Chapter 5 PROMETHEE METHOD

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Chapter 5 PROMETHEE METHOD
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OVERVIEW

• PROMETHEE is an MADM technique
• PROMETHEE Preference Ranking Organization
  METHod for Enrichment Evaluations
• PROMETHEE versions I II, III IV, being
  developed
• Let us consider an MADM problem
• Maxf1(a), , fk(a) a? K (1)
• Where K is a finite set of actions/alternatives
  
• fi, i 1, , k are criteria
• PROMETHEE consists of two phases
• The construction of an outranking relation on K
• The exploitation of this relation in order to
  give an answer to (1)

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PHASE I The PROMETHEE Valued Outranking Relation

• When we compare two alternatives a, b ? K, we
  must be able to express the result of this
  comparison in terms of preference a preference
  function P
• representing the intensity of preference of
  alternative a over alternative b such that
• P( a, b) 0 means indifference
• P( a, b) 0 means weak preference of a over b
• P( a, b) 1 means strong preference of a over
  b
• P( a, b) 1 means strict preference of a over b

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PHASE I The PROMETHEE Valued Outranking Relation

• In practice, P( a, b) will often be a function of
  the difference between two evaluations
• P( a, b) pf(a) f(b)
• It has to be a nondecreasing function, equal to
  zero for negative values of d.
• d f(a) f(b)

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Generalized Preference Function
Preference of a over b
Preference of b over a

• Where,

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Types of Criteria-preference Function

• Usual criterion
• No parameter has to be defined
• Quasi-criterion

Indifference point
Indifference threshold
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Types of Criteria-preference Function

• (3) Criterion with linear preference
• (4) Level criterion
• The DM has 2 thresholds to define
• Indifference threshold q
• Preference threshold p

1/2
Strict Pref.
Strict Pref.
Weak Pref.
Weak Pref.
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Types of Criteria-preference Function

• (5) Criterion with linear preference and
  indifference area
• (6) Gaussian criterion
• Where s parameter

H(d)
1
d
- p -q 0 q p
H(d)
1
d
0
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Multicriteria Preference Index (?)

• It is the weighted average of the preference
  function Pi
• ?( a, b) represents the intensity of preference
  of the DM of alternative a over alternative
  b, when considering simultaneously all the
  criteria. It is a figure between 0 and 1
• ?( a, b) 0 weak preference of a over b for all
  the criteria
• ?( a, b) 1 strong preference of a over b for
  all the criteria

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Phase 2 The PROMETHEE Ranking

• The preference index (Phase 1) determines a
  valued outranking relation on the set of K
  alternatives. This is represented as a valued
  outranking graph
• Defining leaving flow ?(a)
• ?(a) ??( a, b), b ? K
• As the sum of the values of the arcs leaving node
  a and therefore
• provides a measure of the outranking character of
  a

?( a, b)
?( b, a)
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Phase 2

• Symmetrically, we define the entering flow by
• ?-(a) ? ?( b, a), b ? K
• Which measures the outranked character of a
• We also consider the net flow
• ?(a) ?(a) - ?-(a)

?( b, a)
b
a
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PROMETHEE I

• The higher the leaving flow and the lower the
  entering flow, the better the alternative. These
  flows induce the following preorders
• The partial preorder (PI, II, R is then
  obtained considering the intersection of these 2
  preorders a PI b ( a outranks b) iff a P b
  and a P- b or a P b and a I- b or a I b
  and a P- b
• a II b ( a is indifference to b) iff a Ib
  and a I- b
• a R b ( a and b are incomparable), otherwise
• This partial preorder is then proposed to the DM
  in order to achieve his decision problem. By
  using PROMETHEE I some alternatives may remain
  incomparable.

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PROMETHEE II

• The PROMETHEE II complete preorder is included by
  the net flow
• a PII b ( a outranks b) iff ?(a) gt ?(b)
• a III b ( a is indifferent b) iff ?(a) ?(b)

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Example A Location Problem

• Six criteria are considered as relevant by the DM
  to the rank 6 hydroelectric power station
  projects (x1, .., x6)
• These criteria are
• (min) f1 manpower
• (max) f2 power (MW)
• (min) f3 construction cost (billion )
• (min) f4 maintenance cost (million )
• (min) f5 number of villages to evacuate
• (max) f6 security level

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Example

• Evaluation data of 6 alternatives with respect to
  6 criteria
• x1 x2 x3 x4 x5 x6 TYPE
• f1 80 65 83 40 52 94 II (Quasi)
• f2 90 58 60 80 72 96 III (Linear)
• f3 6 2 4 10 6 7 V (Linear Indif.)
• f4 5.4 9.7 7.2 7.5 2.0 3.6 IV (Level)
• f5 8 1 4 7 3 5 I (Usual)
• f6 5 1 7 10 8 6 VI (Gaussian)
• Parameters f1- q 10 f2 - p 30 f3 - q
  0.5, p 5
• f4 q 1, p 6 f5 - f6 s 5
• The criteria are having equal importance

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Preference Index (?)

• ? x1 x2 x3 x4 x5 x6
  ?(x)
• x1 - 0.296 0.25 0.268 0.1 0.185
  1.44
• x2 0.462 - 0.389 0.333 0.296 0.5
  1.98
• x3 0.236 0.18 - 0.333 0.056 0.429
  1.232
• x4 0.399 0.505 0.305 - 0.223 0.212
  1.644
• x5 0.444 0.515 0.487 0.380 - 0.448
  2.274
• x6 0.286 0.399 0.250 0.423 0.133 -
  1.5
• ?-(x) 1.827 1.895 1.681 1.746 0.808
  1.774
• ?(x) -0.728 0.085 -0.447 -0.102 1.466
  -0.274

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PROMETHEE I

• Gives partial preorder

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PROMETHEE II

• Gives complete preorder

x1
x6
x2
x5
x4
x3