Carlos A. Bana e Costa

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International Doctoral School on

Multiple Criteria Decision Analysis (MCDA), Data Mining and Rough Sets Multiple Criteria Decision Analysis (MCDA), Data Mining and Rough Sets

Carlos A. Bana e Costa

Full Professor of Decision & Information

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

Visiting Professor of Decision Sciences Department of Management

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

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

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Development and use of logical methods for the improvement of decision making for 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 groups techniques for facilitating decision making by groups

• computer modeling software and expert systems for decision support

http://decision-analysis.society.informs.org/

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Problem dominated by

i i

Uncertainty Multiple Objectives

EXTEND conversation

•Event tree

•Fault tree

EVALUATE options

•Multi-criteria decision analysis

CHOOSE 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

NEGOTIATE SEPARATE into components

•Credence decomposition

•Risk analysis

•Multi-criteria

bargaining analysis

y

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MULTIPLE CRITERIA DECISION ANALYSIS

Several issues

Famíly of

Multiple (consequences, attributes,

characteristics, objectives, dimensions, etc.)

fundamental Points of view

Multiple criteria

Descriptors of performance

Intracriterion judgments:

Local evaluation models Coherent family of

Evaluation criteria

Intercriteria Multicriteria

aggregation Overall

judgments:

Weighting criteria

aggregation

of local preferences evaluation model

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MULTI

MULTI--CRITERIA VALUE MEASUREMENT CRITERIA VALUE MEASUREMENT

• Measuring the relative value of options in each criterion:

Numerical (e.g. direct rating) and

Non-numerical approaches (e.g. MACBETH) pp ( g )

• Criteria weighting procedures

Numerical techniques (e.g. swing weighting)

N i l t h i ( MACBETH)

Non-numerical techniques (e.g. MACBETH)

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MULTI-CRITERIA VALUE MEASUREMENT

n

Evaluation framework: Additive value model

) a ( v . k )

a (

V

_j

n

1 j

∑

j

=

⎪ ⎪

⎧

∀

=

j 0 )

h l

(

j , 100 )

anchor upper

(

v

_j _j

With:

V(a) overall value of option a

⎪ ⎪

⎨ =

∀

= ) (

100 )

anchors upper

all ( V

j , 0 )

anchor lower

(

v

_j _j

v

_j

(a) local value (score) f ti

⎪⎩ V ( all lower anchors ) = 0

of option a

against criterion j

∑

ⁿ

k

_j

scaling constant (relative weight)

∑

=

1 j

j

1 k

^and

k

_j > 0 (

j

= 1,…,n )

of criterion j

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Non-numerical approach: MACBETH

Measuring Attractiveness by a Categorical Based Evaluation Technique

An interactive pairwise comparison approach

to guide the construction of a quantitative value model g q from qualitative value judgments

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

it becomes a calculation and not a decision.

Elliot Jaques Elliot Jaques Requisite Organization, 1988

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

http://www.m-macbeth.com

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How does it work?

MACBETH i l ti t l

MACBETH uses a simple question-answer protocol that involves only two options in each question:

Ask the evaluator to pairwise compare options by given a qualitative y g q judgement j g

of the difference in attractiveness between each two options p

For x and y such that For x and y such that

x is preferred to y,

the difference in attractiveness the difference in attractiveness

between x and y is:

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MACBETH semantic categories of difference of attractiveness:

C C

₆

C

₅

C

₄

C

₃

C

₃

C

₂

C C

₁

C

₀

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’.

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How many judgements?

For a set X of m options, the number of pairwise

comparisons can vary from a maximum of m(m-1)/2 judgments when all pairwise comparisons are made judgments, 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 1 in 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).

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Larry’s summer holidays in Hawai Kaua’i

Ni’h holidays in Hawai

O’ahu M l k ’i Ni’hau

Moloka’i

Lãna’i M i Lãna i Maui

Hawai’i

In which island?

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Assessing MACBETH intracriterion preference intracriterion 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 presented

presented.

Technically, this is done by a

th ti l mathematical programming algorithm (see

Bana e Costa et al.

2005 for details).

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Molo

Very weak Lana

Strong

Moderate Maui

Moderate

Weak

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For a set of consistent judgements, MACBETH suggestsj g , gg a numerical scale v on X that satisfies the following

measurement rules:

Rule 1 Rule 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 2 Rule 2

∀ k, k’ ∈ { 1, 2, 3, 4, 5, 6 }, ∀ x, y, w, z ∈ X, with (x, y) ∈ C_k and (w, z) ∈ C_k’ :

k ≥ k’ + 1 ⇒ v(x) -v(y) > v(w) -v(z)

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The software The software determines the interval within which each score which 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.

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“Behind” M-MACBETHBehind M MACBETH

João Bana e Costa Carlos Bana e Costa

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A B C D

A Very Weak Moderate Very Strong Extreme 6

Very

St 5

B Moderate Strong

Strong

Strong 4

C Strong Moderate 3

Weak 2

D

Very Weak 1

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A B C D

Very

St 5

1

Strong

Strong 4 1

3

Weak 2

4 D

Very Weak 1

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A B C D

Very

St 5

1 4

Strong

Strong 4

1 4

3

Weak 2

4 D

Very Weak 1

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

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A B C D

Very

St 5

1 4

Strong

Strong 4

1 4

3

Weak 2

4 D

Very Weak 1

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

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A B C D

Very

St 6

1 4

Strong

Strong 5

1 4

3

C Strong Moderate 3-4

Weak 2

5 D

Very Weak 1

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

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A B C D

Very

St 9

1 4

Strong

Strong 5-8

1 4

3 8

C Strong Moderate 3-4

Weak 2

5 D

Very Weak 1

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

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A 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 no

Very Weak 1

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

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MACBETH weighting procedure

strong weak

t very t strong

strong to

moderateto strong very strongto

Qualitative swing judgements

SUM=1

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2) to compare the difference in overall value between any two swings (such that the first is more attractive than the second)

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REFERENCES

B C C A V i k J C D C J M (2003) MACBETH W ki P LSEOR 03 6

Bana 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 use

MACBETH t b ild tit ti l d l b d lit ti l j d t E

MACBETH to build a quantitative value model based on qualitative value judgments. European Journal of Operational Research 153(2) 323-331.

Bana e Costa C.A., J.M. De Corte, J.C. Vansnick. 2005. On the mathematical foundations of MACBETH 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 t MACBETH. J. Figueira, S. Greco, M. Ehrgott, eds. Multiple Criteria Decision Analysis: State of the Art Surveys. 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# id

http://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 i

Portugal,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 of reusable bid evaluation models for the Portuguese Electric Transmission Company", Decision A l i 5 1 (22 42)

Analysis, 5, 1 (22-42).