

∑ π (b, a),

(7.4)

b ∈A

3. The net flow: φ ( a ) = φ+ ( a ) − φ− ( a ) .

(7.5)

In PROMETHEE methods, the higher the leaving flow and the lower the entering flow, the better the alternative. The leaving and entering flow induce, respectively, the following preorders on alternatives on A: ⎧aP + b ⎨ + ⎩ aI b

iff φ + ( a ) > φ + ( b ) ; iff φ + ( a ) = φ + ( b ) ;

(7.6)

⎧aP − b ⎨ − ⎩ aI b

iff φ − ( a ) < φ − ( b ) ; iff φ − ( a ) = φ − ( b ) ,

(7.7)

where P and I represent preference and indifference, respectively.

7.2.1

PROMETHEE I

According to Brans et al. (1984, 1985), PROMETHEE I determines the partial preorder (PI,II,R) on the alternatives of A that satisfied the following principle: ⎧aP + b and aP −b ⎪ aP b ( a outranks b ) , if ⎨ aP + b and aI −b , ⎪ aI + b and aP −b ⎩

(7.8)

aI I b ( a is indifferent to b ) , if aI + b and aI − b,

(7.9)

aRb ( a and b are incomparable ) , otherwise.

(7.10)

I

From the above equations, we can obtain a partial order for alternatives, while some others are not order (i.e., if aRb cases exist incomparable).

PROMETHEE Method

99

7.2.2 PROMETHEE II Furthermore, PROMETHEE II gives a complete preorder (PII, III) induced by the net flow and defined by aP II b ( a outranks b ) , iff φ ( a ) > φ ( b ) ,

(7.11)

aI II b ( a is indifferent to b ), iff φ (a ) = φ( b ),

(7.12)

It seems easier for the decision maker to achieve the decision problem by using the complete preorder in PROMETHEE II instead of the partial one given by PROMETHEE I. However, the partial preorder provides more realistic information by considering only confirmed outranking with respect to the leaving and entering flows. On the other hand, the relation of incomparabilities can also be very useful. In PROMETHEE I and II, the indifference case between two actions only occurs when the corresponding flows are strictly equal. Nevertheless, due to the continuous character of the generalized criteria (as Table 7.1), it may happen that for two actions a and b the flows are very close to each other, then indifference between a and b is considered.

7.2.3

PROMETHEE III

Based on the above reasons, PROMETHEE III associates, to each action a, an interval [xa, ya], and defines a complete interval order (PIII, IIII) as follows: aP III b ( a outranks b )

iff xa > yb ,

aI III b ( a is indifferent to b ) iff xa ≤ yb and xb ≤ ya ,

(7.13) (7.14)

the interval [xa, ya] is given by ⎧ xa = φ ( a ) − ασ a , ⎨ ⎩ ya = φ ( a ) + ασ a

(7.15)

where n is the number of actions (or criteria): φ (a) =

1 n

∑ (π (a, b) − π (b, a)) = n φ (a) 1

b ∈A

(7.16)

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Multiple Attribute Decision Making: Methods and Applications

xc

xb

yc

xa

yb

ya

Relations of a, b, c: aIIIIb, bIIIIc, aPIIIc

FIGURE 7.1 Diagram of intransitive nature of PROMETHEE III.

σ 2a =

1 n

∑ (π (a, b) − π (b, a) − φ (a)) , 2

(7.17)

b ∈A

where α > 0 in general. In other words, [xa, ya] is an interval, the center of which is the net mean flow of a and the length of which is proportional to the standard error of the distribution of the numbers (π(a, b) – π(b, a)). In addition, the smaller the value of α, the greater the number of strict outranking; for α = 0, (PIII, IIII ) coincides with (PII, III ). It is remarkable that IIII is not necessarily transitive while PIII is still transitive. For example, for the outranking of three actions a, b, and c, we have aIIIIb and bIIIIc, but aPIIIc exists (see as Figure 7.1). In fact, the choice of α will depend upon the application. For instance, to avoid too many indifferences, it may be requested that the mean length of the intervals be less than the mean distance between two successive mean flows. This leads in general to a value of about 0.15 for α. In brief, if we utilize PROMETHEE I in our cases, it can help us to determine the partial preorder (PI, II, R ) on the set of alternatives A; if we use PROMETHEE II, we can obtain a complete preorder (PII, III ) induced by the net flow; furthermore, exploiting PROMETHEE III has the advantage of allowing intransitive indifference and distinguishing incomparability from indifference.

7.2.4 PROMETHEE IV Furthermore, PROMETHEE IV extends PROMETHEE II to the case of a continuous set of actions (or alternatives) A. Such a set arises when the actions are, for instance, percentages, dimensions of a product, compositions of an alloy, investments, and so on. The generalized criteria of PROMETHEE IV are defined by the above, from preference functions Ph(a,b) such that: Ph(a,b) = ℘(d), where dh = f h(a) − f h(b), h = 1, 2,...,k. Besides, the leaving flow, the entering flow, and the net flow for continuous set A are defined as follows: φ+ ( a ) =

∫ π (a, b)db,

(7.18)

∫ π (b, a)db,

(7.19)

A

φ− (a) =

A

PROMETHEE Method

101

TABLE 7.2 Information Table in Example 7.1 Preferred Ratings Alternative 1 Alternative 2 Alternative 3 Alternative 4 Alternative 5 Weights

Size

Age

9 6 2 9 5 0.23

Transportation

6 9 7 10 9 0.18

4 4 9 4 3 0.18

Facilities 3 3 6 4 10 0.27

φ( a ) = φ + ( a ) − φ − ( a ) .

Price 8 3 8 2 1 0.14

(7.20)

In fact, it is not always easy to integrate the preference index π(a,b) into the set A. Brans et al. (1984) suggested simplification of Equations 7.18 through 7.20 as follows:

∫ P (a, b) db,

(7.21)

∫ P (b, a) db,

(7.22)

∑ ⎡⎣φ (a) − φ (a)⎤⎦.

(7.23)

φ+ ( a ) =

h

A

φ− (a) =

h

A

and to deduce φ (a) =

1 k

k

+ h

− h

h =1

For example, when A is the real interval [0,1], it is possible to obtain the function ϕ(a) for the generalized criteria of type I to type V (see Table 7.1) when the functions f h are piecewise linear or quadratic, showing that a lot of different situations TABLE 7.3 Leaving, Entering, and Net Flows Alternatives

𝛟+(a)

𝛟−(a)

𝛟(a)

Alternative 1 Alternative 2 Alternative 3 Alternative 4 Alternative 5

0.2727 0.1818 0.4886 0.2614 0.4205

0.3182 0.3636 0.4318 0.2500 0.2614

–0.0455 –0.1818 0.0568 0.0114 0.1591

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Multiple Attribute Decision Making: Methods and Applications Alternative 5

Alternative 1

Alternative 3

Alternative 2

Alternative 4

FIGURE 7.2 The partial ranking of PROMETHEE I.

can be considered. However, in more complicated cases a numerical integration may be used.

7.3 EXAMPLE FOR HOUSE SELECTION Example 7.1 Consider a house-selection problem in Taipei. Fives alternatives with five criteria, size, age, transportation, facilities, and price, are considered and each alternative is evaluated by preferred ratings, as shown in Table 7.2. To choose the best alternative, PROMETHEE I is employed. The parameters of PROMETHEE I can be set as follows: Preference function = linear Indifference threshold = 1 Preference threshold = 2 Then we can calculate the leaving flow, entering flow, and net flow as shown in Table 7.3. Then, we can depict the partial ranking of PROMETHEE I and the complete ranking of PROMETHEE II as shown in Figures 7.2 and 7.3, respectively. Comparing the results with PROMETHEE I and PROMETHEE II it can be seen that we can obtain a similar ranking. However, the main difference is that PROMETHEE I can only derive the kernel solutions and PROMETHEE II can obtain the complete ranking. Alternative 5

Alternative 1

Alternative 3

Alternative 4

FIGURE 7.3 The complete ranking of PROMETHEE II.

Alternative 2