multicriteria decision making
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Multi-Criteria Decision
Making
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“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'
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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)
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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
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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!.
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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
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6ow u))oe #ou have three choice of fruit. *hen the
)air wie com)arion goe a the following
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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 .
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6otice that all the element in the com)arion matri/ are
)oitive, or
.
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*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
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*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
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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.
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%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
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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
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Illustrati!e example
(how to compute in a full hierar chy?)
1
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*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!
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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!
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%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!
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We can 'o the ame for )aire' com)arion with re)ect to
actor C an' D. $owever, the weight of factor C an' D are
ver# mall (loo at *able again, the# are onl# about ? an'0? re)ectivel#!, therefore we can aume the effect of
leaving them out from further coni'eration i negligible. We
ignore thee two weight a et them a 9ero.
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A'"ute' weight for factor A =
A'"ute' weight for factor =
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actor, actor
*om$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
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,&. (,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$s:1. Develo) the rating for each 'eciion alternative for each criterion b#
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2. Develo) the weight for the criteria b#
E 'evelo)ing a )air-wie com)arion matri/ for each
criterionE normali9ing the
reulting matri/
E calculating an' checing the conitenc# ratio
E averaging the value in each row to get the corre)on'ing rating
3. Calculate the weighte' average rating for each
'eciion alternative. Chooe the one with the highet
core.
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+ating and !escri$tion of *riteria:
+ating !escri$tion1 >uall# )referre'
3 &o'eratel# )referre'
0 trongl# )referre'
5 Fer# trongl# )referre'
/tremel# trongl# )referre'
Falue 2, 4, , or 7 ma# alo be aigne' an' re)reent )reference
halfwa# between the integer on either i'e.
n 1 2 3 4 0 5 7
8 8 8.07 .7 1.11 1.20 1.30 1.4
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Example ) Car *election Pro+lem GC cor)oration i )lanning to bu# few car for
it tran)ort ection. *he following four criteria
ha been i'entifie' for the election of the carHE %rice
E &ileage )er litre (&%+!
E Comfort
Et#le
Ie A$% to elect the bet of three car name' A, an' Cfrom the maret.
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%rice
Car election
&%J Comfort t#le
*ar , *ar *ar *
$ierarchical tructure of Car election %roblem
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%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
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%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
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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
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,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
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*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
−= −
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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
= =
= + + =
= =
=
= =
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;<.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.
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;<.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!
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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
=
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5te$s of ;<.5I5 te) 2H Contruct the weighte' normali9e' 'eciion matri/.
E Aume we have a et of weight for each criteria w " for " = 1,…n.
E &ulti)l# each column of the normali9e' 'eciion matri/ b# it aociate'
weight.
E An element of the new matri/ iH
vi" = w " r i"
11 12 111 12 1 1 1 2
1 21 2 1 2
1 21 2 1 2
. . . . . . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . . . . . .
ij n j n j n
i i ij ini i ij in j n
m m mj mnm m mj mn j n
2
! ! ! ! # # # #r r r r
! ! ! ! # # # #r r r r
! ! ! ! # # # #r r r
= = r
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5te$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
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5te$s of ;<.5I5
te) 4H Calculate the e)aration meaure for eachalternative.
*he e)aration from the i'eal alternative iH
iQ = T Σ (v "
Q U vi"!2 V i = 1, …, m "
imilarl#, the e)aration from the negative i'eal
alternative iH Oi = T Σ (v "O U vi"!2 V i = 1, …, m
"
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5te$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.
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,$$lying ;<.5I5 ethod to
?)am$le
eight '%1 '%4 '% '%2
5tyle +eliability uel ?co%
5aturn
ord
9 9 8
8 8
9 @ 8 9
*i#ic
aAda
@ 8 @
*ost
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,$$lying ;<.5I5 to ?)am$le
m = 4 alternative (car mo'el!
n = 4 attributeKcriteria
/i" = core of o)tion i with re)ect to criterion "
= R/i"S 4×4 core matri/.
; = et of benefit attributeH t#le, reliabilit#, fueleconom# (more i better!
;O = et of negative attributeH cot (le i better!
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5te$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
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5te$s of ;<.5I5
te) 1 (b!H 'ivi'e each column b# (Σ/2i" !1K2 to get rij
5tyle +el% uel
5aturn
ord
'%4@ '%@1 '%04 '%0
'%0 '%48 '%48 '%4@
'%09 '%41 '%48 '%09
*i#ic
aAda
'%4' '%48 '%48 '%4'
*ost
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5te$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
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5te$s of ;<.5I5
te) 3 (a!H 'etermine i'eal olution AQ.
AQ = R8.80, 8.244, 8.12, 8.878S
5tyle +el% uel
5aturn
ord
'%'4@ '%244 '%1@2 '%1'@
'%'0 '%192 '%144 '%'92
'%'09 '%1@4 '%144 '%118
*i#ic
aAda
'%'4' '%192 '%144 '%'8'
*ost
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5te$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
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5te$s of ;<.5I5
te) 4 (a!H 'etermine e)aration from i'eal olution AQ =R8.80, 8.244, 8.12, 8.878S i
Q = T Σ (v "
Q U vi"!2 V W
for each row "
5tyle +el% uel
5aturn
ord
(%'4@-%'092 (%244-%2442 ('2 (%'2@2 *i#ic
aAda
*ost
(%'0-%'092 (%192-%2442 (-%'182 (%'122
(%'0-%'092 (%1@4-%2442 (-%'182 (%'82
(%'0-%'092 (%192-%2442 (-%'182 (%'2
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5te$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
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5te$s of ;<.5I5
te) 4 (b!H fin' e)aration from negative i'eal olution AO= R8.848, 8.14, 8.144, 8.117S
iO = T Σ (v "OU vi"!2 V W for each row
"
5tyle +el% uel
5aturn
ord
(%'4@-%'4'2 (%244-%1@42 (%'182 (-%'122*i#ic
aAda
*ost
(%'0-%'4'2
(%192-%1@42
('2
(-%'2@2
(%'0-%'4'2 (%1@4-%1@42 ('2 ('2
(%'0-%'4'2 (%192-%1@42 ('2 (-%'82
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5te$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
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5te$s of ;<.5I5
te) 0H Calculate the relative cloene to the i'eal
olution CiQ = Oi K (i
Q MOi !
Oi K(iQMOi! Ci
Q
5aturn
ord
'%'8B'%112 '%4 ←
?5;
'%'4'B'%'9 '%41
'%'19B'%1'9 '%1
*i#ic
aAda '%'4B'%1'0 '%40
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.+<?;&??( Preference /anking rgani3ation Metho for Enrichment E!aluation4
%<&*$ #iel' a )artial )reor'er
%<&*$ #iel' a uni>ue com)lete )reor'er
- 5rans 6 2incke" 17894
ain 5te$s
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ain 5te$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
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+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
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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
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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
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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
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.+<?;&?? 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
$reorder
a1 %M a2 if ΦM(a1! X ΦM(a2!
a1 M a2 if ΦM(a1! = ΦM( a2!
a1 %- a2 if Φ-(a2! X Φ-(a1!
a1 - a2 if Φ-(a2! = Φ-(a1!
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a1 % a2 Ya1Y outran Ya2Y ifH a1 %M a2 an' a1 %- a2
a1 %M a2 an' a1 - a2
a1 M a2 an' a1 %- a2
a1 a2 Y a1Y an' Y a2Y are in'ifferent ifH
a1 M a2 an' a1 - a2
a1 a2 Ya1Y an' Ya2Y are incom)arableH
in all other cae
.+<?;&?? II
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.+<?;&?? II , uniFue com$lete $reorder is built:
E +an"ing the alternati#es following the decreasingorder of net flows
a1 % a2 Ya1Y outran Ya2Y
if Φ(a1! X Φ(a2!
a1 a2 Ya1Y an' Ya2Y are in'ifferent
if Φ(a1! = Φ(a2!
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a1 % a2 Ya1Y outran Ya2Y if Φ(a1! X Φ(a2!
a1 a2 Ya1Y an' Ya2Y are in'ifferent
if Φ(a1! = Φ(a2!
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*here are i/ )oible bi'H
a1, a2, a3, a4, a0, a
*he following i/ criteria will be taen into accountH
C1(a!H mean maintenance time )er 'a# (in minute!
C2(a!H technical value of the e>ui)ment (core out of 188!
C3(a!H cot (18 !
C4(a!H etimate' maintenance cot for the e>ui)mentO ueful life
(18 ('icounte'!
C0(a!H etimate' mean number of failure )er #ear (bae' on tan'ar' ue!
C(a!H afet# rating of the e>ui)ment offere'
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Selecting the generalized criteria
*riterion ;y$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
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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
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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
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Devising the flow table
*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 '
Devising the flow table
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Devising the flow table
*
(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
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Devising the flow table
*4(a:
1
0
H (d)
d10 60
IV
- 60 -10
1/2
- criterion to be minimiAed
- 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
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Devising the flow table
*0(a:
1
0
H (d)
d
I
- criterion to be minimiAed
- 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
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Devising the flow table
*@(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:
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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
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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
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The ranking obtained using
the romethee ! method
a2
a0 a4 a
a1
a3
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a2 a a1
a0 a4 a3
The ranking obtained using the
romethee !! method
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?>?*;+?
. o# 'eigne' a metho' for choice )roblem in
17H
Zlimination et choi/ tra'uiant la r[alit[\
(?limination and *hoice ;ranslating +eality
ain 5te$s
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ain 5te$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
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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
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'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
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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
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$aving outraning relation, which can be re)reente' b# a'igra)h, a ubet of alternative i ought uch thatH
E an# alternative which i not in the ubet i outrane' b# at leat one alternative of the ubet
E the alternative of the ubet are incom)arable
*hi t#)e of et i calle' a ernel of the gra)h
emarH f the gra)h ha no c#cle, the ernel e/it an' iuni>ue.ach c#cle can be re)lace' b# a uni>ue element
(coni'ering the alternative in the c#cle a tie'!
?)am$le for =ernel
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$
aa0
a1
a2a4
aa@
;he "ernel is subset Ja1G aG a@K
(the set of $referred alternati#es
?)am$le for ?>?*;+?
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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
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<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
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a
a2
a1
a0
a4a
a@
;he "ernels are subsets Ja2G a4G aK
and Ja2G a0G aK
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?>?*;+? ?;&<! ,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
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