Probabilistic aggregation operators and their application in uncertain multi-person decisionmaking/Tikimybiniai sumavimo operatoriai ir ju taikymas priimant grupinius sprendimus neapibreztoje aplinkoje.
Merigo, Jose M. ; Wei, Guiwu
1. Introduction
Decision making problems are very common in our lives because
people are always making decisions. In the literature, we find a wide
range of methods and theories for dealing with the decision process
(Antucheviciene et al. 2010; Brauers and Zavadskas 2010; Kersuliene et
al. 2010; Liu 2009, 2011; Podvezko 2009; Xu 2010; Zavadskas et al. 2009,
2010a; Zavadskas and Turskis 2010; Zhang and Liu 2010). The use of
probabilities and the ordered weighted averaging (OWA) operator (Yager
1988) in the same aggregation process is a very useful method for
considering the probabilistic information and the attitudinal character
of the decision maker in the same formulation. Some studies have already
considered this problem by referring to it as the immediate probability
(Engemann et al. 1996; Merigo 2010; Yager et al. 1995). The main
advantage of this approach is the possibility of underestimate or
overestimate the probabilistic information according to the degree of
orness (or optimism) given in the OWA operator. Thus, we are able to
obtain a parameterized family of aggregation operators (Beliakov et al.
2007) between the maximum and the minimum. For further reading on the
OWA operator, see for example (Chang and Wen 2010; Merigo and Casanovas
2010c; Merigo and Gil-Lafuente 2010, 2011; Wang et al. 2009; Yager 1998;
Yager and Kacprzyk 1997; Zhou and Chen 2010). Note also that there exist
in the literature other approaches that use probabilistic information
and OWA operators in the same formulation including some decision making
methods with Dempster-Shafer belief structure (Merigo and Casanovas
2009; Merigo et al. 2010; Yager 1992).
In this paper, it is worth noting the work by Xu and Da (2002)
regarding the uncertain OWA (UOWA) operator. Basically, it is an
aggregation operator that deals with uncertain information represented
in the form of interval numbers. Since its introduction, several authors
have developed further improvements. For example, Merigo and Casanovas
(2011a) generalized it by using generalized and quasi-arithmetic means
and developed several extensions with fuzzy and linguistic information
(Merigo and Casanovas 2010a, 2010b). Wei (2009) developed a model with
uncertain linguistic information and with intuitionistic fuzzy sets (Wei
2010a, 2010b; Wei et al. 2010).
The concept of immediate probability has some limitations. One of
the most significant problems, as stated by Merigo (2009), is that it is
not able to unify the probability and the OWA operator considering that
sometimes one of them can be more relevant in the aggregation.
Therefore, it is necessary to use another approach that it is able to
unify both concepts but taking into account that they can be more or
less relevant depending on the problem considered. For doing so, Merigo
(2009) has suggested the probabilistic OWA (POWA) operator. It is a new
aggregation operator that unifies the probability and the OWA operator
giving different degrees of importance to each concept according to
their relevance in the specific problem considered.
The POWA operator is very useful to unify the probability with the
OWA operator when using exact numbers in the aggregation process.
However, many situations of the real world cannot be assessed with exact
numbers because the information is uncertain and very complex.
Therefore, it is necessary to use another approach that it is able to
assess this situation such as the use of interval numbers. The interval
numbers (Moore 1966) are a very useful technique for representing the
uncertainty by considering the best and worst possible results that
could happen in the environment and the most possible ones.
The aim of this paper is to present the uncertain probabilistic OWA
(UPOWA) operator. It is an aggregation operator that uses uncertain
information in the aggregation process by using interval numbers in the
POWA operator. Therefore, we are able to assess the POWA operator
considering the best and worst results that could happen in the
aggregation process and some of the most possible ones. The main
advantage of the UPOWA operator is that it provides more complete
information to the decision maker by using interval numbers that
includes a wide range of results and by using probabilities and OWA
operators in the same formulation considering the degree of importance
of each concept in the aggregation. Thus, we are able to consider
objective information (probabilistic) and the attitudinal character of
the decision maker in the same formulation. We study some of its main
properties and particular cases including the UOWA operator, the
uncertain average (UA), the uncertain probabilistic aggregation (UPA),
the uncertain probabilistic maximum and the uncertain probabilistic
minimum. Note that by using interval numbers we can represent all the
possible results that may occur in the uncertain environment. Thus we
can guarantee that at least we are considering all the possible
situations without losing information in the analysis. However, as we
are in uncertainty, we do not know which scenario will occur.
The other objective of this paper is to analyze the applicability
of this new approach and we see that it is very broad because all the
previous studies that use the probability or the OWA operator can be
revised and extended with this new approach. For example, we can apply
it in statistics, economics, engineering, physics and medicine. We focus
in a multi-person decision making problem. We find a more general
aggregation process that considers the opinion of several persons or
experts in the analysis. We call it the multi-person UPOWA (MP-UPOWA)
operator. We see that it also includes a wide range of particular cases
including the multi-person UPA (MP-UPA), the multi-person UA (MP-UA) and
the multi-person UOWA (MP-UOWA) operator. We implement the new approach
in a decision making problem regarding the selection of optimal monetary
policies.
This paper is organized as follows. In Section 2, we briefly review
some basic concepts regarding the interval numbers, the UOWA operator
and the POWA operator. In Section 3, we present the UPOWA operator and
Section 4 analyzes a wide range of particular cases. Section 5
introduces a multi-person decision making application and Section 6 an
illustrative example. Section 7 summarizes the main conclusions found in
the paper.
2. Preliminaries
In this Section, we briefly describe the interval numbers, the UOWA
operator and the POWA operator.
2.1. Interval Numbers
The interval numbers (Moore 1966) are a very useful and simple
technique for representing the uncertainty. They have been used in an
astonishingly wide range of applications.
The interval numbers can be expressed in different forms. For
example, if we assume a 4-tuple ([a.sub.1], [a.sub.2], [a.sub.3],
[a.sub.4]), that is, a quadruplet; we could consider that [a.sub.1] and
[a.sub.4] represents the minimum and the maximum of the interval number,
and [a.sub.2] and [a.sub.3], the interval with the highest probability
or possibility, depending on the use we want to give to the interval
numbers. Note that [a.sub.1] [less than or equal to] [a.sub.2] [less
than or equal to] [a.sub.3] [less than or equal to] [a.sub.4]. If
[a.sub.1] = [a.sub.2] = [a.sub.3] = [a.sub.4], then, the interval number
is an exact number; if [a.sub.2] = [a.sub.3], it is a 3-tuple known as
triplet; and if [a.sub.1] = [a.sub.2] and [a.sub.3] = [a.sub.4], it is a
simple 2-tuple interval number.
In the following, we are going to review some basic interval number
operations as follows. Let A and B be two triplets, where A =
([a.sub.1], [a.sub.2], [a.sub.3]) and B = ([b.sub.1], [b.sub.2],
[b.sub.3]). Then:
A + B = ([a.sub.1] + [b.sub.1], [a.sub.2] + [b.sub.2], [a.sub.3] +
[b.sub.3]);
A - B = ([a.sub.1] - [b.sub.3], [a.sub.2] - [b.sub.2], [a.sub.3] -
[b.sub.1]);
A x k = (k x[a.sub.1], k x[a.sub.2], k x[a.sub.3]); for k > 0;
A x B = ([a.sub.1] x[b.sub.1], [a.sub.2] x[b.sub.2], [a.sub.3]
x[b.sub.3]); for [R.sub.+];
A x B = (Min{[a.sub.1] x[b.sub.1], [a.sub.3] x[b.sub.1], [a.sub.1]
x[b.sub.3], [a.sub.3] x[b.sub.3]}, Max{[a.sub.1] x[b.sub.1], [a.sub.3]
x[b.sub.1], [a.sub.1] x[b.sub.3], [a.sub.3] x[b.sub.3]}); for R;
A / B = ([a.sub.1] / [b.sub.3], [a.sub.2] x [b.sub.2], [a.sub.3] /
[b.sub.1]); for [R.sup.+];
A / B = (Min{[a.sub.1] / [b.sub.1], [a.sub.3] / [b.sub.1],
[a.sub.1] / [b.sub.3], [a.sub.3] / [b.sub.3]}, Max{[a.sub.1] /
[b.sub.1], [a.sub.3] / [b.sub.1], [a.sub.1] / [b.sub.3], [a.sub.3] /
[b.sub.3]}); for R.
Note that other operations could be studied (Moore 1966) but in
this paper we will focus on these ones.
2.2. The Uncertain OWA Operator
The uncertain OWA (UOWA) operator was introduced by Xu and Da
(2002). It is an extension of the OWA operator (Yager 1988) for
uncertain situations where the available information can be assessed
with interval numbers. It can be defined as follows:
Definition 1. Let [OMEGA] be the set of interval numbers. An UOWA
operator of dimension n is a mapping UOWA: [[OMEGA].sup.n] [right arrow]
[OMEGA] that has an associated weighting vector W of dimension n such
that [w.sub.j] [member of] [0, 1] and [[summation].sup.n.sub.j=1]
[w.sub.j] = 1, then:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)
where [b.sub.j] is the jth largest of the [[??].sub.i] and
[[??].sub.i] is the argument variable represented in the form of
interval numbers.
Note also that different families of UOWA operators can be studied
by choosing a different weighting vector such as the step-UOWA operator,
the window-UOWA, the median-UOWA, the olympic-UOWA, the centered-UOWA
and the S-UOWA.
2.3. The Probabilistic OWA Operator
The probabilistic ordered weighted averaging (POWA) operator is an
aggregation operator that unifies the probability and the OWA operator
in the same formulation considering the degree of importance that each
concept has in the analysis (Merigo 2009). It is defined as follows.
Definition 2. A POWA operator of dimension n is a mapping POWA:
[R.sup.n] [right arrow] R that has an associated weighting vector W of
dimension n such that [w.sub.j] [member of] [0, 1] and
[[summation].sup.n.sub.j=1] [w.sub.j] = 1, according to the following
formula:
POWA ([a.sub.1], [a.sub.2],..., [a.sub.n]) = [n.summation over
(j=1)] [[??].sub.j] [b.sub.j], (2)
where [b.sub.j] is the jth largest of the [a.sub.i], each argument
ai has an associated weight (WA) [v.sub.i] with
[[summation].sup.n.sub.j=1] [v.sub.i] = 1 and [v.sub.i] [member of] [0,
1], [[??].sub.j] = [beta][w.sub.j] + (1 - [beta])[v.sub.j] with [beta]
[member of] [0, 1] and [v.sub.j] is the weight (WA) [v.sub.i] ordered
according to [b.sub.j], that is, according to the jth largest of the
[a.sub.i].
By choosing a different manifestation in the weighting vector, we
are able to obtain a wide range of particular types of POWA operators
(Merigo 2009). Especially, when [beta] = 0, we get the probabilistic
aggregation, and if [beta] = 1, we get the OWA operator.
3. The Uncertain Probabilistic OWA Operator
The uncertain probabilistic ordered weighted averaging (UPOWA)
operator is an extension of the OWA operator for situations where we
find probabilistic and uncertain information that can be assessed with
interval numbers. Its main advantage is that it can unify both concepts
considering the degree of importance that they have in the specific
problem considered. Thus, we are able to apply this formulation to all
the previous models that use probabilities or OWAs obtaining a more
complete approach that it is able to consider a wide range of scenarios
that includes the classical approaches. Specially, it is worth noting
that in decision making problems, this approach is able to include
decision making under risk and under uncertainty environments in the
same formulation. This approach seems to be complete, at least as an
initial real unification between OWA operators and probabilities.
However, it is worth noting that some previous models already
considered the possibility of using OWA operators and probabilities in
the same formulation. The main model is the concept of immediate
probability (Engemann et al. 1996; Merigo 2010; Yager et al. 1995; Yager
1999). Although it seems to be a good approach it is not so complete
than the UPOWA because it can unify OWAs and probabilities in the same
model but it can not take in consideration the degree of importance of
each case in the aggregation process. Before studying the UPOWA, we are
going briefly to consider the immediate probabilities with interval
numbers (IP-UOWA). For uncertain situations assessed with interval
numbers, the immediate probability can be defined as follows.
Definition 3. Let [OMEGA] be the set of interval numbers. An IPUOWA
operator of dimension n is a mapping IPUOWA: On [[OMEGA].sup.n] [right
arrow] [OMEGA] that has associated a weighting vector W of dimension n
such that [w.sub.j] [member of] [0, 1] and [[summation].sup.n.sub.j=1]
[w.sub.j] = 1, according to the following formula:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)
where [b.sub.j] is the jth largest of the [[??].sub.i], the
[[??].sub.i] are interval numbers and each one has associated a
probability [p.sub.i] with [[summation].sup.n.sub.j=1] and [p.sub.i]
[member of] [0, 1], [[??].sub.j] = ([w.sub.j] [p.sub.j] /
[[summation].sup.n.sub.j=1] [w.sub.j][p.sub.j]) and [p.sub.j] is the
probability [p.sub.i] ordered according to [b.sub.j], that is, according
to the jth largest of the [[??].sub.i].
Note that the IPUOWA operator is a good approach for unifying
probabilities and OWAs in some particular situations. But it is not
always useful, especially in situations where we want to give more
importance to the OWA operators or to the probabilities. One way to see
why this unification does not seem to be a final model is considering
other ways of representing [[??].sub.j]. For example, we could also use
[[??].sub.j] = [[w.sub.j] + [p.sub.j] / [[summation].sup.n.sub.j=1]
([w.sub.j] + [p.sub.j])] or other similar approaches.
Note that other approaches that could be taken into account are the
hybrid averaging (HA) (Xu and Da 2003; Zhao et al. 2009, 2010) and the
weighed OWA (WOWA) operator (Torra 1997; Torra and Narukawa 2007). These
models unify the OWA operator with the weighted average (WA). Therefore,
they can also be extended for situations with the OWA operator and
probabilities assuming that for some situations the WA can be seen as a
probability, for example, when we use the WA as a subjective
probability. As said before, these an other approaches are useful for
some particular situations but they does not seem to be so complete than
the UPOWA because they can unify OWAs with probabilities (or with WAs)
but they can not unify them giving different degrees of importance to
each case. Note that in future research we will also prove that these
models can be seen as a special case of a general UPOWA operator (or its
respective model with WAs) that uses quasi-arithmetic means. Obviously,
it is possible to develop more complex models of the IP-UOWA, the HA (or
uncertain HA) and the WOWA that takes into account the degree of
importance of the OWAs and the probabilities (or WAs) in the model but
they seem to be artificial and not a natural unification as it will be
shown below. In the following, we are going to analyze the UPOWA
operator. It can be defined as follows.
Definition 4. Let [OMEGA] be the set of interval numbers. An UPOWA
operator of dimension n is a mapping UPOWA: [[OMEGA].sup.n] [right
arrow] [OMEGA] that has associated a weighting vector W of dimension n
such that [w.sub.j] [member of] [0, 1] and [[summation].sup.n.sub.j=1]
[w.sub.j] = 1, according to the following formula:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)
where [b.sub.j] is the jth largest of the [[??].sub.i], the
[[??].sub.i] are interval numbers and each one has an associated
probability [p.sub.i] with [[summation].sup.n.sub.j=1] [p.sub.i] = 1 and
[p.sub.i] [member of] [0, 1], [[??].sub.j] = [beta][w.sub.j] + (1 -
[beta])[p.sub.j] with [beta] [member of] [0, 1] and [p.sub.j] is the
probability [p.sub.i] ordered according to [b.sub.j], that is, according
to the jth largest of the [[??].sub.i].
Note that it is also possible to formulate the UPOWA operator
separating the part that strictly affects the OWA operator and the part
that affects the probabilities. This representation is useful to see
both models in the same formulation but it does not seem to be as a
unique equation that unifies both models.
Definition 5. Let [OMEGA] be the set of interval numbers. An UPOWA
operator is a mapping UPOWA: [[OMEGA].sup.n] [right arrow] [OMEGA] of
dimension n, if it has associated a weighting vector W, with
V[[summation].sup.n.sub.j=1] [w.sub.j] = 1 and [w.sub.j] [member of] [0,
1] and a probabilistic vector V, with [p.sub.i] = 1 and [p.sub.i]
[member of] [0, 1], such that:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)
where [b.sub.j] is the jth largest of the arguments [[??].sub.i],
the [[??].sub.i] are interval numbers and [beta] [member of] [0, 1].
Note that if the weights of the probabilities and the OWAs are also
uncertain, then, we have to establish a method for dealing with these
uncertain weights. Note that in these situations it is very common that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, a very useful
method for dealing with these situations is by using:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)
In the following, we are going to give a simple example of how to
aggregate with the UPOWA operator. We consider the aggregation with both
definitions. For simplicity, we assume that the weights are exact
values.
Example 1. Assume the following arguments in an aggregation
process: ([20, 30], [40, 50], [50, 60], [30, 40]). Assume the following
weighting vector W = (0.2, 0.2, 0.2, 0.4) and the following
probabilistic weighting vector P = (0.4, 0.3, 0.2, 0.1). Note that the
probabilistic information has a degree of importance of 60% while the
weighting vector W a degree of 40%. If we want to aggregate this
information by using the UPOWA operator, we will get the following. The
aggregation can be solved either with the Eq. (4) or (5). With Eq. (4)
we calculate the new weighting vector as:
[[??].sub.1] = 0.4 x 0.2 + 0.6 x 0.2 = 0.2, [[??].sub.2] = 0.4 x
0.2 + 0.6 x 0.3 = 0.26,
[[??].sub.3] = 0.4 x 0.2 + 0.6 x 0.1 = 0.14, [[??].sub.4] = 0.4 x
0.4 + 0.6 x 0.4 = 0.4,
And then, we calculate the aggregation process as follows:
UPOWA = 0.2 x[50, 60] + 0.26 x[40, 50] + 0.14 x[30, 40] + 0.4 x[20,
30] = [32.6, 42.6]. With Eq. (5), we aggregate as follows:
UPOWA = 0.4 x(0.2 x[50, 60] + 0.2 x[40, 50] + 0.2 x[30, 40] + 0.4
x[20, 30]) + 0.6 x(0.4 x[20, 30] + 0.3 x[40, 50] + 0.2 x[50, 60] + 0.1
x[30, 40]) = [32.6, 42.6].
Obviously, we get the same results with both methods.
Note that different types of interval numbers could be used in the
aggregation such as 2-tuples, triplets, quadruplets, etc. When using
interval numbers in the UPOWA operator, we have the additional problem
of how to reorder the arguments because now we are using interval
numbers. Thus, in some cases, it is not clear which interval number is
higher, so we need to establish an additional criteria for reordering
the interval numbers. For simplicity, we recommend the following
criteria. First, we analyze if there is an order between the interval
numbers. That is, if all the values of the interval A = ([a.sub.1],
[a.sub.2], [a.sub.3]) are higher than the values in the interval C =
([c.sub.1], [c.sub.2], [c.sub.3]) such that [a.sub.1] > [c.sub.3]. If
not, we will calculate an average of the interval number. For example,
if n = 2, ([a.sub.1] + [a.sub.2]) / 2; if n = 3, ([a.sub.1] + 2[a.sub.2]
+ [a.sub.3]) / 4; and so on. In the case of tie, we will select the
interval with the lowest increment ([a.sub.2] - [a.sub.1]). For 3-tuples
and more we will select the interval with the highest central value.
From a generalized perspective of the reordering step, it is
possible to distinguish between the descending UPOWA (DUPOWA) and the
ascending UPOWA (AUPOWA) operator by using [w.sub.j] =
[w.sup.*.sub.n-j+1], where is the jth weight of the DUPOWA and
[w.sup.*.sub.n-j+1] the jth weight of the AUPOWA operator.
If B is a vector corresponding to the ordered arguments [b.sub.j],
we shall call this the ordered argument vector and [W.sup.T] is the
transpose of the weighting vector, then, the UPOWA operator can be
expressed as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)
A further interesting result consists in using infinitary
aggregation operators (Mesiar and Pap 2008). Thus, we can represent an
aggregation process where there are an unlimited number of arguments
that appear in the aggregation process. Note that
[[summation].sup.[infinity].sub.j=1] [[??].sub.j] = 1. By using, the
UPOWA operator we get the infinitary UPOWA ([infinity]-UPOWA) operator
as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)
The reordering process is very complex because we have an unlimited
number of arguments, so we never know which argument is the first one to
be aggregated. For further reading on the usual OWA, see Mesiar and Pap
(2008).
The UPOWA is monotonic, commutative, bounded and idempotent. It is
monotonic because if [[??].sub.i] [greater than or equal to]
[[??].sub.i] for all [[??].sub.i], then, [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII]. It is commutative because any permutation of the
arguments has the same evaluation. That is, [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII] is any permutation of the arguments
([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). It is bounded
because the UPOWA aggregation is delimitated by the minimum and the
maximum: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is
idempotent because if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII], for all [[??].sub.i], then, [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII].
4. Families of UPOWA Operators
Different types of UPOWA operators can be studied depending on the
considerations made in the analysis. First of all we are going to
consider the two main cases of the UPOWA operator that are found by
analyzing the coefficient [beta]. Basically, if [beta] = 0, then, we get
the probabilistic approach and if [beta] = 1, the UOWA operator. The
more of [beta] located to the top, the more we use the UOWA operator and
vice versa.
By using a different manifestation in the weighting vector W (or P)
we can analyze a wide range of particular cases. For example, we can
obtain the uncertain probabilistic maximum, the uncertain probabilistic
minimum, the uncertain arithmetic probabilistic aggregation (UAPA), the
uncertain probabilistic weighted average (UPWA) and the uncertain
arithmetic OWA (UAOWA) operator.
Remark 1. The uncertain probabilistic maximum is found when
[w.sub.1] = 1 and [w.sub.j] = 0 for all j [not equal to] 1. The
probabilistic minimum is formed when [w.sub.n] = 1 and [w.sub.j] = 0 for
all j [not equal to] n.
Remark 2. More generally, the step-UPOWA is formed when [w.sub.k] =
1 and [w.sub.j] = 0 for all j [not equal to] k. Note that if k = 1, the
step-UPOWA is transformed into the uncertain probabilistic maximum, and
if k = n, the step-UPOWA becomes the uncertain probabilistic minimum
operator.
Remark 3. The UAPA operator is obtained when [w.sub.j] = 1/n for
all j, and the uncertain probabilistic weighted average is obtained when
the ordered position of i is the same as the ordered position of j. The
UAOWA operator is obtained when [p.sub.i] = 1/n for all i.
Remark 4. For the median-UPOWA, if n is odd we assign [w.sub.(n +
1)/2] = 1 and [w.sub.j*] = 0 for all others. If n is even we assign for
example, [w.sub.n/2] = [w.sub.(n/2) + 1] = 0.5 and [w.sub.j*] = 0 for
all others.
Remark 5. The olympic-UPOWA is generated when [w.sub.1] = [w.sub.n]
= 0, and for all others [w.sub.j*] = 1/(n - 2). Note that it is possible
to develop a general form of the olympic-UPOWA by considering that
[w.sub.j] = 0 for j = 1, 2,..., k, n, n - 1,..., n - k + 1, and for all
others [w.sub.j*] = 1/(n - 2k), where k < n/2. Note that if k = 1,
then this general form becomes the usual olympic-UPOWA. If k = (n -
1)/2, then, this general form becomes the median-UPOWA aggregation. That
is, if n is odd, we assign [w.sub.(n + 1) / 2] = 1, and [w.sub.j*] = 0
for all other values. If n is even, we assign, for example, [w.sub.n/2]
= [w.sub.(n / 2) + 1] = 0.5 and [w.sub.j*] = 0 for all other values.
Remark 6. Note that it is also possible to develop the contrary
case of the general Olympic-UPOWA operator. In this case, [w.sub.j] =
(1/2k) for j = 1, 2,..., k, n, n - 1,..., n - k + 1, and [w.sub.j] = 0,
for all other values, where k < n/2. Note that if k = 1, then we
obtain the contrary case for the median-UPOWA.
Remark 7. Another interesting family is the S-UPOWA operator. It
can be subdivided into three classes: the "or-like," the
"and-like" and the generalized S-UPOWA operators. The
generalized S-UPOWA operator is obtained if [w.sub.1] = (1/n)(1 -
([alpha] + [beta])) + [alpha], [w.sub.n] = (1/n)(1 - ([alpha] + [beta]))
+ [beta], and [w.sub.j] = (1/n)(1 - ([alpha] + [beta])) for j = 2 to n -
1, where [alpha], [beta] [member of] [0, 1] and [alpha] + [beta] [less
than or equal to] 1. Note that if [alpha] = 0, the generalized S-UPOWA
operator becomes the "and-like" S-UPOWA operator, and if
[beta] = 0, it becomes the "or-like" S-UPOWA operator.
Remark 8. Another family of aggregation operator that could be used
is the centered-UPOWA operator. We can define an UPOWA operator as a
centered aggregation operator if it is symmetric, strongly decaying and
inclusive. Note that these properties have to be accomplished for the
weighting vector W of the UOWA operator but not necessarily for the
weighting vector P of the probabilities. It is symmetric if [w.sub.j] =
[w.sub.j+n-1]. It is strongly decaying when i < j [less than or equal
to] (n + 1)/2 then [w.sub.i] < [w.sub.j] and when i > j [greater
than or equal to] (n + 1)/2 then [w.sub.i] < [w.sub.j]. It is
inclusive if [w.sub.j] > 0. Note that it is possible to consider a
softening of the second condition by using [w.sub.i] [less than or equal
to] [w.sub.j] instead of [w.sub.i] < [w.sub.j], then, we get the
softly decaying centered-UPOWA operator. And if we remove the third
condition, we get the non-inclusive centered-UPOWA operator.
Remark 9. A further interesting type is the non-monotonic-UPOWA
operator. It is obtained when at least one of the weights [w.sub.j] is
lower than 0 and [[summation].sup.n.sub.j=1] [w.sub.j] = 1. Note that a
key aspect of this operator is that it does not always achieve
monotonicity.
Remark 10. Other families of UPOWA operators could be used
following the recent literature about different methods for obtaining
OWA weights (Merigo and Gil-Lafuente 2009; Yager 1993, 2009).
5. Multi-Person Decision-Making with the UPOWA Operator
The UPOWA operator can be applied in a wide range of fields because
all the previous studies that use the probability or the OWA operator
can be revised and extended with this new approach. For example, we
could develop a wide range of applications in statistics, economics,
engineering and decision theory. In this paper, we focus on a
decision-making application in the selection of national strategies by
the government, such as the selection of monetary policies, using a
multi-person analysis. A multi-person analysis provides a more complete
representation of the problem because it is based on the opinion of
several people. Therefore, we can aggregate the opinion of different
people to obtain a representative view of the problem. In politics and
national decision-making, this is very useful because usually decisions
are not individual, but are made by a group of people in the parliament
or in the ministries council.
The procedure to select monetary policies with the UPOWA operator
in multi-person decision-making is described in this section. Many other
group decision-making models have been discussed in the literature
(Merigo and Casanovas 2011b; Wei et al. 2010; Xu 2010).
Step 1: Let A = {[A.sub.1], [A.sub.2],..., [A.sub.m]} be a set of
finite alternatives, S = {[S.sub.1], [S.sub.2],..., [S.sub.n]}, a set of
finite states of nature (or attributes), forming the payoff matrix
[([[??].sub.hi]).sub.mxn]. Let E = {[e.sub.1], [e.sub.2], [e.sub.q]} be
a finite set of decision-makers. Let U = ([u.sub.1], [u.sub.2],...,
[u.sub.p]) be the weighting vector of the decision-makers such that
[[summation].sup.q.sub.k=1] [u.sub.k] = 1 and [u.sub.k] [member of] [0,
1]. Each decision-maker provides his own payoff matrix
[([[??].sub.hi.sup.(k)]).sub.mxn].
Step 2: Calculate the weighting vector [??] = [beta] x W + (1 -
[beta]) x P to be used in the UPOWA aggregation. Note that W =
([w.sub.1], [w.sub.2],..., [w.sub.n]) such that
[[summation].sup.n.sub.j=1] [w.sub.j] = 1 and [w.sub.j] [member of] [0,
1] and P = ([p.sub.1], [p.sub.2],..., [p.sub.p]) such that
[[summation].sup.n.sub.i=1] [p.sub.i] = 1 and [p.sub.i] [member of] [0,
1].
Step 3: Use the WA to aggregate the information of the
decision-makers E using the weighting vector U. The result is the
collective payoff matrix [([[??].sub.hi]).sub.mxn]. Thus, [[??].sub.hi]
= [[summation].sup.p.sub.k=1] [u.sub.k] [[??].sup.k.sub.hi]. Note that
it is possible to use other types of aggregation operators instead of
the WA to aggregate this information.
Step 4: Calculate the aggregated results using the UPOWA operator
explained in Eq. (5). Consider different families of UPOWA operators as
described in Section 4.
Step 5: Adopt decisions according to the results found in the
previous steps. Select the alternative (s) that provides the best result
(s). Moreover, establish an ordering or a ranking of the alternatives
from the most- to the least-preferred alternative, enabling
consideration of more than one selection.
This aggregation process can be summarized using the following
aggregation operator that we call the multi-person--UPOWA (MP-UPOWA)
operator.
Definition 6. A MP-UPOWA operator is a mapping MP-UPOWA:
[[OMEGA].sup.n] x [[OMEGA].sup.p] [right arrow] [OMEGA] that has a
weighting vector U of dimension p with [[summation].sup.p.sub.k=1]
[u.sub.p] = 1 and [u.sub.k] [member of] [0, 1] and a weighting vector W
of dimension n with [[summation].sup.n.sub.j=1] [w.sub.j] = 1 and
[w.sub.j] [member of] [0, 1], such that:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)
where [b.sub.j] is the jth largest of the [[??].sub.i], each
argument [[??].sub.i] is an interval number and has an associated weight
(WA) [v.sub.i] with [[summation].sup.n.sub.i=1] [p.sub.i] = 1 and
[p.sub.i] [member of] [0, 1], [[??].sub.j] = [beta][w.sub.j] + (1 -
[beta])[p.sub.j] with [beta] [member of] [0, 1] and [p.sub.j] is the
probability [p.sub.i] ordered according to [b.sub.j], that is, according
to the jth largest of the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] is the argument variable provided by each person (or experts).
Note that the MP-UPOWA operator has similar properties than those
explained in Section 3, such as the distinction between descending and
ascending orders, the aggregation with uncertain weights, and so on.
The MP-UPOWA operator includes a wide range of particular cases
following the methodology explained in Section 4. Thus, it includes the
multi-person--UPA (MP-UPA) operator, the multi-person--UOWA (MP-UOWA)
operator, the multi-person--uncertain average (MP-UA) operator, the
multi-person--arithmetic-UPA (MP-AUPA) operator and the
multi-person--arithmetic-UOWA (MP-AUOWA) operator.
6. Illustrative Example
In the following, we present a numerical example of the new
approach in a multi-person decision-making problem regarding the
selection of national strategies. We analyze an economic problem
concerning the selection of the optimal monetary policy of a country.
Note that other decision-making applications could be developed in other
areas such as construction design (Turskis et al. 2009) and the
selection of project managers (Zavadskas et al. 2010b).
Step 1: Assume the government of a country has to decide on the
type of monetary policy to use the next year. They consider six
alternatives:
[A.sub.1] = Develop an extremely strong expansive monetary policy.
[A.sub.2] = Develop a strong expansive monetary policy.
[A.sub.3] = Develop an expansive monetary policy.
[A.sub.4] = Do not develop any change in the monetary policy.
[A.sub.5] = Develop a contractive monetary policy.
[A.sub.6] = Develop a strong contractive monetary policy.
In order to evaluate these strategies, the government has brought
together a group of experts. This group considers that the key factor is
the economic situation of the world economy for the next period. They
consider 7 possible states of nature that could happen in the future:
[S.sub.1] = Very bad economic situation.
[S.sub.2] = Bad economic situation.
[S.sub.3] = Regular--Bad economic situation.
[S.sub.4] = Regular economic situation.
[S.sub.5] = Regular--Good economic situation.
[S.sub.6] = Good economic situation.
[S.sub.7] = Very good economic situation.
The experts are classified in 3 groups. Each group is led by one
expert and gives different opinions than the other two groups. The
results of the available strategies, depending on the state of nature
[S.sub.i] and the alternative [A.sub.k] that the government chooses, are
shown in Tables 1, 2 and 3.
Step 2-3: In this problem, we assume the following weighting vector
for the three group of experts: U = (0.3, 0.3, 0.4). The experts assume
the following weighting vector for the OWA: W = (0.1, 0.1, 0.1, 0.1,
0.2, 0.2, 0.2); for the probability: P = (0.1, 0.1, 0.2, 0.2, 0.2, 0.1,
0.1); and [beta] = 40%. Thus, with these OWA weights we see that they
are assuming a pessimistic attitude because they give more importance to
the worst results. On the other hand, we see with the probabilities that
they believe that the economic situation for the next year will be
moderate as they give more importance to the central part of the
probabilistic weights. First, we aggregate the information of the three
groups into one collective matrix that represents all the experts of the
problem. The results are shown in Table 4.
Step 4: With this information, we can aggregate the expected
results for each state of nature in order to make a decision by using
Eq. (5). In Table 5, we present the results obtained using different
types of UPOWA operators. Note that we can also obtain these results by
using Eq. (4). Obviously, we get the same results with both methods.
Step 5: If we establish an ordering of the alternatives, then we
get the results shown in Table 6. Note that the first alternative in
each ordering is the optimal choice.
Evidently, the order preference for the monetary policy strategies
may be different, depending on the aggregation operator used. Therefore,
the decision about which strategy to select may be also different.
However, in this example it seems that A5 should be the optimal choice
excepting for some extreme pessimistic situations where [A.sub.1] would
be the optimal one.
7. Conclusions
We have presented the UPOWA operator. It is an aggregation operator
that unifies the OWA operator and the probability in the same
formulation and in an uncertain environment that can be assessed with
interval numbers. The main advantage of this new model is that it is
able to unify the probability and the OWA operator giving different
degrees of importance to them according to the relevance they have in
the specific problem considered. Moreover, by using interval numbers, we
are able to provide more complete information to the decision maker
because we represent the environment considering the best and worst
result that could occur under uncertainty. We have compared this
approach with the concept of immediate probability and we have seen how
the UPOWA operator is able to overcome the main limitations of the
immediate probability by considering the degree of importance that the
probability and the OWA operator has in the aggregation. We have also
studied some of its main properties and particular cases including the
uncertain probabilistic minimum, the uncertain probabilistic maximum,
the UOWA, the UPA, the UAOWA and the UAPA operator.
We have also studied the applicability of this new approach and we
have seen that it is very broad because all the studies that use the
probability or the OWA operator can be revised and extended with this
new approach. The reason is that we can always reduce this new approach
to the classical cases where we only use the probability or the OWA
operator. We have seen that it is possible to apply it in statistics,
economics and engineering. We have focussed on an application in a
multi-person decision making problem. Thus, we have obtained the MPUPOWA
operator that permits to consider the opinion of several experts in the
analysis. It also includes a wide range of particular cases including
the MP-UPA and the MP-UOWA operator. We have developed an example in a
national decision-making problem concerning policy management. We have
analyzed the selection of monetary policies in a country.
In future research, we expect to develop further extensions of the
UPOWA operator by using other techniques for representing the
uncertainty (fuzzy numbers, linguistic variables, etc.) and other
variables such as order inducing variables, generalized means, distance
measures and more complex structures. We will also consider other
applications giving special attention to statistics and decision theory
such as the development of a new variance and covariance measure with
the UPOWA operator and the development of a new linear regression model.
doi: 10.3846/20294913.2011.584961
Acknowledgements
We would like to thank the anonymous reviewers for valuable
comments and suggestions that have improved the quality of the paper.
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Jose M. Merigo (1), Guiwu Wei (2)
(1) Department of Business Administration, University of Barcelona,
Av. Diagonal 690, 08034 Barcelona, Spain
(2) Department of Economics and Management, Chongqing University of
Arts and Sciences, Chongqing 402160, P.R. China
E-mails: (1) jmerigo@ub.edu (corresponding author); (2)
WeiguiWu@163.com
Received 29 November 2010; accepted 29 March 2011
Jose M. MERIGO has a MSc and a PhD degree in Business
Administration from University of Barcelona, Spain. His PhD received the
Extraordinary Award from the University of Barcelona. He also holds a
Bachelor Degree in Economics from Lund University, Sweden. He is an
Assistant Professor in the Department of Business Administration at the
University of Barcelona. He has published more than 100 papers in
journals, books and conference proceedings including journals such as
Information Sciences, International Journal of Intelligent Systems,
International Journal of Uncertainty, Fuzziness and Knowledge-Based
Systems, Cybernetics & Systems, Computers & Industrial
Engineering and International Journal of Fuzzy Systems. He has published
4 books including one edited with World Scientific "Computational
Intelligence in Business and Economics". He is on the editorial
board of several journals including the Journal of Advanced Research on
Fuzzy and Uncertain Systems and the ISTP Transactions of Systems &
Cybernetics. He has participated in several scientific committees and
serves as a reviewer in a wide range of journals including IEEE
Transaction on Fuzzy Systems, Information Sciences and European Journal
of Operational Research. He is currently interested in Aggregation
Operators, Decision Making and Uncertainty.
Guiwu WEI has a MSc in applied mathematics from SouthWest Petroleum
University and a PhD degree in Business Administration from School of
Economics and Management at SouthWest Jiaotong University, China. He is
an Associate Professor in the Department of Economics and Management at
Chongqing University of Arts and Sciences. He has published more than 90
papers in journals, books and conference proceedings including journals
such as Expert Systems with Applications, Applied Soft Computing,
Knowledge and Information Systems, Knowledge-based Systems,
International Journal of Uncertainty, Fuzziness and Knowledge-Based
Systems and International Journal of Computational Intelligence Systems.
He has published 1 book. He has participated in several scientific
committees and serves as a reviewer in a wide range of journals
including Computers & Industrial Engineering, International Journal
of Information Technology and Decision Making, Information Sciences and
European Journal of Operational Research. He is currently interested in
Aggregation Operators, Decision Making and Computing with Words.
Table 1. Opinion of the first group of experts
[S.sup.1] [S.sup.2] [S.sup.3] [S.sup.4]
[A.sup.1] (40,50,60) (30,40,50) (60,70,80) (70,80,90)
[A.sup.2] (60,70,80) (70,80,90) (50,60,70) (40,50,60)
[A.sup.3] (10,20,30) (50,60,70) (70,80,90) (80,90,100)
[A.sup.4] (80,90,100) (60,70,80) (40,50,60) (60,70,80)
[A.sup.5] (40,50,60) (10,20,30) (60,70,80) (20,30,40)
[A.sup.6] (10,20,30) (20,30,40) (40,50,60) (70,80,90)
[S.sup.5] [S.sup.6] [S.sup.7]
[A.sup.1] (80,90,100) (60,70,80) (70,80,90)
[A.sup.2] (60,70,80) (50,60,70) (60,70,80)
[A.sup.3] (30,40,50) (70,80,90) (40,50,60)
[A.sup.4] (20,30,40) (40,50,60) (70,80,90)
[A.sup.5] (30,40,50) (60,70,80) (50,60,70)
[A.sup.6] (30,40,50) (60,70,80) (40,50,60)
Table 2. Opinion of the second group of experts
[S.sub.1] [S.sub.2] [S.sub.3] [S.sub.4]
[A.sub.1] (50,60,70) (30,40,50) (40,50,60) (60,70,80)
[A.sub.2] (70,80,90) (70,80,90) (30,40,50) (40,50,60)
[A.sub.3] (30,40,50) (80,90,100) (70,80,90) (60,70,80)
[A.sub.4] (60,70,80) (30,40,50) (50,60,70) (60,70,80)
[A.sub.5] (40,50,60) (40,50,60) (80,90,100) (60,70,80)
[A.sub.6] (20,30,40) (20,30,40) (30,40,50) (60,70,80)
[S.sub.5] [S.sub.6] [S.sub.7]
[A.sub.1] (70,80,90) (50,60,70) (60,70,80)
[A.sub.2] (60,70,80) (50,60,70) (40,50,60)
[A.sub.3] (30,40,50) (70,80,90) (50,60,70)
[A.sub.4] (20,30,40) (60,70,80) (40,50,60)
[A.sub.5] (40,50,60) (70,80,90) (50,60,70)
[A.sub.6] (50,60,70) (70,80,90) (40,50,60)
Table 3. Opinion of the third group of experts
[S.sub.1] [S.sub.2] [S.sub.3] [S.sub.4]
[A.sub.1] (60,70,80) (30,40,50) (40,50,60) (60,70,80)
[A.sub.2] (40,50,60) (70,80,90) (60,70,80) (50,60,70)
[A.sub.3] (30,40,50) (60,70,80) (70,80,90) (70,80,90)
[A.sub.4] (60,70,80) (20,30,40) (70,80,90) (60,70,80)
[A.sub.5] (40,50,60) (30,40,50) (70,80,90) (60,70,80)
[A.sub.6] (30,40,50) (20,30,40) (40,50,60) (40,50,60)
[S.sub.5] [S.sub.6] [S.sub.7]
[A.sub.1] (70,80,90) (80,90,100) (40,50,60)
[A.sub.2] (40,50,60) (70,80,90) (40,50,60)
[A.sub.3] (30,40,50) (70,80,90) (50,60,70)
[A.sub.4] (30,40,50) (60,70,80) (30,40,50)
[A.sub.5] (40,50,60) (30,40,50) (50,60,70)
[A.sub.6] (40,50,60) (70,80,90) (60,70,80)
Table 4. Collective results
[S.sub.1] [S.sub.2] [S.sub.3] [S.sub.4]
[A.sub.1] (51,61,71) (30,40,50) (46,56,66) (63,73,83)
[A.sub.2] (55,65,75) (70,80,90) (48,58,68) (44,54,64)
[A.sub.3] (24,34,44) (63,73,83) (70,80,90) (70,80,90)
[A.sub.4] (66,76,86) (35,45,55) (55,65,75) (60,70,80)
[A.sub.5] (40,50,60) (27,37,47) (70,80,90) (48,58,68)
[A.sub.6] (21,31,41) (20,30,40) (37,47,57) (55,65,75)
[S.sub.5] [S.sub.6] [S.sub.7]
[A.sub.1] (73,83,93) (65,75,85) (55,65,75)
[A.sub.2] (52,62,72) (58,68,78) (46,56,66)
[A.sub.3] (30,40,50) (70,80,90) (47,57,67)
[A.sub.4] (24,34,44) (54,64,74) (45,55,65)
[A.sub.5] (37,47,57) (51,61,71) (50,60,70)
[A.sub.6] (40,50,60) (67,77,87) (48,58,68)
Table 5. Aggregated results
Max-UPA Min-UPA UA
[A.sub.1] (63.1,73.1,83.1) (45.9,55.9,65.9) (54.7,64.7,74.7)
[A.sub.2] (59.0,69.0,79.0) (48.6,58.6,68.6) (53.2,63.2,73.2)
[A.sub.3] (60.6,70.6,80.6) (42.2,52.2,62.2) (53.4,63.4,73.4)
[A.sub.4] (55.0,65.0,75.0) (38.2,48.2,58.2) (48.4,58.4,68.4)
[A.sub.5] (56.6,66.6,76.6) (39.4,49.4,59.4) (46.1,56.1,66.1)
[A.sub.6] (52,62,72) (33.2,43.2,53.2) (41.1,51.1,61.1)
UPA UOWA UPOWA
[A.sub.1] (56.5,66.5,76.5) (51,61,71) (54.3,64.3,74.3)
[A.sub.2] (51.7,61.7,71.7) (51.1,61.1,71.1) (51.4,61.4,71.4)
[A.sub.3] (54.4,64.4,74.4) (47.5,57.5,67.5) (51.6,61.6,71.6)
[A.sub.4] (47.8,57.8,67.8) (44.3,54.3,64.3) (46.4,56.4,66.4)
[A.sub.5] (47.8,57.8,67.8) (42.7,52.7,62.7) (45.7,55.7,65.7)
[A.sub.6] (42,52,62) (36.6,46.6,56.6) (39.8,49.8,59.8)
Table 6. Ranking of the monetary policies
Ordering
Max-UPA [A.sub.4]} [A.sub.5]} [A.sub.3]} [A.sub.2]}
[A.sub.3]} [A.sub.6]
Min-UPA [A.sub.1]} [A.sub.5]} [A.sub.2]} [A.sub.4]}
[A.sub.3]} [A.sub.6]
UA [A.sub.5]} [A.sub.1]} [A.sub.2]} [A.sub.4]}
[A.sub.3]} [A.sub.6]
Ordering
UPA [A.sub.5]} [A.sub.1]} [A.sub.2]} [A.sub.4]}
[A.sub.3]} [A.sub.6]
UOWA [A.sub.5]} [A.sub.1]} [A.sub.2]} [A.sub.4]}
[A.sub.3]} [A.sub.6]
UPOWA [A.sub.5]} [A.sub.1]} [A.sub.2]} [A.sub.4]}
[A.sub.3]} [A.sub.6]
