multicriteria decision making

7/18/2019 Multicriteria decision making

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1

Multi-Criteria Decision

Making



2

“It has become more and more difficult to see the world

around us in a unidimensional way and to use only asingle criterion when judging what we see”

- Zeleny (1982

“!ecision ma"ing is the study of identifying and choosingalternati#es based on the #alues and $references of thedecision ma"er%”

- &arris (198'



3

C- criteria, W-weight,A- alternative

C1W1

C2W2

C3W3

C4W4

A1

A2A3

A4

a11

a21a31

a41

a12

a22a32

a42

a13

a23a33

a43

a14

a24a34

a44

i = 1,…, m. (number of alternative!,

"= 1,…, n. (no. of criteria!

!ecision atri)



Analytic hierarchy Process Anal#tic $ierarch# %roce (A$%! i one of &ulti Criteria 'eciion

maing metho' that wa originall# 'evelo)e' b# %rof. *homa +.

aat#.

n hort, it i a metho' to 'erive ratio cale from )aire'

com)arion. *he in)ut can be obtaine' from actual meaurement uch a )rice,

weight etc., or from ub"ective o)inion uch a atifaction feeling

an' )reference.

A$% allow ome mall inconitenc# in "u'gment becaue human i

not alwa# conitent. *he ratio cale are 'erive' from the )rinci)al igen vector an' the

conitenc# in'e/ i 'erive' from the )rinci)al igen value.

4



Example

0

which fruit #ou lie better than the other an' how much #ou

lie it in com)arion with the other. +et u mae a relative

cale to meaure how much #ou lie the fruit on the left

(A))le! com)are' to the fruit on the right (anana!.



f #ou lie the a))le better than banana, #ou thic a mar

between number 1 an' on left i'e, while if #ou favor

banana more than a))le, then #ou mar on the right i'e.

or intance trongl# favor banana to a))le then give

mar lie thi



5

6ow u))oe #ou have three choice of fruit. *hen the

)air wie com)arion goe a the following



7

1.If the jugment !alue is on the left sie of 1" #e put theactual judgment !alue.

$.If the jugment !alue is on the right sie of 1" #e put

the reciprocal !alue .



6otice that all the element in the com)arion matri/ are

)oitive, or

.



18



11

*he normali9e' )rinci)al igen vector can be

obtaine' b# averaging acro the row

*he normali9e' )rinci)al igen vector i alo

calle' $riority #ector



12

*onsistency Inde) and *onsistency +atio

What i the meaning that our o)inion i conitent: $ow

'o we meaure the conitenc# of ub"ective "u'gment:

irt he )refer anana to A))le. *hu we a# that for;ohn, anana ha greater value than A))le. We write it a

http://people.revoledu.com/kardi/tutorial/AHP/Consistency.htm

http://people.revoledu.com/kardi/tutorial/AHP/Consistency.htm



13

ince an' , logicall#, we ho)e thator anana mut be )referable than Cherr#. *hi logic of

)reference i calle' tranitive )ro)ert#. f ;ohn anwer in

the lat com)arion i tranitive (that he lie anana more

than Cherr#!, then hi "u'gment i conitent. <n thecontrar#, if ;ohn )refer Cherr# to anana then hi anwer

i inconitent.



14

%rof. aat# )ro)oe' that we ue thi in'e/ b# com)aring

it with the a))ro)riate one. *he a))ro)riate Conitenc#

in'e/ i calle' an'om Conitenc# n'e/ ( !.

n 1 2 3 4 0 5 7 18

8 8 8.07 8. 1.12 1.24 1.32 1.41 1.40 1.4



10

f the value of Conitenc# atio i maller or e>ual to

18?, the inconitenc# i acce)table. f the Conitenc#

atio i greater than 18?, we nee' to revie the

ub"ective "u'gment.

or n=3 i o.07



Illustrati!e example

(how to compute in a full hierar chy?)

1

http://people.revoledu.com/kardi/tutorial/AHP/AHP-Example.htm








15

*riteria , * !.riority/ector

, 1.88 3.88 5.88 .88 0%93 8.33 1.88 0.88 5.88 29%13

* 8.14 8.28 1.88 3.88 9%'3

! 8.11 8.14 8.33 1.88 4%403

5um 1.0 4.34 13.33 28.88 188.88?

=4.22, C = 8.875, C = 7.5? @ 18?

(acce)table!



17

Choice 6 7 Z .riority /ector

1.88 1.88 5.88 01%'03

B 1.88 1.88 3.88 8%93 8.14 8.33 1.88 1'%'13

um 2.14 2.33 11.88 188.88?

%aire' com)arion matri/ level 2 with re)ect to actor A

=3.184, C = 8.80, C = 7.5? @ 18? (acce)table!



1

%aire' com)arion matri/ level 2 with re)ect to actor

*hoice 6 7 Z .riority /ector

6 1.88 8.28 8.08 11%493

7 0.88 1.88 0.88 '%283

Z 2.88 8.28 1.88 18%223

5um 7.88 1.48 .08 188.88?

=3.877, C = 8.84, C = 5.07? @ 18? (acce)table!



28

We can 'o the ame for )aire' com)arion with re)ect to

actor C an' D. $owever, the weight of factor C an' D are

ver# mall (loo at *able again, the# are onl# about ? an'0? re)ectivel#!, therefore we can aume the effect of

leaving them out from further coni'eration i negligible. We

ignore thee two weight a et them a 9ero.



21

A'"ute' weight for factor A =

A'"ute' weight for factor =



22

actor, actor

*om$ositeeight

(A'"ute'! eight %.&&' %.''(

*hoice 6 01.80? 11.4? %23

*hoice 7 37.3? 58.27? 49%493*hoice Z 18.81? 17.22? 12%83



23

,&. (,nalytic &ierarchy .rocessasic .rocedure:

1. Decom)oe the )roblem into a hierarch#

2. &ae )air-wie com)arion an' etablih )rioritie among the

element in the hierarch#3. #nthei9e the reult (to obtain the overall raning of alternative

w.r.t. goal!

4. valuate the conitenc# of "u'gement

E 'evelo)ing a )air-wie com)arion matri/ for

each criterion

E normali9ing the reulting matri/

E calculating an' checing the

conitenc# ratio

E averaging the value in each row to get the corre)on'ing rating

5te$s:1. Develo) the rating for each 'eciion alternative for each criterion b#



24

2. Develo) the weight for the criteria b#

E 'evelo)ing a )air-wie com)arion matri/ for each

criterionE normali9ing the

reulting matri/

E calculating an' checing the conitenc# ratio

E averaging the value in each row to get the corre)on'ing rating

3. Calculate the weighte' average rating for each

'eciion alternative. Chooe the one with the highet

core.



20

+ating and !escri$tion of *riteria:

+ating !escri$tion1 >uall# )referre'

3 &o'eratel# )referre'

0 trongl# )referre'

5 Fer# trongl# )referre'

/tremel# trongl# )referre'

Falue 2, 4, , or 7 ma# alo be aigne' an' re)reent )reference

halfwa# between the integer on either i'e.

n 1 2 3 4 0 5 7

8 8 8.07 .7 1.11 1.20 1.30 1.4



2

Example ) Car *election Pro+lem GC cor)oration i )lanning to bu# few car for

it tran)ort ection. *he following four criteria

ha been i'entifie' for the election of the carHE %rice

E &ileage )er litre (&%+!

E Comfort

Et#le

Ie A$% to elect the bet of three car name' A, an' Cfrom the maret.



25

%rice

Car election

&%J Comfort t#le

*ar , *ar *ar *

$ierarchical tructure of Car election %roblem



27

%rice &%J Comfort t#le %riorit#

%rice 1 3 2 2 8.37

&%J 1K3 1 1K4 L 8.870

Comfort 1K2 4 1 1K2 8.217

t#le 1K2 4 2 1 8.2

%riorit# of Criteria



2

%rice Car A Car Car C %riorit#

Car A 1 1K3 1K4 8.123

Car 3 1 1K2 8.328

Car C 4 2 1 8.055

&%J Car A Car Car C %riorit#

Car A 1 1K4 1K 8.875

Car 4 1 1K3 8.254

Car C 3 1 8.3



38

Comfort Car A Car Car C %riorit#

Car A 1 2 7 8.03

Car 1K2 1 8.341

Car C 1K7 1K 1 8.8

t#le Car A Car Car C %riorit#

Car A 1 1K3 4 8.20

Car 3 1 5 8.00

Car C 1K4 1K5 1 8.878



31

,sing AP to De!elop an !erall Priority /anking

*he )roce'ure ue' to com)ute the overall )rioritie for each 'eciion

alternative b# uing the )riorit# for each criterion a a weight that

reflect it im)ortance.

*he overall )riorit# for each 'eciion alternative i obtaine' b#

umming the )ro'uct of the criterion )riorit# time the )riorit# of it

'eciion alternative.

or e/am)le overall car A )riorit# = .37 (.123!

M .870 (.875! M 8.217 (.03! M .2 (.20!

Alternative %riorit#

Car A 8.20

Car 8.421

Car C 8.314



32

*onsistency Inde)

1. &ulti)l# each column of the )air-wie com)arion matri/ b# the

corre)on'ing weight.

2. Divi'e of um of the row entrie b# the corre)on'ing weight.

3. Com)ute the average of the value from te) 2, 'enote it b# 0ma/.

4. *he a))ro/imate C i

*onsistency +atio:

C = C K

En )ractice, a C of 8.1 or below i coni'ere' acce)table.

EAn# higher value at an# level in'icate that the "u'gement warrant re-

e/amination.

ma/1

nCI n

λ

−= −



33

Consistency of Comfort Matrix1 2 7 8.03 8.72 8.027 1.783

8.03 1K 2 8.341 1 8.8 8.25 8.341 8.3 1.834

1K 7 1K 1 8.854 8.805 8.8 8.15

1.783K 8.03 3.848, 1.834K8.341

+ + = + + =

= 3.832, 8.15K8. 2.70

(3.848 3.832 2.70!K3 3.81

C (3.81-3!K2 8.818

= 8.07C= 8.81K8.07 8.815

λ

= =

= + + =

= =

=

= =



34

;<.5I5(*echni>ue of <r'er %reference b# imilarit# to 'eal olution!

-=wangsun 7oon and &wang *hing->ai(198'

*hi metho' coni'er three t#)e of attribute or criteria

E Nualitative benefit attributeKcriteria

E Nuantitative benefit attributeE Cot attribute or criteria

n thi metho' two artificial alternative are h#)othei9e'H

E 'eal alternativeH the one which ha the bet level for all attribute

coni'ere'.

E 6egative i'eal alternativeH the one which ha the wort attribute

value.



30

;<.5I5 ?;&<! *<% elect the alternative that i the cloet to the

i'eal olution an' farthet from negative i'eal alternative.

*<% aume that we have m alternative (o)tion! an'

n attributeKcriteria an' we have the core of each o)tion

with re)ect to each criterion.

+et /i" core of o)tion i with re)ect to criterion "

We have a matri/ = (/i"! m×n matri/.

+et ; be the et of benefit attribute or criteria (more i

better! +et ;O be the et of negative attribute or criteria (le i

better!



3

5te$s of ;<.5I5

te) 1H Contruct normali9e' 'eciion matri/.

E *hi te) tranform variou attribute 'imenion into

non-'imenional attribute, which allow com)arion

acro criteria.

E 6ormali9e core or 'ata a follow

P 1,..., P 1,...,

2

1

xijr #herei M an j ij

M xiji

= = =

∑=

1 2 3

1 11 12 13 1

2 21 22 23 2

3 31 32 33 3

1 2 3

. . .

. . .

. . .

. . .. . . . .

. . . . .

. . . . .

. . .

n

n

n

n

m m m m mn

D

x x x x

x x x x A x x x x A

x x x x A

x x x x A

=



35

5te$s of ;<.5I5 te) 2H Contruct the weighte' normali9e' 'eciion matri/.

E Aume we have a et of weight for each criteria w " for " = 1,…n.

E &ulti)l# each column of the normali9e' 'eciion matri/ b# it aociate'

weight.

E An element of the new matri/ iH

vi" = w " r i"

11 12 111 12 1 1 1 2

1 21 2 1 2

1 21 2 1 2

. . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . . . . . .

ij n j n j n

i i ij ini i ij in j n

m m mj mnm m mj mn j n

2

! ! ! ! # # # #r r r r

! ! ! ! # # # #r r r r

! ! ! ! # # # #r r r

= = r



37

5te$s of ;<.5I5 te) 3H Determine the i'eal an' negative i'eal olution.

'eal olution.

AQ = R v1Q , …, vn

QS, where

v "Q =R ma/ (vi"! if " ∈ ; P min (vi"! if " ∈ ;O S

i i

6egative i'eal olution.

AO = R v1O , …, vnO S, where

vO = R min (vi"! if " ∈ ; P ma/ (vi"! if " ∈ ;O S i i



3

5te$s of ;<.5I5

te) 4H Calculate the e)aration meaure for eachalternative.

*he e)aration from the i'eal alternative iH

iQ = T Σ (v "

Q U vi"!2 V i = 1, …, m "

imilarl#, the e)aration from the negative i'eal

alternative iH Oi = T Σ (v "O U vi"!2 V i = 1, …, m

"



48

5te$s of ;<.5I5

te) 0H Calculate the relative cloene to

the i'eal olution CiQ

CiQ

= Oi K (iQ MOi ! , 8 < Ci

Q < 1

elect the o)tion with CiQ cloet to 1.



41

,$$lying ;<.5I5 ethod to

?)am$le

eight '%1 '%4 '% '%2

5tyle +eliability uel ?co%

5aturn

ord

9 9 8

8 8

9 @ 8 9

*i#ic

aAda

@ 8 @

*ost



42

,$$lying ;<.5I5 to ?)am$le

m = 4 alternative (car mo'el!

n = 4 attributeKcriteria

/i" = core of o)tion i with re)ect to criterion "

= R/i"S 4×4 core matri/.

; = et of benefit attributeH t#le, reliabilit#, fueleconom# (more i better!

;O = et of negative attributeH cot (le i better!



43

5te$s of ;<.5I5

te) 1(a!H calculate (Σ/2i" !1K2 for each column

5tyle +el% uel

5aturn

ord

49 81 81 @4

@4 49 @4 49

81 @ @4 81

*i#ic

aAda

*ost

Σ/i"2i

(Σ/2

!1K2

@ 49 @4 @

2' 210 2 2'

10%1 14%@@ 1@%02 10%1



44

5te$s of ;<.5I5

te) 1 (b!H 'ivi'e each column b# (Σ/2i" !1K2 to get rij

5tyle +el% uel

5aturn

ord

'%4@ '%@1 '%04 '%0

'%0 '%48 '%48 '%4@

'%09 '%41 '%48 '%09

*i#ic

aAda

'%4' '%48 '%48 '%4'

*ost



40

5te$s of ;<.5I5

te) 2 (b!H multi)l# each column b# w " to get vi".

5tyle +el% uel

5aturn

ord

'%'4@ '%244 '%1@2 '%1'@

'%'0 '%192 '%144 '%'92

'%'09 '%1@4 '%144 '%118

*i#ic

aAda

'%'4' '%192 '%144 '%'8'

*ost



4

5te$s of ;<.5I5

te) 3 (a!H 'etermine i'eal olution AQ.

AQ = R8.80, 8.244, 8.12, 8.878S

5tyle +el% uel

5aturn

ord

'%'4@ '%244 '%1@2 '%1'@

'%'0 '%192 '%144 '%'92

'%'09 '%1@4 '%144 '%118

*i#ic

aAda

'%'4' '%192 '%144 '%'8'

*ost



45

5te$s of ;<.5I5

te) 3 (a!H fin' negative i'eal olution AO.

AO = R8.848, 8.14, 8.144, 8.117S

5tyle +el% uel

5aturn

ord

'%'4@ '%244 '%1@2 '%1'@

'%'0 '%192 '%144 '%'92

'%'09 '%1@4 '%144 '%118

*i#ic

aAda

'%'4' '%192 '%144 '%'8'

*ost



47

5te$s of ;<.5I5

te) 4 (a!H 'etermine e)aration from i'eal olution AQ =R8.80, 8.244, 8.12, 8.878S i

Q = T Σ (v "

Q U vi"!2 V W

for each row "

5tyle +el% uel

5aturn

ord

(%'4@-%'092 (%244-%2442 ('2 (%'2@2 *i#ic

aAda

*ost

(%'0-%'092 (%192-%2442 (-%'182 (%'122

(%'0-%'092 (%1@4-%2442 (-%'182 (%'82

(%'0-%'092 (%192-%2442 (-%'182 (%'2



4

5te$s of ;<.5I5

te) 4 (a!H 'etermine e)aration from i'eal olution iQ

Σ(v "Q Uvi"!

2 iQ = T Σ (v "

Q U vi"!2 V

5aturn

ord

'%'''840 '%'29

'%''2'8 '%'0

'%''818@ '%'9'

*i#ic

aAda '%''89 '%'08



08

5te$s of ;<.5I5

te) 4 (b!H fin' e)aration from negative i'eal olution AO= R8.848, 8.14, 8.144, 8.117S

iO = T Σ (v "OU vi"!2 V W for each row

"

5tyle +el% uel

5aturn

ord

(%'4@-%'4'2 (%244-%1@42 (%'182 (-%'122*i#ic

aAda

*ost

(%'0-%'4'2

(%192-%1@42

('2

(-%'2@2

(%'0-%'4'2 (%1@4-%1@42 ('2 ('2

(%'0-%'4'2 (%192-%1@42 ('2 (-%'82



01

5te$s of ;<.5I5

te) 4 (b!H 'etermine e)aration from negative i'eal

olution iO

Σ(v "OUvi"!2 iO = T Σ (v "OU vi"!2 V

5aturn

ord

'%''@9'4 '%'8

'%''1@29 '%'4'

'%'''@1 '%'19

*i#ic

aAda '%''2228 '%'4



02

5te$s of ;<.5I5

te) 0H Calculate the relative cloene to the i'eal

olution CiQ = Oi K (i

Q MOi !

Oi K(iQMOi! Ci

Q

5aturn

ord

'%'8B'%112 '%4 ←

?5;

'%'4'B'%'9 '%41

'%'19B'%1'9 '%1

*i#ic

aAda '%'4B'%1'0 '%40



03

.+<?;&??( Preference /anking rgani3ation Metho for Enrichment E!aluation4

%<&*$ #iel' a )artial )reor'er

%<&*$ #iel' a uni>ue com)lete )reor'er

- 5rans 6 2incke" 17894

ain 5te$s



04

ain 5te$s

1. uil'ing the outraning relation

E D& chooe a generali9e' criterion an' fi/e the necear#

)arameter relate' to the electe' criterionH a )referencefunction i 'efine' for each attribute

E &ulticriteria )reference in'e/ i 'efine' a the weighte'average of the )reference function

E *hi )reference in'e/ 'etermine a value' outraning

relation on the et of alternative.

2. /)loitating the outraning relation with regar' to the choentatement of the )roblem

E or each alternative, a leaving an' an entering flow are'efine'. A net flow i alo coni'ere'

E A )artial )reor'er (%<&*$ ! or a com)lete )reor'er(%<&*$ ! can be )ro)oe' to the D&

+ecommended CeneraliAed *riteria



00

+ecommended CeneraliAed *riteria

E Pk (ai,aj! =

8 8( , !

1 8

,sualCriterion

P a ak i j

∀ ≤= ∀ >

88 8( , !

1 8 1

:uasiCriterion

; k P a ak i j

;k

∀ ≤∀ ≤ = ∀ > ∀ >

. . 8 8

8 8( , ! K 8

1 8 1

Crit #ithlinear pref

P a a p pk i j k k

pk

≤∀ ≤

= ≤ ≤ ∀ > ≥

8

8 8( , ! 8.01 8

1

0e!el Criterion

;k

P a a ; pk i j k k

pk

≤

∀ ≤ = < ≤ ∀ > >

.

8

8 8( , !

1 8

1

Crit #ithinifferencearea

;k ;k P a a ; pk i j k k

p ;k k

pk

≤

∀ ≤ −= ≤ ≤

∀ > − ≥

8 8 8 8( , ! 2 2 1 8 1 e/)( K2 ! 8

<aussianCriterion

P a ak i j

k

σ

≤∀ ≤ =

∀ > − − ≥

here d D *" (ai - *" (a j

CeneraliAed *riteria



0

CeneraliAed *riteria

Criterion

1

P k (ai,a j!

'

;k Criterion

1

P k (ai,a j!

'

P k (ai,a j!

pk Criterion

1

'

Criterion F ;k pk

8.0

1

P k (ai,a j!

'

;k pk Criterion F

1

P k (ai,a j!

'

σ k Criterion F

1

P k (ai,a j!

'

Eecessary *alculations



05

Eecessary *alculations

&ultiattribute %reference n'e/

+eaving low

ntering low

6et low

( , ! ( , !a a # P a ai j k k i j

k

π = ∑

( ! ( , !a a ai i ja A j

π +Φ = ∑

∈

( ! ( , !a a ai j ia A j

π −Φ = ∑

∈

( ! ( ! ( !a a ai i i+ −Φ = Φ − Φ

?#aluating $references



07

a. n)utH Φ M (a 1 ! =

1

1 k −

Σ Π ( a 1 , a i !

b. <ut)utH Φ - (a 1 ! = 1

1 k − Σ Π ( a i , a 1 !

c. 6et flowH Φ (a 1 ! = Φ M (a 1 ! - Φ - (a 1 !

?#aluating $references

.+<?;&?? I



0

.+<?;&?? I *wo com)lete )reor'er are builtH

E aning the alternative following the 'ecreaing or'er

of leaving flow

E aning the alternative following the increaing or'er

of entering flow

*he interection of the )reor'er #iel' the $artial

$reorder

a1 %M a2 if ΦM(a1! X ΦM(a2!

a1 M a2 if ΦM(a1! = ΦM( a2!

a1 %- a2 if Φ-(a2! X Φ-(a1!

a1 - a2 if Φ-(a2! = Φ-(a1!



8

a1 % a2 Ya1Y outran Ya2Y ifH a1 %M a2 an' a1 %- a2

a1 %M a2 an' a1 - a2

a1 M a2 an' a1 %- a2

a1 a2 Y a1Y an' Y a2Y are in'ifferent ifH

a1 M a2 an' a1 - a2

a1 a2 Ya1Y an' Ya2Y are incom)arableH

in all other cae

.+<?;&?? II



1

.+<?;&?? II , uniFue com$lete $reorder is built:

E +an"ing the alternati#es following the decreasingorder of net flows

a1 % a2 Ya1Y outran Ya2Y

if Φ(a1! X Φ(a2!

a1 a2 Ya1Y an' Ya2Y are in'ifferent

if Φ(a1! = Φ(a2!



2

a1 % a2 Ya1Y outran Ya2Y if Φ(a1! X Φ(a2!

a1 a2 Ya1Y an' Ya2Y are in'ifferent

if Φ(a1! = Φ(a2!



3

*here are i/ )oible bi'H

a1, a2, a3, a4, a0, a

*he following i/ criteria will be taen into accountH

C1(a!H mean maintenance time )er 'a# (in minute!

C2(a!H technical value of the e>ui)ment (core out of 188!

C3(a!H cot (18 !

C4(a!H etimate' maintenance cot for the e>ui)mentO ueful life

(18 ('icounte'!

C0(a!H etimate' mean number of failure )er #ear (bae' on tan'ar' ue!

C(a!H afet# rating of the e>ui)ment offere'



4

Selecting the generalized criteria

*riterion ;y$e .arameters

C1(a! -- > = 18

C2(a! --- ) = 38

C3(a! F > = 08P ) = 088

C4(a! -F > = 18 P ) = 8

C0(a! -

C(a! F- σ = 0



0

The data

*riterion inBma) a1 a2 a a4 a0 a@C1(a! &in 78 0 73 48 02 4

C2(a! &a/ 8 07 8 78 52

C3(a! &in 88 288 488 1888 88 588

C4(a! &in 04 5 52 50 28 3

C0(a! &in 7 1 4 5 3 0

C(a! &a/ 0 1 5 18 7

D i i h fl bl



Devising the flow table

*1(a:

1

- 10 0 + 10

H (d)

d

II

-criterion to be minimiAed

-a1 is thus worse than a2: *1 (a1 - *1(a2 D 10

and therefore

.1 (a1G a2 D '

.1 (a2G a1 D 1

D i i th fl t bl



5


*2(a:

1

0

H (d)

d-30 +30

III

-criterion to be ma)imiAed

-a1 is thus better than a2: *2 (a1 - *2 (a2 D 2

and therefore

.2(a1G a2 D 1

.2(a2G a1 D '




7


*

(a:

1

0

H (d)

d50 500- 500 - 50

V

- criterion to be minimiAed

- a1 is thus worse than a: *(a1 - *(a2 D 4''

and therefore

. (a1G a2 D '

. (a2G a1 D '%8

D i i h fl bl




*4(a:

1

0

H (d)

d10 60

IV

- 60 -10

1/2


- a1 is better than a2: *4 (a1 - *4 (a2 D - 4

and therefore

.4 (a1G a2 D '%0

.4 (a2G a1 D '

D i i h fl bl



58


*0(a:

1

0

H (d)

d

I


- a1 is thus worse than a2: *0 (a1 - *0 (a2 D

and therefore

.0(a1G a2 D '.0(a2G a1 D 1

D i i th fl t bl



51


*@(a:

1

0

H (d)

d

VI

- criterion to be ma)imiAed

&@(d D 1 - e

d

2 σ2

2

- a1 is better than a2:

and therefore

.@ (a1G a2 D 1 - e

42

22.5

D '%24

.@(a2G a

1 D '

5ummary:



52

5ummary:

,#erage of .i(a1G a2: Π(a1,a2! =1

(8M1M8M8,0M8M8.254! = 8.2

,#erage of .i(a2G a1: Π(a2,a1! =

1

(1M8M8,557M8M1M8! = 8.42

# a))l#ing the above reaoning to all the )air (ai, a !, we obtain the following

tableH

a1 a2 a a4 a0 a@

a1 8.2 8.208 8.27 8.188 8.170

a2 8.42 8.37 8.333 8.2 8.088

a 8.23 8.178 8.333 8.80 8.42

a4 8.3 8.080 8.380 8.223 8.212

a0 8.444 8.010 8.475 8.378 8.447a@ 8.27 8.3 8.208 8.432 8.133



53

a1 a2 a a4 a0 a@

H

D H-

-

a1 8.2 8.208 8.27 8.188 8.170 8.228 -8.14

a2 8.42 8.37 8.333 8.2 8.088 8.3 M8.815

a 8.23 8.178 8.333 8.80 8.42 8.245 -8.87

a4 8.3 8.080 8.380 8.223 8.212 8.32 -8.828

a0 8.444 8.010 8.475 8.378 8.447 8.400 M8.23

a@ 8.27 8.3 8.208 8.432 8.133 8.388 -8.800

- 8.3 8.35 8.33 8.34 8.12 8.300



54

The ranking obtained using

the romethee ! method

a2

a0 a4 a

a1

a3



50

a2 a a1

a0 a4 a3

The ranking obtained using the

romethee !! method



5

?>?*;+?

. o# 'eigne' a metho' for choice )roblem in

17H

Zlimination et choi/ tra'uiant la r[alit[\

(?limination and *hoice ;ranslating +eality

ain 5te$s



55

ain 5te$s

te) 1H 6ormali9ing the 'eciion matri/

P 1,..., P 1,...,2

1

aijr #herei M an j ij M

aiji

= = =

∑=

. . .1 2 3

. . .1 11 12 13 1

. . .2 21 22 23 2

. . .3 31 32 33 3

. . . . .

. . . . .

. . . . .

. . .1 2 3

# # # #n

A r r r r n

A r r r r n

A r r r r n

/

A r r r r m m m m mn

=

C- criteria, W-weight,

A- alternative

C1

W1

C2

W2

C3

W3

C4

W4

A1

A2

A3

A4

a11

a21

a31

a41

a12

a22

a32

a42

a13

a23

a33

a43

a14

a24

a34

a44

te) 2H &eaure the argument in favor of the



57

tatement Zai outran a"\H

where w i the relative im)ortance of attribute

an' W i the total im)ortance of all attribute.

Concor'ance matri/ i 'efine' a followH

( , ! ( ! K

H

C A A # =

i j k k a aik jk

= ∑≥

12 1

21 2

31 3

1

...

...

...

... ...

M

c c

c cC

c c

c

−

− = −

−

te) 4H Among the attribute in favor of a", ome ma# have ome

'oubt u)on the tatement Zai outran a"\ *hi )henomenon i



5

'oubt u)on the tatement ai outran a" . *hi )henomenon i

re)reente' b# a 'icor'ance in'e/H

where r i the normali9e' )erformance value an'

i the ma/imum 'ifference between the normali9e'

)erformance value of an# two alternative w.r.t. all attribute.

Dicor'ance matri/ i 'efine' a followH

8,( , !

ma/ ( ! K ,ik jk

i j jk ik

k

if a a A A

r r other#iseδ

≥= −

12 1

21 2

31 3

1

...

...

...

... ...

M

D

− −

= −

−

δ

te) 0H ;hresholds <utran"ing +elation



78

te) 0H ;hresholds <utran"ing +elation

A (relativel# large! concor'ance threhol' an',

if necear#, a (relativel# mall! 'icor'ancethrehol' are 'efine' b# the D& or b# calculating the

average value of the in'ice.

Iing the concor'ance threhol' an' 'icor'ance

threhol' or et, the outraning relation i 'efine'H

]( , !

]( , !

i j

i ji j

c a a ca *a

a a

≥⇔

≤

]c

]

]( , !

( , ! ,

i j

i jik jk k

c a a ca *a

x x D k

≥⇔

∉ ∀

or

=ernel



71

$aving outraning relation, which can be re)reente' b# a'igra)h, a ubet of alternative i ought uch thatH

E an# alternative which i not in the ubet i outrane' b# at leat one alternative of the ubet

E the alternative of the ubet are incom)arable

*hi t#)e of et i calle' a ernel of the gra)h

emarH f the gra)h ha no c#cle, the ernel e/it an' iuni>ue.ach c#cle can be re)lace' b# a uni>ue element

(coni'ering the alternative in the c#cle a tie'!

?)am$le for =ernel



72

$

aa0

a1

a2a4

aa@

;he "ernel is subset Ja1G aG a@K

(the set of $referred alternati#es

?)am$le for ?>?*;+?



73

uil'ing the elationH Concor'ance in'ice

a 1 a 2 a 3 a 4 a 0 a a 5

a 1 18K10 18K10 18K10 18K10 18K10 18K10

a 2 12K10 12K10 5K10 18K10 5K10 18K10

a 3 11K10 11K10 18K10 18K10 18K10 18K10

a 4 7K10 7K10 12K10 12K10 12K10 18K10

a 0 7K10 11K10 12K10 12K10 12K10 18K10

a 11K10 11K10 11K10 11K10 11K10 18K10

a 5 0K10 7K10 0K10 7K10 7K10 K10

a1 i better than a2 w.r.t. acceleration (3!,a2 i better than a1 w.r.t. )rice (0!,

a1 i a goo' a a2 w.r.t. comfort (4! an' 'eign (3!

c(a1, a2! = (3M4M3!K10P c(a2, a1! = (0M4M3!K10

,ssume concordance threshold as 12B10



74

<utraning relation

Concor'ance in'ice which are greater than or e>ual to concor'ance

threhol' are foun'.<utraning relation are obtaine' for the or'ere' )air aociate' b#

thee in'ice if the )air are not the element of the 'icor'ance et

a 1 a 2 a 3 a 4 a 0 a a 5

a1

18K10 18K10 18K10 18K10 18K10 18K10

a 2 12B10 12B10 5K10 18K10 5K10 18K10

a 3 11K10 11K10 18K10 18K10 18K10 18K10

a 4 7K10 7K10 12B10 12B10 12B10 18K10

a 0 7K10 11K10 12B10 12B10 12B10 18K10

a 11K10 11K10 11K10 11K10 11K10 18K10

a 5 0K10 7K10 0K10 7K10 7K10 K10

e)reentation of outraning relation b# a gra)h



70

a

a2

a1

a0

a4a

a@

;he "ernels are subsets Ja2G a4G aK

and Ja2G a0G aK



?>?*;+? ?;&<! ,I>7

+C* (! i 'eigne' for choice )roblem

+C* aim to ran the alternative

+C* concern raning )roblem involving >uai

an'Kor )eu'o criteriaP bae u)on a value' outraning

relation

+C* F ran action without intro'ucing an#

weighting of criteria

multicriteria decision making

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