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Superiority Index Method for Multiple
Attribute Decision-Making under
Uncertainty
Jiuping Xu a,1, Shouyang Wang b,2, Jianming Shi c,3
aInstitute of Information and Decision Science, Sichuan University Chengdu
610065, P. R. China
bInstitute of Systems Science, Academy of Mathematics and Systems Sciences
Chinese Academy of Sciences, Beijing 100080, P. R. China Institute of Policy and
Planning Sciences, University of Tsukuba, Japan
cSchool of Management, Science University of Tokyo, Kuki City, Saitama
346-8512, Japan
Abstract
In this paper, we propose a superiority index method for multiple attribute decision-
making under uncertainty. A superiority index is defined for each alternative by
comparing interval numbers with the minimum cost and the maximum profit. With
the order of superiority indices, the alternatives can be ranked in such a manner that
optimal alternatives will have the largest index and be chosen first and those with
the largest index are either optimal or at least non-inferior alternatives. Finally, we
illustrate the superiority index method via an application.
Key words: Multiple attribute decision-making; Uncertainty; Interval numbers;
Superiority index
1 Introduction
How to rank alternatives under uncertainty is fundamentally important in
decision-making process, especially at the multiple criteria decision-making
Supported by NSFC, MADIS, AMSS, Chinese Academy of Sciences and Atsumi
International Scholarship Foundation.
1E-mail: xuejping@mail.sc.cninfo.net
2The corresponding author. E-mail: sywang@iss02.iss.ac.cn
3E-mail: j.shi@ms.kuki.sut.ac.jp
Preprint submitted to Elsevier Preprint 14 June 2001
situations. In the last few decades, A number of approaches have been devel-
oped to deal with multiple criteria decision-making under uncertainty. Among
them, four types should be mentioned. One type is to ask the decision-maker
to compare the pairs of alternatives over and over again in time under the same
experimentally essential conditions [7,25,26]. Another type is to consider the
preferences of a group of decision-makers on the same set of alternatives. Ba-
sically, this type of approaches are based on aggregating the opinions of the
decision-makers to obtain a fuzzy relation [15,20]. The third type is based on
the preferences along with the attributes which can be aggregated to get a
degree of the preferences for the alternatives. And the degree is exploited to
establish a fuzzy relation if the alternatives have multiple attributes [6]. The
fourth is based on vague set theory, which allows the degrees of satisfaction
and non-satisfaction of each alternative with respect to a set of criteria to
be vague values [4]. Staring with a crisp relation R(Ai,A
k) that represents
the fact that the alternative Aiis globally not worse than the alternative
Akwith R(Ai,A
k) = 0 or 1, a number of methods have been proposed to
determine subsets of the set of alternatives which correspond to the “best”
elements in A, for instance, von Neumann-Morgenstern [22], Roth [16], Fish-
burn [5], Miller [10], Aizerman [1], Moulin [12], Schwartz [19]. Kitainik [9] was
the first to translate the previous decision rules in terms of fuzzy implication
and inclusions. He studied those concepts to determine which combinations
of Lukasiewicz t-norm should be in contention under the most general condi-
tions. Recently, Bisdorff-Roubens [3] used a constraint programming system
supporting a finite domain solver [21] to determine efficient fuzzy kernels (min
t-norm) and showed the links between their results and the crisp kernels de-
rived from the strict median cut in the initial valued relation [17]. Orlovski [13]
was the first to introduce the concept of a fuzzy choice function, and consid-
ered the set of unfuzzy-undominated alternatives. Ovchinnikov-Roubens [14],
Fodlor-Roubens [6] and others solved some particular problems. In his book
[18], Roubens surveyed recent advances in the multiple attributes decision-
making methods dealing with fuzzy or ill-defined information. Some aggrega-
tion procedures, choice problems and the treatment of interactive attributes
are covered in the book; trends in research and some open problems are also
presented in the book.
In this paper, we propose a model to solve multiple criteria decision-making
problems under uncertainty based on a given subjective weak preference re-
lation of the decision-maker. A new concept of strict preference is defined in
terms of weak preference and an order relation is also defined in terms of the
interval-valued representation.
We organize the paper as follows. In Section 2, we briefly introduce some basic
concepts of interval numbers. We also introduce a new order relation of interval
numbers representing the decision-maker’s preference between interval profits
and costs which are defined respectively by a maximization problem and a
2
minimization problem. In Section 3, we introduce the concept of a superiority
index of an alternative and present a few lemmas and theorems. A superiority
index method is proposed in Section 4. In Section 5, a concrete example is
given to illustrate our superiority index method.
2 A class of interval-valued orders
Let Abe a set of alternatives and Cbe a set of criteria, where
A={A1,A
2,···,A
n},(1)
C={C1,C
2,···,C
m}.(2)
Each alternative Aiin Ais an objective having the following form
Ai={[L1(Ai),R
1(Ai)]),[L2(Ai),R
2(Ai)],···,[Lm(Ai),R
m(Ai)]},(3)
where Lj(Ai) and Rj(Ai) define respectively the minimal degree and the max-
imal degree of alternative Aiunder the criterion Cj. The criterion Cjfor the
alternative Aiis associated with an interval defined by an ordered pair in a
bracket as
Ai
j=[Lj(Ai),R
j(Ai)] = {x:Lj(Ai)≤x≤Rj(Ai),x∈R}.(4)
The center and width of Ai
jare
C(Ai
j)=(Rj(Ai)+Lj(Ai))/2(5)
and
W(Ai
j)=(Rj(Ai)−Lj(Ai))/2.(6)
The Ai
jcan be given by its center and width as
Ai
j=C(Ai
j),W(Ai
j)
={x:C(Ai
j)−W(Ai
j)≤x≤C(Ai
j)+W(Ai
j),x∈R}.
(7)
Therefore, for a given Ai
j, one can choose the interval form or the center-
width form to represent it. For simplicity, we only discuss the interval form in
3
this paper. For the case of the center-width form, one can also establish the
corresponding theory and method. The generalization of ordinary arithmetic
to closed intervals is known as interval arithmetic. For details, one can refer
to, for instance, Alefeld-Herzberger [2], and Moore [11]. We only quote a basic
concept here. Let + and ×be two binary operations on the set of real numbers.
Ai
j+Ak
j:= [Lj(Ai),R
j(Ai)] + [Lj(Ak),R
j(Ak)]
=[Lj(Ai)+Lj(Ak),R
j(Ai)+Rj(Ak)],(8)
ρAi
j:= ρ[Rj(Ai),L
j(Ai)]
=
[ρRj(Ai),ρL
j(Ai)] for ρ≥0,
[ρLj(Ai),ρR
j(Ai)] for ρ<0,
(9)
where ρis a real number. Suppose that the uncertain profit and cost from two
alternatives Aiand Ajare given by Ai
jand Ak
j, respectively. It is assumed that
the real profit and real cost from the alternatives are only known to lie in the
corresponding intervals. Let Ai
jand Ak
jbe the same as in (4). Order relation
LR between interval Ai
jand Ak
jis defined as
Ai
jLR Ak
jif and only if Lj(Ai)≤Lj(Ak) and Rj(Ai)≤Rj(Ak),
Ai
j≺LR Ak
jif and only if Ai
jLR Ai
jand Ai
j=Ak
j.
(10)
The order relation above implies that the decision-maker has a preference for
the alternative with a higher minimum cost and maximum profit. Note that
the order LR is a partial order, therefore there are many pairs of intervals,
which cannot be compared by LR . We use ∗
LR, the same order relation as
LR, to represent that the decision-maker has a preference for the alternative
with the lower minimum cost and maximum profit, that is, if Ai
j∗
LR Ak
j, then
Aiis preferred to Ak. In the sequel, we use the order relation ab(a≺b)if
and only if aLR b(a≺LR b), and the order relation a∗b(a≺∗b) if and
only if b∗
LR a(b≺∗
LR a), where aand bare interval numbers.
3 Superiority index
To deal with a multiple attribute decision-making problem in the form of
max1≤i≤nAi, we rank the alternatives by comparing them under the “larger
is better” principles. For this purpose, we define an order of alternative Aiof
(1) via the order of their attributes Cjof (3).
Definition 1 For every two alternatives Ai,Akin Aand Cjin C, denote
4
aj
ik :=
1,if Ak
j≺Ai
jor Ak
j≺∗Ai
j,
1
2,if Ak
j=Ai
j,1≤j≤m
0,otherwise.
(11)
Let aik := m
j=1 aj
ik, which is called the superiority index of Aiover Ak; let
Si:= n
k=1 aik, which is called the superiority index of Ai(1 ≤i≤n).
From the above definition, one can observe that the superiority indices have the
following properties: (1) aj
ii =1
2(1 ≤j≤m) and aii =1
2m, (2) 0 ≤aik ≤m,
(3) 1
2m≤Si≤m(n−1) + 1
2m.
Definition 2 For every two alternatives Aiand Akin A, we define (1)
AiAkif and only if Ai
jAk
jor Ai
j∗Ak
j,1≤j≤m; (2) Ai≤Akif and
only if Ai
jAk
jor Ai
j∗Ak
j,1≤j≤m, and Ai
j0≺Ak
j0or Ai
j0≺∗Ak
j0for
some j0. (3) Ai<A
kif and only if Ai
j≺Ak
jor Ai
j≺∗Ak
j,1≤j≤m.
With the order of alternatives, we introduce the following concepts.
Definition 3 Let Aebe an alternative in A. Then (1) Aeis an inferior al-
ternative in Aif Ae≤Aifor some Ai∈A; (2) Aeis a non-inferior alternative
in Aif there is no Ai∈Asuch that Ae≤Ai; (3) Aeis an optimal alternative
in Aif AiAefor all Ai∈A; (4) Aeis a strongly optimal alternative in A
if Ai<A
efor all Ai∈A(i=e).
Lemma 4 Let Aeand Afbe two alternatives in A. Then Ae>A
fif and
only if aef =m.
PROOF. Suppose Ae>A
f. Then Af
j≺Ae
jor Af
j≺∗Ae
j,1≤j≤mby
Definition 2 (3). Hence, aj
ef =1(1≤j≤m). Thus, aef =m
j=1 aj
ef . Suppose
that aef =m=m
j=1 aj
ef . Since 0 ≤aj
ef ≤1, we have aj
ef = 1 such that
Af
j≺Ae
jor Af
j≺∗Ae
jfor each jby Definition 1. From Definition 2 (3),
Ae>A
f.
Lemma 5 Aeis a strongly optimal alternative in Aif and only if Se=
m(n−1) + 1
2m.
PROOF. Suppose Aeis a strongly optimal alternative in A. Then Ae>A
i
and aei =m(for i=e) by 4. Since aee =1
2m, we have
Se=
n
i=1
m
j=1
aj
ei =m(n−1) + 1
2m. (12)
5
Conversely, suppose Se=m(n−1) + 1
2m. Since 0 ≤aj
ei ≤1(1≤i≤n, 1≤
j≤m) and aee =1
2m, we have aei =mso that Ae>A
i(i=e) by 4. Hence,
Aeis a strongly optimal alternative in A.
Lemma 6 Let Akbe alternatives in A.IfAfAe, then (1) aj
ek ≥aj
fk
(1 ≤j≤m)for any Ak∈A; (2) aek ≥afk for any Ak∈A; (3) Se≥Sf.
PROOF. (1) If aj
fk = 0, the inequality holds. If aj
fk =0.5, then Af
j=Ak
j.
Since Af
jAe
jor Af
j∗Ae
j, we have Ak
jAe
jor Ak
j∗Ae
j. Hence, aj
ek ≥
0.5=aj
fk. Finally, if aj
fk = 1, then Ak
j≺Af
jor Ak
j≺∗Af
j. Since Af
j<A
k
j,we
have Ak
j≺Ae
jor Ak
j≺∗Ae
jwhich implies aj
ek = 1. The assertions (2) and (3)
are direct consequences of (1).
Lemma 7 If Aeis an optimal alternative in A, then Se= max1≤i≤nSi.
PROOF. This follows immediately from Definition 3 and Lemma 6.
Lemma 8 Let Aeand Afbe alternatives in A.IfAf≤Ae, then (1)
aff <a
ef ; (2) Sf<S
e.
PROOF. (1) Since Af≤Ae,soaj
ef ≥0.5=aj
ff (1 ≤j≤m) and there exists
at least one j0such that Af
j0Ae
j0(or Af
j0∗Ae
j0). Hence, aj0
ef =1>a
j0
ff.
Therefore,
aef =
m
j=1
aj
ef >
m
j=1
aj
ff =aff.(13)
(2) By 6, aek ≥afk (1 ≤k≤m) and by (1), aff ≤aef .Thus
Se=
n
k=1
aek >
n
k=1
afk =Sf.(14)
Lemma 9 Let Aebe an alternative in A.IfSe= max1≤i≤nSi, then Ae
is a non-inferior alternative in A.
PROOF. This follows immediately from Definition 3 and 8.
Theorem 10 Let Aeand Afbe alternatives in A.IfSf= max1≤i≤n,i=eSi,
then Afis a non-inferior alternative in A=A\{Ae}.
6
PROOF. Suppose that Afis not a non-inferior alternative in A. Then there
exists some Ag∈Asuch that Af≤Ag. Hence, Sf<S
gby 8, which is a
contradiction.
Theorem 11 Let Afbe a non-inferior alternative in Aand Af≥Ak0
for some k0∈{1,2,···,m}.IfSfis the superiority index of the alternative
Af, then Sf≥n−1
2.
PROOF. Suppose that Afis a non-inferior alternative in A. For each Ai(i=
f)inA, there may occur two cases: (1) Af=Ai, and we let N1={j|Af=Aj}
and the cardinality |N1|=n1; (2) Af=Ai, and we let N2={j|Af=Aj}and
|N2|=n2.
Obviously, {f}∪N1∪N2={1,2,···,n},n1+n2=n−1. In the first case,
because aj
fi =1
2for each j∈{1,2,···,m}, therefore afi =1
2m≥1. In the
second case, we denote the number of inequalities Af
jAi
j(Af
j∗Ai
j)or
Af
j≺Ai
j(Af
j≺∗Ai
j)bym0i, the number of Af
jwhich is incomparable with
Ai
jby m1iand the number of Af
jsatisfying Aj
f=Ai
jby m2i, respectively. We
have m0i≥1,m
1i≥0, m2i≥0, and m0i+m1i+m2i=m. Therefore,
afi =
m
j=1 aj
fi =1
2m≥1 when f=i, i ∈N1,
m
j=1 aj
fi =1×m0i+0×m1i+1
2×m2i
=m0i+1
2m2i≥1 when f=i, i ∈N2
1
2when f=i.
(15)
So,
Sf=
n
i=1
afi =
i∈N1
afi +
i∈N2
afi +aff ≥n1+n2+1
2=n−1
2.(16)
Theorem 12 Let Ae,Akand Afbe alternatives in A,Se= max1≤i≤nSi,
Skand Sfbe superiority indices of Akand Af, respectively. If |Sk−Sf|>
m−1, then the order of the elements between Akand Afin Ais invariant in
A=A\{Ae}.
PROOF. Because Se= max1≤i≤nSi,Aeis a non-inferior alternative in Aby
9. Next we prove that aie ≤m−1 for each i∈{1,2,···n}. If it was not the
case, then there would exist an i0such that ai0e=mor ai0e=m−1
2. It implies
that aj
i0e≥1
2for each jand that there exists some j0such that aj0
i0e= 1. Hence
Ae
jAi0
j(or Ae
j∗Ai0
j) for all jand Ae
j0≺Ai0
j0(or Ae
j0≺∗Ai0
j0) for some j0.It
7
leads to Ai0≥Ae.By8,Sj0>S
e. This is a contradiction to the assumption
Se= max1≤i≤nSi. Note that aie ≥0. Hence, we have
1−m≤ale −afe ≤m−1.(17)
Therefore, for any Aland Afin A,
S
l−S
f=(Sk−ale)−(Sf−afe)=(Sl−Sf)+(afe −ale),(18)
where S
land S
fare superiority indices in A=A\{Ae}, respectively. So,
associated with the assumption that |Sk−Sf|>m−1, we have
S
l>S
fwhen Sl>S
f;S
f>S
lwhen Sf>S
l.
This is the assertion of the theorem.
4 Precedence order method
Now we can propose our method for solving multiple attribute decision-making
problems in the form of max1≤i≤nAi. All the alternatives in Aare ranked
according to the magnitude of their superiority indices. The theorems given
in Section 3 show that the optimal alternative in A(if any) will rank in the
first place followed by the second optimal one, which is optimal among the
remaining alternatives. Generally, there may not exist any optimal alternative
and so we will look for a non-inferior alternative. Moreover, if Aiis superior to
Ak(Ak<A
ior Ak≤Ai), then Aiwill rank in front of Ak. Theorem 10 suggests
that if Aihas the largest superiority index among the set of alternatives in
A, then Aiis non-inferior in the set. This ranking of alternatives provides a
precedence order method for the decision-maker to choose the most preferred
one among the alternatives. Theorem 11 implies that the necessary condition
for Aiin Ato be a non-inferior alternative is Si≥n−1
2. Meanwhile, if
there are only two attributes for all the alternatives in A, then the necessary
and sufficient condition for an alternative Aito be non-inferior is Si=n−1
2,
i=1,2,···,n. From Theorem 12, we can easily deduce the sufficient condition
for the order between Akand Afto be invariant in Aand A=A\{Ae}, where
Se= max1≤i≤nSi,is|Sk−Sf|>m−1.
8
Type
Attribute A1A2A3A4A5A6
Price C111 13 9 10 10.4 9.8
Degree of Luxury C2[3,4] [4,5] [1,2] [1.5,2.5] [1.5,2.5] [1.5,2.5]
Max Speed C3200 250 180 220 210 190
Max Load C42000 1800 2100 2000 2200 2050
Reliability C5[6,8] [8,10] [5,7] [6,8] [6,7] [7,9]
Sensitivity C6[7,10] [6,8] [5,7] [4,6] [3,6] [4,5]
Smoothness C7[4,5] [6,6] [5,7] [5,6] [4,5] [4,5]
Loudness C8[10,12] [11,12] [10,13] [9,12] [9,11] [10,13]
Table 1
types and attributes
Type
Attribute A1A2A3A4A5A5
C15 6 1 3 4 2
C22 1 4 3 3 3
C34 1 6 2 3 5
C44 5 2 4 1 3
C53 1 5 3 4 2
C61 2 3 4 6 5
C74 1 2 3 4 4
C83 4 5 2 1 5
Table 2
sequence for attribute of each alternative
5 Example of application
Let us to consider to purchase cars. Suppose there are six types of cars for us to
choose from. Each type (alternative) is evaluated by eight attributes as shown
in Table 1. Remark 5.1. Some of the attributes (C1,C
3and C4) are definite in
quantity and are expressed by real numbers. Other attributes (C2,C
5,C
6,C
7
and C8) are indefinite (uncertain) in nature and expressed by appropriate
interval numbers resulted from the experts’ judgments. Remark 5.2. Fo r
the attributes C1(price) and C8(loudness), the smaller values are desirable.
Therefore, when two values of the attribute are compared, we can assume
9
aj
ik A1A2A3A4A5A6aj
i•aj
ik A1A2A3A4A5A6aj
i•
0.5 1 0 0 0 0 1.50.5 0 1 1 1 1 4.5
0 0.5 0 0 0 0 0.51 0.5 1 1 1 1 5.5
1 1 0.5 1 1 1 5.50 0 0.5 0 0 0 0.5
C11 1 0 0.5 1 0 3.5C20 0 1 0.5 0.5 0.52.5
1 1 0 0 0.5 0 2.50 0 1 0.5 0.5 0.52.5
1 1 0 1 1 0.54.50 0 1 0.5 0.5 0.52.5
0.5 0 1 0 0 1 2.50.5 1 0 0.5 0 0 2
1 0.5 1 1 1 1 5.50 0.5 0 0 0 0 0.5
0 0 0.5 0 0 0 0.51 1 0.5 1 0 1 4.5
C31 0 1 0.5 1 1 4.5C40.5 1 0 0.5 0 0 2
1 0 1 0 0.5 1 3.51 1 1 1 0.5 1 5.5
0 0 1 0 0 0.51.51 1 0 1 0 0.53.5
0.5 0 1 0.5 1 0 30.5 1 1 1 1 1 5.5
1 0.5 1 1 1 1 5.50 0.5 1 1 1 1 4.5
0 0 0.5 0 0 0 0.50 0 0.5 1 1 1 3.5
C50.5 0 1 0.5 1 0 3C60 0 0 0.5 1 1 2.5
0 0 1 0 0.5 0 1.50 0 0 0 0.5 0 0.5
1 0 0 1 0 0.54.50 0 0 0 1 0.51.5
0.5 0 0 0 0.5 0.51.50.5 1 1 0 0 1 3.5
1 0.5 1 1 1 1 5.50 0.5 1 0 0 1 2.5
1 0 0.5 1 1 1 4.50 0 0.5 0 0 0.51
C71 0 0 0.5 1 1 3.5C81 1 1 0.5 0 1 4.5
0.5 0 0 0 0.5 0.51.51 1 1 1 0.5 1 5.5
0.5 0 0 0 0.5 0.51.50 0 0.5 0 0 0.51
Table 3
superiority index of each alternative - a
that the smaller value contributes to the superiority index. By calculating the
sequence for the attributes of each alternative with the definitions in Section
2, we obtain Table 2. By calculating the superiority index of each alternative
according to the definitions in Section 2, we have Table 3 and 4 and
S1=24,S
2=30,S
3=20.5,S
4=26,S
5=23,S
6=20.5.
The precedence order, from the best to the worst, is
A2A4A1A5A3≈A6.
Therefore, the best alternative is A2followed closely by A4. These two types
of cars can be chosen for further consideration.
10
aj
ik A1A2A3A4A5A6aj
1•aj
2•aj
3•aj
4•aj
5•aj
6•Si
a1k4 4 5 3 3.5 4.5− − − − − − 24
a2k4 4 6 5 5 6 − − − − − − 30
a3k3 2 4 4 3 4.5− − − − − − 20.5
a4k5 3 4 4 5.5 4.5− − − − − − 26
a5k4.5 3 5 2.5 4 4 − − − − − − 23
a6k3.5 2 3.5 3.5 4 4 − − − − − − 20.5
Sk− − − − − − 24 30 20.5 26 23 20.5−
Table 4
superiority index of each alternative - b
6 Conclusions
In this paper, we proposed a superiority index method for multiple attribute
decision-making under uncertainty. The proposed method was illustrated with
a concrete example. The method can be applied to solving many practical de-
cision problems under uncertainty. A decision problem under uncertainty can
viewed as a fuzzy optimization problem. To solve the fuzzy optimization prob-
lem, one would make the assumption that the membership function and the
probability distribution function are known or given. Actually, the member-
ship function and the probability distribution play crucial roles in designing
methods based on fuzzy sets and stochastic programming. However, the mem-
bership function and the probability distribution are not always easily given
by the decision-maker. Our method has some advantages over other fuzzy and
multiple criteria decision-making methods, such as fuzzy mathematics AHP,
and order number methods. The simplicity of our method and the efficiency
are two among the advantages.
References
[1] N.A. Aizerman, New problem in the general choice theory, Social Choice and
Welfare 2 (1985) 235-285.
[2] G. Alefeld, J. Herzberger, Introduction to Interval Computations, Academic
Press, New York, 1983.
[3] R. Bisdorff, M. Roubens, Defining kernels on L-fuzzy binary relations, Proc.
IPMU’96, Granada, Spain, July 1-5, 1996.
[4] S.M. Chen, J.M. Tan, Handling multi-criteria fuzzy decision-making problems
based on vague set theory, Fuzzy Sets and Systems 67 (1994) 163-172.
11
[5] P.C. Fishburn, Concordant social choice functions, SIAM J. Appl. Math. 33
(1977) 469-487.
[6] J.C. Fodor, M. Roubens, Fuzzy Preference Modeling and Multi-Criteria Decision
Support, Kluwer Academic Publishers, Dordrecht, 1994.
[7] C.L. Hwang, K. Yoon, Multiple Attribute Decision Making, Springer-Verlag,
Berlin, 1981.
[8] L.C. Jin, J.F. Gu, G.F.Shu,Order number methods for MCDM, J. Sys. Sci. &
Math . Scis. 4 (1984) 281-293.
[9] L. Kitaink, Fuzzy Decision Procedures with Binary Relation, Kluwer Academic
Publisher, Dordrecht, 1993.
[10] N.R. Miller, A new solution set for tournaments and majority voting:further
graph-theoretical approaches to the theory of voting, Amer. J. Polit. Scis. 24
(1980) 68-96.
[11] R.E. Moore, Method and Applications of Interval Analysis, SIAM, Philadelphia,
1979.
[12] H. Moulin, Choosing from a tournament, Social Choice and Welfare 3 (1986)
271-291.
[13] S.A. Orlovski, Decision-making with a fuzzy preference relation, Fuzzy Sets
and Systems 1 (1978) 15-167.
[14] S. Ovchinnikov, M. Roubens, On fuzzy strict preference, indifference and
incomparability relations, Fuzzy Sets and Systems 49 (1992) 15-20.
[15] F.S. Roberts, Measurement Theory, Addison Wesley, MA, 1979.
[16] A.E. Roth, Sub-solutions and the super-core of cooperatives games, Math. Oper.
Res. 1 (1976) 43-49.
[17] M. Roubens, Choice procedures, Fuzzy Sets and Systems 84 (1996) 15-142.
[18] M. Roubens, Fuzzy sets and decision analysis, Fuzzy Sets and Systems 90
(1997) 199-206.
[19] T. Schwartz, The Logic of Collective Choice, Columbia Univ. Press, New York,
1986.
[20] I.B. T¨urksen, Measurement of membership functions and their assessment,
Fuzzy Sets and Systems, 40 (1991), 5-38.
[21] P. Van Hentenrck, H. Simonis, Solving large combination problem in logic
programming, J. Logic Programming 8 (1990) 75-93.
[22] J. von Neumann, O. Morgenstern, Theory of Games and Economic Behavior,
Princeton Univ. Press., Princeton, 1953.
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[23] J. Xu, A ranking method to solve multiple criteria decision problems under
uncertainty, New Developments of MCDM, pp.135-138, International Academic
Press, Beijing, 199x (in Chinese).
[24] J. Xu, Interval valued strict preference for multiple criteria decision-making
problems under uncertainty, Advances in Operations Research and Systems
Engineering, pp.134-141, Global-Link Informatics Ltd., Hong Kong, 1998.
[25] P.L. Yu, Multiple Criteria Decision-Making:Concepts, Techniques and
Extensions, Plenum, New York, 1985.
[26] M. Zeleny, Multiple Criteria Decision Making, McGraw-Hill, New York, 1982.
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