Chapter 6 GAIA: MCDM VISUAL INTERACTIVE MODELLING

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Chapter 6 GAIA MCDM VISUAL INTERACTIVE
MODELLING
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Development of the Net Flow

• GAIA Geometrical Analysis for Interactive Aid
• Net flow ?(a) ?(a) - ?-(a)
• Single criterion net flows

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Development of the Net Flow

• Matrix of the single criterion flows
• This matrix is richer in information than the
  evaluation table
• GAI procedure Geometrical Interpretation of the
  information included in this matrix

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Development of the Net Flow

• Representation in a Rk space each action can be
  represented as a point in a Rk space
• ai ?1(ai), ?2(ai), , ?j(ai), , ?k(ai)
• n actions ? cloud of n points
• Projection of the cloud on a plane
• GAIA plane plane (u, v) on which the cloud is
  projected and so that
• Loss on information minimum
• Obtained by PCA techniques
• (Eigen values eigen vectors)
• d quality of projection
• ( information in GAIA plane)

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Development of the Net Flow
GAIA plane
ai
?1

• d

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PCA Principle Components Analysis

• Projection on a vector OPi aiu
• Projection on a plane
• Max (uCu vCv) s.t. uu 1, vv 1, u-v
• Max (uCu vCv) ?1 ?2
• ?1, ?2 best and second best eigen vector
• u, v associated eigen vector
• Quality of the projection on the (u, v) plane

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Geometrical Representation of the Criteria

• Cjj variance of criterion j
• Cpq covariance between criteria p and q
• The variances Cjj
• Cjj large ? ?j long criterion j differentiates
  strongly the actions
• The covariances Cpq
• Cpqgt0 and large ? similar criteria
• large Cpq ? large (?p,?q)? cos? 1
• Cpq 0 independent criteria
• Cpq 0 ? (?p,?q) 0 ? cos? 0
• Cpqlt0 conflictual criteria
• Cpqlt0 ? (?p,?q) lt 0 ? cos? -1

?p
?
?q
?q
?
?p
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Geometrical Representation of the Criteria
GAIA Plane
?2
?5
?7
?1
?8
?3
?4
?10
?6
?9
d
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Geometrical Representation of the Criteria
GAIA Plane
?2
?10
?5
a1
a6
?1
a2
?6
a7
a3
a8
?3
?5
?4
?7
a4
a5
?8
d
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Geometrical Representation of the Actions

• Good actions on particular criteria
• Cluster of similar actions small distances
  between a1, a2, a3, a4 similar values ?(a1),
  ?(a2), ?(a3), ?(a4) ?cluster similar actions (a1,
  a2, a3, a4)
• Discrepancy between actions clusters of the
  conflictual actions (a1, a2, a3, a4) and (a6, a7,
  a8, a9)
• Good compromises in case of conflictual criteria
  actions close to the origin. Never good, but also
  never bad on each criterion

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PROMETHEE Decision Analysis

• Geometrical interpretation of the weights
• Decision stick! w (w1, w2, , wj, , wk)
• Net flow of action a
• Projection of on net flow of a
• The larger this projection the better the action
• w axis on which the net flows are projected
• e unit vector on w
• p projection of e on the GAIA plane, PROMETHEE
  decision axis. The more the action is represented
  in the direction of p, the better it is.

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w
a6
a4
a5
e
a3
a1
a2
a4
p
GAIA plane

• d

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Analysis of Two Particular Cases
GAIA Plane
a1
a2
a5
a3
p
a6
a4
a7
d
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• Criteria not too much conflictual
• PROMETHEE Decision axis
• p very long
• Strong decision power
• Recommended actions
• As far as possible in the direction p
• a2, a3, a4

GAIA plane
w
p
d
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Case 2
GAIA Plane
a3
a2
a4
a8
p
a7
a9
a5
a1
a6
d
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• Criteria very conflictual
• PROMETHEE Decision axis
• p short
• Weak decision power
• Recommended actions
• close to the origin
• Good compromises, never good, never bad on the
  criteria set
• a1, a2, a7, a9

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Modification of the Weights

• The position of the criteria and the actions in
  the GAIA plane remains unmoved for every
  modification of the weights
• Only the PROMETHEE decision axis p moves
• weights ? decision stick
• Clear interpretation of the weights

w
w
GAIA plane
p
p
d
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Sensitivity Analysis ? DSS

• Variation of the weights within the limits agreed
  by the decision-maker
• Appreciation of the new position of the decision
  axis p ? new ranking of the actions
• Acquirement of more maturity on the problem
• Finalize the decision

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Classification of the Hard and Soft MCDM Problems

• Data authorized variations on the weights
• wj wj wj j 1, 2, ,k
• Determination of the hypersphere of the set ?
  corresponding to the authorized weight vectors
• Determination of the projection ? of ? on the
  GAIA plane

w
?
GAIA plane
?
p
d
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Classification of the Hard and Soft MCDM Problems

• Soft MCDM Problems
• ? doesnt include the origin
• Each modification of the weights does not lead to
  a fundamental modification of the direction of
  the PROMETHEE decision axis

• Hard MCDM Problems
• ? includes the origin
• An authorized modification of the weights can
  lead to a fundamental modification of the
  direction of the PROMETHEE decision axis

GAIA Plane
GAIA Plane
p
?
p
p
?
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Example A Location Problem

• Six criteria are considered as relevant by the DM
  to the rank 6 actions (a1, .., a6)
• 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
• a1 a2 a3 a4 a5 a6 TYPE
• f1 80 65 83 40 52 94 II (Quasi)
• f2 90 58 60 80 72 96 III (Linear)
• f3 600 200 400 1000 600 700 V (Linear Indif.)
• f4 54 97 72 75 20 36 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
  50, p 500
• f4 q 10, p 60 f5 - f6 s
  5
• The criteria are having equal importance

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Example
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Sensitivity Analysis Modification of the Weights

• Actions and criteria unmoved
• New decision axis
• New PROMETHEE II ranking
• Appreciation ? decision

• New Weights

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Sensitivity Analysis Modification of the Weights
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Sensitivity Analysis Modification of the Weights
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PROMETHEE V MCDM Analysis With Clustering
Constraints
A

• n actions evaluated on k criteria
• fj(ai), i 1, , n j 1, , k
• In addition to these data, the actions are also
  grouped in clusters
• Clusters ai, i ? Sr, r 1, , R ?Sr
  1,2,,n
• The problem is to select several actions
• Clustering constraints within and between the
  clusters must hold (cardinality, financial,
  manpower, etc.)
• PROMETHEE V
• Step1 PROMETHEE-GAIA Analysis. ?a F(a)
• Step 2 Resolution of the following BIP

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PROMETHEE V MCDM Analysis With Clustering
Constraints

• Where hold for ?, ? or
• xi 1 if ai selected 0, otherwise

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Example Location of Distribution Centers

• Basic data 12 actions 5 criteria and 4
  segments
• S1 1,2 (Antwerp) S2 3,4,5(Bruges)
• S3 6, 7, 8, 9 (Brussels) S410, 11, 12
  (Namur)
• PROMETHEE V
• Step 1 PROMETHEE-GAIA Analysis
• Step 2 BIP including clustering constraints
• Objective function net flows obtained by
  PROMETHEE II
• Constraint K1, K2 the total number of selected
  clusters must be between 5 and 9
• Constraint K3 Total expected return must exceed
  4000
• Constraint K4 Total man power employed in S1 and
  S2 must exceed 200
• Constraint K5 the wages paid in Brussels may not
  exceed those paid jointly in Antwerp, Bruges and
  Namur

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Example Location of Distribution Centers

• Constraint K6, K7, K8, K9 Cardinality
  constraints within th clusters Antwerp ( 1),
  Bruges (? 2), Brussels (? 2), Namur (?1)
• Constraint K10, K11, K12 Exclusion constraints
• Constraint K13, K14, K15, K16 Maximum available
  man power in each cluster Antwerp (300), Bruges
  (200), Brussels (500), Namur (150)
• Solution of the BIP
• PROMETHEE V proposes 7 locations
• The clustering constraints holds
• A Total net flow of 126 has been collected
• Sensitivity analysis on the BIP is possible

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Example
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Example
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Example
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Example
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Example Location of Distribution Centers
Antwerp1 (-184)
Antwerp2 (-92)
Bruges1 (327)
Brussels1 (-336)
Bruges2 (354)
Namur1 (61)
Brussels2 (-365)
Namur2 (177)
Bruges3 (341)
Namur3 (161)
Brussels3 (-274)
Brussels4 (-171)
BELGIUM
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Software PROMCALC- GAIA

• Demo versions
• PROMCALC Examples. 6 actions. 6 criteria
• GAIA Examples. 6 actions. 6 criteria
• Full versions
• PROMCALC 60 actions. 30 criteria
• GAIA 60 actions. 30 criteria
• PROMCALC-GAIA actionscriteria upto 3600!
• Other dimensions possible
• Author Information Professor J.P. Brans
• References
• B. MARESCHAL and J.P.Brans, 1988. Geometrical
  Representations for MCDA. European Journal of
  Operational Research, 34, 69-77
• J.P.Brans and B. Mareschal, 1992. PROMETHEE V
  MCDM Problems with Segmentation Constraints.
  INFOR, 30, 2, 85- 95