ON METHODS FORMULTIDIMENSIONAL RANKING

By Shrinivas Vasala

OBJECTIVE:

To Study different ranking methods

Implementing these method on agiven data set.

THE MULTIDIMENSIONALRANKING PROBLEM

m candidates (or “alternatives”) M = {1,…,m}: set of candidates n criteria (or “agents” or “judges”)N = {1,…,n}: set of criteria Each voter i, has an ranking i on Mi(a) < i(b) means i-th voter prefers a to b

The rank aggregation problem:Combine 1,…,n into a single ranking on M, whichrepresents the “social choice” of the voters. Rankaggregation function: f(1,…,n) =

METHODS

For finding thewe rank

bestuse with

optionsome

fromsound

all the feasiblealternatives,aggregate attributes.

approaches toonerespect to more than

Positional Rank Aggregation Methods

TOPSIS METHOD (Technique for Orderby Similarity to an Ideal Solution)

Preference

KEMENEY’S METHOD

POSITIONAL RANK AGGREGATIONMETHODS

Pluralityscore(a) = # of voters who chose a as #1: order candidates by decreasing scores

Top-k approvalscore(a) = # of voters who chose a as one: order candidates by decreasing scores

Borda’s rule [Borda, 1770]score(a) = i i(a)

=[sum of rank value]: order candidates by increasing scores

of the top k

TOPSIS METHOD[HWANG AND YOON, 1981]

The factors involved are: Ideal and Negative ideal location Distances from Ideal and negative Weights of item Sum of square of each items.

ideal location.

Relative closeness of Ideal and Negative ideallocation.

EXAMPLE:”A SURVEY FOR DEMAND DRAFTCHARGES IN INDIA”

S.No Bank name Category avg <1K avg 1-5K avg 5-10Kavg >10K (for

every additional 1K)

Cancell- ation fees

1 Allahabad Bank Public 25 35 40 3 1002 Andhra Bank Public 22.5 25 27.5 2 1003 Bank of Baroda Public 20 35 35 3 754 Bank of India Public 32.5 42.5 42.5 4 52.55 Bank of Maharashtra Public 25 30 35 3 1006 Canara Bank Public 25 40 40 3.25 5257 Central Bank of India Public 22.5 32.5 32.5 3.5 758 Corp. Bank Public 30 30 40 2.5 1009 Dena Bank Public 35 35 35 3.25 85

10 Indian Bank Public 30 30 30 3 42.511 Indian Overseas Bank Public 10 15.5 20 1.8 0

42.512 Oriental Bank of Commerce Public 17.5 17.5 22.5 2.2513 Punjab National Bank Public 35 35 35 3.5 10014 Punjab & Sind Bank Public 25 25 25 3 015 State Bank of India Public 30 30 30 2.5 100

27016 Syndicate Bank Public 22.5 35 35 2.2517 UCO Bank Public 37.5 37.5 37.5 2.88 10018 Union Bank of India Public 21.88 21.88 43.75 3.85 55.6319 United Bank of India Public 19.5 33.5 42.5 2.25 5020 Vijaya Bank Public 24.5 34 34 2.25 3921 Axis Bank Limited Private 50 50 50 2.5 5522 Catholic Syrian Bank Private 12.5 20 30 2.25 5023 Dhanlakshmi-B Private 12.5 20 25 1.25 1024 Federal Bank Ltd. Private 25 25 25 2.5 87.5

BASIC COMPUTATION

1/ 2

n m

i

1

i=1,…,mj =1,…,nS

w2

(x)2 x2 u /i ij j ijj

j 1

1/ 2

n m

i 1

i=1,…,mS w2 ( x )2 x2 v /i j ij j ij j =1,…,n j

1 Si C

i=1,…,mi Si Si

# Ranking according to increasing order of Ci

TOPSIS RANK TABLE

S.No Bank name CategoryComposite Index(5 var)34

232811

Citibank N.A. Foreign 0.035992Dhanlakshmi-B Private 0.046986Karnataka Bank Ltd. Private 0.050524Indian Overseas Bank Public 0.064452

30 The Bank of Rajasthan Ltd.

Private 0.07398431122220

The South Indian Bank Ltd.

Private 0.082675Oriental Bank of Commerce

Public 0.086082Catholic Syrian Bank Private 0.088519Vijaya Bank Public 0.089792

19 United Bank of India Public 0.0933772 Andhra Bank Public 0.102432

14 Punjab & Sind Bank Public 0.10589632 ABN-AMRO Bank N.V. Foreign 0.10723229 Kotak Mahindra Bank Private 0.10934624 Federal Bank Ltd. Private 0.1095

725 HDFC Bank Ltd. Private 0.11258910 Indian Bank Public 0.11274521 Axis Bank Limited Private 0.11494633 Abu Dhabi Commercial

BankForeign 0.1152

715 State Bank of India Public 0.116578 Corp. Bank Public 0.118078

3 Bank of Baroda Public 0.120726

KEMENY’S METHOD

Kendall tau distance (or “bubble sort distance”) K(,) = # of pairs of candidates (a,b) on which

and mismatchEx: K( (a b c d), (a d c b)) = 0 + 2 + 1 = 3

Kemeny Optimal aggregation[Kemeny 1959] Optimal aggregation w.r.t. Kendall tau distance

For a collection of given ordering τ1, τ2,…, τn on candidate a Kemeny optimal ordering minimizes the sum of the kendall tau distance

k

K , i i 1

APPLICATION PROCEDURE:

We apply KEMENY’S method in two different ways:

Direct application of KEMENY’S method

Stepwise application of KEMENY’S method

DIRECT APPLICATION OF KEMENY’S METHOD

Step 1:o we first decide number of candidate for applying

KEMENY.o we choose m=4(Banks A=1, B=2,C=3, D=4). We work out the 4! = 24 permutations of 1,2,3,4. A permutation2314 means that the ranking is BCAD.

Step 2:o we take our data matrix as 4 rows and n=6 columnso Construct rank position matrix with respect to each

of the 6 columns.

DIRECT APPLICATION(CONT.)

Step 3:o we calculate the number of non matching pair

between each of the permutations of 1,2,3,4 &position ordering for 9 variable.

Step 4:o next calculate Kendall-distance for permutations.

o Sum kendall distance over permutation. And find permutation corresponding minimum kendall sum. This will be our ranked candidate.

EXAMPLE

Position Rank Matrix:2 2 3 1 1 13 3 4 3 2 21 1 2 4 3 34 4 1 2 4 4

A=1

Indian Overseas Bank

0.06445

0.069847

0.094086

0.092438

13.5 12B=

2Dhanlakshmi-B 0.04

6990.050308

0.070184

0.226337

17.5 14.5C=

3Karnataka Bank Ltd.

0.05052

0.05131

0.126883

0.118069

27

22D=

4Oriental Bank of Commerce

0.08608

0.086876

0.198224

0.198181

34

25.5

EXAMPLE(CONT.)

1 2 3 4

21

13

17

25

15

#Ranking is 1 2 3 4 (* min kendall sum)

Tranpose Position Matrix2 3 1 42 3 1 43 4 2 11 3 4 2

1 2 3 4

Kendall Tau Distance Kendall sum3 3 2 5 3 32 2 3 6 4 43 3 4 5 5 54 4 5 4 6 64 4 3 4 4 45 5 4 3 5 53 3 2 5 3 32 2 5 4 4 41 1 4 3 3 32 2 3 2 2 24 4 5 2 4 43 3 4 1 3 35 5 2 3 3 36 6 3 2 4 45 5 4 1 3 34 4 3 0 2 24 4 1 2 2 23 3 2 1 1 12 2 1 4 2 21 1 2 5 3 30 0 3 4 2 21 1 2 3 1 13 3 0 3 1 12 2 1 2 0 0

19212529232719

15

23

21

21

15111315119

117*

S. No. Permutation123456789101112131415161718192021222324

2 3 4 13 2 4 13 4 2 14 3 2 12 4 3 14 2 3 14 3 1 23 4 1 23 1 4 21 3 4 24 1 3 21 4 3 22 4 1 34 2 1 34 1 2 31 4 2 32 1 4 31 2 4 32 3 1 43 2 1 43 1 2 41 3 2 42 1 3 41 2 3 4

STEPWISE OF KEMENY’SAPPLICATION METHOD

Example:

Step1: We, first fix the no. of rows(=k) to be taken at a time(say, 4). After fixing, we proceed as direct application.

Step2: The optimal Kemeny, thus obtained, is used toidentify the Rank 1 bank. In the next step, we remove rank

5th1 and add the(n-k+1).

bank then apply step 1. and go upto

A CitibankN.A. 0.03599

0.03346

0.329862

0.357041

51 45B IndianOverseasBank 0.0644

50.06985

0.094086

0.092438

13.5 12C Dhanlakshmi-B 0.0469

90.05031

0.070184

0.226337

17.5 14.5D KarnatakaBankLtd. 0.0505

20.05131

0.126883

0.118069

27 22E OrientalBankofCommerc

e0.08608

0.08688

0.198224

0.198181

34 25.5F CatholicSyrianBank 0.0885

20.0874 0.24697 0.21870

945 32

G TheSouthIndianBankLtd. 0.08268

0.08043

0.294886

0.280081

56 43

STEPWISE APPLICATION(CONT.)

Step3:Repeat step1 & step2 the process tillranking stabilize.

Example(cont.): Here, four Kemeny’s steps in each iteration. After four iteration, rank of banks is stabilized.

1st iteration:

i1st Step 2nd Step 3rd Step 4th Step

SelectRow Rank

SelectRow Rank

SelectRow Rank

SelectRow RankBank

A A D . . . . . .B B A A E . . . .C C B B A A F . .D D C C B B A A AE . . E C C B B BF . . . . F C C CG . . . . . . G G

STEPWISE APPLICATION(CONT.)2nd iteration:

3rd iteration:

iii1st Step 2nd Step 3rd Step 4th Step

SelectRow Rank

SelectRow Rank

SelectRow Rank

SelectRow RankBank

D D F . . . . . .F F D D C . . . .C C C C D D D . .G G G G G G G G BE . . E E E E E EA . . . . A A A AB . . . . . . B G

ii1st Step 2nd Step 3rd Step 4th Step

SelectRow Rank

SelectRow Rank

SelectRow Rank

SelectRow RankBank

D D D . . . . . .E E E E F . . . .F F F F A A C . .A A A A B B A A GB . . B E E E E EC . . . . C B B AG . . . . . . G B

STEPWISE APPLICATION(CONT.)4th iteration:

4thOptimum combination(result) for iteration:B C D A E

,… iteratio

n.

F G5th 6thWe get same result after ,

iv1st Step 2nd Step 3rd Step 4th Step

SelectRow Rank

SelectRow Rank

SelectRow Rank

SelectRow RankBank

F F B . . . . . .C C F F C . . . .D D C C F F D . .B B D D D D F F AE . . E E E A A EA . . . . A E E FG . . . . . . G G

STEPWISE APPLICATION(CONT.)

Ranking By Kemeny’s method:

# Here Indianthis method.

Overseas Bank is best according to

Kemeny's MethodBanks Categor

yB IndianOverseasBank Public

C Dhanlakshmi-B PrivateD KarnatakaBankLtd. PrivateA CitibankN.A. Foreign

E OrientalBankofCommerce

PrivateF CatholicSyrianBank PrivateG TheSouthIndianBankLtd. Public

APPLICATION:

Meta search: Combine results of different web searchengines into a better over all rankingRank items in a database according to multiple criteriaOverall ranking of educational institute

o

ranking of Banks by no of customer, no of branches,o

different facilities etc.pollution ranking of citieslevel of poisonous gas like

on the basis of atmosphericCO2 ,SO2, NO2 etc.

o

Ex: Choose of Flight ticket by price, # of stops, date &time.Choosing car by style, reliability, fuel economy, cost.

o

o

.

Thank You