Matteo Brunelli | Robert Fullér | József Mezei
Olympic OWA Operators for Modeling
Group Decisions
TUCS Technical Report
No 936, August 2009
Olympic OWA Operators for Modeling
Group Decisions
Matteo Brunelli
IAMSR, Åbo Akademi University,
Joukahaisenkatu 3-5, FIN-20520 Åbo,
matteo.brunelli@abo.fi
Robert Fullér
IAMSR, Åbo Akademi University,
Joukahaisenkatu 3-5, FIN-20520 Åbo,
robert.fuller@abo.fi
József Mezei
Turku Centre for Computer Science,
Joukahaisenkatu 3-5, FIN-20520 Åbo,
jmezei@abo.fi
TUCS Technical Report
No 936, August 2009
Abstract
Olympic OWA operators are mainly used in group decision making when we want
to exclude the two extreme judgments/opinions from consideration. In this paper
we will introduce the principle of Olympic-ideal decision and find the constrained
group decision by minimizing the Euclidean distance between the set of feasible
decisions and the Olympic-ideal decision.
Keywords: Olympic OWA operator, Euclidean distance, Group decision
TUCS Laboratory
1
Olympic OWA Operators Modeling Group Decisions
Mich et al. [4] introduced a new measure of consensus depending on a function estimated for each expert according to her/his aversion to opinion changing.
These functions have become known in the literature as opinion changing aversion (OCA) functions, and have been further developed in [1, 3]. Fedrizzi et al.
[2] used fuzzy linguistic quantifiers and OWA operators to define a fuzzy majority and to derive a degree of consensus under fuzzy preferences. Suppose that a
certain amount of money exists and it has to be assigned to 3 different projects (alternatives) according to their importance. Moreover, let us also assume that n experts have been asked to evaluate these projects and they express their preferences
under the form of normalized priority vectors, i.e. their components (portions)
sum up to one [5]. The preferences of the i-th expert can be modeled by a triple
(ai , bi , ci ), where ai denotes how much portion of the money should be assigned
to the first project, and so on. For simplicity, we will now consider only three
alternatives/criteria, but our results can easily be derived for the general case.
Yager [6] introduced a new aggregation technique based on the ordered weighted
averaging (OWA) operators. An OWA operator of dimension n is a mapping
F : Rn → R, that has an associated n vector W = (w1 , w2 , . . . , wn )T such as wi ∈
[0, 1], 1 ≤ i ≤ n, w1 + w2 + · · · + wn = 1 Furthermore
n
F(a1 , . . . , an ) =
∑ w jb j
j=1
where b j is the j-th largest element of the bag < a1 , . . . , an >. Olympic type OWA
operators of dimension n, denoted by Fn , were introduced by [7] and defined by
the weighting vector
w = (0, 1/(n − 2), · · · , 1/(n − 2), 0),
where n ≥ 3. That is, an Olympic OWA operator computes the arithmetic average
of all aggregates except the smallest and the largest ones.
Consider the portions the experts want to allocate for the first project a =
(a1 , a2 , . . . , an ). In our model we will assume that the fair value of this distribution
is computed by Yager’s OWA operator,
∑ni=1 ai − mini ai − maxi ai
Fn (a) = Fn (a1 , a2 , . . . , an ) =
n−2
(1)
That is, the fair value is defined as the arithmetic mean of the opinions without the
1
two extreme opinions. We will call

 n
∑i=1 ai − mini ai − maxi ai


 
n−2

 n
Fn (a)
 ∑ bi − mini bi − maxi bi 

Fn (b) =  i=1


n
−
2


Fn (c)
 ∑ni=1 ci − mini ci − maxi ci 
n−2
as an Olympic-ideal solution the group assignment problem. To find a minimaldistance solution from the Olympic-ideal group assignment problem we have to
solve the following nonlinear optimization problem
(x − Fn (a))2 + (y − Fn (b))2 + (z − Fn (c))2 → min; subject to x, y, z ∈ X
where X is a set of budget constraints.
Example 1. Suppose that a certain amount of money exists and it has to be assigned to 6 different projects (alternatives) according to their importance. Moreover, let us also assume that 6 experts have been asked to evaluate these projects
and they express their preferences under the form of normalized priority vectors
[5], i.e. their components (portions) sum up to one. Their preferences can be
summarized in 6 weighting vectors in the following form
(i)
(i)
(i)
w(i) = (w1 , w2 , . . . , w6 )
(i)
where w j denotes how much portion of the money should be assigned to the j-th
projects according to the i-th expert. The 6 vectors of portions are actually given
by
w(1) = (0.2, 0.3, 0.1, 0.05, 0.15, 0.2)
w(2) = (0.1, 0.1, 0.15, 0.2, 0.25, 0.2)
w(3) = (0.4, 0.1, 0.05, 0.2, 0.3, 0.05)
w(4) = (0.15, 0.1, 0.05, 0.2, 0.25, 0.25)
w(5) = (0.2, 0.1, 0.2, 0.1, 0.3, 0.1)
w(6) = (0.05, 0.05, 0.1, 0.4, 0.25, 0.15)
The olympic-ideal solution is computed by (1) as,


0.1625
 0.1 


 0.1 


 0.175  .


 0.2625 
0.1625
2
Let x j denote the portion of money to be assigned to project j. Then we have
the natural constraint x1 + · · · + x6 = 1. Let us further assume that there exists a
constraint for each of the portions,
x j ≥ 1/8,
for j = 1, . . . , 6. To find a minimal-distance solution from the Olympic-ideal group
assignment problem we have to solve the following nonlinear optimization problem
(x1 − 0.1625)2 + (x2 − 0.1)2 + (x3 − 0.1)2 + (x4 − 0.175)2 +
(x5 − 0.2625)2 + (x6 − 0.1625)2 → min
6
subject to x j ≥ 1/8, j = 1, . . . , 6 ,
∑ xj = 1
j=1
The unique solution of this six-expert six-alternatives problem is
 ∗ 

x1
0.159375
x∗   0.125 
 2∗  

x   0.125 
3
 ∗ = 

x  0.171875
 4∗  

x  0.259375
5
x6∗
0.159375
and the optimal distance is 0.00128906.
2
Modeling Group Decisions Under Fuzzy Budget
Constraints
Let us consider the optimal settlement problem with quadratic opinion changing
aversion functions when budget constraints are flexible (fuzzy) and the group has
a target level for the maximal Euclidean distance from the Olympic-ideal solution,
denoted by q0 > 0.
Definition 1. A fuzzy set of the real line given by the membership function
A(t) =


1−

0
|a − t|
if |a − t| ≤ d,
d
otherwise,
where d > 0 will be called a symmetrical triangular fuzzy number with center
a ∈ R and tolerance level d and we shall refer to it by the pair (a, d).
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If the budget constraints are fuzzy and the group has a target level for the maximal Euclidean distance from the Olympic-ideal solution then we can state the following fuzzy mathematical programming (FMP) problem [8]: Find (x∗ , y∗ , z∗ ) ∈
R3 such that it satisfies the following inequalities as much as possible
τn (a, b, c) ≤ (q0 , d0 )
w11 x + w12 y + w13 z ≤ (q1 , d1 )
..
.
wm1 x + wm2 y + wm3 z ≤ (qm , dm )
(2)
where
τn (a, b, c) = (x − Fn (a))2 + (y − Fn (b))2 + (z − Fn (c))2
q0 is the target level for the maximal Euclidean distance from the Olympic-ideal
solution, (qi , di ) are fuzzy numbers of symmetric triangular form with center qi
and tolerance level di > 0, for i = 0, 1, . . . , m, and the inequalities are understood
in a possibilistic sense. That is, the degree of satisfaction of the i-th constraint by
a point (x, y, z) ∈ R3 , denoted µi (x, y, z), is defined by
µi (x, y, z) =

1



1−



if wi1 x + wi2 y + wi3 z ≤ qi ,
wi1 x + wi2 y + wi3 z − qi
di
if qi < wi1 x + wi2 y + wi3 z ≤ qi + di
if wi1 x + wi2 y + wi3 z > qi + di
0
for i = 1, . . . , m. The degree of satisfaction of the fuzzy goal (q0 , d0 ) by a point
(x, y, z) ∈ R3 is defined by
µ0 (x, y, z) =

1



1−



if τn (a, b, c) ≤ q0 ,
τn (a, b, c) − q0
.
d0
if q0 < τn (a, b, c) ≤ q0 + d0
if τn (a, b, c) > q0 + d0
0
If for a vector (x, y, z) ∈ R3 the value of wi1 x + wi2 y + wi3 z is less or equal than
qi then (x, y, z) satisfies the i-th budget constraint with the maximal conceivable
degree: one. If qi < wi1 x + wi2 y + wi3 z < qi + di then (x, y, z) is not feasible in
classical sense, but the group can still tolerate the violation of the crisp budget
constraint, and accept (x, y, z) as a solution with a positive degree, however, the
bigger the violation the less is the degree of acceptance. Finally, if wi1 x + wi2 y +
wi3 z > qi + di then the violation of the i-th constraint is untolerable by the group,
that is, µi (x, y, z) = 0.
Furthermore, if for a vector (x, y, z) ∈ R3 the value of Euclidean distance,
τn (a, b, c), is less or equal to q0 then (x, y, z) satisfies the target level for the Euclidean distance with the maximal conceivable degree: one. If q0 < τn (a, b, c) <
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q0 + d0 then (x, y, z) is not feasible in classical sense, but the group can still tolerate the exceeded target level, and accept (x, y, z) as a solution with a positive
degree, however, the bigger the overstepping the less is the degree of acceptance.
Finally, if τn (a, b, c) > qi + di then the exceed of the target level is untolerable by
the group, that is, µ0 (x, y, z) = 0.
Then the (fuzzy) solution of the FMP problem (2) is defined as a fuzzy set on
R3 whose membership function is given by
µ(x, y, z) = min{µ0 (x, y, z), µ1 (x, y, z), . . . , µm (x, y, z)},
In this setup µ(x, y, z) denotes the degree to which all inequalities are satisfied at
point (x, y, z) ∈ R3 . To maximize µ on R3 we have to solve the following crisp
quadratic programming problem
max λ
λ d0 + τn (a, b, c) ≤ q0 + d0 ,
λ d1 + w11 x + w12 y + w13 z ≤ q1 + d1 ,
···
λ dm + wm1 x + wm2 y + wm3 z ≤ qm + dm ,
0 ≤ λ ≤ 1, x, y, z ∈ R.
that is,
max λ
λ d0 + (x − Fn (a))2 + (y − Fn (b))2 + (z − Fn (c))2 ≤ q0 + d0 ,
λ d1 + w11 x + w12 y + w13 z ≤ q1 + d1 ,
···
λ dm + wm1 x + wm2 y + wm3 z ≤ qm + dm ,
0 ≤ λ ≤ 1, x, y, z ∈ R.
References
[1] Fedrizzi, M. (1993). Fuzzy approach to modeling consensus in group decisions. In: Proceedings of First Workshop on Fuzzy Set Theory and Real
Applications, Milano, May 10, 1993, Automazione e strumentazione, Supplement to November 1993 issue, 9–13.
[2] Fedrizzi, M., Kacprzyk, J., & Nurmi, H. (1993). Consensus degree under
fuzzy majorities and fuzzy preferences using OWA operators. Control and
Cybernetics, 22, 77–86.
[3] Fedrizzi, M. (1995). Fuzzy consensus models in GDSS. In: Proceedings
of the 2nd New Zealand Two-Stream International Conference on Artificial
Neural Networks and Expert Systems (pp. 284–287).
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[4] Mich, L., Fedrizzi, M., & Gaio, L. (1993). Approximate Reasoning in the
Modeling of Consensus in Group Decisions. In: E.P.Klement and W.Slany
(Eds.), Fuzzy Logic in Artificial Intelligence (pp. 91–102). Berlin: SpringerVerlag.
[5] Saaty, T.L. (1980). The Analythic Hierarchy Process, New York, McGrawHill.
[6] Yager, R.R. (1988). Ordered weighted averaging aggregation operators in
multi-criteria decision making. IEEE Trans. on Systems, Man and Cybernetics, 18, 183–190.
[7] Yager, R.R. (1993). Families of OWA operators. Fuzzy Sets and Systems, 59,
125–148.
[8] Zimmermann, H.-J. (1978). Fuzzy programming and linear programming
with several objective functions. Fuzzy Sets and Systems, 1, 45–55.
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