Decision making with dempster-shafer belief structure and the Owawa operator.
Merigo, Jose M. ; Engemann, Kurt J. ; Palacios-Marques, Daniel 等
JEL Classification: C44, C49, D81, D89.
Introduction
The Dempster-Shafer (D-S) theory of evidence was introduced by
Dempster (1967) and by Shafer (1976). Since its introduction, this
theory has been studied and applied in a lot of situations (Srivastava,
Mock 2002; Yager, Liu 2008). It provides a unifying framework for
representing uncertainty because it includes as special cases the
situations of risk (probabilistic uncertainty) and ignorance
(imprecision). One of the key application areas of the D-S theory is in
decision making because it allows the use of risk and uncertain
environments in the same framework. This model can be carried out with a
lot of aggregation operators (Merigo, Casanovas 2009; Reformat, Yager
2008). Some authors (Engemann et al. 1996a; Merigo et al. 2010; Yager
1992) have considered the possibility of using the ordered weighted
averaging (OWA) operator.
The OWA operator (Yager 1988) is an aggregation operator that
provides a parameterized family of aggregation operators between the
maximum and the minimum. Since its introduction, it has been applied in
a wide range of situations (Yager, Kacprzyk 1997; Yager et al. 2011).
For example, Yager (2004) developed a generalization by using
generalized means and Fodor et al. (1995) by using quasi-arithmetic
means. Merigo and Gil-Lafuente (2009) extended the previous approaches
by using induced aggregation operators. Other extensions have considered
problems with imprecise information in the analysis by using interval
numbers (Merigo, Casanovas 2011a, b), fuzzy numbers (Liu 2011; Wei et
al. 2010; Zhao et al. 2010) and linguistic variables (Wei 2011). Other
developments have considered the use of distance measures in the
aggregation process (Merigo, Gil-Lafuente 2010; Zeng, Su 2011). Zhou and
Chen (2010, 2011, 2012) have considered the use of continuous,
logarithmic and power aggregation operators.
Recently, Merigo (2011) has introduced the ordered weighted
averaging--weighted average (OWAWA) operator. It is an aggregation
operator that unifies the weighted average (WA) and the OWA operator in
the same formulation considering the degree of importance that each
concept has in the aggregation. Thus, we can provide a parameterized
family of aggregation operators between the minimum and the maximum that
also considers the importance of the subjective information given by the
weighted average. Note that this model has been further extended by
adding more concepts in the formulation including the use of order
inducing variables (Merigo 2011), probabilities (Merigo, Wei 2011) and
generalized aggregation operators (Merigo et al. 2010).
The aim of this paper is to present a new decision making model
with D-S theory by using the OWAWA operator. The main advantage of this
approach is that we are able to consider probabilistic information with
WAs and OWAs in the same formulation. Thus, we are able to consider a
decision making problem with objective and subjective information and
considering the attitudinal character of the decision maker. For doing
so, we present a new aggregation operator, the belief structure--OWAWA
(BS-OWAWA) operator. It is a new aggregation operator that aggregates
the belief structures with the OWAWA operator. We study some of its main
properties and particular cases.
We generalize this approach by using generalized aggregation
operators. We focus on the use of the quasi-arithmetic mean obtaining
the quasi-arithmetic BS-OWAWA (BS-Quasi-OWAWA) operator. It includes a
wide range of particular cases including the generalized BS-OWAWA
(BS-GOWAWA) operator because the quasi-arithmetic mean includes the
generalized mean in its formulation. It also includes a lot of other
aggregation operators including the quadratic BS-OWAWA (BS-OWAWQA)
operator and the geometric BS-OWAWA (BS-OWAWGA) operator.
We further extend D-S belief structure by using group decision
making techniques. Thus, we are able to obtain more complete information
of the problem because usually the opinion of several persons is better
than the opinion of one. We introduce the multi-person BS-OWAWA
(MP-BS-OWAWA) operator. Its main advantage is that it can aggregate the
information of several persons in the same formulation. We generalize
this approach by using quasi-arithmetic means obtaining the
quasi-arithmetic MP-BS-OWAWA (MP-BS-Quasi-OWAWA) operator.
We also develop an application of the new approach in a decision
making problem concerning the selection of policies. We study a problem
where a government is planning the fiscal policy for the next year. The
main advantage of using this approach is that we are able to consider a
wide range of scenarios and select the one closest with our interests.
Moreover, we can represent in a more complete way the information of the
problem because in a government we find different groups that give
different opinions regarding the available information. And in order to
properly assess it we need to use collective results that correctly
represent the different opinions.
This paper is organized as follows. In Section 1, we briefly review
some basic concepts regarding the D-S theory, the WA, the OWA and the
OWAWA operator. In Section 2 we present the new decision making
approach. Section 3 introduces the use of group decision making
techniques in D-S Framework. In Section 4 we develop an application of
the new approach in political decision making. The final section
summarizes the main conclusions of the paper.
1. Preliminaries
In this section, we briefly review some basic concepts to be used
throughout the paper. We analyse the Dempster-Shafer belief structure,
the weighted average (WA), the OWA operator and the OWAWA operator.
1.1. Dempster-Shafer belief structure
The D-S theory (Dempster 1967; Shafer 1976) provides a unifying
framework for representing uncertainty as it can include the situations
of risk and ignorance as special cases. Note that the case of certainty
is also included as it can be seen as a particular case of risk and
ignorance. It can be defined as follows.
Definition 1. A D-S belief structure defined on a space X consists
of a collection of n nonnull subsets of X, [B.sub.j] for j = 1, ..., n,
called focal elements and a mapping m, called the basic probability
assignment, defined as, m: [2.sup.x] [right arrow] [0, 1] such that:
1) m([B.sub.j]) [member of] [0, 1];
2) [n.summation over (j=1)] m([B.sub.j]) = 1;
3) m(A) = 0, [for all]A [not equal to] B.
As said before, the cases of risk and ignorance are included as
special cases of belief structure in the D-S framework. For the case of
risk, a belief structure is called Bayesian belief structure if it
consists of n focal elements such that B = {x}, where each focal element
is a singleton. Then, we can see that we are in a situation of decision
making under risk environment as m([B.sub.j]) = [P.sub.j] = Prob
{[x.sub.j]}.
The case of ignorance is found when the belief structure consists
in only one focal element B, where m(B) essentially is the decision
making under ignorance environment as this focal element comprises all
the states of nature. Thus, m(B) = 1. Other special cases of belief
structures such as the consonant belief structure or the simple support
function are studied by Shafer (1976). Note that two important
evidential functions associated with these belief structures are the
measures of plausibility and belief.
1.2. The OWA operator and the weighted average
The OWA operator (Yager 1988) is an aggregation operator that
provides a parameterized family of aggregation operators between the
minimum and the maximum. In decision making it is very useful for
representing the degree of optimism/pessimism of the decision maker. It
can be defined as follows:
Definition 2. An OWA operator of dimension n is a mapping OWA:
[R.sup.n] [right arrow] R that has an associated weighting vector W of
dimension n with [w.sub.j] [member of] [0, 1] and [n.summation over
(j=1)] = 1, such that:
OWA([a.sub.1], ..., [a.sub.n]) = [n.summation over (i=1)]
[w.sub.j][b.sub.j], (1)
where [b.sub.j] is the jth largest of the [a.sub.j].
Note that different properties could be studied such as the
distinction between descending and ascending orders, different measures
for characterizing the weighting vector and different families of OWA
operators (Yager, Kacprzyk 1997; Yager et al. 2011; Zhou et al. 2012b).
The weighted average (WA) is one of the most common aggregation
operators found in the literature. It has been used in a wide range of
applications (Xu 2010). It can be defined as follows:
Definition 3. A WA operator of dimension n is a mapping WA:
[R.sup.n] [right arrow] R that has an associated weighting vector V,
with [v.sub.j] [member of] [0, 1] and [n.summation over (i=1)] = 1, such
that:
WA ([a.sub.1], ..., [a.sub.n]) = .., (2)
where [a.sub.1] represents the argument variable.
The WA operator accomplishes the usual properties of the
aggregation operators. For further reading on different extensions and
generalizations of the WA, see for example (Beliakov et al. 2007; Han,
Liu 2011; Merigo 2012; Podvezko 2011; Zhang, Liu 2010; Zhou et al.
2011).
1.3. The OWAWA operator
The ordered weighted averaging--weighted average (OWAWA) operator
(Merigo 2011) is a new model that unifies the OWA operator and the
weighted average in the same formulation. Therefore, both concepts can
be seen as a particular case of a more general framework that considers
the degree of importance that each concept has in the aggregation. It
can be defined as follows:
Definition 4. An OWAWA operator of dimension n is a mapping OWAWA:
[R.sup.n] [rigt arrow] R that has an associated weighting vector W of
dimension n such that [w.sub.j] [member of] [0, 1] and [n.summation over
(j=1)] [w.sub.j] = 1, according to the following equation: j=1
OWAWA([a.sub.1], ..., [a.sub.n]) = [n.summation over (j=1)]
[[??].sub.j][b.sub.j], (3)
where: b. is the jth largest of the a., each argument a. has an
associated weight (WA) v. with [n.summation over (i=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].
As we can see, if [beta] = 1, we get the OWA operator and if p = 0,
the WA. The OWAWA operator accomplishes similar properties than the
usual aggregation operators including monotonicity, idempotency and the
boundary condition. Note that we can distinguish between descending and
ascending orders, extend it by using mixture operators, and so on
(Merigo 2011). Note also that some previous models already considered
the possibility of using the WA and the OWA in the same formulation
(Torra 1997; Xu, Da 2003; Yager 1998) although they did not consider the
degree of importance of each concept in the analysis. Some other methods
considered the use of the OWA operator with the probability (Engemann et
al. 1996b, 2004; Yager et al. 1995).
Note that in order to measure the degree of optimism or pessimism
of the aggregation we can use the degree of orness measure suggested by
Yager (1988). Note that if we only use it in the OWA operator, we assume
that we only use the degree of optimism in the OWA part, while in the
weighted average we assume a neutral position based on other aspects. In
this case we use:
[alpha](W) = [beta][n.summation over (j=1)][w.sub.j](n - j/n - 1).
(4)
However, it is also possible to formulate the degree of orness of
the OWAWA operator as a general measure that analyzes the general
tendency of the aggregation to the maximum or to the minimum. In this
case we get:
[alpha]([??]) = [beta][n.summation over (j=1)] [w.sub.j](n - j/n -
1) + (1 - [beta]) [n.summation over (j=1)][v.sub.j](n - j/n - 1) (5)
The OWAWA operator can be applied in a wide range of fields because
all the previous studies that use the weighted average or the OWA
operator can be revised and extended with this new approach. The reason
is that we can always reduce this model to the classical cases but
usually it will add an additional interpretation of the aggregation
problem.
2. Decision making with Dempster-Shafer theory using the OWAWA
operator
In this section we present the new decision making approach by
using D-S theory and the OWAWA operator. Moreover, we also analyze the
aggregation process formed and study some of its main properties.
2.1. Decision making approach
A new method for decision making with D-S theory is possible by
using the OWAWA operator. The main advantage of this approach is that we
can use probabilities, WAs and OWAs in the same formulation. Thus, we
are able to represent the decision problem in a more complete way
because we can use subjective and objective information in the analysis
and the attitudinal character (degree of optimism) of the decision
maker. The decision process can be summarized as follows.
Assume we have a decision problem in which we have a collection of
alternatives {[A.sub.1], ..., [A.sub.q]} with states of nature
{[S.sub.1], ..., [S.sub.n]}. [a.sub.ih] is the payoff if the decision
maker selects alternative [A.sub.i] and the state of nature is
[S.sub.h]. The knowledge of the state of nature is captured in terms of
a belief structure m with focal elements [B.sub.1], ..., [B.sub.r] and
associated with each of these focal elements is a weight m([B.sub.k]).
The objective of the problem is to select the alternative which gives
the best result to the decision maker. In order to do so, we should
follow the following steps:
Step 1: Calculate the results of the payoff matrix.
Step 2: Calculate the belief function m about the states of nature.
Step 3: Calculate the attitudinal character (or degree of orness)
of the decision maker a(W) (Yager 1988).
Step 4: Calculate the collection of weights, w, to be used in the
OWAWA aggregation for each different cardinality of focal elements. Note
that it is possible to use different methods depending on the interests
of the decision maker (Merigo 2011; Xu 2005; Yager 1993). Note also that
for the WA aggregation we have to calculate the weights according to a
degree of importance (or subjective probability) of each state of
nature. This can be carried out by using the opinion of a group of
experts that has some information about the possibility that each state
of nature will occur.
Step 5: Determine the results of the collection, [M.sub.ik], if we
select alternative A. and the focal element B, occurs, for all the
values of i and k. Hence [M.sub.ik] = {[a.sub.ih] | [S.sub.h] [member
of] [B.sub.k]}.
Step 6: Calculate the aggregated results, [V.sub.ik] =
OWAWA([M.sub.ik]), using Eq. (3), for all the values of and k.
Step 7: For each alternative, calculate the generalized expected
value, [C.sub.i], where:
[C.sub.i] = [r.summation over (k=1)] [V.sub.ik] m([B.sub.k]). (6)
Step 8: Select the alternative with the largest [C.sub.i] as the
optimal. Note that in a minimization problem, the optimal choice is the
lowest result.
From a generalized perspective of the reordering step, it is
possible to distinguish between descending and ascending orders in the
OWAWA aggregation. This is useful for example when distinguishing
between minimization and maximization problems.
2.2. The BS-OWAWA operator
Analyzing the aggregation in Steps 6 and 7 of the previous
subsection, it is possible to formulate in one equation the whole
aggregation process. We call this process the belief structure OWAWA
(BS-OWAWA) aggregation. It can be defined as follows:
Definition 5. A BS-OWAWA operator is defined by:
BS - OWAWA = [r.summation over (k=1)][q.summation over
(j=1)]m([B.sub.k])[[??].sub.jk][b.sub.jk], (7)
where: [[??].sub.jk] is the weighting vector of the kth focal
element such that [q.summation over (j=1)][[??].sub.jk] = 1 and
[[??].sub.jk] [member of] [0, 1], [b.sub.jk] is the jkth largest of the
[a.sub.ik], each argument [a.sub.ik] has an associated weight (WA)
[v.sub.ik] with [q.summation over (i=1)] [v.sub.ik] = 1 and [v.sub.ik]
[member of] [0, 1], and a weight (OWA) [w.sub.jk] with [q.summation
over (j=1)][w.sub.jk] and [w.sub.jk] [member of] [0, 1], [[??].sub.jk] =
[beta][w.sub.jk] + (1 - [beta])[v.sub.jk] 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.ik], and
m([B.sub.k]) is the basic probability assignment.
Note that [q.sub.k] refers to the cardinality of each focal element
and r is the total number of focal elements. The BS-OWAWA operator is
monotonic, bounded and idempotent. By choosing a different manifestation
in the weighting vector of the OWAWA operator, we are able to develop
different families of BS-OWAWA operators (Merigo 2011; Merigo, Casanovas
2009). As it can be seen in Definition 5, each focal element uses a
different weighting vector in the aggregation step with the OWAWA
operator. Therefore, for each focal element, we can use a different type
of OWAWA operator. For example, if [beta] = 1, we get the BS-OWA
operator and if [beta] = 0, the BS-WA operator.
Remark 1. Some other cases could be used following the OWA
literature (Merigo 2011; Yager 1993). For example:
--The maximum-WA if [w.sub.1] = 1 and [w.sub.j] = 0, for all j [not
equal to] 1;
--The minimum-WA if [w.sub.n] = 1 and [w.sub.j] = 0, for all j [not
equal to] n;
--The average when [w.sub.j] = 1/n and [v.sub.i] = 1/n, for all
[a.sub.i];
--The step-OWAWA operator when [w.sub.k] = 1 and [w.sub.j] = 0, for
all j [not equal to] k;
--The arithmetic-WA when [w.sub.j] = 1/n for all j;
--The arithmetic-OWA (A-OWA) when [v.sub.i] = 1/n, for all i;
--The olympic-OWAWA 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-OWAWA by considering that w, = 0 for j = 1, 2, ..., k, n, n - 1,
..., n - k + 1, and for all others w,, = 1/ (n - 2k), where k < n/2;
--The centered-OWAWA when it is symmetric, strongly decaying and
inclusive. It is symmetric if [[??].sub.j], = [[??].sub.j+n-1]. It is
strongly decaying when i < j [less than or equal to] (n + 1)/2 then
[[??].sub.i] < [[??].sub.j] and when i > j [greater than or equal
to] (n + 1)/2 then [[??].sub.i] < v j. It is inclusive if
[[??].sub.j] > 0,
--And many others (Merigo 2011; Xu 2005; Yager 1993).
2.3. Generalized aggregation operators in D-S framework
A further generalization of the previous model can be developed by
using generalized aggregation operators in the analysis by using
generalized means (Merigo, Gil-Lafuente 2009; Yager 2004; Zhou et al.
2012a) and quasi-arithmetic means (Fodor et al. 1995; Merigo, Casanovas
2011a). The main advantage of using these generalizations is that we can
represent the information in a more complete way including a wide range
of particular cases. By using generalized means in the analysis we use
the generalized OWAWA (GOWAWA) operator in D-S framework. In this case,
the decision making process is very similar to the framework shown in
Section 3.1 with the following differences.
In Steps 3-4, when calculating the collection of weights, w, we
have to consider that we are using the GOWWA operator in the aggregation
for each different cardinality of focal elements.
In Step 6, when calculating the aggregated payoff, we should use
[V.sub.ik] = GOWAWA([M.sub.ik]), for all the values of i and k.
In this case, we could also formulate in one equation the whole
aggregation process as follows. We call it the BS-GOWAWA operator. Note
that the formulation is the same than the BS-OWAWA with the difference
that now we add an additional parameter l that represents the use of the
generalized mean in the analysis. That is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)
where l is a parameter such that [lambda] [member of] (-[infinity],
[infinity]) - {0}.
If we use quasi-arithmetic means in the model, we are using the
quasi-arithmetic OWAWA (Quasi-OWAWA) operator in D-S belief structure.
Thus, the decision process is very similar than the previous one with
the difference that now instead of using generalized means, we use
quasi-arithmetic means. Thus, in Step 6 we should use [V.sub.ik] =
Quasi-OWAWA([M.sub.ik]), for all the values of i and k.
If we formulate this approach in one equation, the model is the
same with the difference that we replace the parameter [lambda] by a
strictly continuous monotonic function g(b) obtaining the BS-Quasi-OWAWA
operator. That is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)
where g(b) is a strictly continuous monotonic function.
As said before, the main advantage of these models is that they
include a wide range of particular cases including the BS-OWAWA
operator, the BS-WA and the BS-OWA operator. Remark 2. For example, the
BS-OWAWA operator is found when [lambda] = [delta] = 1 or g(b) = h(a) =
b. That is:
BS - OWAWA = [r.summation over (k =
1)]m([B.sub.k])([beta]([q.summation over (j=1)][w.sub.jk][b.sub.jk]) +
(1 - [beta])([q.summation over (i=1)] [v.sub.ik][a.sub.ik])). (10)
Note that Eq. (10) is equivalent to Eq. (7).
Remark 3. If [lambda] = [delta] = 2 or g(b) = h(a) = [b.sup.2], we
get the quadratic BS-OWAWA (BS OWAWQA) operator:
BS-OWAWQA = [r.summation over (k=1)] m([B.sub.k])
([beta][([q.summation over (j = 1)]
[w.sub.jk][b.sup.2.sub.jk]).sup.1/2] + (1 - [beta])[([q.summation over
(i = 1)][v.sub.ik][a.sup.2.sub.ik]).sup.1/2]). (11)
Remark 4. If [lambda] = [delta] = 3 or g(b) = h(a) = [b.sup.3], we
get the cubic BS-OWAWA (BS-OWAWCA) operator:
BS-OWAWQA = [r.summation over (k=1)] m([B.sub.k])
([beta][([q.summation over (j = 1)]
[w.sub.jk][b.sup.3.sub.jk]).sup.1/3] + (1 - [beta])[([q.summation over
(i = 1)][v.sub.ik][a.sup.3.sub.ik]).sup.1/3]). (12)
Remark 5. If [lambda] = [delta] [right arrow] 0 or g(b) = h(a)
[right arrow] [b.sup.0], we get the geometric BS-OWAWA (BS-OWAWGA)
operator:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)
Note that more complex situations could be formed by using
different values in the parameter l (OWA) and 8 (WA) and in the function
g (OWA) and h (WA). Moreover, we could also assume that the
probabilities given by the focal elements can also be extended with
generalized and quasi-arithmetic means.
3. Group decision making with Dempster-Shafer theory
In the previous framework, we assume that the information is
assessed by one decision maker. However, in real-world problems it is
very common that the decisions are assessed by a group of persons. This
is especially relevant when considering macrodecisions with very strong
implications that need very serious assessments in order to obtain the
most appropriate selection such as decisions concerning the variation of
the interest rate of the European Central Bank, the European
Constitution and variations in the taxes of a country.
Note that in the next section we will analyze a problem in
political decision making where it is very common to make group
decisions because generally, the decision of a group is better than the
decision of a person because the knowledge provided by a group is
higher. Typical examples are the decisions made in the parliament of a
country or in the ministries council.
The procedure to follow when making group decisions with
Dempster-Shafer theory of evidence and the OWAWA operator can be
summarized as follows. Note that many other decision-making models have
been discussed in the literature (Antucheviciene et al. 2010; Brauers,
Zavadskas 2010; Engemann, Miller 2009; Kersuliene et al. 2010; Liu 2009;
Podvezko 2009; Zavadskas, Turskis 2011; Zavadskas et al. 2010a, b).
Assume we have a decision problem in which we have a collection of
alternatives {[A.sub.1], [A.sub.q]} with states of nature {[S.sub.1],
..., [S.sub.n]} forming the payoff matrix [([a.sub.hi]).sub.mxn]. Let E
= {[e.sub.1], [e.sub.2], ..., [e.sub.p]} 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 [P.summation over
(t=1)] [u.sub.t] = 1 and [u.sub.t] [member of] [0, 1]. Each
decision-maker provides his own payoff matrix
[([a.sub.hi.sup.(t)]).sub.mxn]. The knowledge of the state of nature is
captured in terms of a belief structure m with focal elements B1, Br and
associated with each of these focal elements is a weight m([B.sub.k]).
The objective of the problem is to select the alternative that gives the
best result to the decision maker. In order to do so, we should follow
the following steps:
Step 1: Constructp individual payoff matrices according to the
information given by each decision-maker of the group.
Step 2: Use the WA to aggregate the information of the
decision-makers E using the weighting vector U. The result is the
collective payoff matrix [([a.sub.hi]).sub.mxn]. Thus, [a.sub.hi] =
[p.summmation over (t=1)] [u.sub.t][a.sup.t.sub.hi].
Note that it is possible to use other types of OWAWA operators
instead of the WA to aggregate this information.
Step 3: Calculate the belief function m about the states of nature.
Step 4: Calculate the attitudinal character (or degree of orness)
of the decision maker [alpha](W) (Yager 1988) using Eq. (4) and Eq. (5).
Step 5: Calculate the collection of weights, w, to be used in the
OWAWA aggregation ([??] = [beta] x W + (1 - [beta]) x V) for each
different cardinality of focal elements. Note that W = ([w.sub.1],
[w.sub.2], ..., [w.sub.n]) such that [n.summation over (j=1)] [w.sub.j]
= 1 and [w.sub.j] [member of] [0, 1] and V = ([v.sub.1], [v.sub.2],
..., [v.sub.n]) such that [n.summation over (i=1)] [w.sub.i] = 1 and
[v.sub.i] [member of] [0, 1].
Step 6: Determine the results of the collection, [M.sub.ik], if we
select alternative A. and the focal element [B.sub.k] occurs, for all
the values of i and k. Hence [M.sub.ik] = {[a.sub.ih] | [S.sub.h]
[member of] [B.sub.k]}.
Step 7: Calculate the aggregated results, [V.sub.ik] =
OWAWA([M.sub.ik]), using Eq. (3), for all the values of i and k.
Consider different families of OWAWA operators as described in Section 3
in order to provide a complete representation of the information.
Step 8: For each alternative, calculate the generalized expected
value, [C.sub.i], where:
C = [r.summation over (k=1)][V.sub.ik]m([B.sub.k]). (14)
Step 9: Select the alternative with the largest C as the optimal.
Note that in a minimization problem, the optimal choice is the lowest
result.
This aggregation process can be summarized using the following
aggregation operator
that we call the multi-person - BS-OWAWA (MP-BS-OWAWA) operator.
Definition 6. A MP-BS-OWAWA operator is a mapping MP-BS-OWAWA:
[R.sup.n] x [R.sup.p] x [R.sup.r] [right arrow] R that has a weighting
vector U of dimension p with [p.summation over (k=1)] [u.sub.p] = 1 and
[u.sub.k] [member of] [0, 1], such that:
MP - BS - OWAWA(([a.sup.1.sub.1], ..., [a.sup.p.sub.1]), ...,
([a.sup.1.sub.n], [a.sup.p.sub.n])) = [r.summation over (k =
1)][q.summation over (j=1)]m([B.sub.k])[[??].sub.jk][b.sub.jk], (15)
where: [[??].sub.jk] is the weighting vector of the kth focal
element such that [q.summation over (j=1)][[??].sub.jk] = 1 and
[[??].sub.jk] [member of] [0, 1], [b.sub.jk] is the jkth largest of the
[a.sub.ik], each argument [a.sub.ik] has an associated weight (WA)
[v.sub.ik] with [k.summation over (i=1)][v.sub.ik] = 1 and [v.sub.ik]
[member of] [0, 1], and a weight (OWA) [w.sub.jk] with [q.summation
over (j=1)][w.sub.jk] = 1 and [w.sub.jk] [member of] [0, 1],
[[??].sub.jk] = [beta][w.sub.jk] + (1 - [beta])[v.sub.jk] with [beta]
[member of] [0, 1] and [v.sub.i] is the weight (WA) [v.sub.i] ordered
according to [b.sub.j], that is, according to the jth largest of the
[a.sub.ik], [a.sub.jk] = [p.summation over
(t=1)][u.sub.t][a.sup.t.sub.ik], [a.sup.t.sub.ik] is the argument
variable provided by each person (or expert), and m([B.sub.k]) is the
basic probability assignment.
Note that the MP-BS-OWAWA operator has similar properties to those
explained in Section 3, such as the distinction between descending and
ascending orders, and so on.
The MP-BS-OWAWA operator includes a wide range of particular cases
following the methodology explained in Section 3. Thus, it includes:
The multi-person - BS-WA (MP-BS-WA) operator: When [beta] = 0.
The multi-person - BS-OWA (MP-BS-OWA) operator: When [beta] = 1.
The multi-person - BS-arithmetic mean (MP-BS-AM) operator: When
[w.sub.j] = 1/n and [v.sub.i] = 1/n, for all [a.sub.i].
The multi-person - BS-arithmetic-WA (MP-BS-AWA) operator: When
[w.sub.j] = 1/n for all j.
The multi-person - BS-arithmetic-OWA (MP-BS-AOWA) operator: When
[v.sub.i] = 1/n, for all i.
Note that if t = 1, we obtain the BS-OWAWA operator because we
assume that we only have one decision maker in the aggregation.
Note that it is possible to consider more complex situations by
using different types of aggregation operators to aggregate the
experts' opinions (Merigo 2011; Merigo, Gil-Lafuente 2009) and by
analyzing different types of belief structures (Shafer 1976; Yager, Liu
2008).
Furthermore, it is possible generalize the MP-BS-OWAWA operator by
using generalized aggregation operators. By using quasi-arithmetic means
(Fodor et al. 1995; Merigo, Casanovas 2011a) we obtain the
quasi-arithmetic MP-BS-OWAWA (MP-BS-Quasi-OWAWA) operator. It is very
similar to the MP-BS-OWAWA operator with the difference that we add a
strictly continuous monotonic function for the WA and the OWA that
includes a wide range of particular cases including quadratic and
geometric aggregations. It can be formulated as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)
where g and h are strictly continuous monotonic functions.
Remark 6. Some interesting particular cases of the
MP-BS-Quasi-OWAWA operator can be formed as follows:
Note that if g(b) = [b.sup.[lambda]] and h(a) = [a.sup.[delta]], we
obtain the generalized MP-BS-OWAWA (MP BS-GOWAWA) operator.
Quadratic MP-BS-OWAWA (MP-BS-OWAWQA) operator: When g(b) =
[b.sup.2] and h(a) = [a.sup.2].
Cubic MP-BS-OWAWA (MP-BS-OWAWCA) operator: When g(b) = [b.sup.2]
and h(a) = [a.sup.2].
Geometric MP-BS-OWAWA (MP-BS-OWAWGA) operator: When g(b) [right
arrow] [b.sup.0] and h(a) [right arrow] [a.sup.0].
4. Application in political management
This new approach can be implemented in a wide range of decision
making problems including strategic decision making, investment
selection, political management and juridical decision making. In this
paper, we focus on an application in political decision making regarding
the selection of the optimal fiscal policy in a country/region by using
the OWAWA operator and Dempster-Shafer belief structure.
Assume a government that it is planning his fiscal policy for the
next year and considers five possible alternatives:
--[A.sub.1] = Develop a strong expansive fiscal policy;
--[A.sub.2] = Develop an expansive fiscal policy;
--[A.sub.3] = Do not make any change;
--[A.sub.4] = Develop a contractive fiscal policy;
--[A.sub.5] = Develop a strong contractive fiscal policy.
In order to evaluate these fiscal policies, the group of experts of
the government considers that the key factor is the economic situation
of the world for the next year. After careful analysis, the experts have
considered five possible situations that could happen in the future:
--[S.sub.1] = Very bad economic situation;
--[S.sub.2] = Bad economic situation;
--[S.sub.3] = Regular economic situation;
--[S.sub.4] = Good economic situation;
--[S.sub.5] = Very good economic situation.
The group of experts of the government can be divided in 3 groups,
each providing its own opinion. Depending on the situation that could
happen in the future, each group of experts establish its opinion
concerning the payoff matrix. The results are shown in Tables 1, 2 and
3.
In this example, we assume the following weighting vector for the
three groups of experts: U = (0.4, 0.3, 0.3) representing the degree of
importance they have in the analysis. Thus, we can aggregate their
opinions obtaining a single collective payoff matrix that represents the
aggregated information of the previous Tables. The results are shown in
Table 4.
After careful analysis of the information, the experts have
obtained some general probabilistic information about which state of
nature will happen in the future although there is no specific
probability for each state of nature. This information is based on
historical data and several experiments made by the experts. Due to the
high degree of uncertainty involved, they can only provide the
probabilistic information in the form of a belief structure. This
information is represented by the following belief structure about the
states of nature.
Focal element
[B.sub.1] = {[S.sub.1], [S.sub.2], [S.sub.3]} = 0.3;
[B.sub.2] = {[S.sub.1], [S.sub.3], [S.sub.5]} = 0.3;
[B.sub.3] = {[S.sub.3], [S.sub.4], [S.sub.5]} = 0.4.
The attitudinal character of the government is very complex because
it involves the opinion of several political parties with different
interests. After careful evaluation, the experts establish the following
weighting vectors for both the WA and the OWA operator: W = (0.2, 0.4,
0.4) and V = (0.3, 0.3, 0.4). It is worth noting that for the OWA we
assume a weighting vector that tends to be a bit pessimistic with the
assumption that the government wants to make a safety decision. Note
that they assume that the OWA has a degree of importance of 30% and the
WA a degree of 70%. With this information, we can obtain the aggregated
results. In this example, apart from considering the results obtained
with the OWAWA operator, we also consider the results obtained with the
maximum, minimum, the Min-WA, the MaxWA, the arithmetic mean, the
weighted average and the OWA operator. Thus, we can get a more complete
picture of the potential situations that may occur in the future. They
are shown in Table 5.
Once we have the aggregated results, we have to calculate the
generalized expected value. The results are shown in Table 6.
As we can see, depending on the aggregation operator used, the
results and decisions may be different. Note that in this case, our
optimal choice is the same for all the aggregation operators but in
other situations we may find different decisions between each
aggregation operator.
A further interesting issue is to establish an ordering of the
policies. Note that this is very useful when the decision maker wants to
consider more than one alternative. The results are shown in Table 7.
It is worth noting that different alternatives may be optimal
depending on the assumptions we assume regarding the uncertainty. Note
that in this example the optimal choice seems to be [A.sub.1], although
in some extreme optimistic situations we could find that A4 is optimal.
Moreover, we also find differences in the ranking process since each
alternative may be ranked in a different position according to the
aggregation process used in the analysis.
Conclusions
We have presented a new decision making approach with D-S belief
structure by using the OWAWA operator. The main advantage of this
approach is that it deals with probabilities, WA s and OWA s in the same
framework. Therefore, we are able to consider subjective and objective
information and the attitudinal character of the decision maker. For
doing so, we have developed the BS-OWAWA operator. It is a new
aggregation operator that uses belief structures with the OWAWA
operator. We have studied several families of BS-OWAWA operators and we
have seen that it contains the OWA and the WA aggregation as particular
cases. Moreover, by using the OWAWA we can consider a wide range of
inter medium results giving different degrees of importance to the WA
and the OWA.
We have further generalized this approach by using generalized
aggregation operators obtaining the BS-GOWAWA and the BS-Quasi-OWAWA
operators. Their key advantage is that they include a wide range of
particular cases including the BS-OWAWA operator, the BS-OWAWQA operator
and many others. We have also extended this approach to group decision
making problems where the decisions are taken by a group instead of an
individual person. We have introduced the MP-BS-OWAWA operator and the
MP-BS-Quasi-OWAWA operator. We have seen that they include a wide range
of cases including the MP-BS-WA and the MP-BS-OWA.
We have also developed an application in political management by
using the new approach. We have focused on a decision making problem
regarding the selection of fiscal policies in a country. The main
advantage of this approach is that it provides a more complete
representation of the decision process because the decision maker can
consider many different scenarios depending on his interests by dealing
with probabilities, weighted averages and OWA operators.
In future research, we expect to develop further extensions of this
approach by considering more complex aggregation operators such as those
that use uncertain information, order-inducing variables or unified
aggregation operators. We will also consider other decision making
applications including strategic management and investment selection.
doi: 10.3846/20294913.2013.869517
Acknowledgements
We would like to thank the anonymous reviewers for valuable
comments that have improved the quality of the paper. Support from the
Spanish Ministry of Education under project "JC2009-00189",
the University of Barcelona (099311) and the European Commission
(PIEF-GA-2011-300062) is gratefully acknowledged.
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Jose M. MERIGO (a), Kurt J. ENGEMANN (b), Daniel PALACIOS-MARQUES
(c)
(a) Department of Business Administration, University of Barcelona,
Av. Diagonal 690, 08034 Barcelona, Spain
(b) Hagan School of Business, Iona College, 10801 New Rochelle, New
York, USA
(c) Department of Business Organisation, Universitat Politecnica de
Valencia, Camino Vera s/n. 46022, Valencia, Spain
Received 16 November 2011; accepted 29 September 2012
Corresponding author Jose M. Merigo
E-mail: jmerigo@ub.edu
Jose M. MERIGO has a MSc and a PhD degrees in Business
Administration from University of Barcelona, Spain. His PhD received the
Extraordinary Award from the University of Barcelona. He also holds a
Bachelor's Degree in Economics and a Master's Degree in
European Business Administration and Business Law 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 200 papers in journals, books and conference proceedings including
journals such as Information Sciences, International Journal of
Information Technology and Decision Making, Technological and Economic
Development of Economy, Expert Systems with Applications, International
Journal of Intelligent Systems, International Journal of Uncertainty,
Fuzziness and Knowledge-Based Systems, Cybernetics & Systems,
Computers & Industrial Engineering, Applied and Computational
Mathematics, International Journal of Computational Intelligence Systems
and International Journal of Fuzzy Systems. He has published 8 books
including one edited with World Scientific "Computational
Intelligence in Business and Economics" and three with Springer. 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. He is currently interested in aggregation operators,
decision making and uncertainty.
Kurt J. ENGEMANN is a Full Professor in the Department of
Information Systems at Hagan School of Business at Iona College (New
Rochelle, New York, USA). He holds a PhD in Operations Research from New
York University. He is also the Director of the Center for Business
Continuity and Risk Management (BCRM) at Iona College. He has consulted
professionally over the past thirty years in the area of risk management
and decision modelling for major organizations and has been instrumental
in the development and implementation of comprehensive business
continuity management programs. He is a Certified Business Continuity
Professional (CBCP) with the Disaster Recovery Institute International.
He has published more than 150 publications in journals, books and
conference proceedings including journals such as International Journal
of Intelligent Systems, Interfaces, International Journal of General
Systems and International Journal of Uncertainty, Fuzziness and
Knowledge-Based Systems. He has published several books and has
participated in a wide range of scientific committees. He is the
editor-in-chief of the International Journal of Technology, Policy and
Management and the International Journal of Business Continuity and Risk
Management. He has served as a reviewer in a wide range of journals.
Daniel PALACIOS-MARQUES is an Associate Professor of Management at
the Technical University of Valencia, Spain. He has an MBA and a PhD in
Quality Management and Business Administration. He has published
articles in journals such as Tourism Management, Annals of Tourism
Research, Small Business Economics, Management Decision, International
Journal of Technology Management, Cornell Quarterly Management, Services
Industries Journal, Service Business, International Entrepreneurship and
Management Journal, Journal of Knowledge Management, Journal of
Intellectual Capital, International Journal of Innovation Management and
International Journal of Contemporary Hospitality Management. He has
been Editor of the book Connectivity and Knowledge Management in Virtual
Organizations, Networking and Developing Interactive Communications.
Table 1. Payoff matrix--Expert 1
[S.sub.1] [S.sub.2] [S.sub.3] [S.sub.4] [S.sub.5]
[A.sub.1] 30 70 90 30 30
[A.sub.2] 20 60 70 50 50
[A.sub.3] 60 50 60 90 80
[A.sub.4] 40 90 90 70 40
[A.sub.5] 60 50 30 50 70
Table 2. Payoff matrix--Expert 2
[S.sub.1] [S.sub.2] [S.sub.3] [S.sub.4] [S.sub.5]
[A.sub.l] 90 20 60 70 60
[A.sub.2] 30 60 90 50 80
[A.sub.3] 20 40 30 30 80
[A.sub.4] 40 50 90 70 30
[A.sub.5] 10 50 30 80 80
Table 3 Payoff matrix--Expert 3
[S.sub.1] [S.sub.2] [S.sub.3] [S.sub.4] [S.sub.5]
[A.sub.1] 90 80 80 30 60
[A.sub.2] 50 60 90 50 80
[A.sub.3] 60 20 50 70 80
[A.sub.4] 40 10 90 70 50
[A.sub.5] 70 50 70 80 60
Table 4. Payoff matrix--Collective result
[S.sub.1] [S.sub.2] [S.sub.3] [S.sub.4] [S.sub.5]
[A.sub.1] 70 60 80 40 50
[A.sub.2] 30 60 80 50 70
[A.sub.3] 50 40 50 70 80
[A.sub.4] 40 60 90 70 40
[A.sub.5] 50 50 40 70 70
Table 5. Aggregated results
Min Max Min-WA Max-WA AM WA OWA OWAWA
[V.sub.11] 60 80 67.7 73.7 70 71 68 70.1
[V.sub.12] 50 80 60.5 69.5 66.6 65 64 64.7
[V.sub.13] 40 80 51.2 63.2 56.6 56 52 54.8
[V.sub.21] 30 80 50.3 65.3 56.6 59 52 56.9
[V.sub.22] 30 80 51.7 66.7 60 61 56 59.5
[V.sub.23] 50 80 61.9 70.9 66.6 67 64 66.1
[V.sub.31] 40 50 44.9 47.9 46.6 47 46 46.7
[V.sub.32] 50 80 58.4 67.4 60 62 56 60.2
[V.sub.33] 50 80 62.6 71.6 66.6 68 64 66.8
[V.sub.41] 40 90 58.2 73.2 63.3 66 58 63.6
[V.sub.42] 40 90 50.5 65.5 56.6 55 50 53.5
[V.sub.43] 40 90 56.8 71.8 66.6 64 62 63.4
[V.sub.51] 40 50 44.2 47.2 46.6 46 46 46
[V.sub.52] 40 70 50.5 59.5 53.3 55 50 53.5
[V.sub.53] 40 70 54.7 63.7 60 61 58 60.1
Table 6. Generalized expected value
Min Max Min-WA Max-WA AM WA OWA OWAWA
[A.sub.1] 49 80 58.94 68.24 63.62 63.2 60.4 62.36
[A.sub.2] 38 80 55.36 67.96 61.62 62.8 58 61.36
[A.sub.3] 47 71 56.1 63.3 58.62 60 56.2 58.79
[A.sub.4] 40 90 55.33 70.33 62.61 61.9 57.2 60.49
[A.sub.5] 40 64 50.29 57.49 54 54.7 52 53.89
Table 7. Ranking of the policies
Ranking
Min [A.sub.1][??][A.sub.3][??][A.sub.4] = [A.sub.5][??][A.sub.2]
Max [A.sub.4][??][A.sub.1] = [A.sub.2][??][A.sub.3][??][A.sub.5]
Min-wa [A.sub.1][??][A.sub.3][A.sub.2][??][A.sub.3][??][A.sub.5]
Max-wa [A.sub.4][??][A.sub.1][??][A.sub.2][??][A.sub.3][??][A.sub.5]
am [A.sub.1][??][A.sub.4][??][A.sub.2][??][A.sub.3][??][A.sub.5]
wa [A.sub.1][??][A.sub.2][??][A.sub.4][??][A.sub.3][??][A.sub.5]
owa [A.sub.1][??][A.sub.2][??][A.sub.4][??][A.sub.3][??][A.sub.5]
owawa [A.sub.1][??][A.sub.2][??][A.sub.4][??][A.sub.3][??][A.sub.5]
