E LECTRE T RI -C:AMultipleCriteriaSortingMethodBasedonCentralReferenceActions

E

T

-C: A Multiple Criteria Sorting Method Based on Central Reference Actions

Juscelino ALMEIDADIAS^12∗, José Rui FIGUEIRA¹², Bernard ROY¹

1LAMSADE, Université Paris-Dauphine, France

2CEG-IST, Technical University of Lisbon, Portugal

∗Supported by FCT-Portugal (Grant: SFRH/BD/22985/2005)

Doctoral School: Algorithmic Decision Theory MCDA, Data Mining, and Rough Sets

Troina, Italy 11-16 April, 2008

Contents

1 Introduction

2 Concepts, assumptions, and requirements

3 ELECTRE TRI-C: Assignment rules and properties

4 Comparison with ELECTRE TRI-B

5 Conclusions

6 References

Introduction

Sorting problematic

Sorting problematic: the set of categories emerges naturally from thedecision aiding contextthrough an interaction process with the decision-makers.

Nature of the categories: each category is defined in order to assign actions which will be subject to thesame treatment or analysis.

Absolute evaluation: the assignment of an action only takes into account theintrinsic evaluationof this action on all the criteria and does not depend on nor influencethe category to which another action should be assigned.

Introduction

Motivation examples

Example (Credit analysis) Actions: Credit demand files Categories:

Accepted without additional information Accepted with additional information

Sent to a particular department further analysis Rejected under certain conditions

Rejected with no conditions at all

Example (Medical diagnosis)

Actions: Patients waiting for treatment Categories: Set of pathologies studied

Introduction

Reference actions

Profile limits

Each category is bounded by a lower and an upper profile.

The well-known method called up to nowELECTRE TRIbased on profile limits, orboundary actions, will be designated here by ELECTRE TRI-B.

Central reference actions

Each category is defined by a central reference action.

ELECTRE TRI-Cis, therefore, the designation of the procedures based oncentral reference actions.

Key concepts

Actions, criteria, and credibility index

a₁,a₂, . . .are thepotential actions. The set of such actions,A, can be partially known a priori.

F ={g₁, . . . ,g_j, . . . ,gn}is acoherent family of criteria, with n≥2.

Ωj(a,a^′)is theadvantageofaovera^′ on criterion g_j ∈F , Ω_j(a,a^′) =

g_j(a)−g_j(a^′) ifg_j is to be maximized g_j(a^′)−g_j(a) ifg_j is to be minimized

Each criterion g_j ∈F will be considered as apseudo-criterion.

Definition

σ(a,a^′)is thecredibility of the comprehensive outrankingofa overa^′ when taking all the criteria from F into account. ^Definition

Basic assumptions

ELECTRETRI-C

C={C1, . . . ,C_h, . . . ,C_q}is the set ofpre-defined and ordered categories, where C₁is the worst category, and Cq the best one, with q≥2.

Each categoryC_his defined by acentral reference actionb_h, h=1, . . . ,q.

B={b0,b₁, . . . ,b_h, . . . ,b_q,b_q+1}is the set of(q+2)reference actions.

b₀is a particular reference action with the worst possible evaluation g_j(b0)on criterion g_j, for all g_j ∈F .

b_q+1is a particular reference action with the best possible evaluation g_j(b_q+1)on criterion g_j, for all g_j ∈F .

Basic assumptions

Strict separability condition

Definition (Dominance)

The set of reference actions, B, fulfills the (strict) dominance relation if and only if

∀j, Ωj(b_h+1,b_h)≥0and∃j, Ωj(b_h+1,b_h)>0;h=0, . . . ,q.

Definition (Strict separability condition)

The set of reference actions, B, fulfills the strict separability condition if and only if

Ωj(b_h+1,b_h)>p_j; j =1, . . . ,n; h=0, . . . ,q.

Weak

Structural requirements

1 Conformity: Each central reference action,b_h, must be assigned to the category,C_h, h=1, . . . ,q.

2 Monotonicity: If an actionastrictly dominatesa^′, thenais assigned to a category at least as good as the categorya^′ is assigned to.

3 Homogeneity: Two actions must be assigned to the same category when they compare themselves in an identical manner with the reference actions.

4 Stability: After a modification of the set B by applying either a mergingor asplitting procedure, the non-adjacent categories to the modified ones will remain with the same actions as before the modification.

Stability requirement

Definition (Basic modification procedures)

1 Merging procedure: The distinction between two consecutive categories,C_h−1andC_h, will be ignored by introducing a new central reference action,b^′_h, such that:

Ωj(b^′_h,b_h−1)≥0, for all g_j ∈F Ωj(bh,b_h^′)≥0, for all g_j ∈F

2 Splitting procedure: The categoryC_hcan be split into two new consecutive categories by introducing two new central reference actions,b_h^′ andb_h^′′, such that:

bh+1∆F b^′′_h, b^′′_h ∆F b^′_h, and b_h^′ ∆F b_h−1 Ωj(b^′′_h,b_h)≥0, for all g_j ∈F

Ωj(bh,b_h^′)≥0, for all gj ∈F

Slackness functions

Definition (Slackness functions)

Letλ∈[0.5,1]denote the chosen majority level:

1 Direct slackness function:

ξ_h⁺(a, λ) =σ(a,b_h)−λ, h = (q+1), . . . ,0.

2 Reverse slackness function:

ξ_h⁻(a, λ) =σ(bh,a)−λ, h =0, . . . ,(q+1).

Proposition

1 ξ_h⁺(·)does not decreasewhen moving from a given category to a worst one.

2 ξ_h⁻(·)does not decreasewhen moving from a given category to a best one.

Assignment rules of E

T

-C

Definition (Descending assignment rule) Choose a majority levelλ(0.5≤λ≤1).

Decrease h from(q+1)until the first value such that ξ_h⁺(a, λ)≥0.

Ifξ_h⁺(a, λ)≤ |ξ⁺_h+1(a, λ)|, then assign actionato categoryC_h. Otherwise, assignatoC_h+1.

Definition (Ascending assignment rule) Choose a majority levelλ(0.5≤λ≤1).

Increase h from 0 until the first value such that ξ_h⁻(a, λ)≥0.

Ifξ_h⁻(a, λ)≤ |ξ⁻_h−1(a, λ)|, then assign actionato categoryC_h. Otherwise, assignatoC_h−1.

Example Comparing ELECTRETRI-C rules ELECTRETRI-B rules

Numerical example

ELECTRETRI-C assignment results

Example

Actions b1 b2 b3 b4 Descending Ascending a₁ ≻ ≻ R^λ R^λ C₂ C₄ a₂ R^λ R^λ R^λ R^λ C₁ C₄

a₃ ≻ ≻ ≺ ≺ C₃ C₃

a₄ ≻ ≻ ≺ ≺ C₂ C₃

a₅ I^λ ≺ ≺ ≺ C₂ C₁

a₆ ≻ ≻ R^λ R^λ C₂ C₄

a₇ ≻ ≺ ≺ ≺ C₁ C₂

a₈ ≻ ≺ ≺ ≺ C₁ C₁

a₉ R^λ R^λ R^λ R^λ C₁ C₄

a₁₀ ≻ ≺ ≺ ≺ C₁ C₁

Note:λ=0.70; Source: Data adapted from Merad et al., 2004

ELECTRETRI-C rules Comparison ELECTRETRI-C rules

Properties of the assignment rules I

ELECTRETRI-C

Theorem

a) Themonotonicity,homogeneity, andstabilityrequirements hold.

b) If thestrict separability conditionis fulfilled, then theconformity requirement holds.

Remark

The properties ofuniquenessandindependenceof the assignments are also fulfilled.

Basic assumptions

Weak separability condition

Definition (Dominance)

The set of reference actions, B, fulfills the (strict) dominance relation if and only if

∀j, Ωj(b_h+1,b_h)≥0and∃j, Ωj(b_h+1,b_h)>0;h=0, . . . ,q.

Definition (Weak separability condition)

The set of reference actions, B, fulfills the weak separability condition if and only if

∀j, Ωj(b_h+1,b_h)≥0and∃j, Ωj(b_h+1,b_h)>p_j;h=0, . . . ,q.

Strict

Properties of the assignment rules II

ELECTRETRI-C

Theorem

If theweak separability conditionis fulfilled, then there exists a compatible majority level,λ^c, for which theconformityrequirement holds, whenever the chosen majority levelλ≥λ^c, such that

λ^c = 1 2+ 1

2 max

h=0, ...,q

nσ(bh,b_h+1)o

Example

Numerical example

Compatible majority level

Example

b₁ b₂ b₃ b₄

b₁ 1.0000 0.3696 0.0000 0.0000 b₂ 1.0000 1.0000 0.0217 0.0217 b₃ 1.0000 1.0000 1.0000 0.0652 b₄ 1.0000 1.0000 1.0000 1.0000 Compatible majority level:

λ^c = 1 2 +1

2 max

h=0,...,4{σ(bh,b_h+1)}=0.69

Source: Data adapted from Merad et al., 2004

Theorem

An overview of E

T

-B

Basic assumptions

Cb ={Cb₁, . . . ,Cb_h, . . . ,Cb_q}is the set ofpre-defined and ordered categories, whereCb₁is the worst category andCbq the best one, with q≥2.

Each categoryCb_his defined by alower profile limit,bˆ_h−1, and anupper profile limit,ˆb_h, such thatbˆ_h∆_Fbˆ_h−1, h=1, . . . ,q.

Bb ={ˆb₀,bˆ₁, . . . ,bˆ_h, . . . ,bˆ_q−1,bˆ_q}is the set of the(q+1)profile limits.

bˆ₀andbˆq play a similar role as b₀and b_q+1when using central reference actions.

An overview of E

T

-B

The most well-known assignment rules

Definition (Pseudo-conjunctive assignment rule) Choose a majority levelλ(0.5≤λ≤1).

Decrease h from q until the first value such that ξ_h⁺₋₁(a, λ) ≥0.

Assign actionato categoryCb_h.

Definition (Pseudo-disjunctive assignment rule) Choose a majority levelλ(0.5≤λ≤1).

Increase h from 0 until the first value such that ξ_h⁻(a, λ)≥0 andξ_h⁺(a, λ)<0.

Assign actionato categoryCb_h.

Comparing ELECTRETRI-B rules ELECTRETRI-C rules

Comparing results I

ELECTRETRI-C and ELECTRETRI-B

Theorem

Consider(q+2)reference actionsdefined to apply ELECTRE TRI-C with q categories. When such reference actions are used asprofile limitsof the(q+1)categories in ELECTRE TRI-B,

a) if an actionais assigned toC_hby the ELECTRE TRI-C descending rule, thenais assigned toCb_horCb_h+1by the ELECTRE TRI-B pseudo-conjunctive rule.

b) if an actionais assigned toC_t by the ELECTRE TRI-C ascending rule, thenais assigned toCb_k, withk ≥t, by the ELECTRE TRI-B pseudo-disjunctive rule.

Example Comparing results II

Numerical example

Comparing assignment results

Example

ELECTRETRI-C ELECTRETRI-B

Actions Descending Ascending Pseudo-conjunctive Pseudo-disjunctive

a1 C2 C4 Cb3 Cb5

a2 C1 C4 Cb1 Cb5

a3 C3 C3 Cb3 Cb3

a4 C2 C3 Cb3 Cb3

a5 C2 C1 Cb2 Cb2

a6 C2 C4 Cb3 Cb5

a7 C1 C2 Cb2 Cb2

a8 C1 C2 Cb1 Cb1

a9 C1 C4 Cb1 Cb5

a10 C1 C1 Cb1 Cb1

Note:λ=0.70; Source: Data adapted from Merad et al., 2004

Conclusions

InELECTRE TRI-Cthe categories are defined throughcentral reference actionsinstead of profile limits.

Central reference actions and profile limits aretwo alternative waysof defining ordered categories.

ELECTRE TRI-Cfulfills the properties ofuniqueness, independence,conformity,monotonicity,homogeneity, and stability.

When the set of reference actionsdoes not fulfill the strict separability condition, but only a weak separability condition, then acompatible majority levelis required.

A comparison withELECTRE TRI-Bshows the main similarities of the two methods.

References

Almeida Dias, J.; Figueira, J.R. and Roy, B. (2008). ELECTRETRI-C: A Multiple Criteria Sorting Method Based on Central Reference Actions.

Technical Report 5/2008, CEG-IST, Tecnical University of Lisbon, Lisboa, Portugal.

Merad, M.; Verdel, T.; Roy, B. and Kouniali, S. (2004). Use of

multi-criteria decision-aids for risk zoning and management of large area subjected to mining-induced hazards. Tunnelling and Underground Space Technology, 19:165–178.

Roy, B. (1996). Multicriteria Methodology for Decision Aiding. Kluwer Academic Publishers, Dordrecht.

Roy, B. and Bouyssou, D., (1993). Aide Multicritère à la Décision : Méthodes et Cas. Economica, Paris, France.

Yu, W. (1992). Aide Multicritère à la Décision dans le Cadre de la Problématique du Tri : Concepts, Méthodes et Applications. Thèse de Doctorat. LAMSADE, Université Paris-Dauphine, Paris, France.

Appendix

7 Appendix: Additional results Pseudo-criterion model Credibility index

Binary relations

Comparing ELECTRE TRI-C assignment rules Comparing ELECTRE TRI-B assignment rules Comparing results II

ELECTRE TRI-C particular results

Pseudo-criterion model

Definition (Pseudo-criterion)

Apseudo-criterionis a function g_j associated with two threshold functions,q_j(·)andp_j(·), satisfying the following condition: for all actionsain the sets of actions,g_j(a) +p_j and g_j(a) +q_j are

non-decreasing monotone functions of g_j(a). (Roy, 1996)

Key concepts

Remark

Consider an ordered pair of actions(a,a^′), and the two thresholds associated to the pseudo-criterion model,

a) a P_j a^′ ⇔Ωj(a,a^′)>p_j(·)

b) a Q_j a^′⇔q_j(·)<Ω_j(a,a^′)≤p_j(·) c) a I_j a^′ ⇔ −q_j(·)≤Ω_j(a,a^′)≤q_j(·)

Key concepts

Credibility index of ELECTREmethods

Definition (Credibility index)

Thecredibility of the comprehensive outrankingof an actionaovera^′, which means thatamay be judged at least as good asa^′ when taking all the criteria from F into account, is defined by aggregating the comprehensive concordance index,c(a,a^′), and the partial discordance indices,d_j(a,a^′), as follows:

σ(a,a^′) =c(a,a^′) Yn j=1

T_j(a,a^′)

where,

T_j(a,a^′) = ( ₁

−d_j(a,a^′)

1−c(a,a^′) ifd_j(a,a^′)>c(a,a^′)

1 otherwise

Key concepts

Binary relations

Definition (λ-binary relations)

1 λ-outranking:aS^λa^′⇔σ(a,a^′)−λ≥0

2 λ-indifference:aI^λa^′ ⇔σ(a,a^′)−λ≥0 ∧ σ(a^′,a)−λ≥0

3 λ-incomparability:aR^λa^′ ⇔σ(a,a^′)−λ <0 ∧ σ(a^′,a)−λ <0

4 λ-preference:a≻a^′ ⇔σ(a,a^′)−λ≥0 ∧ σ(a^′,a)−λ <0

Comparing assignment rules

ELECTRETRI-C

Theorem

a) If an actionaisλ-indifferentto at least one reference action, thenais assigned by the descending rule to a category at least as good as the oneais assigned to when using the ascending rule.

b) If an actionaisλ-incomparableto at least one reference action, thenais assigned by the descending rule to a category at most as good as the oneais assigned to when using the ascending rule.

c) Otherwise, both rules assign the actionato the same category or to two different but consecutive categories.

Example ELECTRETRI-C rules

Comparing assignment rules

ELECTRETRI-B

Roy and Bouyssou, 1993, p. 395

If an actionais assigned to categoryCb_k by the

pseudo-conjunctive rule and toCb_hby the pseudo-disjunctive rule, thenk ≤h.

Furthermore, the two assignment rules provide the same results if and only if there is no t such thatξ_t⁺(a, λ)<0 andξ_t⁻(a, λ)<0 or there is at most one t such thatξ_t⁺(a, λ)≥0 andξ⁻_t (a, λ)≥0.

ELECTRETRI-B rules

Comparing results II

ELECTRETRI-C and ELECTRETRI-B

Theorem

Consider(q+1)profile limitsdefined to apply ELECTRE TRI-B with q categories. When such profile limits are used asreference actionsof the(q−1)categories in ELECTRE TRI-C,

a) if an actionais assigned toCb_hby the ELECTRE TRI-B pseudo-conjunctive rule, thenais assigned toC_horC_h−1by the ELECTRE TRI-C descending rule.

b) if an actionais assigned toCb_t by the ELECTRE TRI-B

pseudo-disjunctive rule, thenais assigned toC_k, withk ≤tby the ELECTRE TRI-C ascending rule.

Comparing results I

E

T

-C particular results

Proposition

a) Ifaλ-outranksb_h, thenais assigned at least toC_h

by the descending rule.

b) Ifb_hλ-outranksa, thenais assigned at most toC_h

by the ascending rule.

c) Ifaisλ-preferredtob_h, thenais assigned at least toC_h by both rules.

d) Ifb_hisλ-preferredtoa, thenais assigned at most toC_h

by both rules.

ELECTRE methods with interaction between criteria: An extension of the concordance index

JOSÉ RUIFIGUEIRA¹, SALVATORE GRECO², BERNARDROY³

1CEG-IST, Instituto Superior Técnico, Portugal

2Faculty of Economics, University of Catania, Italy

3LAMSADE, Université Paris-Dauphine, France

Cost Action 0602, Troina, Sicily (11-16 April 2008)

Contents

1 Introduction

2 Illustrative Examples

3 Concepts: Definitions and notation

4 Types of interactions considered

5 Extensions of the concordance index

6 Conclusions

7 References

Introduction

This presentation is devoted to an extension of the comprehensive concordance indexof ELECTRE methods.

Such an extension have been considered to take into account the interaction between criteria.

Three types of interaction effects has been considered,mutual strengthening,mutual weakening, andantagonistic.

In real-world decision-making situations is reasonable to consider the interaction between asmall number of pairs of criteria.

Various conditions,boundary,monotonicity, andcontinuityhave been imposed.

Introduction

This presentation is devoted to an extension of the comprehensive concordance indexof ELECTRE methods.

Such an extension have been considered to take into account the interaction between criteria.

Three types of interaction effects has been considered,mutual strengthening,mutual weakening, andantagonistic.

In real-world decision-making situations is reasonable to consider the interaction between asmall number of pairs of criteria.

Various conditions,boundary,monotonicity, andcontinuityhave been imposed.

Introduction

This presentation is devoted to an extension of the comprehensive concordance indexof ELECTRE methods.

Such an extension have been considered to take into account the interaction between criteria.

Three types of interaction effects has been considered,mutual strengthening,mutual weakening, andantagonistic.

In real-world decision-making situations is reasonable to consider the interaction between asmall number of pairs of criteria.

Various conditions,boundary,monotonicity, andcontinuityhave been imposed.

Introduction

This presentation is devoted to an extension of the comprehensive concordance indexof ELECTRE methods.

Such an extension have been considered to take into account the interaction between criteria.

Three types of interaction effects has been considered,mutual strengthening,mutual weakening, andantagonistic.

In real-world decision-making situations is reasonable to consider the interaction between asmall number of pairs of criteria.

Various conditions,boundary,monotonicity, andcontinuityhave been imposed.

Introduction

This presentation is devoted to an extension of the comprehensive concordance indexof ELECTRE methods.

Such an extension have been considered to take into account the interaction between criteria.

Three types of interaction effects has been considered,mutual strengthening,mutual weakening, andantagonistic.

In real-world decision-making situations is reasonable to consider the interaction between asmall number of pairs of criteria.

Various conditions,boundary,monotonicity, andcontinuityhave been imposed.

Introduction

...

It allows for the representation of some

very important preference information, one that could not be modeled by the existing MCDA methodologies.This crucial preference

information, is theinteraction between criteriaexpressing preferences of the same sign (synergyandredundancy), or opposite sign (the power of the opposing criteria).

...

in Greco and Figueira (2003)

Introduction

...

It allows for the representation of some

very important preference information, one that could not be modeled by the existing MCDA methodologies. This crucial preference

information, is theinteraction between criteriaexpressing preferences of the same sign (synergyandredundancy),or opposite sign (the power of the opposing criteria).

...

in Greco and Figueira (2003)

Introduction

...

It allows for the representation of some

very important preference information, one that could not be modeled by the existing MCDA methodologies. This crucial preference

information, is theinteraction between criteriaexpressing preferences of the same sign (synergyandredundancy), or opposite sign (the power of the opposing criteria).

...

in Greco and Figueira (2003)

Introduction

B. Roy. Apropos de la signification des dépendancesentre critères : Quelle place et quels modèles de prise en compte pour laide à la décision ? Cahier du LAMSADE N. 244, 2007.

Introduction

Application areas:

Environmental problems

Constructions of indices (in this case several pairs ... of interaction criteria)

Introduction

Application areas:

Environmental problems

Constructions of indices (in this case several pairs ... of interaction criteria)

Introduction

Application areas:

Environmental problems

Constructions of indices (in this case several pairs ... of interaction criteria)

Illustrative Example 1

CHOOSING THE SITE FOR CONSTRUCTING A NEW HOTEL

Criteria:

g₁: land purchasing and construction costs (investment costs) [min];

g2: annual operating costs (annual costs)[min];

g₃: personnel recruitment possibilities (recruitment)[max];

g₄: target client perceptions of the city district (image)[max];

g5: facility of access for the target clients (access)[max].

Mutual strengtheningbetween criteria: investmentandannual costs.

Mutual weakeningbetween criteria: imageandaccess.

Illustrative Example 1

CHOOSING THE SITE FOR CONSTRUCTING A NEW HOTEL

Criteria:

g₁: land purchasing and construction costs (investment costs) [min];

g2: annual operating costs (annual costs)[min];

g₃: personnel recruitment possibilities (recruitment)[max];

g₄: target client perceptions of the city district (image)[max];

g5: facility of access for the target clients (access)[max].

Mutual strengtheningbetween criteria: investmentandannual costs.

Mutual weakeningbetween criteria: imageandaccess.

Illustrative Example 1

CHOOSING THE SITE FOR CONSTRUCTING A NEW HOTEL

Criteria:

g₁: land purchasing and construction costs (investment costs) [min];

g2: annual operating costs (annual costs)[min];

g₃: personnel recruitment possibilities (recruitment)[max];

g₄: target client perceptions of the city district (image)[max];

g5: facility of access for the target clients (access)[max].

Mutual strengtheningbetween criteria: investmentandannual costs.

Mutual weakeningbetween criteria: imageandaccess.

Illustrative Example 1

CHOOSING THE SITE FOR CONSTRUCTING A NEW HOTEL

Criteria:

g₁: land purchasing and construction costs (investment costs) [min];

g2: annual operating costs (annual costs)[min];

g₃: personnel recruitment possibilities (recruitment)[max];

g₄: target client perceptions of the city district (image)[max];

g5: facility of access for the target clients (access)[max].

Mutual strengtheningbetween criteria: investmentandannual costs.

Mutual weakeningbetween criteria: imageandaccess.

Illustrative Example 1

Thecomparisonof site a with sites b, c, and d ,in terms of the two financial criteriag₁and g₂

b c d

g₁ a is better than b a is worse than c a is better than d g₂ a is worse than b a is better than c a is better than d According to theclassic definition of the concordance index, the role that g₁and g₂should have for supporting the answer to the assertion

“a is at least as good as b (or c or d )” is characterized by the following weights,

k₁in the comparison with b, k₂in the comparison with c, k₁+k₂in the comparison with d .

Illustrative Example 1

Thecomparisonof site a with sites b, c, and d ,in terms of the two financial criteriag₁and g₂

b c d

g₁ a is better than b a is worse than c a is better than d g₂ a is worse than b a is better than c a is better than d According to theclassic definition of the concordance index, the role that g₁and g₂should have for supporting the answer to the assertion

“a is at least as good as b (or c or d )” is characterized by the following weights,

k₁in the comparison with b, k₂in the comparison with c, k₁+k₂in the comparison with d .

Illustrative Example 1

Thecomparisonof site a with sites b, c, and d ,in terms of the two financial criteriag₁and g₂

b c d

g₁ a is better than b a is worse than c a is better than d g₂ a is worse than b a is better than c a is better than d According to theclassic definition of the concordance index, the role that g₁and g₂should have for supporting the answer to the assertion

“a is at least as good as b (or c or d )” is characterized by the following weights,

k₁in the comparison with b, k₂in the comparison with c, k₁+k₂in the comparison with d .

Illustrative Example 1

Thecomparisonof site a with sites b, c, and d ,in terms of the two financial criteriag₁and g₂

b c d

g₁ a is better than b a is worse than c a is better than d g₂ a is worse than b a is better than c a is better than d According to theclassic definition of the concordance index, the role that g₁and g₂should have for supporting the answer to the assertion

“a is at least as good as b (or c or d )” is characterized by the following weights,

k₁in the comparison with b, k₂in the comparison with c, k₁+k₂in the comparison with d .

Illustrative Example 1

DMR considers theweightsk₁and k₂appropriate,when only one criterion,supports a decisionthat one action is better than another one.

However, he/she judges thatthe sumk₁+k₂is not sufficientto characterize the role of this criteria pairwhen bothsupports the decision

Because in this caseeach criterion is strengthened by the other given thedegree of complementaritybetween them.

If one action is better than another one with respect to criteria g₁ and g₂conjointly, it would be interesting to be able to take this mutual strengthening effectinto account.

This effect can be taken into account byincreasing the weightsk₁ and k₂in the concordance index.

Illustrative Example 1

DMR considers theweightsk₁and k₂appropriate,when only one criterion,supports a decisionthat one action is better than another one.

However, he/she judges thatthe sumk₁+k₂is not sufficientto characterize the role of this criteria pairwhen bothsupports the decision

Because in this caseeach criterion is strengthened by the other given thedegree of complementaritybetween them.

If one action is better than another one with respect to criteria g₁ and g₂conjointly, it would be interesting to be able to take this mutual strengthening effectinto account.

This effect can be taken into account byincreasing the weightsk₁ and k₂in the concordance index.

Illustrative Example 1

DMR considers theweightsk₁and k₂appropriate,when only one criterion,supports a decisionthat one action is better than another one.

However, he/she judges thatthe sumk₁+k₂is not sufficientto characterize the role of this criteria pairwhen bothsupports the decision

Because in this caseeach criterion is strengthened by the other given thedegree of complementaritybetween them.

If one action is better than another one with respect to criteria g₁ and g₂conjointly, it would be interesting to be able to take this mutual strengthening effectinto account.

This effect can be taken into account byincreasing the weightsk₁ and k₂in the concordance index.

Illustrative Example 1

DMR considers theweightsk₁and k₂appropriate,when only one criterion,supports a decisionthat one action is better than another one.

However, he/she judges thatthe sumk₁+k₂is not sufficientto characterize the role of this criteria pairwhen bothsupports the decision

Because in this caseeach criterion is strengthened by the other given thedegree of complementaritybetween them.

If one action is better than another one with respect to criteria g₁ and g₂conjointly, it would be interesting to be able to take this mutual strengthening effectinto account.

This effect can be taken into account byincreasing the weightsk₁ and k₂in the concordance index.

Illustrative Example 1

DMR considers theweightsk₁and k₂appropriate,when only one criterion,supports a decisionthat one action is better than another one.

However, he/she judges thatthe sumk₁+k₂is not sufficientto characterize the role of this criteria pairwhen bothsupports the decision

Because in this caseeach criterion is strengthened by the other given thedegree of complementaritybetween them.

If one action is better than another one with respect to criteria g₁ and g₂conjointly, it would be interesting to be able to take this mutual strengthening effectinto account.

This effect can be taken into account byincreasing the weightsk₁ and k₂in the concordance index.

Illustrative Example 1

Thecomparisonsof site a with sites b, c, d^′ in terms of thetwo purely ordinal criteria, g₄and g₅

b c d^′

g₄ a is better than b a is worse than c a is better than d^′ g₅ a is worse than b a is better than c a is better than d^′ The DMR judges thatthe sumk₄+k₅is too highto characterize the role of this criteria pair when both supports the decision that one action is better than another one, because in this caseeach criterion is weakenedby the other due to thedegree of redundancybetween them.

Illustrative Example 1

Thecomparisonsof site a with sites b, c, d^′ in terms of thetwo purely ordinal criteria, g₄and g₅

b c d^′

g₄ a is better than b a is worse than c a is better than d^′ g₅ a is worse than b a is better than c a is better than d^′ The DMR judges thatthe sumk₄+k₅is too highto characterize the role of this criteria pair when both supports the decision that one action is better than another one, because in this caseeach criterion is weakenedby the other due to thedegree of redundancybetween them.

Illustrative Example 1

Thecomparisonsof site a with sites b, c, d^′ in terms of thetwo purely ordinal criteria, g₄and g₅

b c d^′

g₄ a is better than b a is worse than c a is better than d^′ g₅ a is worse than b a is better than c a is better than d^′ The DMR judges thatthe sumk₄+k₅is too highto characterize the role of this criteria pair when both supports the decision that one action is better than another one, because in this caseeach criterion is weakenedby the other due to thedegree of redundancybetween them.

Illustrative Example 2

LAUNCHING A NEW DIGITAL CAMERA MODEL

Criteria:

g1: purchasing costs (cost)[min];

g₂: weaknesses (fragility)[min];

g₃: user friendliness of the controls (workability)[max];

g4: image quality (image)[max];

g₅:aesthetics[max];

g₆:volume[min];

g7:weight[min].

Antagonisticeffect between criteria: costandfragility.

Illustrative Example 2

LAUNCHING A NEW DIGITAL CAMERA MODEL

Criteria:

g1: purchasing costs (cost)[min];

g₂: weaknesses (fragility)[min];

g₃: user friendliness of the controls (workability)[max];

g4: image quality (image)[max];

g₅:aesthetics[max];

g₆:volume[min];

g7:weight[min].

Antagonisticeffect between criteria: costandfragility.

Illustrative Example 2

LAUNCHING A NEW DIGITAL CAMERA MODEL

Criteria:

g1: purchasing costs (cost)[min];

g₂: weaknesses (fragility)[min];

g₃: user friendliness of the controls (workability)[max];

g4: image quality (image)[max];

g₅:aesthetics[max];

g₆:volume[min];

g7:weight[min].

Antagonisticeffect between criteria: costandfragility.

Illustrative Example 2

Comparisonsof digital camera model a with models b, c, and d , according to criteria g₁and g₂:

b c d

g₁ a is better than b a at least as good as c a is better than d g₂ a is better than b a is better than c a is worse than d According to theclassic definition of the concordance indexthe role these criteria should play in supporting the assertion “model a is at least as good as model b (or c or d )” is characterized by the following weights,

k₁+k₂in the comparison with b, k₂in the comparison with c, k₁in the comparison with d .

Illustrative Example 2

Comparisonsof digital camera model a with models b, c, and d , according to criteria g₁and g₂:

b c d

g₁ a is better than b a at least as good as c a is better than d g₂ a is better than b a is better than c a is worse than d According to theclassic definition of the concordance indexthe role these criteria should play in supporting the assertion “model a is at least as good as model b (or c or d )” is characterized by the following weights,

k₁+k₂in the comparison with b, k₂in the comparison with c, k₁in the comparison with d .

Illustrative Example 2

Comparisonsof digital camera model a with models b, c, and d , according to criteria g₁and g₂:

b c d

g₁ a is better than b a at least as good as c a is better than d g₂ a is better than b a is better than c a is worse than d According to theclassic definition of the concordance indexthe role these criteria should play in supporting the assertion “model a is at least as good as model b (or c or d )” is characterized by the following weights,

k₁+k₂in the comparison with b, k₂in the comparison with c, k₁in the comparison with d .

Illustrative Example 2

DMR considers thatweightsk₁and k₂adequatelycharacterize the role these two criteria should play when comparing a with b and a with c

However, he/she considers thatthe same is not true when comparinga with d

Based on a customer survey, it seems that when one model is less fragile than another,the benefit derived from the lower cost is partially masked by the fact the model is less fragile.

This phenomenon can be modeled bydecreasing the weightof criterion g₁in the concordance index of the assertion “a is at least as good as b”.

Illustrative Example 2

DMR considers thatweightsk₁and k₂adequatelycharacterize the role these two criteria should play when comparing a with b and a with c

However, he/she considers thatthe same is not true when comparinga with d

Based on a customer survey, it seems that when one model is less fragile than another,the benefit derived from the lower cost is partially masked by the fact the model is less fragile.

This phenomenon can be modeled bydecreasing the weightof criterion g₁in the concordance index of the assertion “a is at least as good as b”.

Illustrative Example 2

DMR considers thatweightsk₁and k₂adequatelycharacterize the role these two criteria should play when comparing a with b and a with c

However, he/she considers thatthe same is not true when comparinga with d

Based on a customer survey, it seems that when one model is less fragile than another,the benefit derived from the lower cost is partially masked by the fact the model is less fragile.

This phenomenon can be modeled bydecreasing the weightof criterion g₁in the concordance index of the assertion “a is at least as good as b”.

Illustrative Example 2

DMR considers thatweightsk₁and k₂adequatelycharacterize the role these two criteria should play when comparing a with b and a with c

However, he/she considers thatthe same is not true when comparinga with d

Based on a customer survey, it seems that when one model is less fragile than another,the benefit derived from the lower cost is partially masked by the fact the model is less fragile.

This phenomenon can be modeled bydecreasing the weightof criterion g₁in the concordance index of the assertion “a is at least as good as b”.

Concepts: Definitions and notation

Notation

Pseudo criterion

The criteria weights and the concordance index

Partial concordance index c_i(a,b)

Properties of the comprehensive concordance index c(a,b)

Concepts: Definitions and notation

Notation

Pseudo criterion

The criteria weights and the concordance index

Partial concordance index c_i(a,b)

Properties of the comprehensive concordance index c(a,b)

Concepts: Definitions and notation

Notation

Pseudo criterion

The criteria weights and the concordance index

Partial concordance index c_i(a,b)

Properties of the comprehensive concordance index c(a,b)

Concepts: Definitions and notation

Notation

Pseudo criterion

The criteria weights and the concordance index

Partial concordance index c_i(a,b)

Properties of the comprehensive concordance index c(a,b)

Concepts: Definitions and notation

Notation

Pseudo criterion

The criteria weights and the concordance index

Partial concordance index c_i(a,b)

Properties of the comprehensive concordance index c(a,b)

Concepts: Definitions and notation

Notation

Pseudo criterion

The criteria weights and the concordance index

Partial concordance index c_i(a,b)

Properties of the comprehensive concordance index c(a,b)

Notation

- F ={g₁,g₂, . . . ,g_i, . . . ,gn}denote a coherent set or family of criteria; for the sake of simplicity we shall use also F as the set of criteria indices (the same will apply later on for subsets of F );

- A={a,b,c, . . .}denote a finite set ofactionswith cardinality m;

- g_i(a)∈E_i denote theevaluationof action a on criterion g_i, for all a∈A and i ∈F , where E_i is the scale associated to criterion g_i (no restriction is imposed to the scale type).

- k_i is the relative importance orweightof criterion g_i.

- C(aTb)represents thecoalition of criteriain favor of the assertion

“aTb”, where T ∈ {P,Q,S}(introduced later).

- C(aTb)¯ denote thecomplementof C(aTb).

Notation

- F ={g₁,g₂, . . . ,g_i, . . . ,gn}denote a coherent set or family of criteria; for the sake of simplicity we shall use also F as the set of criteria indices (the same will apply later on for subsets of F );

- A={a,b,c, . . .}denote a finite set ofactionswith cardinality m;

- g_i(a)∈E_i denote theevaluationof action a on criterion g_i, for all a∈A and i ∈F , where E_i is the scale associated to criterion g_i (no restriction is imposed to the scale type).

- k_i is the relative importance orweightof criterion g_i.

- C(aTb)represents thecoalition of criteriain favor of the assertion

“aTb”, where T ∈ {P,Q,S}(introduced later).

- C(aTb)¯ denote thecomplementof C(aTb).