This paper proposes a method of ranking the elements of a set of objects by multi-criteria analysis on multiple principal components derived from a multiple variable dataset pertaining to the objects. Concordance-discordance method of outranking and derivation of preference relations has been suggested as a technique of multi-criteria decision-making. The method has been illustrated by conducting the taxonomical analysis of regions according to their level of development.Taxonomical analysis; levels of regional development; principal components analysis; multi-criteria decision-making; concordance analysis; outranking relations; Bihar

Mishra, SK

English

Research Papers in Economics

1 
 
Taxonomical analysis of regional development by 
outranking relations on multiple principal components 
 
SK Mishra 
Department of Architecture  
and Regional Planning 
Indian Institute of Technology 
 Kharagpur 
 
Introduction: For taxonomical analysis (TA) of levels of development of economies, use of the 
index of per capita income was almost unchallenged till the recent past. It was so due to several 
reasons. Historically, since development economics was an offshoot of the classical economics 
which  flourished  in  developed  economies  of  the  West  and  adored  competitive  general 
equilibrium, it was not unexpected that the level of development of an economy was believed 
to be represented by per capita income. For institutional reasons, since development economics 
and planning had initially been a concern of central agencies and sovereign states and much was 
not taken care of the economies at regional levels, the question of representativeness of per 
capita  income  as  a  measure  of  level  of  development  drew  little  attention.  For  operational 
reasons, social accounting at national and state levels was both practicable and practiced, and 
hence, there was no urge for any measure of development level other than per capita income. 
 
With the recent emergence of interest in and need for regional planning, especially in 
under-developed economies, economists experienced uneasiness with regard to all the three 
norms mentioned above, and hence, there is observed a marked interest of the regional planner 
in searching for some new index or measure of development of regional economies. The quest is 
justified,  but  yet  not  achieved.  The  failure  led  the  regional  planner  to  seek  help  of  some 
statistical-mathematical techniques by which a number of variables, each one representing level 
of development of some particular aspect of the economy, could somehow be aggregated to 
make a composite index, which, in turn, could be used for TA of overall level of development of 
regional economies. The principal component analysis was found quite promising in making 
such a composite index (3, 4) and hence is its popularity among regional planners. 
 
No doubt, attempts to construct composite indices and using them for TA of regional 
level of development successfully solved the last two problems that one was faced with in using 
per capita income. However, the first problem remains intact. If regional economies are not 
well-evolved and integrated and do not characterize competitive general equilibrium such that 
price mechanism in product and factor markets generate income stream proportional to the 
level  of  development,  there  cannot  be  any  possibility,  by  the  same  token,  of  satisfactorily 
representing several aspects of development by a single composite index. It has been frequently 
reported  that  correlations  among  different  indicators  of  development  across  sectors  (e.g. 
primary and secondary sectors) are weak and sometimes non-conformal (1, 5, 6, 11, 15). As a 
result, the recourse taken by the regional planner is to avoid construction of composite index of 
overall development using full information (i.e. full matrix of intersectoral and intrasectoral 
correlation coefficients) and instead, attempt is made to construct several sectoral indices (each 
one  a  first  principal  component  derived  from  the  submatrix  of  intrasectoral  correlation 
coefficients). These sectoral indices are, at the second stage, subjected to principal component 
analysis once more and the index of overall development is derived, which, in turn is the first 
principal component of the sectoral indices. It is obvious that such an approach will lead to 2 
 
construction  of  highly  inefficient  index  as  it  is  based  on  partial  information.  It  has  been 
suggested by the author elsewhere (15) that while making the index of the level of overall 
development, the use of ordinary correlation coefficients (among sectoral composite indices) 
should  be  replaced  by  the  use  of  canonical  correlation  coefficients  (among  several  sets  of 
variables) so that the index of the level of overall development is more efficient. However, the 
use of canonical correlation alleviates inefficiency but does not remove it. 
 
There is yet another aspect of this problem of inefficiency. It has been reported by many 
researchers that they dropped a number of variables from further analysis after they found 
them to be non-conformal with the retained set of variables (1, 5, 6, 11). It is obvious that the 
purpose in doing so is to construct a composite index that is more representative. But it is 
illusive. The representativeness of an index derived from partial information is always less than 
that of another index which uses full information. 
 
The  problem  of  inefficiency  would  not  have  been  acute,  could  the  first  principal 
component of each sector explain a very large portion of the total variance of the original set of 
variables, as we urged above. But we have also seen that in underdeveloped economies, due to 
reasons already discussed, any single index cannot represent numerous un-integrated forces of 
development. Hence TA of regional levels of development cannot be efficient unless we use 
many  composite  indices  –  i.e.  multiple  principal  components  together.  It  is  remarkable, 
however, that most of the studies on taxonomical analysis of regional level of development have 
avoided  using  multiple  principal  components,  relying  on  the  first  principal  component, 
howsoever inefficient it might be. 
 
No doubt, it has been partially due to the illusion about representativeness (based on 
several  unwarranted  manipulations)  and  partly  due  to  lack  of  awareness  regarding  the 
possibilities  of  using  multiple  principal  components  for  taxonomical  analysis.  The  lack  of 
awareness has been due to the inertia of technological culture. We have been tuned to cardinal 
mathematics and single criterion TA since ages, and hence, our readiness to accept multi-criteria 
analysis and non-cardinal mathematics must be low. This led to our restricted vision. A user of 
cardinal  mathematics  knows  that  two  principal  components  (derived  from  the  same  set  of 
original variables) cannot be subjected to ordinary mathematical operations and they cannot be 
used  for making  any  composite  index.  Hence,  better  to  be  satisfied  with  the  first  principal 
component, howsoever inefficient it might be. 
 
Recent  developments  in  multi-criteria  analysis  has,  however,  opened  before  us  the 
possibilities  of  using  multiple  mutually  non-commensurable  criteria  for  deciding  dominance 
relations of paired comparisons (2, 7, 9, 12, 13, 14). In this context, techniques like concordance 
analysis (9) ELECTRE I, II and III (12, 13) have proven their mettle. These techniques can be 
applied for multi-criteria TA, in which multiple principal components might be used as criteria 
for deriving dominance relations among sub-regions of a region. 
 
The  methodology  of  TA  by  outranking  relations  on  multiple  principal  components:  We 
propose here a method of TA of regional levels of development by outranking relations on 
multiple principal components. We suppose that multiple principal components are, among 
themselves, non-commensurable, and in themselves, semi-cardinally measurable. 
 3 
 
Let there be N sub-regions belonging to a region and let there be M variables each of 
which represents some particular aspect of the regional development. Thus,  ij x  is the value of 
the variable j recorded in sub-region i. Given X(N,M), we could go in for multi-criteria analysis to 
establish dominance relation of sub-region i over sub-region k (i, k є N). But since we do not 
know the relative weights and the one variable is not independent of the other (rij≠0; j≠i; i,j є M) 
and relative contribution of each principal component in explaining the total variance in X is 
obtained.  We  use  these  relative  contributions  as  weights  w(M).  Now,  how  many  principal 
components we should use is based on our judgment. We should, however, see to it that P0 
represents X as exhaustibly as possible (P0⊆ P; P0(N, M0)). 
 
With  P0(N,M0)  and  w(M0)  we  carry  out  multi-criteria  outranking  analysis.  Using 
outranking relations we construct two matrices, C(N,N) and D(N,N), named as concordance and 
discordance matrices respectively. With C and D we construct E(N,N) by any suitable method 
detailed out in the appendix of this paper and use E for establishing dominance relations of i 
over  j  (i,  j  є  N).  These  dominance  relations  readily  give  the  taxonomy  of  the  levels  of 
development of the sub-regions. 
 
An illustrative application: We illustrate the use of the proposed method in classifying the 
districts of Bihar according to their levels of overall development. We have seventeen districts 
and five indicators of sectoral levels of development. (We treat these indicators as variables for 
purely illustrative purpose). Thus, X is for 17 regions and 5 variables, or X(17,5). We derive five 
principal components P(17,5), to exhaustively represent X(17,5). Explanatory contributions of 
P’s are: P1 (0.65527), P2 (0.26696), P3 (0.04174), P4 (0.02685), P5 (0.00918). We could use the first 
three P’s also for an effective classification, but we preferred to use all the five. 
 
Next, we construct C and D, and compare them with varying Tc and Td to obtain logically 
stable classes. Our finding is that by using multiple principal components it is not possible to 
establish  strong  dominance  relation  which  was  possible  to  do  by  using  the  first  principal 
component (single criterion) only. On the criterion of the first principal component our ordering 
would have been as follows: 16 p 1 p 17 p 9 p 8 p 3 p 7 p 4 p 6 p 2 p 5 p 15 p 11 p 14 p 10 p 12 p 13, 
where p means ‘dominates over’ and numerals are the codes for the districts. But on all the five 
criteria our dominance relations are as follows (where q means ‘does not dominate over’):  
(16 q 1) p (17 q 9 q 8 q 3) p (7 q 4) p (6 q 2 q 5) p (15 q 11) p (14 q 10) p (12 q 13).  
 
Conclusion: It is remarkable that multi-criteria analysis simply denies the undue dominance 
relations established by a single-criterion analysis. Relevance of this denial is immense. To recall 
back, we urged that per capita income as a criterion of taxonomy may establish an unwarranted 
dominance of sub-region i over sub-region j (i ≠j), and hence, we searched for some other 
criterion.  We  found  that  the  criterion  of  the  1
st  principal  component,  especially  when  its 
explanatory contribution is low (which is quite expected in less-developed economies), also 
established unwarranted dominance. It is true that the 1
st principal component provides us with 
a numerical measure (and we have a bias for numerical measures howsoever illegitimate they 
might  be),  while  outranking  method  proposed  here  provides  us  with  logical  relations  of 
dominance and non-dominance and none of the numerical measures, yet the former is illusive 
and the latter is logically sound. The former is based on partial information and therefore apt to 
be inefficient, while the latter is based on full information and hence more efficient. 
 4 
 
 
Table-1. Principal Components of X  P as weighted aggregation of X 
Sl No.  Districts  P1  P2  P3  P4  P5  P1=0.971x2+0.933x4+0.863x3+0.846x4-0.055x1 
P2=0.976x1+0.428x3+0.052x4+0.012x2-0.444x5 
P3=0.281x4+0.154x1+0.141x5-0.176x2-0.235x3 
P4=0.257x5+0.141x1+0.020x2-0.029x3-0.218x4 
P5=0.161x2+0.033x1-0.002x4-0.048x5-0.130x3 
 
For identification of x ref (1), 
Chattopadhyay and Mishra. 
1  Patna  2.168  0.983  2.313  1.417  0.405 
2  Gaya  -0.436  0.517  1.071  1.088  0.570 
3  Sahabad  0.176  1.046  1.952  1.543  0.664 
4  Saran  -0.060  0.621  1.814  1.344  0.569 
5  Champaran  -0.678  0.670  1.784  1.490  0.724 
6  Muzaffarpur  -0.134  -0.045  0.079  0.133  0.025 
7  Darbhanga  -0.001  0.108  0.680  0.501  0.135 
8  Monghyr  0.571  0.402  1.315  0.901  0.171 
9  Bhagalpur  0.707  0.393  1.182  0.536  0.196 
10  Saharsa  -1.167  -0.111  0.432  0.143  0.437 
11  Purnea  -0.819  -0.081  -0.877  0.344  0.214 
12  S. Pargana  -1.187  -0.802  -1.871  -1.044  -0.338 
13  Palamau  2.636  -0.409  -2.090  -1.111  0.308 
14  Hazaribagh  -1.129  -0.419  -1.832  -0.956  -0.203 
15  Ranchi  -0.713  -0.938  -2.908  -2.102  -0.843 
17  Dhanbad  4.134  -0.664  -1.049  -1.659  -1.446 
17  Sighbhum  1.203  -1.269  -2.787  -2.568  -1.589 
 
 
Table-2. Concordance and Discordance Matrices (CD) 
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17 
1  -  B  C  B  B  O  O  O  O  A  O  O  O  O  O  N  O 
2  M  -  N  N  I  I  N  N  N  O  O  O  O  O  O  N  M 
3  J  O  -  O  B  O  O  J  B  O  O  O  O  O  O  N  H 
4  M  O  N  -  N  O  E  N  N  O  O  O  O  O  O  N  K 
5  M  J  M  J  -  L  N  N  N  O  O  O  O  O  O  N  N 
6  N  G  N  N  G  -  N  N  N  E  E  O  O  O  O  N  N 
7  N  G  N  G  G  O  -  N  N  C  B  O  B  O  O  N  M 
8  N  F  G  G  G  O  O  -  N  B  B  O  B  O  O  N  G 
9  N  F  G  G  G  O  O  G  -  B  A  O  B  O  O  N  G 
10  M  N  N  N  N  M  M  M  M  -  N  O  O  E  I  N  N 
11  N  N  N  N  N  M  M  M  M  C  -  O  B  O  E  N  N 
12  N  N  N  N  N  N  N  N  N  N  N  -  F  N  N  N  N 
13  N  N  N  N  N  N  M  M  M  N  M  K  -  N  N  N  N 
14  N  N  N  N  N  N  N  N  N  G  N  O  F  -  L  N  N 
15  N  N  N  N  N  N  N  N  N  G  G  G  G  G  -  N  N 
16  G  G  G  G  G  G  G  G  G  G  G  C  F  F  C  -  O 
17  N  G  G  G  G  G  G  G  G  G  G  G  G  G  F  N  - 
The  matrix  CD  contains  Concordance  and  Discordance  matrices.  If  i<j,  CDij=Dij  and  if  i>j  CDij=Cij.  For 
compactness  we  have  used  symbols  for  numerical  values.  Hence,  in  the  cells,  the  letters  represent 
numerical in the following scheme (exclusive of the lower class limits). O=0.0; N=1.0; A=(0.000-0.0005); 
B=(0.0005 – 0.005); C=(0.005-0.05); D=(0.05 – 0.1); E=(0.1-0.2); F=(0.2-0.3); G=0.3-0.4); H=)0.4-0.5); I=(0.5-
0.6); J=(0.6-0.7), K=(0.7-0.8) L=(0.8-0.9); M=(0.9-0.99999) 
 
 
 
 
 
 5 
 
Appendix 
Given  ) var ; ( , 1 ; , 1 ; iables M regions sub N M j N i xij − = =  
M j N i x x x j ij ij , 1 ; , 1 ; / ) ( = = − ⇐ σ where 
2
1
2 2
1 ) / 1 ( ; ) 1 ( j
N
i ij j
N
i ij x x N x N x − = = ∑ ∑ = = σ  
M k j x x N R ik
N
i ij jk , 1 , ; ) / 1 (
1 = = ∑ =  
Set  0 = l  
1.  M j p j , 1 ; 1 = =  
1 + ⇐l l  
2.  M k p R q j
M
j jk k , 1 ;
1 = =∑ =  
M k qk l , 1 ); max( = = λ  
M k q q l k k , 1 ; / = ⇐ λ  
If( 3 000001 . 0 |) | to Go p q k k ≤ −  
 
Else 
M k q p k k , 1 ; = ⇐  
Go to 2 
3.  M k j q q T j k kj , 1 , ; = =  
∑ = =
M
k k q s
1
2  
l s g λ / =  
M k g q w k k , 1 / = =  
N i w x V j
M
j ij li , 1 ;
1 = =∑ =  
M k s T R R l jk jk jk , 1 ); / ( = − ⇐ λ  
If( M l < ) Go to 1 
Else 
M l M l l , 1 ; / = ⇐ λ λ  
Comment: By now we have computed M principal components ( li V ) and weights ( l λ ) and now 
proceed for concordance analysis 
N i M j V V V
N
i ji ji ji , 1 ; , 1 ; /
1
2 = = ⇐ ∑ =  
M j i C l
M
l l
j i
ij , 1 , ;
1 = =∑ =
≠
β λ where  0 = l β  if( lj li V V − ) < 0 else  1 = l β ; M l , 1 =  
M j i V U j ji ji , 1 , ; = = λ  
 



= ≥ − =
≠ − =
1 0 ) ( 0
); | max(|
l lj li l
l lj li ij
else V V if where
j i U U D
β β
β
N j i
M l
, 1 ,
, 1
=
=
 
N j i M l
N j i D D
j i U U
ij ij ij
li lj l ij
, 1 , ; , 1 ;
, 1 , ; /
|); (| max
= =
 



= ∆ ⇐
≠ − = ∆
 6 
 
Comment: Computation of Concordance and Discordant matrixes is over. 
Given arbitrary values of Tc =(0, 1) and Td=(0, 1) 
N j i Eij , 1 , ; = = β where  ) ( 0 d ij c ij T D and T C if > > = β else  1 = β  
Comment: Decision may be taken now.  
Sub-region iis preferred to sub-region  j if 1 = ij E  
End 
 
Note: (a) For computation of  ij E based on min operator and gamma operator, vide Singh, D. (14, 1983). 
(b) The symbol ⇐means ‘is replaced by’. 
 
References:  
 
1.  Chattopadhyay, R.N. and S.K. Mishra (1979) “Towards Balanced Development of Bihar: A Case 
Study in Inter-regional Allocation of Resources”, Geographical Review of India, Vol. 41:3. 
2.  Delft, A. and P. Nijkamp (1976) “A Multi-Objective Decision Model for Regional Development, 
Environmental  Quality  Control  and  Industrial  Land  Use”,  Papers  of  the  Regional  Science 
Association, Vol. 36. 
3.  Hagood, M.J. et al. (1941) “An Examination of the Use of Factor Analysis in the Problem of Sub-
Regional Delineation”, Rural Sociology, Vol. 6. 
4.  Hotalling, H. (1933) “Analysis of a Complex of Statistical Variables into Principal Components”, 
Journal of Educational Psychology, Vol. 26. 
5.  Mishra,  S.K.  and  A.  Chopra  (1978)  “Dimensions  of  Inter-District  Disparities  in  Levels  of 
Development in MP – 1971”, The Indian Geographical Journal, Vol. 53:1. 
6.  Mishra, S.K. and S.B. Gaikwad (1979) “Impacts of Economic Development on Welfare and Living 
Conditions of People of MP”, Indian Journal of Regional Science, Vol. 11:1. 
7.  Mishra,  S.K.  and  A.P.  Subudhi  (1982)  “Concordance  Analysis  for  Multi-Criteria  Evaluation  of 
Competing Projects; A Computer-Oriented Approach”, National Geographical Journal of India, 
Vol. 28: 3-4. 
8.  Mishra, S.K. (1984) Evaluation of Public Policies for Agricultural Development in Less Developed 
Regions, Deptt. Of Architecture and Regional Planning, IIT, Kharagpur. 
9.  Nijkamp,  P.  (1975)  “A  Multi-Criteria  Analysis  of  Project  Evaluation”,  Papers  of  the  Regional 
Science Association, Vol. 35. 
10.  Pal,  M.N.  (1974)  “A  Method  of  Regional  Analysis  of  Economic  Development  with  Special 
Reference to South India”, Indian Journal of Regional Science, Vol. 5. 
11.  Pal,  M.N.  (1975)  “Regional  Disparities  in  Levels  of  Development  in  India”,  Indian  Journal  of 
Regional Science, Vol. 7. 
12.  Roy,  B.  (1973)  “How  Outranking  Relation  Helps  Multiple  Criteria  Decision  Making”,  in  J.L. 
Cochrane and M. Zeleny (eds.) Multiple Criteria Decision-Making. Univ. of South Carolina Press, 
Columbia. 
13.  Roy, B. and Ph. Vinke (1981) “Multi-Criteria  Analysis: Survey and New Directions”, European 
Journal of Operations Research, Vol. 8. 
14.  Singh, D. (1983) Multiple Criteria Decision Rating: Preference Modelling using Fuzzy Sets (unpubs) 
Ph.D. Thesis, Deptt. Of Mathematics, I.I.T., Kharagpur. 
15.  Subudhi,  A.P.  and  S.K.  Mishra  (1982)  “Dimensions  of  Regional  Disparities  in  Levels  of 
Development: A Case Study of Uttar Pradesh”, in L.R. Singh (ed.) Regional Planning and Rural 
Development, Thinkers’ Library. 
 
                 Published in Hill Geographer, a Journal of the Geographical Society of the North-Eastern Hill Region (India) 
        Shillong; 1984: vol III(1): pp. 20-28. 

A Method of Regional Analysis of Economic Development with Special Reference to South India”,

A Multi-Criteria Analysis of Project Evaluation”,

A Multi-Objective Decision Model for Regional Development, Environmental Quality Control and Industrial Land Use”,

An Examination of the Use of Factor Analysis in the Problem of SubRegional Delineation”,

Analysis of a Complex of Statistical Variables into Principal Components”,

Concordance Analysis for Multi-Criteria Evaluation of Competing Projects;

Dimensions of Regional Disparities in Levels of Development: A Case Study of Uttar Pradesh”,

Evaluation of Public Policies for Agricultural Development in Less Developed Regions, Deptt. Of Architecture and Regional Planning,

How Outranking Relation Helps Multiple Criteria Decision Making”,

Impacts of Economic Development on Welfare and Living Conditions of People of MP”,

Multi-Criteria Analysis: Survey and New Directions”,

Multiple Criteria Decision Rating: Preference Modelling using Fuzzy Sets (unpubs)

Regional Disparities in Levels of Development in India”,

Towards Balanced Development of Bihar: A Case Study in Inter-regional Allocation of Resources”,

Taxonomical analysis of regional development by outranking relations on multiple principal components

Taxonomical analysis of regional development by outranking relations on multiple principal components

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