Generalizing and Integrating TOPSIS and Cook

Generalizing and Integrating TOPSIS and Cook-Seiford method for
multi-criteria group decision-making with both cardinal and ordinal data
1
1
2
Wu Li (李 武 ), Guanqi Guo (郭观七), and Xiaoqiang Zhou (周小强)
1
School of Information and Communication Eng ineering, Hunan Institute of Science and Technology,
Yueyang 414006, China
2
School of Computer, Hunan Institute of Science and Technolog y, Yueyang 414006, China
Correspondence should be addressed Wu Li; [email protected]
Abstract: The T OPSIS and Cook-Seiford social choice function are generalized and integrated for multi-criteria grou p
decision-making (MCGDM) with both cardinal evaluations and ordinal preferen ces o f the alternatives. Unlike traditional
T OPSIS, at first, the group’s positive ideal solution and negative ideal solution under cardinal and ordinal preferen ces are
defined respectively. T hus the group rankings of the alternatives with respect to each criteri a are derived from the individual
preferences by the modified group T OPSIS considering the weights of decision makers under each criteria. Then the weighted
distance function representing the total inconsistency between the comprehensive rankings of all alternatives and the ones under
all criteria is presented after the criteria w eights are taken into account. Form the respective of minimizing the criteria-weighte d
distance of the rankings, a nonlinear integer programming is developed and transform ed into an assignment problem to obtain
the final rankings o f all alternatives. An illustrative c ase is presented and some comparisons on the results show the develop ed
approach is practical and effective. T his study extends T OPSIS to group decision-making with ordinal preferences and
generalizes Cook-Seiford social choice fun ction to multi-criteria decision-making considering the criteri a weights, and can be a
novel benchmark for MCGDM with both cardinal and ordinal data.
Keywords: Multi-criteria group decision-making (MCGDM); TOPSIS; Cook-Seiford social choice fun ction; Group’s ideal
solution; Minimum distance method
1. Introduction
Multi-criteria group decision-making (MCGDM) has
wide applications in engineering, economics, management,
military fields and so on. Its essence is assembling
decision-making information, sorting and selecting the
outcome through definite alternatives by a group o f
decision makers according to multiple criteria [1-4]. As a
typical MCGDM problem, for example, supplier sel ection
in supply chain management need take into account
multiple criteria including quality, delivery, perform ance
history, warranties, price, technical capability and financial
position [5] and is usually made by multiple experts in
quality control, financial management or supply chain
management, and related administrative o fficials.
MCGDM often requires the experts to provide their
qualitative/quantitative assessments for determining the
perform ance of each alternative with respect to each
criterion, and thus leads to the concurrent appearance o f
both cardinal and ordinal preferences in the same MCGDM.
For example, the experts usually are willing or easy to give
their cardinal/ordinal preferences of alternatives for those
qualitative/quantitative criteria. Also, it is quite natural to
think that different experts can provide their evaluations by
means of such different preference structures as preference
orderings, utility functions, multiplicative preference
relations, fuzzy preference relations, and so on [6, 7]. For
group decision-making (GDM), fortunat ely, Chiclana and
Delgado initiated a notable model with multiple preferences
represented by means o f preference orderings, utility
fun ctions and fuzzy preferen ce relations early in the
late-1990s [8, 9]. Following them, Herrera [10], Mata [11],
Dong [12, 6], Fan [13], Wang [14] and Xu [15] et al. have
made great progress in the GDM with different preference
structures. T heir methodologies can be divided into two
categori es: trans formation function [8-12 ] and direct
aggregation [6, 13-15] method. T he former transfo rms all
preferen ces in different structures into a uniform fo rm and
then aggregates the preferences unified to obtain the
rankings o f all alternatives. Unlike the trans formation
method, the latter obtain directly the rankings o f all
alternatives by developing and solving some optimization
models [13-15] or aggregating the individual preferences
determinated firstly and separately. In spite of their success
in dealing with the complexity resulted from multiple
preferen ce form, the transformation method may lose
decision info rmation and even cause preferen ce distortion,
and in the meanwhile, the direct framework is a bit
convoluted [6]. Moreover, almost all these work have not
involved multiple criteria, i.e., little attention has been paid
to MCGDM, especially with both cardinal and o rdinal
preferen ces. In fact, the technique for o rder p erfo rmance by
similarity to ideal solution (TOPSIS) [16] and Cook-Seiford
social choice fun ction [17] can be ext ended and combinat ed
fo r this problem, and this is just what we shall investigate in
this paper.
As a useful technique in dealing with multi-criteria
decision-making (MCDM), T OPSIS argues that the ranking
of alternatives will be based on the shortest distance from
the positive ideal solution (PIS) and the farthest from the
negative ideal solution (NIS). It simultaneously considers
the distances to both PIS and NIS, and a preferen ce ord er is
ranked according to their relative closeness, and a
combination of these two distance measur es [16].
According to Shih et al. [18], T OPSIS has advantages such
as: (i) a sound logic that represents the rationale of human
choice; (ii) a scalar value that accounts fo r both the best and
worst alternatives simultaneously; (iii) a simple
computation process that can be easily programmed into a
spreadshe et; and (iv) the perfo rmance measures o f all
alternatives on attributes can be visualized on a polyhedron,
at least for any two dimensions. Moreover, Zanakis et al.
used the simulation comparison to show that T OPSIS has
the fewest rank reversals among the eight methods in the
category [19]. T hese advantages mentioned above make
T OPSIS a major MCDM technique as compared with other
related te chniques such as AHP and ELECT RE. In recent
years, T OPSIS has been extended to GDM problems with
interval data, fuzzy data or linguistic data [20-25]. T hese
works generalizing TOPSIS for GDM can b e classi fied into
two categories: ext ernal aggregation and intern al
aggreg ation [18]. Outside the traditional TOPSIS procedure,
the form er utilizes some operations to manipulate the
alternative ratings and weight ratings [20, 21], or uses a
social welfare function to obtain a final ranking from
individual decision makers of the group [22]. T he latter
aggreg ates the pr eference o f individuals within the T OPSIS
procedu re [23 ]. Besides, in the external aggregation class,
we can further distinguish the methods as pre-operation (i.e.,
mathematical operators fo r cardinal info rmation, refer to
[20, 21]) and post-operation (i.e., Borda’s count or function
fo r ordinal information, see [2]), which depend on whether
the aggregation is done before or afte r the T OPSIS
procedu re. It seems that extern al aggregation aims to
provide more information to support a complex decision,
and the internal aggregation focus es on an integrat ed
decision making procedure.
In spite of the success of T OPSIS in GDM, there are
several problems we need to discuss fu rther. Firstly, these
above works based on T OPSIS only dealt with the
decision-making problem with crisp numbers, fuzzy data,
interval data or linguistic information of alternatives. How
to extend T OPSIS to decision-making with ordinal
preferences has remain ed unknown. Secondly, these works
either used mathematical operators to manipulate the
alternative ratings and weight ratings given by individual
decision makers before the T OPSIS procedure or c alculat ed
the separation measures o f PIS and NIS for the group afte r
obtaining the separation measures from individual positive
ideal solution (IPIS) and individual negative ideal solution
(INIS) within the T OPSIS procedure. Like traditional
T OPSIS, that is to say, these works were bas ed on IPIS and
INIS rather than group positive ideal solution (GPIS) and
group negative ideal solution (GNIS). T hirdly, these works
only considered the whole weights o f the decision makers
without taking into account the weights o f them under each
criteria. For the same criteria, in fact, the authoritativeness
and validity of evalu ations of alternatives given by different
decision maker may be different since the knowledge,
ability, position, and familiarity with the decision-making
problem o f di fferent decision maker are different. Thus, the
weights of d ecision makers with respe ct to each criteri a
have to be considered respe ctively. Lastly, it should be
noted that traditional T OPSIS doubles the effects o f
attribute weighting on the separation measures. Thus, the
priorities of alternatives are overly controlled by attribute
weights. For these reasons, we shall modify and generalize
T OPSIS for MCGDM with both cardinal and ordinal
preferen ces.
As a good social choice method, Cook and Seiford [17 ]
first defined a distance fun ction on the set of all preference
orders given by multiple decision makers and which was b e
proved to satisfy certain desirable properties. Then they
developed a nonlinear integ er prog ramming minimizing the
distance between the final rankings o f the alternatives and
the ones given by all decision makers, and trans formed it to
an assignment problem to obtain the rankings of the
alternatives fo r the group. However, the method was used
fo r GDM in single criteria setting. In fact, the Cook-Seifo rd
method, although proposed initially for aggregating
individual prefe rences in GDM, can be extended for
MCDM to integrating the rankings of all alternatives under
each criteri a into the comprehensive rankings. More
importantly, it regarded the sum of the difference o f every
alternative’s ranking in two preference orders as the
distance o f the two preference orders and did not consider
the decision makers’ weights. So the method proposed by
Cook and Seiford may yi eld multiple results for the sam e
decision-making problem. In fact, its distance measurement
approach can be modifi ed.
For these reasons above, in this paper we propose a
novel methodology for MCGDM with both cardin al and
ordinal data. Firstly, the GPIS and GNIS with respect to
each criteria under cardinal and ordinal preferences are
defined respe ctively, and then the alternatives’ rankings for
the group under each criteria are derived by ou r modifi ed
group T OPSIS. Secondly, considering all the criteria
weights, we present a distance fun ction to denote the
inconsistency between the compreh ensive rankings o f all
alternatives and the ones unde r all criteri a. According to the
idea that the ideal comprehensive ranking is the one which
minimizes the criteria-weighted distance between itself and
the ones under all criteria, a nonlinear integer prog ramming
is developed. We transform it into an assignment problem to
obtain the final rankings of all alternatives. Lastly, an
illustrative case is presented and some comparisons on the
results show the developed appro ach is practical and
effective.
T he remainder of this paper is organized as follows. T he
proposed approach is introduced in the following section.
Section 3 presents an illustrative cas e and some discussions
on the results. Finally, several conclusions will be drawn in
Section 4.
2.2. The proposed algorithm
2. Proposed framework
To obtain the final rankings of all alternatives fo r the
group, there are two processes to implement, namely (1) the
aggregation process for aggregating the individual
preferen ces under each criteri a into group rankings of the
alternatives by modified group T OPSIS, and (2) the
integration process fo r integrating the group rankings of th e
alternatives under each crite ria into the final comp rehensiv e
rankings of them by generalized Cook-Seifo rd method.
(1) The aggregation process based on modifi ed group
T OPSIS
We present firstly the concepts o f GPIS and GNIS as
follows.
Definition 1. If the perform ance ratings o f a altern ative
with respect to certain criteria Cn given by all decision
makers are all the highest one, then the alternative is the
GPIS with respect to criteria Cn , named GPISn .
Definition 2. If the perform ance ratings o f a altern ative
with respect to certain criteria Cn given by all decision
makers are all the lowest one, then the alternative is the
GNIS with respect to criteria Cn , named GNISn .
Now we detail the proposed modi fied group TOPSIS.
T he algorithm involves the following steps.
Step 1. Determine GPI S n and GN ISn under criteri a
In this section we shall develop a novel hybrid MCGDM
model based on the combination o f the extended T OPSIS
and Cook-Seiford social choice function in the presence o f
both cardinal evaluations and ordinal preferences. Before
presenting the proposed method we define and formulat e
the MCGDM first.
2.1. The MCGDM problem formulation
Without loss of generality and for the sake of simplicity,
we put forward two reason able assumptions as follows.
Firstly, we assume the individual cardinal preferen ces o f
the alternatives are express ed by crisp numbers since our
proposed approach can also work well fo r other cardinal
preference fo rms (for example, interval data, fuzzy data,
linguistic information and so on) as long as the following
group T OPSIS is modified acco rdingly.
Secondly, the criteria evaluated by cardinal preferences
are set be benefit type and the cardinal evaluations have
been standardized. Otherwise, they can be transfo rmed into
benefit indexes by some appropriate normalization method.
So we consider a MCGDM problem with L decision
makers evalu ating M alternatives according to N
Dl (l  1,2, , L)
criteria.
is
decision
maker,
Am ( m  1, 2, , M ) is alternative, and Cn ( n  1,2, , N )
is criteria. n ( n  1, 2, , N ) is the weight of criteria C n
N
which
satisfy
n  1
and
n  0
.
n1
ln (l  1,2, , L; n  1, 2, , N ) is the weight of decision
L
maker Dl under criteria
Cn
which satisfy
 ln  1
l1
l
is the performance rating of alternative
and ln  0. rnm
Am with respect to criteria C n given by decision maker
l
is a crisp number and fo r
Dl . For cardinal evaluations rnm
ordinal preferences it is the rank position of Am and
r  1,2, , M  . For certain n ( n  1, 2, , N ) , all r
l
nm
l
nm
l

constitute the group decision matrix Rn   rnm
as
M L
follows.
 rn11 rn21  rnL1 
 1

2
L
 r r  rn 2 
Rn   n 2 n 2


 


1
2
L
 rnM

rnM
 rnM
Below our concern is to obtain the final rankings o f all
alternatives in the opinion of the group of the decision
makers.
Cn ( n  1, 2, , N ) .



GPISn  rn1 , rn2  , , rnL  , GNISn  rn1 , rn2  , , rnL 

where
l
max rnm
, for cardinal evaluations;
rnl    m
, l  1,2, , L .
1, for ordinal preferences.
l
min rnm
, for cardinal evaluations;
rnl    m
, l  1, 2,, L .
M , for ordinal preferences.


Step 2. Calculate dnm
and dnm
respectively, i.e., the
separation measure o f Am ( m  1, 2, , M ) from GPI S n
and GNISn
group.

dnm

under criteria
Cn ( n  1, 2, , N ) for the
L
L
l 1
l 1
  nl ( rnml  rnl ) 2 , dnm    nl ( rnml  rnl  ) 2
Note: T he traditional T OPSIS doubles the effects o f
attribute weighting on the separation measures. Thus, the
priorities of alternatives are overly controlled by attribute
weights. So we adopt the weighted Minkowski distance
fun ction above to overcome this problem and also to
enhance the reliability of decision.
Step 3. Calculate d nm , the relative closeness of
Am ( m  1,2, , M )
to
criteria C n for the group.
dnm 
GPISn ( n  1, 2, , N )

dnm


d nm
 dnm
under
G
Step 4. Obtain rnm
, the group ranking position of
M
(P2 )
Am ( m  1, 2,, M ) under criteria Cn ( n  1, 2, , N ) . We
can rank the alternatives using dnm in descending order. In
j 1 k 1
M
N
  n (rmG  rnmG ) 2 , where
m 1
rmG is the final group ranking
n1
position of Am ( m  1, 2, , M ) .
Now the integration process is composed of the
following two steps.
Step 1. Develop the optimization model to
determinat e rmG , the final ranking position of
Am ( m  1, 2, , M ) .
As mentioned above, the comprehensive rankings o f all
alternatives should minimize the weighted distance in
definition 3. So the final rankings of them in the view of the
group can be obtained by solving a nonlinear integer
programming as follows.
M
(P1 )
N
 n (rmG  rnmG ) 2
min 
m 1
s.t .
n1
G
m
r  1,2,  , M  ( m  1, 2,  , M )
G
G
rj  rk ( j  k; j , k  1,2,  , M )
Step 2. Solve the optimization model above and obtain
the rmG , the final ranking position of Am ( m  1, 2,, M ) .
Since the nonlinear integer programming above is hard to
solve by traditional approach to nonlinear programming or
integer programming, we have to turn to other method. In
fact, each alternative must be rank ed at a location among 1
to M , and each alternative can be located only one
location in the above rang e, so the final rankings o f all
alternatives can be obtained by solving the following
assignment problem.
N
n ( k  rnjG )2
(1)
n 1
M
s.t .  x jk  1 ( j  1,2, , M )
other words, the larger d nm is, the more superior Am is
with respect to criteria C n for the group.
(2) T he integration process based on generalized
Cook-Seiford method.
Cook-Seiford social choice function calculates firstly the
sum of the difference o f each alternative’ ranking position
between the group and a certain individual ordering vector
without considering decision maker weight, and regards the
sum as the inconsistency of the group preference and the
individual rankings. T hen it adds up all these sums to obtain
the total inconsistency of the group and all individual
ordinal preferences. Inspired by the Cook-Seiford method
and unlike it, we first consider the criteria weights and
calculat e the
weighted devi ation
between th e
comprehensiv e ranking position and the one under all
criteria o f single alternative (not ordering vector), then add
up the weighted deviations of all alternatives to the total
inconsistency between the compreh ensive rankings o f all
alternatives and the ones under all criteria as follows.
Definition 3. T he weighted distance representing the total
inconsistency between the compreh ensive rankings o f all
alternatives and the ones under all criteria is expr essed by
M
min  x jk
(2)
k 1
M
 x jk  1 ( k  1,2, , M )
(3)
j 1
x jk  0,1 ( j , k  1, 2, , M )
(4)
N
 n (k  rnjG ) 2
in formula (1) denotes the weighted
n1
deviation between the compreh ensive ranking position and
the ones under all criteria o f S j when its final ranking
position is set at k . Formula (2) and (4) indicate that each
alternative can be loc ated only one location, and similarly
fo rmula (3) and (4) say that each location can also be
located by only one alternative. x jk  1( j, k  1, 2, , M )
in the solution of (P2 ) implies that r Gj  k , that is to say,
the final group ranking position of S j is k .
3. Illustrative case and discussions on the results
In this section we utilize a supplier selection example in
hydroelectric project supply chain to illustrate the
application of the developed app roach and then present
some comparisons and analyses on the results.
3.1. An illustrative case
Consider a MCGDM problem with 5 decision makers
evaluating 4 altern atives according 4 criteri a. T he weights
of the criteria and the decision makers with respect to each
criteria are listed in table 1.
Table1 T he weights of all criteria and decision makers
under each criteria
n
n
1n
 2n
3n
4n
5n
1
0.3
0.3 0.15 0.15 0.15 0.25
2
0.3
0.15 0.3 0.25 0.15 0.15
3
0.2
0.2
0.2
0.2
0.2
0.2
4
0.2
0.2
0.2
0.2
0.2
0.2
Suppose the decision makers g ave their cardinal
evaluation fo r the fi rst and second criteria, and ordinal
preferen ces for the latter two crite ria. The perform ance
rating o f alternatives in respect to each criteria giv en by the
decision makers constitute the group decision make matrix
Rn ( n  1,2,3,4) as follows.
0.3

0.6
R1  
0.2

0.7
0.5 0.4 0.3 0.2 
0.2


0.3 0.3 0.7 0.5
0.4
, R2  
0.6 0.7 0.4 0.4 
0.8


0.5 0.6 0.5 0.6 
0.6
0.3 0.6 0.6 0.4 

0.7 0.9 0.4 0.8
0.2 0.6 0.5 0.6 

0.8 0.3 0.8 0.5 
 4 3 3 4 4
3 3 2 3 4 




3 2 1 2 2
2 1 4 1 1
R3  
, R4  
2 4 4 3 3
4 4 3 4 3 




1
1
2
1
1


 1 2 1 2 2
Now we use the developed approach to rank the
alternatives and the procedu re is as follows.
Step 1. Determine GPI S n and GNISn where
n  1, 2, 3, 4
GPIS1  (0.7, 0.6,0.7, 0.7, 0.6), GNIS1  (0.2,0.3,0.3,0.3, 0.2);
GPIS2  (0.8, 0.8, 0.9,0.8, 0.8), GNIS2  (0.2,0.2, 0.3, 0.4,0.4);
GPIS3  (1,1,1,1,1), GNIS 3  (4,4, 4, 4,4);
GPIS4  (1,1,1,1,1), GNIS4  (4, 4,4, 4, 4).
Step 2. Calculate dnm , the relative closeness o f
Am ( m  1, 2, 3, 4) to GPIS n (n  1, 2, 3, 4) . T he results are
listed in table 2.
Table 2 T he relative closeness of all alternatives to the
corresponding GPIS
m
d1m
d2m
d3m
d4m
1
2
3
4
0.223330
0.597171
0.413591
0.798026
0.294086
0.661637
0.428023
0.545316
0.192927
0.639355
0.320377
0.863473
0.360645
0.637771
0.192927
0.759747
G
Step 3. Obtain rnm
, the group ranking position of
Am ( m  1, 2, 3, 4) under criteria Cn ( n  1, 2, 3, 4) . The
result can be derived from the data in table 2 and listed in
table 3.
Table 3 T he rankings of all suppliers with respect to each
criteria for the group
m
r1m
G
r2 m
G
r3m
G
r4 m
1
2
3
4
4
2
3
1
4
1
3
2
4
2
3
1
3
2
4
1
T hat is to say, the final group rankings of 4 alternatives are:
A4  A2  A3  A1
So the best alternative is A4 .
3.2. Discussions on the results
To show the effectiven ess of the proposed method, we
shall solve the illustrative MCGDM problem again by other
existing methods.
For the cardinal evaluations, at first, we can use weighted
sum method to rank the alternatives for the group.
According to the weighted sum method, we can set
L
l
eGnm    lnm rnm
(n  1, 2 ; m  1, 2,3, 4) , and then rank the
l 1
alternatives using the index in descending order. In other
words, the larger eGnm is, the more superior Am is for the
group. All eGnm under the first and second criteria are as
follows:
e11G  0.32, e12G  0.5, e13G  0.415, e14G  0.6;
eG21  0.42, eG22  0.675, e23G  0.495, e24G  0.6.
So the rankings of the alternatives under these two criteria
fo r the group are as follows:
r11G  4 , r12G  2 , r13G  3, r14G  1;
r21G  4 , r22G  1, r23G  3, r24G  2.
Obviously the results are identical with the ones obtained
by the proposed method above.
Secondly, we utilize Borda’s count method [2] to obtain
the rankings of the alternatives under the criteria evaluat ed
by ordinal preferences. T he Borda’s count is defined as
L
G
l
bnm
  (M  rnm
) ( n  3,4; m  1, 2, 3, 4) and similarly the
G
l 1
larger bGnm is, the more superior Am is for the group.
T hus we have
G
b31G  2 , b32G  10 , b33G  4 , b34
 14;
Step 4. Determine rmG , the final group ranking position
of Am ( m  1, 2, 3, 4) .
N
Let
e jk 
n (k  rnjG )2 ( j, k  1, 2,3,4)
and
then
n1
constitute E   e jk 
where ejk is located in the jth
4 4
row and kth column as follows.
2.82843 1.84391 0.89443 0.44721 


0.83666 0.54772 1.37840 2.34521
E
2.23607 1.26491 0.44721 0.89443 


0.54772 0.83666 1.51658 2.73861
Substituting E into (P2 ) , we obtain the solution:
x14  x 22  x33  x41  1;
x jk  0, for other j , k .
G
G
b41G  5 , b42
 11, b43
 2 , b44G  12.
So the rankings of the suppliers with respect to the criteria
of supply capacity and aft er servic e fo r the group are as
follows and they are also identical with the ones obtained
by the proposed method above.
r31G  4 , r32G  2 , r33G  3, r34G  1;
r41G  3, r42G  2 , r43G  4 , r44G  1.
It must be noted that the Borda’s count above do not take
into account the weights of the decision makers since the
weights of them in the illustrative case are just the same.
Otherwise the weighted Borda’s count can be expressed as
L
G
l
bnm
  ln  ( M  rnm
) (n  3,4; m  1, 2, 3, 4) which shall
l 1
brings about the same results.
Now we calculate the final rankings o f all alternatives fo r
the group by Bernardo method [2] which involves the
following steps:
Step 1. Defin e the consistency matrix Fn   f jkn 
M M
under criteria Cn ( n  1,2,, N ) where
1, when rnjG  k ;
f jkn  
( j , k  1, 2,  , M )
0, otherwise.
Step 2. Calculate the weighted consistency matrix
N
F   f jk M  M  n  Fn .
defined, and thus the Cook-Seirford social choice function
is extended for MCGDM.
(3) T he proposed frame work can be a benchmark
solution for MCGDM with both cardinal and o rdinal
preferen ces since it can work w ell for other cardinal
preferen ce forms as long as the group T OPSIS is modified
accordingly.
However, we do not involve the situation where there are
multiple different preference structures even under one
criteria. T hese problems will be left fo r our future study.
n 1
Step 3. Obtain rmG , the comprehensive ranking position
of Am ( m  1, 2, , M ) for the group by solving the
following 0-1 programming problem:
M
(P3 )
M
max   f jk x jk
Conflict of Interests
T he authors declare that there is no con flict of interests
regarding the publication of this paper.
Acknowledgements
j 1 k 1
M
s.t .
x
jk
 1, j  1, 2,..., M
jk
 1, k  1, 2,..., M
T his work was supported by the National Natural Science
Foundation of China (No. 61473118), Natural Science
Foundation of Hunan Provin ce (No. 2015JJ2074) and
Scientific Research Fund o f Hunan Provincial Edu cation
Department (No. 13K102).
k 1
M
x
j 1
x jk   0,1 , j , k  1, 2,..., M
x jk  1( j, k  1, 2, , M )
in
the solution
of
(P3 )
G
j
indicates r  k , that is to say, the final ranking position
of Aj is k .
According to the procedur e above, we have
0 0.2 0.8
 0


0.3
0.7
0
0 
F
 0
0 0.8 0.2 


0 
 0.7 0.3 0
Substituting F into (P3 ) , we obtain the solution:
x14  x22  x33  x41  1; x jk  0 , for other j , k .
T he solution indicates the fin al rankings o f 4 alte rnatives in
the view of the group are:
A4  A2  A3  A1
The final result is also identical with the one obtained by
the proposed method in this paper and this just indicates it
is effective and rational.
4. Conclusions
In this paper we investigate the MCGDM problem with
both cardinal and ordinal preferences and develop a
approach based on combination o f our extended g roup
T OPSIS and generalized Cook-Seifo rd social choic e
method. T he main contribution of this study may be
summarized as follows.
(1) Group ideal solution concepts fo r cardinal and ordinal
preferences are presented, and thus a modified g roup
T OPSIS without doubling the effects of attribute weighting
on the separation measures is developed for MCGDM.
(2) A weighted distance fun ction of ranking vectors is
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