carlos a. bana e costa - université paris dauphinetsoukias/download/tutorials/shortvaria/... ·...

International Doctoral School onMultiple Criteria Decision Analysis (MCDA), Data Mining and Rough SetsMultiple Criteria Decision Analysis (MCDA), Data Mining and Rough Sets

Carlos A. Bana e CostaFull Professor of Decision & Information

School of Engineering (IST), Technical University of LisbonMember of the Centre of Management Studies of IST (CEG-IST)g ( )

Visiting Professor of Decision SciencesDepartment of Management

L d S h l f E i d P liti l S iLondon School of Economics and Political Science

http://web.ist.utl.pt/carlosbana

Troina, Sicily, Italy, 12 April 2008

Development and use of logical methodsfor the improvement of decision makingfor the improvement of decision-making

in public and private enterprise.

Such methods include:

• models for decision-making under conditions of

uncertainty or multiple objectives

• techniques of risk analysis and risk assessment;

i l d d i i di f d i i ki b h i• experimental and descriptive studies of decision-making behavior

• economic analysis of competitive and strategic decisions

• techniques for facilitating decision-making by groupstechniques for facilitating decision making by groups

• computer modeling software and expert systems

for decision support

http://decision-analysis.society.informs.org/© 2008 Carlos A. Bana e Costa

Problem dominated by

i i

Uncertainty Multiple Objectives

EXTEND conversation•Event tree•Fault tree

EVALUATE options•Multi-criteria decision analysisCHOOSE option

REVISE opinion

Fault tree•Influence diagram

decision analysis

ALLOCATE resources•Multi criteria

p•Payoff matrix•Decision tree

•Bayesian nets•Bayesian statistics

SEPARATE into components

•Multi-criteria commons dilemma

NEGOTIATESEPARATE into components•Credence decomposition•Risk analysis

•Multi-criteria bargaining analysis

y

© 2005 Larry PhillipsReference: L.D.Phillips, Decision Analysis in 2005

MULTIPLE CRITERIA DECISION ANALYSIS

Several issuesFamíly of

Multiple(consequences, attributes, characteristics, objectives,

dimensions, etc.)

fundamental

Points of view

Multiple

criteria

Descriptorsof performance

Intracriterion judgments:Local evaluation

models

Coherent family ofEvaluation criteria

Intercriteria Multicriteriaaggregation Overall

judgments:Weighting criteria

aggregationof local preferences evaluation

model


MULTIMULTI--CRITERIA VALUE MEASUREMENTCRITERIA VALUE MEASUREMENT

• Measuring the relative value of options in each criterion:

Numerical (e.g. direct rating) andNon-numerical approaches (e.g. MACBETH)pp ( g )

• Criteria weighting procedures• Criteria weighting proceduresNumerical techniques (e.g. swing weighting)N i l t h i ( MACBETH)Non-numerical techniques (e.g. MACBETH)


MULTI-CRITERIA VALUE MEASUREMENT

n

Evaluation framework: Additive value model

)a(v.k)a(V j

n

1jj∑

=

=

⎪⎪⎧

∀

∀=

j0)hl(j ,100)anchorupper(v jjWith:

V(a) overall value of option a⎪⎪

⎪⎪

⎨=

∀=

)(100)anchorsupperall(Vj,0)anchorlower(v jj

vj(a) local value (score)f ti

⎪⎩ = 0)anchorslowerall(V

of option aagainst criterion j

∑n

kj scaling constant(relative weight)

∑=

=1j

j 1k and kj > 0 ( j = 1,…,n )

of criterion j


Non-numerical approach: MACBETH

Measuring Attractiveness by a Categorical Based Evaluation Technique

An interactive pairwise comparison approachto guide the construction of a quantitative value modelg q

from qualitative value judgments

When you take all non-verbal judgment out of a decision it becomes a calculation and not a decisionit becomes a calculation and not a decision.

Elliot JaquesElliot JaquesRequisite Organization, 1988


Download the Demo Version of M-MACBETH http://www m-macbeth comhttp://www.m-macbeth.com


How does it work?MACBETH i l ti t lMACBETH uses a simple question-answer protocolthat involves only two options in each question:Ask the evaluator to pairwise compare optionsby given a qualitative judgementy g q j gof the difference in attractivenessbetween each two optionsp

For x and y such thatFor x and y such thatx is preferred to y,

the difference in attractivenessthe difference in attractivenessbetween x and y is:


MACBETH semantic categories ofdifference of attractiveness:

CC6

C5

C4

C3C3

C2

CC1

C0

Note: the ‘weak’, ‘strong’ and ‘extreme’ were initially called the fundamental categories, but the M-MACBETH software that implements the MACBETH approach does not make this distinction and even allows for group judgments that do not


does not make this distinction and even allows for group judgments that do not distinguish between several consecutive categories, such as ‘strong or very strong’.

How many judgements?

For a set X of m options, the number of pairwisecomparisons can vary from a maximum of m(m-1)/2 judgments when all pairwise comparisons are madejudgments, when all pairwise comparisons are made,to a minimum acceptable number of m–1 judgments, as when comparing only each two consecutive options in the ranking or one options with all of the other m 1in the ranking or one options with all of the other m–1 (however, it is recommended to ask for some additional judgments to perform several consistency checks)


checks).

Larry’s summerholidays in Hawai

Kaua’iNi’h holidays in HawaiO’ahu

M l k ’iNi’hau

Moloka’iLãna’i M iLãna i Maui

Hawai’i

In which island?


Assessing MACBETH intracriterion preferenceintracriterion preference information

As each judgement is entered in the matrix, its consistency with the judgments already inserted is checked and possible

if an inconsistency is detected

already inserted is checked and possible inconsistencies are detected.

is detected, suggestions to overcome it are presentedpresented. Technically, this is done by a

th ti lmathematical programming algorithm (see Bana e Costa et al. 2005 for details).


Molo

Very weak

Lana

Strong

ModerateMaui

Moderate

Weak

Oahu© 2008 Carlos A. Bana e Costa

For a set of consistent judgements, MACBETH suggestsj g , gga numerical scale v on X that satisfies the followingmeasurement rules:Rule 1Rule 1∀ x, y ∈ X : v(x) = v(y) iff x and y are equally attractive∀ x, y ∈ X : v(x) > v(y) iff x is more attractive than y;Rule 2Rule 2∀ k, k’ ∈ { 1, 2, 3, 4, 5, 6 }, ∀ x, y, w, z ∈ X,with (x, y) ∈ Ck and (w, z) ∈ Ck’ :

k ≥ k’ + 1 ⇒ v(x) -v(y) > v(w) -v(z)


The softwareThe software determines the interval within which each scorewhich each score of each option can vary when the other m-1 scores are fixed and still remain compatible with the matrix of judgments.

This allows the adjustment of the scale by comparing differences of scores, to arrive to a cardinal scale.


“Behind” M-MACBETHBehind M MACBETH

João Bana e Costa Carlos Bana e Costa

A B C DA B C D

A Very Weak Moderate Very StrongExtreme 6

Very St 5

B Moderate StrongStrong

Strong 4

C StrongModerate 3

Weak 2

DVery Weak 1

© 2008 João Bana e Costa & Carlos A. Bana e Costa

A B C DA B C D


Very St 51


Strong 4

1

3

C StrongModerate 3

Weak 24

DVery Weak 1


A B C DA B C D


Very St 51 4


Strong 4

1 4

3

C StrongModerate 3

Weak 24

DVery Weak 1

v(A)-v(C) = v(A)-v(B) + v(B)-v(C)


A B C DA B C D


Very St 51 4


Strong 4

1 4

3

C StrongModerate 3

Weak 24

DVery Weak 1

v(A)-v(C) < v(C)-v(D)


A B C DA B C D


Very St 61 4


Strong 5

1 4

3

C StrongModerate 3-4

Weak 25

DVery Weak 1

Increase v(C)-v(D) of 1


A B C DA B C D


Very St 91 4


Strong 5-8

1 4

3 8

C StrongModerate 3-4

Weak 25

DVery Weak 1

v(B)-v(D) = v(B)-v(C) + v(C)-v(D)


A B C DA B C D

A no Very Weak Moderate Very Strong

1 4 9

Extreme 10

Very St 9

B no Moderate Strong

1 4

3 8

9 Strong

Strong 5-8

C no Strong

5

Moderate 3-4

Weak 2

D noVery Weak 1

v(A)-v(D) = v(A)-v(B) + v(B)-v(C)+ + v(C)-v(D)


MACBETH weighting procedure

strong weakt

veryt

strongto

strongto

moderatestrongto

very strong

Qualitative swing judgementsSUM=1


2) to compare the difference in overallvalue between any two swings (suchthat the first is more attractive thanthe second)


REFERENCES

B C C A V i k J C D C J M (2003) MACBETH W ki P LSEOR 03 6Bana e Costa, C.A., Vansnick, J.C., De Corte, J.M. (2003), MACBETH, Working Paper LSEOR 03.56,London School of Economics, London.

Bana e Costa C.A., M.P. Chagas. 2004. A career choice problem: an example of how to useMACBETH t b ild tit ti l d l b d lit ti l j d t EMACBETH to build a quantitative value model based on qualitative value judgments. EuropeanJournal of Operational Research 153(2) 323-331.

Bana e Costa C.A., J.M. De Corte, J.C. Vansnick. 2005. On the mathematical foundations ofMACBETH J Fi i S G M Eh tt d M lti l C it i D i i A l i St t f th A tMACBETH. J. Figueira, S. Greco, M. Ehrgott, eds. Multiple Criteria Decision Analysis: State of the ArtSurveys. Springer, New York, 409-442.

Bana e Costa C.A., J.M. De Corte, J.C. Vansnick. 2005. M-MACBETH Version 1.1 User’s Guide,htt // b th /d l d ht l# idhttp://www.m-macbeth.com/downloads.html#guide.

Bana e Costa J. 2007. Behind MACBETH. Presented at the Nato Advanced Research on Risk,Uncertainty and Decision Analysis for Environmental Security and Non-chemical Stressors, Lisbon,P t l htt // b th / f ht l#b iPortugal, http://www.m-macbeth.com/references.html#basic.

Bana e Costa, C.A., Lourenço, J.C., Chagas, M.P., Bana e Costa, J.C. (2008), "Development ofreusable bid evaluation models for the Portuguese Electric Transmission Company", DecisionA l i 5 1 (22 42)


Analysis, 5, 1 (22-42).

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