csm 2006, laxenburg, 28-30 august 2006 1 hierarchical reference approach to multi-criteria analysis...

CSM 2006, Laxenburg, 28-30 August 20061

Hierarchical reference approach to multi-criteria

analysis of discrete alternatives

JANUSZ GRANATNational Institute of Telecommunications,Warsaw, and Warsaw

University of Technology, PolandMAREK MAKOWSKI

International Institute for Applied System Analysis, Laxenburg, Austria

ANDRZEJ P. WIERZBICKICenter for Strategic Development of Science and Technology, Japan

Advanced Institute of Science andTechnology, Ichikawa, Japan, and National Institute of

Telecommunications,Warsaw, Poland


Outline

Motivation The limitation of the existing

approaches Hierarchical criteria aggregations Applications Conclusions


The criteria for selection of energy technologies

level 1 level 2 level 3

criteria criteria/indicators indicators

Economy

Financial requirements

Production cost

Investment

Fuel Price

Resources Availability

Generation potential

Environment

Global warming

Total waste

SocialEmployment

Risk aversion


Hierarchical weighting

0,6 0,4

0,2 0,8 0,1 0,10,8

0,12 0,48 0,04 0,32 0,04


Bottom-up weighting

0,3 0,7

0,2 0,1 0,5 0,10,1

0,2 0,1 0,5 0,1 0,1


Compensatory versus noncompensatory criteria Compensatory criteria – an improvement

of a criterion can be rationally substantiated to compensate a deterioration of another criterion.

e.g. operational costs and investment costs

Noncompensatory criteria are such that no rational substantiation exists for defining weighting coefficients.

e.g. costs and loss of human life


Ranking„ranking

” „classification” „partial ordering”


Subjective versus objective ranking

Full objectivity is obviously – after Heisenberg and Quine – not attainable, but in many situations we must try to be as much objective as possible.


Objective ranking

Weighting coefficients and/or aspiration and reservation levels should be determined in some objective or intersubjectively fair fashion. We shall consider three possible ways of achieving this goal: neutral statistical voting


Neutral weights - objective weighting

coefficients for compensatory criteria and weighting coefficients equal in size for all noncompensatory criteria

aspirations/reservations - a neutral definition of reference points e.g. all aspiration levels equal to 67% of criteria ranges, all reservation levels equal to 33% of these ranges


Voting

A voting procedure between a group of decision makers.

Many voting procedures, see H.Nurmi (1999).

Voting results actually only in intersubjective aggregation.


Statistical Based on some meaningful statistics. weights - it is very difficult to find

statistical data to substantiate weighting coefficients

aspirations/reservations - the average score of all options, e.g.:

qai = qm

i+(qmaxi –qm

i)/2; qri = qm

i-(qmi –

qmini)/2

qmi - is average value of the i-th criterion for all decision

options qa

i, qri - aspiration and the reservation levels, respectively


Approaches to hierarchical criteria aggregation

Compensatory aggregation on lower level, noncompensatory analysis on upper level.

Noncompensatory aggregation both on lower and on upper level

Noncompensatory aggregation with weighting coefficients as importance factors


Compensatory aggregation on lower level, noncompensatory analysis on upper level.

qC = ∑i є C wi qi for all C = A,…H

qA qB

q1 q2 q1 q3q2

⌠ 1 + α (qC - qaC)/(qup

C - qaC), if qa

C ≤ qC ≤ qupC

σC(qC, qaC, qr

C) = ┤ (qC – qrC)/(qa

C – qrC), if qr

C ≤ qC < qaC

│ β (qC – qrC)/(qr

C – qloC), if qlo

C ≤ qC < qrC

σ(q, qa,qr, ε) = minC=A,…H σC(qC, qaC, qr

C)+ ε ∑ C=A,…H σC(qC, qaC, qr

C)



qA qB

q1 q2 q1 q3q2

⌠ 1 + α (qC - qaC)/(qup

C - qaC), if qa

C ≤ qC ≤ qupC

σC(qC, qaC, qr

C) = ┤ (qC – qrC)/(qa

C – qrC), if qr

C ≤ qC < qaC

│ β (qC – qrC)/(qr

C – qloC), if qlo

C ≤ qC < qrC

σ(q, qa,qr, ε) = minC=A,…H σC(qC, qaC, qr

C)+ ε ∑ C=A,…H σC(qC, qaC, qr

C)

qC = σC(qC, qaC,qr

C, ε) = miniєC σi(qi, qa

i, qri)+ ε ∑ iєC σi(qi, q

ai, q

ri)


Noncompensatory aggregation with weighting coefficients treated as importance factors

qianw = qi

lo + wi (qiup – qi

lo) qi

rnw = qilo + 0.5 wi (qi

up – qilo)

The weights are interpreted as importance factors and are used for modification of neutral aspiration and reservation levels e.g.:


Noncompensatory aggregation with weighting coefficients treated as importance factors - weights

b

c

d

e

a

q2

q2


Noncompensatory aggregation with weighting coefficients treated as importance factors – neutral aspiration

b

c

d

e

a

q2

q2

(0.73, 0,73)

(0.23, 0,23)


Noncompensatory aggregation with weighting coefficients treated as importance factors – weighted aspiration

b

c

d

e

a

q2

q2

(0.51, 0,22)

(0.16, 0,07)

w=(0.7, 0.3)


Preservation of Pareto optimality after hierarchical aggregation

Theorem. In a hierarchical aggregation of criteria, suppose that the functions used to aggregate criteria in groups on the lower level are strictly monotone with respect to the partial orders defining the vector optimization problems on lower level. Then any decision option that is Pareto optimal in the space of aggregated criteria is also Pareto optimal in the original space of all lower level criteria (with respect to the overall partial order induced by the partial orders for all groups of criteria).


Electricity supply technologies - hierarchical weighting

Economy (1, 0, 0)

Environment (0, 1, 0) Social (0, 0, 1)

Equal weights (0.5,0.5,0.5)

00,10,20,30,40,50,60,70,80,9

Hydro Wind Nuclear NG PV Lignite HardCoal

Oil

00,10,20,30,40,50,60,70,80,9

Nuclear HardCoal

Lignite Hydro NG Oil Wind PV

0

0,2

0,4

0,6

0,8

1

1,2

Nuclear Wind Hydro PV NG Lignite Oil HardCoal

0

0,2

0,4

0,6

0,8

1

PV NG Hydro Wind HardCoal

Lignite Nuclear Oil


Compensatory aggregation on lower level, noncompensatory analysis on upper level

Aspirations/reservations

-1-0,8-0,6-0,4-0,20

0,20,40,60,81

1,2

Hydro Wind NG Nuclear Lignite Oil HardCoal

PV

noncompensatory upper level compensatory lower

level

Equal weights (0.5,0.5,0.5)

00,10,20,30,40,50,60,70,80,9

Hydro Wind Nuclear NG PV Lignite HardCoal

Oil



-1-0,8-0,6-0,4-0,20

0,20,40,60,81

1,2

Hydro Wind NG Nuclear Lignite Oil HardCoal

PV

-1,5

-1

-0,5

0

0,5

1

Hydro Wind NG PV Lignite Nuclear HardCoal

Oil

-1,5

-1

-0,5

0

0,5

1

1,5

2

2,5

Hydro PV NG Wind HardCoal

Nuclear Lignite Oil

compensatory upper level noncompensatory lower

level

noncompensatory upper level noncompensatory

lower level

noncompensatory upper level compensatory lower

level


Conclusions (I) Distinction between subjective and

objective ranking Distinction between compensatory and

noncompensatory groups of criteria. Approaches to hierarchical

aggregation of criteria: Compensatory aggregation on lower level,

noncompensatory analysis on upper level; Noncompensatory aggregation both on

lower and on upper level; Noncompensatory aggregation with

weighting coefficients treated as importance factors.


Conclusions (II) The discussion and a theorem on the

preservation of Pareto optimality after hierarchical aggregation with strictly monotone aggregating functions.

The resulting approaches will be used on the problem of the selection of electricity supply technologies

csm 2006, laxenburg, 28-30 august 2006 1 hierarchical reference approach to multi-criteria analysis...

Documents

criteria ranges

weighting coefficients

aspiration levels equal

reservation levels equal

qmiqmi qmini2 qmi

qri aspiration

qmi qmaxi qmi2 qri

decision options qai