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Mathematical Ranking (and Consensus Forming) Method
Hiroaki Ishii Graduate School of Information Science and
Technology Osaka University, Japan
Decision making based on Mathematical evaluation
Data envelopment analysis ( DEA) --- mathematical evaluation method for measuring the efficiency of decision-making units (DMU) on the basis of the observed data practiced in comparable DMUs, such as public departments (governments, universities, libraries, hospitals, etc), banking, etc.
DEA is originated by Charnes et al. and extended by Banker et al. The basic DEA models are known as CCR and BCC named after the authors’ initials.
input
Virtual Input=
1 2( , , , )j j mjx x x
1 1 2 2j j m mjv x v x v x
Output weight
Virtual output
1 2( , , , )j j sjy y y
1 1 2 2j j s sju y u y u y virtual output
efficiencyvirtual input
Weight , 1, 2,...,iv i m
, 1, 2,...,ru r s
11 12 1
21 22 2
1 2
n
n
m m mn
x x x
x x xX
x x x
11 12 1
21 22 2
1 2
n
n
s s sn
y y y
y y yY
y y y
1 1 2 20
, 1 1 2
1 1 2 2
1 1 2
1 2 1 2
max
1, 1,2,...,
, , , 0, , ,...., 0
r i
o o s so
u v o o m mo
j j s sj
j j m mj
m s
u y u y u yFP
v x vx v x
u y u y u ysubject to j n
v x vx v x
v v v u u u
0 1 1 2 2,
1 1 2 20
1 1 2 2 1 1 2 2
1 2 1 2
max
1
,
1,2,...,
, , , 0, , ,...., 0
r io o s so
u v
o m mo
j j s sj j j m mj
m s
LP u y u y u y
subject to v x v x v x
u y u y u y v x v x v x
j n
v v v u u u
DEFINITION (CCR EFFICIENCY)
1. is CCR-efficient if and there exist at least one
optimal with
DMUo * 1
( *, *)v u *, * 0v u
2. Otherwise, is CCR-inefficient.DMUo
PRODUCTION POSSIBILITY SET P
1. The observed activities belong to P( , ), 1, 2,...,j jx y j n
2. If an activity (x, y) belongs to P, then activity (tx, ty) belongs to P
for any positive scalar t.
3. For an activity (x, y) in P, any semi-positive activity withˆ ˆ( , )x yˆ ˆ,x x y y is included in P.
4. Any semi-positive linear combination of activities in P belongs to P.
For ranking of alternatives, one of the most familiar methods is to compare the weighted sum of their votes, after determining suitable weights of each alternative. Borda initially proposed the “Method of Marks” more than two hundred years ago so as to obtain an agreement among different opinions. His
method is surely a useful method evaluating consumers’ preferences of commodities in marketing, or in ranking social policies in political sciences, for instance. It is, however, difficult to determine suitable weight of each alternative a priori. In this context, Cook and Kress formulated the measure to automatically decide on the total rank order weight in order to hold the advantage using Data Envelopment Analysis model.
Later, Green et al. evolved the measure so as to make it possible to decide on the total rank order of all candidates.
And so, the weight of first, second, and third place are set to be5,3, and 1, respectively, where these are given a priori. The totalScore of i-th candidate, , is given as follows: . Here, denote the number of i-th place votes earned by candidate i.Table 1 Voting data, MVP of CL (Central League) in JPB
i321 135 iiii vvv
ijv
Candidates First Second Third (5) (3) (1)
Total score
1.Noguchi (Dragons) 2.Uehara (Giants) 3.Sekikawa (Dragons)
59 72 41 80 35 33 55 62 43
552 538 504
6 59 3 72 1 41 611´ + ´ + ´ =6 80 3 35 1 33 618´ + ´ + ´ =
In this example, if different weights are given to places, then the result of ranking becomes different. Then, an important issue is how to determine proper weights of first, second, and third place.
Since all candidates want to be ranked first place, they wish each weight to be assigned so as to maximize their own composite score. So, Green’s method by using LP which can determine the value of weights, is a very useful method. However, their method has undesirable points from the viewpoint of application.
Rank ordering method by Green et al Each candidate j=1,2,…, m obtained number yj1 of vote as first place, yj2 as second
place, yjk of k-th place the weight of k-th place (k=1,2,…,K). Assign the each weight so as to maximize the weighted sum to her/his vote . (1)
1
k
jj j jw y
2jy
jiw
jiw
Total Ranking Method based on DEA
Among m objects, n persons select till k ranks according to their preference
1
1
max
1,
1,..., , 1,..., ,
k
ji jijji
k
jq ji qii
w y
w y
q m i k
s.t.
1 1 2
1 2 3
0, 0
12 3 ,
( 1)
ji ji j j jk
j j j jk jk
w w w w w
w w w kw wk m
+- ³ ³ ³ ³ ³
³ ³ ³ ³ ³+ + ´
L
LL
Caseⅰ (Green et al.)
Caseⅱ (Noguchi et al.)
Weight×captured votes
1DMU
2DMU
・・・
1DMU 2DMU mean・・・・・・・・・・・・
・・・ ・・・・・・ ・・・・・・・
1121 22
1221
1
1 2 , 1, 2,...,mq q q qm q m
Geometric mean
mDMU 1m 2m mm mf
1mf
2m
mDMU
Cross evaluation
A B C D E F
A 1.000 1.000 1.000 1.000 1.000 1.000
B 1.000 0.980 1.000 1.000 0.980 0.980
C 0.961 0.939 0.961 0.967 0.939 0.939
D 0.716 0.714 0.716 0.710 0.714 0.714
E 0.480 0.490 0.480 0.484 0.490 0.490
F 0.353 0.367 0.353 0.355 0.367 0.367
Geometric mean
1.000
0.990
0.951
0.714
0.486
0.360
rank
1
2
3
4
5
6
Example Six DMU (A ~ F)
1
1
1 2
max
1, 1,2,...,
2 0
ks sjj j j
ks sjq j q
j j jk
w y
Subject to w y q m
w w kw
In case of many categories, we solve the following linear programming problems
1
1 2( )s s s mos o o mo
1
1
max
1 ,
1,2,...,
S
oo
S
l j
w
w
j m
f q
q
=
=
=
£
=
å
å
l ll
ll
s.t.
Evaluation of DMU based on various data
m DMU : S categories Efficiency of DEA
1 21
1 2 3
0, 0
12 3 ,
( 1)
S
S S
w w w w w
w w w Sw wS m
- - ³ ³ ³ ³ ³
³ ³ ³ ³ ³+ + ´
l l L
LLⅱ)
ⅰ)
Multiple choice
(Changing weight order)
Different type
Candidates are chosen.
1DMU
2DMU
・・・
1DMU 2DMU mean・・・・・・・・・・・・
・・・ ・・・・・・ ・・・・・・・
1121 22
1221
1DMU
2DMU
mDMU
・・・
1 2 S1w 2w Sw
・・・・・・・・・・・・
・・・
・・・ ・・・・・・ ・・・・
1121 k2
k1
2mq1mq
12
mSq
22
weight
category
Efficiency of DEA
1
S
o l oll
wf q=
= å
mDMU
mDMU
1m 2m mf
1m2m
mm
AHP
maxTheconsistency index C.I.=
1
n
n
max (consistent)n
Example Application to Apparel Maker Selction Problem
1. Criteria Selection, candidate of maker,
Hierarchy construction
2. Using DEA, making scores of makers from characteristics
3 . Based on AHP, subjective evaluation with respect to Priority
4. Total Preference of maker
Selection of maker
Hierarchy of Apparel Maker Selection Problem -1
Level 1
Purpose
Level 2
Evaluation Points
Level 3
Objects
Maker D
Selection of apparel maker
Sewing Design
Maker A
Maker B
Maker C
Assortment
AHP Applicable of AHP directly
AHP? Not Applicable
of AHP
Selection of Suit Brand (DEA+AHP)
Hierachy-2
Level 1
Purpose
Level 2
Evaluation Points
Level 3
Objects
Maker D
Selection of Apparel maker
Sewing Design
Maker A
Maker B
Maker C
Assortment
AHP Applicable of AHP directly
Vote by staff (Piecewise
comparison may be replaced by
Voting
Vote,First rank, second rank DEA
Voting data (Input DATA)
1st 2nd 1st 2nd 1st 2ndMaker A 8 9 7 7 1 0Maker B 6 7 7 8 2 2Maker C 5 3 3 3 4 8Maker D 1 1 3 2 13 10
Sewing Design Assortment
Max 8*wsa1+9* wsa2
Subject to 8*wsa1+9* wsa2 1,≦
6*wsa1+7* wsa2 1,≦
5*wsa1+3* wsa2 1,≦
1*wsa1+1* wsa2 1,≦
wsa1 2≧ * wsa2.,
wsa2 2/{20*2(2+1)}.≧
Linear sum for maker A maximize by two weughts wsa1 wsa2
1st 2ndMaker A 8 9Maker B 6 7Maker C 5 3Maker D 1 1
Sewing
( obtained votes as the first rank)×weight for the first rank ) + ( obtained votes as the second rank) x weight for the 2nd ra
nk
1.000.760.520.12
Voting Data Analysis from Sewing point of view
maximize A B C Dw1 0.08 0.08 0.1063 0.1063 meanw2
0.04 0.04 0.017 0.017Maker A 1.0000 1.00001.0000 1.000 1.000Maker B 0.76 0.76 0.757 0.7
570.758
Maker C 0.52 0.52 0.583 0.583
0.551Maker D 0.12 0.12 0.123 0.123 0.1215
Geometric mean of Preference ratejjji
Sewing Design AssortmentMaker A 1.0000 0.9761 0.0597Maker B 0.758 1.0000 0.1641Maker C 0.551 0.4183 0.4160Maker D 0.1215 0.3941 1.0000
Piecewise Comparison
Sewing Design Assortment PrioritySewing 1 5 3 0.6495Design 1/5 1 3 0.2295
Assortment 1/3 1/3 1 0.121
A maker :0.650*1.0000+0.223*0.9761+0.121*0.0597=0.87
49
B maker :0.650*0.758+0.223*1.0000+0.121*0.1641=0.735
6
C maker :0.650*0.551+0.223*0.4183+0.121*0.4160=0.501
8
D maker :0.650*0.1215+0.223*0.3941+0.121*1.0000=0.28
69
Selection of A maker is the most preferable
jppjiij xxx 11
Conjoint analysis is a scaling method developed in mathematical psychology by American psychologist Luce and Turkey in 1964. A model of consumer’s preference formation in common use is the simple additive model. In this model, we think that each possible level of an attribute has a “part worth” to a level of an attribute, and the sum of the part worthies of its attributes is the “total worth” to a consumer of a product. Generally, conjoint analysis introduces a part worth value of each attribute of each product based upon some goodness fit criterion from preference rank ordinal data. The rank ordinal data in that case is a result of people’s selection by individuals.
Part worth value
6(1)
5(2)
4(3)
3(4)
2(5)
1(6)
0
1
0
1
0
1
1
0
1
0
1
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
1
1
0
0
A
B
C
D
E
F
Condominium
Detached
house
Totally
European
Semi-European
style
Totally
Japanese
Forms Styles Scores z
(ranking
y)
Factors
Real estate
11b 13b 21b 22b
Example of Conjoint Analysis
12b
P Q
A 0 1 0 1 0
B 0 1 0 0 1
C 1 0 0 1 0
D 1 0 0 0 1
E 0 0 1 1 0
F 0 0 1 0 1
1p 2p 3p 1q 2q Rank
1
2
3
4
5
6
Monotonic TransformationZ
6
5
4
3
2
1
Part worth11b 12b 13b 21b 22b
Factor Analysis by Conjoint analysis
0-1 design matrix to indicate each level of products
:,,
:ˆ
:ˆ
:,
:
:,,
21
,21
1
111
21
n
Tn
nmn
m
Tn
bbb
zzz
dd
dd
yyy
b
cz
Dbz
z
D
y Ordinal scale of consumer’s preference for products
Order preserving transformation of y
An additive conjoint model
Average vector of z
The part worth values to be estimated
Fitting criterion for conjoint analysis
Quadratic fractional programmingQuadratic fractional programming
NumeratorNumerator
denominatordenominator
Difference between estimation and actual Difference between estimation and actual datadata
Variation of actual dataVariation of actual data
2 ,S f b To be minimizedTo be minimized
2 ,S f b Optimal bOptimal b
minimize
maximize
)()(
)()(
)ˆˆ()ˆˆ(
)ˆ()ˆ(),(2
cDbcDb
zDbzDb
zzzz
zzzzb
T
T
T
T
fS
2( , )S f b
Parametric quadratic function with
• Theorem
)()()()()( cDbcDbzDbzDbF TT
cczzcDbzDbDbDb TTTTTTTT 22)1(
Let
If
then
*
* 2
min{ ( )
min{
}
( |
|
0
) }
a
S
t
F F b
b b
F
Further, the minimizer of
solution of
)( *F is also an optimal
)(2 bS
Optimality conditions
0)(
b
F )(F is convex , Since
)2(0)( zDbzzzzDbF TTTT
b )(FLet b, be , substitute into
From (1), (2), obtain the optimal part worth value
)1()1( zDcDDbD TTT
P Q
A 0 1 0 1 0
B 0 1 0 0 1
C 1 0 0 1 0
D 1 0 0 0 1
E 0 0 1 1 0
F 0 0 1 0 1
Part Worth 0 2 -2 0.5 -0.5
1p 2p 3p 1q 2q
2.5
1.5
0.5
-0.5
-1.5
-2.5
422 Pb 15050 ..bQ
Method B
RESEARCH PURPOSE
Conjoint Conjoint AnalysisAnalysis
Many Multi-purpose Problem ActuallyMany Multi-purpose Problem Actually
Conjoint Analysis combined with Conjoint Analysis combined with DEADEA
More ApplicableMore Applicable
MarketingConsumer Preference
Single Objective
Multi-purpose
DEA ・ Conjoint Analysis
DEA ・・・
Conjoint Analysis ・・
multiple total evaluationmultiple total evaluationWeighting Total OrderingWeighting Total Ordering
Voting Data
from total evaluationfrom total evaluationFactor—part worth valueFactor—part worth value
Ordinal Data
Conjoint Analysis combined with DEA
sample category Colour First Second
Th
ird
Pencil SharpPencil
Ball point pen
red blue
A 1 0 0 1 0 2 2 6
B 0 1 0 0 1 5 3 2
C 0 0 1 1 0 3 5 2
〈〈 ExampleExample 〉〉 Voting Data Voting Data
MultipleMultiple evaluationevaluationTotal evaluationTotal evaluation
DEA
Conjoint AnalysisConjoint Analysis
Part worth value
sample order
A 3
B 1
C 2
DEA model
constraintconstraint
Objective functionObjective function
1
1k
iq ij qjj
y w v
1,2, ,q m
1 2 32 3 0i i i ikw w w kw ;ijw
;ijv
Choose k preferences among m alternatives with rankingChoose k preferences among m alternatives with ranking
Weight of each rankWeight of each rank
Captured votesCaptured votes
Preferable Weighting For each alternative
1
k
ii ij ijj
y w v Maximize
Ranking by Cross-valuation
Application
DEA -Conjoint Analysis
Evaluation Method for Voting Data
Application to development plan of new medicine
From activation values of 40 samples, From activation values of 40 samples, we find promising we find promising
Compounds from various aspect.Compounds from various aspect.
(single objective case )Mother compound
• Artificial example
NHCNHSO2
O N
N
CH
1R 2R
3R
The combination of substituents expected as the new medicines
5.410
6.49
4.98
5.17
5.16
5.95
6.34
6.63
7.12
7.61
Sample 1R 2R 3R
COOEt
COOEt
Cl
3CH
3COOCH
COOEt
2NO
3CH
Cl
COOEt
Cl
3OCH
3OCH
3OCH
3OCH
3CH
3CH
3CH
3CH
3OCH
3OCH
3OCH
H
H
H
H
H
H
H
H
Activationvalue
Application to the design matrix
5.4100100100010
6.410100100009
4.910100010008
5.110100000017
5.110100000106
5.910010100005
6.310010000014
6.610010001003
7.101010000102
7.601001000011
Activation value
Z3COOCH 2NO3CH Cl Cl 3OCH 3CH 3OCH HCOOEt
b 11b 12b 13b 14b 15b 21b 22b 23b 31b 32b
1R 2R 3R
Calculation Result
COOEt3CH 3COOCH Cl 2NO Cl 3OCH 3CH 3OCH H
44.5 00.5 22.6 83.4 94.5 55.0 44.0 0 77.1 0b
1R 2R 3R
Estimate a new medicine with combination of substituents of high part worth value
: : :
3COOCHCl
3OCH3
2
1
R
R
R Activation value
(estimation)8.54
Theme
Extending Distance Measure to Construct Joint Ballot Model
Single Ballot
Choosing “special and
the best” objects
Joint Ballot
Choosing “the most
favorable pairs”
Theme
constructing a model to reflect voter’s favorable pairs when they place objects in the order
Distance FunctionMinimizing “distance” which indicates degree of disagreement of voters
Cook&Kress’s Relative Distance
Position j Forward Indicator Vector
“degree of differences”
are expressed
Relative Distance
Position j Backward Indicator Vector
Consensus Formation
RelativeDistance
1 2 3 4Cocktail
0 1 0 0
Beer 0 0 1 0焼酎 1 0 0 0Wine 0 0 0 1
Definition
Relative Distance shows the Degree of Differences
Relative Distance Function
Ranking of Voter A
Joint Ballot ~ Basic Idea ~
Model to reflect “priority of viewpoints” and “patterns of objects”
1.Signifying viewpoints
2.Comparing objects relatively in each viewpoint
BrandDesign
Price
Functionality
2 steps
Joint Ballot
sital
~ Joint Ballot Model ~
1.Joint Ballot Model
Adding Weights wsj of rankings and viewpoints and Viewpoint to relative distance measure
2.Weights
(i) Viewpoints ws : applying AHP
(ii) Rank wj
(iii) Integrating these 2 weights
Every object is not placed in the same standing
Every object is placed only in a standing
ssj jw ww=
Joint Ballot~ Procedure to Rank Ordering ~
Voters’ task
Specifying Viewpoints
Integrating Weights of Ranks and Viewpoints
Overall Ranking
pairwise Comparison of Importance between Viewpoints
Rank Ordering Objects in Each Viewpoint
Joint Ballot to
Choose n Objects:
Top n of Overall Ranking
Validation
~ Questionnaire ~
1.Situation
Chain restaurant “X” sticks to catering “fresh”, “tasty” dinning and “cozy” space at affordable price. “X” will target young people and open 3 new shops around “Y” University. It sends out a questionnaire to search the most effective combination of stores.
2.Objects
Bar ・ Chinese Restaurant ・ European Restaurant ・ Japanese Restaurant ・Burger Shop ・ Bakery ・ Café ・ Cake Shop ・ Taiwanese Traditional Tea Shop3.Viewponts
Occasion ・ Menu Item ・ Number of People ・ Freshness
4.Joint Ballot
In advance usual Joint ballot is given to compare our method. 3 stores are chosen.
Validation ~ Coincidence with Joint Ballot ~
Average of Data Spread :4.2
Standard Deviation:3.17
Proposed Method Coincides with Joint Ballot to Some Degree
Cause
Similarity Effect
Difficulty to Weigh Viewpoints
Bar
Bakery
Cafe
Tea
European R
R27Burger
R4
Japanese R Chinese
R
Cake Shop
CausalSpecial
Combination of Restaurant and Light Meal
Combination of Light Meals
Light Meal
Full Meal
Validation ~ Ogawa and Ishii’s Count ~
Ogawa and Ishii’ Count
1. Priority as of viewpoints with AHP
2. Preference Rate with DEA
AHP Applied to Cook and Kress’s Count
3. Overall Ranking
Overall score of an object is product-sum of priority as and preference rate Zi.
Preference Rate Zi
Preference Rate Ziq to which Object q (q=1,2,…,m) is Applied
1
1
1 2 3
max
. . 1 ( 1,2,... )
2 3
1 2
(1 2 ) ( 1)
k
ii ij ijj
k
iq ij qjj
i i i ik
ik
Z w v
s t Z w v q m
w w w kw
wk n nk k
Validation ~ Comparison to Ogawa and Ishii’s Count ~
Meaning of Winner
Coincidence with Rank of Accumulated Ballots
Accumulated Ballots
OccasionMenu Item
PeopleFreshness
Ogawa
9 th 0.0167 0.7000 1.0000 0.9667
3 rd 0.9833 0.9667 0.9000 0.9500
1 st 0.9667 0.9333 0.9833 0.8667
Our Metho
d
9 th 0.1083 0.9333 0.8667 0.9000
3 rd 0.7250 0.7000 0.9083 0.9833
1 st 0.7250 0.5667 0.8417 0.9667
In Ogawa and Ishii’s count, ranking is likely to depend on number of ballots in high rank.
Ranking to Advantage of the Strong
Average of n th
rank
Bar
Chinese R
European R
Japanese R
Burger Shop
Bakery
Café
Cake Shop
Tea
Accu
mu
late
dB
allo
ts
Validation
Meaning of Winner Comparison to ranking of Neutral Point
Neutral Ranking which Covers Both of high and low Opinion
Approximately Consistent with Proposed Count
~ Comparison to Ogawa and Ishii’s Count ~
in Neutral
Agreement Disagreement
Bar
Chinese R
RankA
ccu
mu
late
d
Ballo
ts
Validation
Ogawa
Ours
Bar 2 9
Chinese R 4 3
European R 1 1
Japanese R 3 2
Burger Shop 7 4
Bakery 6 5
Café 5 6
Cake Shop 8 7
Tea Shop 9 8
Ranking
Quite Different in ranks of Bar and Burger Shop
~ Comparison to Ogawa and Ishii’s Count ~
Overall Ranking between Different Groups of Preferential Manner
BarBakery
Cafe
Tea
European R
R27 Burger
R4
Japanese R Chines
e R
Cake Shop
Group who Prefers Light Meal and Casual Restaurant
Group who Prefers Exclusive Restaurant
Validation
1 st
2nd
3rd
4 th
5 th
6 th
7 th
8 th
9 th
Occasion Bar 2 0 0 1 1 0 0 1 5
Occasion Burger Shop 0 2 2 1 1 3 0 0 1
Menu Item Bar 0 2 0 0 1 0 0 0 7
Menu Item Burger Shop 0 1 0 3 0 2 0 3 1
People Bar 5 1 0 0 2 1 0 1 0
People Burger Shop 0 1 2 6 0 0 0 1 0
Freshness Bar 4 0 1 1 0 0 0 0 4
Freshness Burger Shop 4 0 1 1 0 0 0 0 4
Burger : Many Ballots at 3rd and 7th
Low Rank in Ogawa’s Count
Middle Rank in Proposed Count
Bar : Many Ballots at 1st and 9th
Ballots at 1st Ogawa’s Count
Ballots at 9th Proposed Count
~ Comparison to Ogawa and Ishii’s Count ~
Overall Ranking between Different Groups of Preferential Manner
Validation
Ranking
Overall Ranking between Different Groups of Preferential Manner
Ogawa Ours
Bar 2 9
Chinese R 4 3
European R 1 1
Japanese R 3 2
Burger Shop 7 4
Bakery 6 5
Café 5 6
Cake Shop 8 7
Tea Shop 9 8
BarBakery
Cafe
Tea
European R
R27 Burger
R4
Japanese R Chines
e R
Cake Shop
~ Comparison to Ogawa and Ishii’s Count ~
Group who Prefers Light Meal and Casual Restaurant
Proposed Count Covers 2 Groups’ Preferential Zone
Approximate Ranking Covering Voter’s Favorable Combination
Group who Prefers Exclusive Restaurant
Summary
Proposed Method Coincides with Joint Ballot to Some Degree
Conclusion
Reasonable Approach with Basic Idea and Distance Function
Proposed Count Ishii and Ogawa’s Count
Neutral Ranking which covers Both of high and low Opinion
Ranking to Advantage of the Strong
Consensus Formation
Problem
Difficulty to Weight Viewpoints Pairwise Comparison with Cook&Kress’s Object Based Distance
Ranking which Takes Importance to Majority