The multi-attribute group decision making method based on the interval grey linguistic variables weighted harmonic aggregation operators.
Jin, Fang ; Liu, Peide ; Zhang, Xin 等
Introduction
Multiple attribute decision making (MADM) has been extensively
applied to various areas such as society, economics, management,
military and engineering technology, etc. Since the objective things are
complex, uncertainty and human thinking is ambiguous, the majority of
multi-attribute decision-making problems are uncertain and fuzzy, so
fuzziness is the major factor which should be considered in the process
of decision making. On the other hand, decision-making problems have the
greyness in the process of dealing with the incomplete information.
"Greyness" means amount of information is smaller' and
inadequate. For example, in agriculture planting, even if the sown area,
seed, fertilizer, irrigation and other information are completely clear,
it is still difficult to accurately predict the productive output. The
productive output is grey. Another example, in 2050, China's total
population will be controlled between 15 and 16 billion. This
"between 15 and 16 billion" is a concept of grey, we cannot
know the accurate value. So, the "greyness" is a concept about
"quantity". However, "fuzziness" means a concept is
not clear. For example, about "young people", it is very
difficult to designate an exact range in which they are young people and
out which they are not, so it is fuzzy. Other examples, such as
"hot water", "wet" etc. are fuzzy. So
"fuzziness" is a concept about "quality". Obviously,
"greyness" and "fuzziness" don't mean that some
information is "grey information" and the other part of the
information is "fuzzy information" for the same problem
because they don't describe the same concept (Bu, Zhang 2002). In
general, we can simply interpret "greyness" and
"fuzziness" as width and depth of an evaluation object. In
reality, the decision making problems have not only the fuzziness, but
also the greyness, which are called the grey fuzzy multi-attribute
decision making problems. For example, about ability of innovation
management of enterprises, it has the fuzziness and greyness
simultaneously, because the concept of ability of innovation management
is unclear, i.e., it has the fuzziness; at the same time, we cannot get
all information about ability of innovation management of enterprises,
so it has the greyness. There are similar examples, such as moral
evaluation, working ability assessment, evaluation of a person's
level of knowledge, etc.
About fuzzy theory, Zadeh (1965) firstly proposed the theory of
Fuzzy Sets, The core idea is to extend membership function to any value
in the closed interval [0,1]. Then fuzzy sets had been extended to
interval numbers, triangular fuzzy numbers, trapezoidal fuzzy numbers,
linguistic variables, intuitionistic fuzzy numbers, etc. and widely used
in the field of decision-making (Herrera et al. 1996; Liu 2011; Liu, Su
2010; Liu, Zhang 2010; Zhang, Liu 2010; Yu 2013; Razavi Hajiagha et al.
2013). About grey theory, Deng (1982) firstly proposed the theory of
Grey Systems, then grey theory has been the rapid development, and a
series of grey decision-making methods were proposed (Deng 2002; Liu, W.
L., Liu, P. D. 2010). The grey fuzzy theory, which combined the fuzzy
theory and the grey theory, takes into account the greyness and fuzzynes
of decision making problems, and is more in line with the objective
reality of things. At present, there have been consistent efforts to the
research on grey fuzzy decision making problems. Bu and Zhang (2002),
Choobineh and Li (1993a, b), Jin and Lou (2003, 2004), Luo and Liu
(2004) studied the ranking method of grey fuzzy number. Bu and Zhang
(2002) transformed the grey fuzzy number into the interval number, and
then utilized the ranking method of interval number to rank the order of
alternatives. For the grey fuzzy multiple attribute decision making
problems which both the fuzzy part and the grey part took the form of
real number, Jin and Lou (2003) proposed the decision making model which
utilized the difference between the alternatives and the fuzzy positive
ideal solution, and between the alternatives and the negative ideal
solution to rank the orders based on the Hamming distance. Jin and Lou
(2004) utilized the distance between each alternative and the grey fuzzy
ideal solution to rank the orders of alternatives. In order to solve the
grey fuzzy decision making problems, Luo and Liu (2004) utilized the
maximum entropy formula to determine attribute weights, then ranked the
orders of alternatives based on the linear combination of fuzzy
information and grey information. Zhu et al. (2006) constructed the
evaluation model in which the fuzzy part and the grey part took the form
of interval number and the real number respectively. Meng et al. (2007)
proposed the interval numbers to present greyness and fuzziness of grey
fuzzy decision making problems, and the mathematical model of interval
valued grey fuzzy comprehensive evaluation is established, and the
application to the selection of the preferred project is given. Wang and
Wang (2008) extended the fuzzy part and the grey part of grey fuzzy
decision making problems to interval numbers, and ranked the order of
alternatives based on the ordered weight aggregation (OWA) operator.
Zhang (2013) proposed the interval grey linguistic variables ordered
weighted aggregation (IGLOWA) operator, and then use the Choquet
integral to develop the interval grey linguistic correlated ordered
arithmetic aggregation (IGLCOA) operator and the interval grey
linguistic correlated ordered geometric aggregation (IGLCOGA) operator.
Because the linguistic variables are easier to express fuzzy
information, and the research on multi-attribute decision making based
on the linguistic variables has made great achievements (Alonso et al.
2009; Cabrerizo et al. 2010a, b; Herrera et al. 2009; Herrera -Viedma et
al. 2003; Kim, Ahn 1999; Martinez et al. 2009; Xu 2004, 2007, 2008). So,
this paper proposes the concept of interval grey linguistic variables in
which the fuzzy part and the grey part adopt linguistic variables and
interval numbers respectively, and then studies the operation rules and
the multiple attribute decision making method based on interval grey
linguistic variables.
The remainder of the paper is organized as follows: Section 1
introduces some relative knowledge; Section 2 defines the interval grey
linguistic variables and proposes some weighted harmonic aggregation
operators; Section 3 gives a method based on the interval grey
linguistic variables hybrid weighted harmonic aggregation operators to
solve the multiple attribute group decision making problems; Section 4
presents an illustrative example to verify effectiveness of this method
and to illustrate its decision making steps; Finally, conclusions are
given in the final section.
1. Preliminaries
1.1. Grey number (Deng 2002; Luo 2005; Lu 2009)
Grey number is the basic unit to express the greyness. We can call
only knowing the ranges roughly and not knowing the exact value as grey
number. In the application, the grey number generally refers to a range
or an uncertain number, and it can be expressed by "[cross
product]". Grey number can be divided into the following
categories:
1) The grey number only with a lower bound
The grey number in this type can be expressed as [[mu].sub.A](x)
[right arrow] [0,1], where [a.bar] is the lower bound of the grey number
[cross product] and it's also a certain number.
2) The grey number only with a upper bound
The grey number in this type can be expressed as [cross product]
[member of] (-[infinity], [bar.a]], where [bar.a] is the upper bound of
the grey number [cross product] and it's also a certain number.
3) The grey number with interval number
The grey number in this type can be expressed as [cross product]
[member of] [[a.bar], [bar.a]], where [a.bar] and [bar.a] are a certain
number, and [a.bar] is the lower bound, [bar.a] is the upper bound of
the grey number [cross product].
4) The grey number with three-point interval number
The grey number in this type can be expressed as [cross product]
[member of] [[a.bar], a, [bar.a]], where [a.bar], [bar.a] and a are a
certain number; [a.bar] and [[nabla].sub.i]are the lower bound and upper
bound of the grey number [cross product] respectively, and a is the
center of gravity which can be get most likely.
5) The black number and white number
When grey number [cross product] [member of] (-[infinity],
+[infinity]), we can call [cross product] as a black number. It shows
that information is completely unknown; when grey number [cross product]
[member of] [[a.bar], [bar.a]] and [a.bar] = [bar.a], we can call [cross
product] as a white number. It shows information is completely known.
1.2. Grey fuzzy math
(Chen 1994; Li and Wang 1994; Wang and Song 1988; Wang 1996)
Definition 1: Let [??] be the fuzzy subset in the space X = {x}, if
the membership degree [[mu].sub.A](x) of x to A has the greyness
[v.sub.A](x) in the interval [0, 1], then [??] is called the grey fuzzy
set in space X:
[??] = {(x, [[mu].sub.a](x), [v.sub.a](x))| x [member of] X}. (1)
The set pair mode is [??] = (A, [??]), where A = {(x,
[[mu].sub.A](x))| x [member of] x} is called the fuzzy part of [??], and
[??] = {(x, [v.sub.A](x))| x [member of] x} is called the grey part of
A.
So the grey fuzzy set is regarded as the generalization of the
fuzzy set and the grey set.
Definition 2: Let X = {x} and Y = {y} be the given space, if
[v.sub.R](x,y) is the greyness of the membership function uR (x, y) of R
which is the fuzzy relationship between x and y, then grey fuzzy set
[??] = {((x,y), [[mu].sub.R](x,y), [v.sub.R](x,y)) | x [member of] X, y
[member of] Y} is called the grey fuzzy relationship in direct product
space X x Y, which is represented as the grey fuzzy matrix mode:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
And [??] = (R, [??]) represents the grey fuzzy relationship in
direct product space X x Y, where R = {((x,y), [[mu].sub.A](x,y))| x
[member of] X, y [member of] Y} represents the fuzzy relationship in
direct product space X x Y, and [??] = {((x, y), [v.sub.A](x, y))| x
[member of] X, y [member of] Y} represents the grey relationship in
direct product space X x Y.
1.3. Linguistic evaluation set and its extension
Suppose that S = ([s.sub.l], [ s.sub.2], ..., [s.sub.l]) is a
finite and totally ordered discrete term set, where I is the odd number.
In real situation, l is equal to 3, 5, 7, 9 etc. In this paper, I =7.
For example, a set S could be given as follows:
S = ([s.sub.1], [s.sub.2], [s.sub.3], [s.sup.4], [s.sub.5],
[s.sub.6], [s.sub.7]) = {very poor, poor, slightly poor, fair, slightly
good, good, very good}.
Usually, in these cases, it requires that [s.sub.i] and [S.sub.j]
must satisfy the following additional characteristics (Herrera,
Herrera-Viedma 2000):
1) The set is ordered: [s.sub.i] < [s.sub.j], if and only if i
< j;
2) There is the negation operator: neg([s.sub.i]) = [s.sub.l-i];
3) Maximum operator: max([s.sub.i], [s.sub.J]) = [s.sub.i], if i
[greater than or equal to] j;
4) Minimum operator: min([s.sub.i], [s.sub.j]) = [s.sub.i], if i
[less than or equal to] j.
For any linguistic labels S = ([s.sub.1], [s.sub.2], ...,
[s.sub.l]), the relationship between the element [s.sub.i] and its
subscript i is strictly monotone increasing (Herrera et al. 1996; Xu
2006a), so the function can be defined as follows:
f : [s.sub.i] = f(i).
Obviously, the function f(i) is the strictly monotone increasing
function about subscript i. To preserve all the given information, the
discrete linguistic label S = ([s.sub.1], [s.sub.2], ..., [s.sub.l]) is
extended to a continuous linguistic label [bar.S] = {[s.sub.[alpha]] |
[alpha] [epsilon] R} which satisfied the above characteristics. If
[s.sub.[alpha]] [epsilon] S, then [s.sub.[alpha]] is called an original
linguistic label, otherwise, [s.sub.[alpha]] is called a virtual
linguistic label. In general, the decision maker uses the original label
to evaluate attributes and alternatives, and the virtual labels can only
appear in the course of operation.
Let [s.sub.i], [s.sub.j] [member of] [bar.S] and [[lambda].sub.1],
[[lambda].sub.2] [member of] [0,1], n is a positive integer, then
operational laws of linguistic variables are given as follows (Xu
2006b):
1) [beta][s.sub.i] = [s.sub.[beta]xi]; (3)
2) [s.sub.i] [direct sum] [s.sub.j] = [s.sub.i+j]; (4)
3) [s.sub.i]/[s.sub.j] = [s.sub.i/j], if j [not equal to] 0; (5)
4) [(Si).sup.n] = [s.sub.[i.sup.n]]; (6)
5) [[lambda].sub.1]([s.sub.i] [direct sum] [s.sub.j]) =
[[lambda].sub.1][s.sub.i] [direct sum][[lambda].sub.1][s.sub.j],
([[lambda].sub.1] + [[lambda].sub.2])[s.sub.i] =
[[lambda].sub.1][s.sub.i] [direct sum] [[lambda].sub.2][s.sub.i]. (7)
Definition 3: Let [s.sub.[alpha]], sp be the two linguistic
variables, then the distance between [s.sub.[alpha]] and [s.sub.[beta]]
is defined as follows:
d([s.sub.[alpha]], [s.sub.[beta]]) = [absolute value of
[alpha]-[beta]]/l. (8)
2. Interval grey linguistic variables
2.1. The definition of interval grey linguistic variables
Definition 4: Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] be the grey fuzzy number, if its fuzzy part A is a linguistic
variable [s.sub.[alpha]] [member of] [bar.S], and its grey part [??] is
a closed interval [g.sup.L.sub.A], [g.sup.U.sub.A] [subset or equal to]
[0,1], where [g.sup.L.sub.A] is the interval lower limit,
[g.sup.U.sub.A] is the interval upper limit, and [g.sup.L.sub.A] [less
than or equal to] [g.sup.U.sub.A], then [??] is called the interval grey
linguistic variables.
Because the linguistic variables are easier to express fuzzy
information, it is more reasonable to utilize the linguistic variables
to represent the fuzzy part, and for the grey part which indicates the
amount of information obtained, it is more accurate to reflect the
information obtained by decision maker using the interval numbers. The
larger the greyness of the grey part is, the less information obtained
and the lower credibility of the obtained information is. The lower the
credibility of the obtained value is, the lower the usage value of the
information is. When the greyness rises, the obtained information
becomes useless. On the other hand, the smaller the greyness is, the
more information obtained is, which causes higher credibility of the
obtained value. That finally leads to higher usage value of the obtained
information.
2.2. The operation of the interval grey linguistic variables
Supposed that [??] = ([s.sub.[alpha]], [[g.sup.L.sub.A],
[g.sup.U.sub.A]]), [??] = ([s.sub.[beta]], [[g.sup.L.sub.B],
[g.sup.U.sub.B]]) and [??] = ([s.sub.[lambda]], [[g.sup.L.sub.C],
[g.sup.U.sub.C]]) are the three interval grey linguistic variables.
Based on the concept of the interval grey linguistic variables, the
linguistic operational rules and extension principle, the operation
rules of interval grey linguistic variables are defined as follows:
1) [??] + [??] = ([s.sub.[alpha]+[beta]] max([g.sup.L.sub.A],
[g.sup.L.sub.b]), max([g.sup.U.sub.A], [g.sup.U.sub.B])]); (9)
2) [??] - [??] = ([s.sub.[alpha]-[beta]], [max([g.sup.L.sub.A],
[g.sup.L.sub.b]), max([g.sup.U.sub.A], [g.sup.U.sub.B])]); (10)
3) [??] x [??] = ([s.sub.[alpha]x[beta]], [max([g.sup.L.sub.A],
[g.sup.L.sub.b]), max([g.sup.U.sub.A], [g.sup.U.sub.B])]);
4) [??] / [??] = ([s.sub.[alpha]/[beta]], [max([g.sup.L.sub.A],
[g.sup.L.sub.b]), max([g.sup.U.sub.A], [g.sup.U.sub.B])]); (12)
5) k[??] = ([s.sub.kx[alpha]], [[g.sup.L.sub.A], [g.sup.U.sub.A]]);
(13)
6) ([??]) = ([s.sub.[[alpha].sup.k]], [[g.sup.L.sub.A],
[g.sup.U.sub.A]]). (14)
2.3. The distance between the two interval grey linguistic
variables
Definition 5: Let [??], [??], [??] be the interval grey linguistic
variables, [??] be the set of the interval rey linguistic variables, f
be the mapping, f: [??] x [??] [right arrow] [??]. If d ([??], [??])
satisfies the following equation:
1) 0 [less than or equal to] d ([??], [??]) [less than or equal to]
1, d ([??], [??]) = 0;
2) d([??], [??]) = d ([??], [??]);
3) d ([??], [??]) + d ([??], [??]) [greater than or equal to]
[greater than or equal to] d ([??], [??])..
Then d 9[??], [??]) is called the distance between the interval
grey linguistic variable A and B.
Definition 6: Let [??] = ([s.sub.[alpha]], [[g.sup.L.sub.A],
[g.sup.U.sub.A]]) and [??] = ([s.sub.[beta]], [[g.sup.L.sub.B],
[g.sup.U.sub.B]) be the interval grey linguistic variables, then the
Hamming distance d f A, B 1 between the interval grey linguistic
variable [??] and [??] is defined as follows:
d ([??], [??]) = 1/2(l -1)([absolyte value of [alpha](1 -
[g.sup.L.sub.A]) - [beta](1 - [g.sup.L.sub.B])] + [absolute value of
[alpha](1 - [g.sup.U.sub.A]) - [beta](1 - [g.sup.U.sub.B])]). (15)
It is easy to verify that the equation (15) satisfies the three
equation of definition 5.
Specially, if [g.sup.L.sub.A] = [g.sup.U.sub.A] = [g.sup.L.sub.B] =
[g.sup.U.sub.B] = 0, then the interval grey linguistic variable is
reduced to linguistic variable, and the equation (15) is transformed
into equation (8). That is, the equation (8) is the special case of
equation (15).
2.4. The comparing method of interval grey linguistic variables (1)
C-OWA aggregate operator
Definition 7 (Yager 2004): function [rho] : [0,1] [right arrow]
[0,1] satisfied that:
1) [rho](0) = 0 ;
2) [rho](1) = 1 ;
3) if x > y, then [rho](x) > [rho](y),
then [??] is called the basic unit-interval monotonic (BUM)
function.
Definition 8 (Yager 2004): Let [a,b] be the interval number, and
[f.sub.[rho]] ([a,b]) = [[integral].sup.1.sub.0] d[rho](y)/dy (b -
y(b - a))dy, (16)
then f is called the continuous interval number OWA
(C-OWA)operator.
If [rho](y) = [y.sup.[delta]]([delta][greater than or equal to] 0),
then [f.sub.[rho]]([a,b]) = b + [delta]a/[delta] + 1.
(2) The expectation value and the rank method of interval grey
linguistic variables
Let [??] = ([s.sub.[alpha]], [[g.sup.L.sub.A], [g.sup.U.sub.A]]) be
the interval linguistic variable, the expectation value of interval grey
linguistic variable [??] is defined as follows:
I([??]) = [s.sub.[alpha]] x [f.sub.[rho]] ([(1 - [g.sup.U.sub.A]),
(1 - [g.sup.L.sub.A]]). (17)
Suppose [??] = ([s.sub.[alpha]], [[g.sup.L.sub.A],
[g.sup.U.sub.A]]) and [??] = ([s.sub.[beta]], [[g.sup.L.sub.B],
[g.sup.U.sub.B]]) are two interval linguistic variables, if I([??]) >
I([??]), then [??] > [??], and If I([??]) = I([??]) and
[s.sub.[alpha]] > [s.sub.[beta]], then [??] > [??].
2.5. Interval grey linguistic variables hybrid weighted harmonic
aggregation (IGLHWHA) operator
Definition 9: Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] be the group of interval grey linguistic variables, and IGBWHU :
[[OMEGA].sup.n] [right arrow] [OMEGA], if
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
where [OMEGA] is the set of all the interval grey linguistic
variables, W = ([w.sub.1], [w.sub.2], ..., [w.sub.n]) is the weight
vector of [??] (j = 1,2, ..., n), and [N.summation over (j=1)] [w.sub.j]
= 1, then IGBWHU is called the interval grey linguistic weighted
harmonic aggregation (IGBWHA)operator.
Based on the operation rules of interval grey linguistic variables,
the equation (18) is deduced to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
Example 1. Assume W = (0.2,0.3,0.1,0.4) and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
By Definition 9, we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
It's easy to prove that the IGBWHU operator has the following
properties.
1) Theorem 1 (Commutativity)
If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is any
permutation of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
2) Theorem 2 (Idempotency)
If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all j,
then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
3) Theorem 3 (Monotonicity)
If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all j,
then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Definition 10: Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] be the group of interval grey linguistic variables, and IGBOWHU :
[[OMEGA].sup.n] [right arrow] [OMEGA], if
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
where [OMEGA] is the set of all the interval grey linguistic
variables, [omega] = ([[omega].sub.1], [[omega].sub.2], ...,
[[omega].sub.n]) is the weight vector associated with the function
IGBOWHU, and [n.summation over (j=1] [[omega].sub.j] = 1. Supposed that
the possible permutation of (1,2, ..., n) is ([[sigma].sub.1],
[[sigma].sub.2], ..., [[sigma].sub.n]), and for any j, we can get
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then IGBOWHUis
called the interval grey linguistic ordered weighted harmonic
aggregation (IGBOWHU)operator.
Based on the operation rules of interval grey linguistic variables,
the equation (20) is deduced to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
The characteristic of IGBOWHU operator is that interval grey
linguistic variables U (j = 1,2,***,n) is ranked in descending order and
aggregated with weights. [[omega].sub.j] is associated with the jth
position of the aggregation process, and [??] isn't associated with
[[omega].sub.j]. So [omega] is called the position weighted vector.
According to the actual situation, the position weight vector
[omega] = ([[omega].sub.1], [[omega].sub.2], ..., [[omega].sub.n]) is
determined by the following method:
1) The method by Wang and Xu (2008):
[[omega].sub.1] = 1 - [alpha]/n + [alpha], [[omega].sub.j] = 1 -
[alpha]/n, j [not equal to] 1, [alpha] [member of] [0,1]. (22)
2) The position weighted vector [omega] is determined by the method
which proposed by Wang and Xu (2008). The equation is shown as follows:
[[omega].sub.i+1] = [C.sup.i.sub.n-1] i = 0,1, ..., n-1. (23)
Example 2. Assume [omega] = (1/8,3/8,3/8,1/8) (calculated by Eq.
23) and [??] =([s.sub.2], [0.2,0.4]), [??] = ([s.sub.1], [0.1,0.3]),
[??] =([s.sub.5], [0.4,0.5]), [??] =([s.sub.4], [0.3,0.6]).
Firstly, calculate the expectation value of interval grey
linguistic variables [??], [??], [??] and [??] according to Eq. (17). 2
Suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Similarly,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Secondly, we rank the [??], [??], [??] and [??], the results are
shown as follows:
[[sigma].sub.1] = 3, [[sigma].sub.2] = 4, [[sigma].sub.3] = 1,
[[sigma].sub.4] = 2.
Finally, by definition 10, we have:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
It's easy to prove that the IG OWHU operator has the following
properties.
1) Theorem 1 (Commutativity).
If I [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is any
permutation of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
2) Theorem 2 (Idempotency)
If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all j,
then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
3) Theorem 3 (Monotonicity).
If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all j,
then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The IGBWHU operator only focuses on the importance of each interval
grey linguistic variable itself, and IGBOWHU operator only weights the
position of each interval grey linguistic variable, therefore, both of
them have certain one-sidedness. In order to overcome the above
weaknesses, the interval grey linguistic variable hybrid weighted
armonic aggregation operator is defined as follows:
Definition 11: Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] be the group of interval grey linguistic variables, and IGBHWHU :
[[OMEGA].sup.n] [right arrow] [OMEGA], if
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
where [OMEGA] is the set of all the interval grey linguistic
variables, [omega] = ([[omega].sub.1], [[omega].sub.2], ...,
[[omega].sub.n]) is the weight vector associated with the function
IGBHWHU, and [n.summation over (j=1)] [[omega].sub.j] = 1. [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] is the weight vector of [??] (j =
1,2, ..., n), and [??] is the interval grey linguistic variable
represented as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
([[sigma].sub.1], [[sigma].sub.2], ..., [[sigma].sub.n]) is a
permutation of (1,2, ..., n), and [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII], for any j, we can get [MATHEMATICAL EXPRESSION
NOT REPRODUCIBLE IN ASCII], then IGBHWHU is called the interval grey
linguistic hybrid weighted harmonic aggregation (IGLHWHA)operator.
Example 3. Assume [omega] = (1/8,3/8,3/8,1/8) (calculated by Eq.
23) and
[??] =([s.sub.2], [0.2,0.4]), [??] =([s.sub.1], [0.1,0.3]), [??]
=([s.sub.5], [0.4,0.5]), [??] =([s.sub.4], [0.3,0.6]); [??]
=([s.sub.3][0.3,0.4]), [??] =([s.sub.4], [0.4,0.4]), [??] =([s.sub.2],
[0.5,0.7J), [??] =([S.sub.4], [0.2,0.6]).
Firstly, calculate the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Secondly, calculate the expectation value of interval grey
linguistic variables [??], [??], [??] and [??] according to Eq. (17).
Suppose [rho](y) = [y.sup.2], then [f.sub.[rho]]([a,b]) = b + 2a/3.
So,
I([??]) = [s.sub.0.67] x [f.sub.[rho]] ([(1 - 0.4), (1 - 0.3)]) =
[s.sub.0.67] x + 0.7 + 2 x 0.6/3 = [s.sub.0.42];
I([??]) = [s.sub.0.25] x [f.sub.[rho]] ([(1 - 0.4), (1 - 0.4)]) =
[s.sub.0.25] x + 0.6 + 2 x 0.6/3 = [s.sub.0.15];
I([??]) = [s.sub.0.25] x [f.sub.[rho]] ([(1 - 0.7), (1 - 0.5)]) =
[s.sub.0.25] x + 0.7 + 2 x 0.6/3 = [s.sub.0.92];
I([??]) = [s.sub.1] x [f.sub.[rho]] ([(1 - 0.6), (1 - 0.3)]) =
[s.sub.1] x + 0.7 + 2 x 0.6/3 = [s.sub.0.5];
Thirdly, we rank the [??], [??], [??] and [??], the results are
shown as follows:
[[sigma].sub.1] = 3, [[sigma].sub.2] = 4, [[sigma].sub.3] = 1,
[[sigma].sub.4] = 2.
Finally, By Definition 9, we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Obviously, IGBWHU operator and IGBOWHU operator are the special
cases of the IGBHWHU operator, which not only shows the importance of
interval grey linguistic variables themselves, but also shows the
importance of the position of the interval grey linguistic variables.
3. The multi-attribute group decision making method based on the
IGLHWHA operator
3.1. The description of multiple attribute group decision making
problem based on the interval grey linguistic variables
Let E = {[e.sub.1], [e.sub.2], ..., [e.sub.p]} be the experts set
in the group decision making, A = {[A.sub.1], [A.sub.2], ..., [A.sub.m]}
be the set of alternatives, and C = ([C.sub.1], [C.sub.2], ...,
[C.sub.n]} be the attribute set with respect to the alternatives.
Supposed that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the
attribute value in the attribute [C.sub.j] with respect to the
alternative [A.sub.i], given by expert [e.sub.k], and [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] is the decision making matrix
given by the expert [e.sub.k], and [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII] is the attribute weight, where [t.sub.kij],
[[eta].sub.jk] [member of] S, S is the linguistic label. Let [lambda] =
[[[lambda].sub.1], [lambda].sub.2], ..., [[lambda].sub.p]) be the
experts weight, and [p.summation over (k=1)][[lambda].sub.k] = 1. The
attribute weight is unknown. We can rank the order of the alternatives
based on the given information.
3.2. Decision making steps
1) Aggregate the evaluation information of each expert
According to the different attributes' attribute values and
weights which were given by different experts under different
alternatives, we can aggregate the attribute values and weights into
group decision making information. Based on the [IGBWHU.sub.[lambda]]
operator, we can get the group decision making matrix [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] and the group decision making
weight vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
2) Calculate the comprehensive evaluation value of each alternative
We utilize the IGBHWHU operator to calculate the comprehensive
evaluation value of each alternative [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII], where [omega] = ([[omega].sub.1],
[[omega].sub.1], ..., [[omega].sub.n]) is the weight vector associated
with the function, and [n.summation over (j=1)] [[omega].sub.j] = 1;
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the weight
vector of [??] (j = 1,2, ..., n), ([[omega].sub.1], [[omega].sub.2],
..., [[omega].sub.n]) is a permutation of (1,2, ..., n),
and[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for any j, we
can get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
3) Rank the alternatives
Because Z[??] is the interval grey linguistic variables, we can get
the ranking alternatives by the expectation value I(Z). The larger the
value I(Z) is, the better the alternative is. The flowchart of this
method is shown in Fig. 1.
[FIGURE 1 OMITTED]
4. Practical examples
To evaluate the performance of the proposed method, practical
examples are presented as follows: now there are four enterprises
{[A.sub.1], [A.sub.2], [A.sub.3], [A.sub.4]}, the object is to evaluate
the technological innovation ability of the enterprises.
The first step is to develop the evaluation criterion for the
project. The criterion is shown as follows: the ability of innovative
resources investment ([C.sub.1]), the ability of innovation management
([C.sub.2]), the ability of innovation tendency ([C.sub.3]) and the
ability of research and development ([C.sub.4]). Based on the four
criterions, the three experts {[e.sub.1], [e.sub.2], [e.sub.3]} are
invited to evaluate the technological innovation ability of the four
enterprises. Supposed that X = (0.4, 0.32, 0.28) be the weight vector
about the three experts, and the evaluating values given by the experts
adopting interval grey linguistic variables are shown in Tables 1, 2 and
3, and the criterion weight values are shown in Table 4. Let S =
([s.sub.1], [s.sub.2], [s.sub.3], [s.sub.4], [s.sub.5], [s.sub.6],
[s.sub.7]) be the linguistic label. The problem is to rank the four
enterprises based on their technological innovation ability.
The evaluation steps used in this paper are proposed as follows:
(1) Based on the Eq. (19), to aggregate the evaluation information
(shown in Tables 1, 2, 3 and 4) about the experts {[e.sub.1], [e.sub.2],
[e.sub.3]}, then we can get the group decision making matrix [??] and
attribute weight vector [??]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(2) Calculate [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
3) Calculate the comprehensive evaluation values of each
alternative
If the BUM function is [rho](y) = [y.sup.2], then
[f.sub.[rho]]([a,b]) = b + 2a/3. The position vector is determined by
the equation (23), and [omega] = (1/3,3/8,3/8,1/8). According to the
equation (24), we can get the comprehensive evaluation values of each
alternative:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
4) Rank the alternatives.
We select the BUM function [rho](y) = [y.sup.2]; the expectation
value I([??]) is calculated by Eq. (17).
I([??]) = [s.sub.50.596], I([??]) = [s.sub.50.588], I([??]) =
[s.sub.50.632], I([??]) = [s.sub.50.490].
So the orders of technological innovation ability of the four
enterprises {[A.sub.1], [A.sub.2], [A.sub.3], [A.sub.4]} are shown as
follows:
[A.sub.3] > [A.sub.1] > [A.sub.2] > [A.sub.4].
5) Discuss.
In order to illustrate the effect of the position weight vector co
on decision making of this example, we use the different value [omega]
to rank the alternatives, and the ranking results are shown as follows:
1) For [omega] = (1/4,1/4,1/4,1/4), the orders are [A.sub.3] >
[A.sub.2] > [A.sub.1] > [A.sub.4].
2) For [omega] = (2/5,1/5,1/5,1/5) which can be calculated by Eq.
(22) in [alpha] = 0.2, the orders are [A.sub.3] > [A.sub.1] >
[A.sub.2] > [A.sub.4].
3) For [omega] = (3/8,1/8,1/8,3/8), the orders are [A.sub.3] >
[A.sub.2] > [A.sub.1] > [A.sub.4].
These show that the position weight vector [omega] has a certain
impact on ranking.
In addition, in order to verify the validity of the method proposed
in this paper, we use the method proposed by Meng et al. (2007) to sort
this example. Because the method proposed by Meng et al. (2007) is for
grey fuzzy decision making problems in which the fuzzy part and the grey
part of grey fuzzy numbers take the form of the interval numbers, and in
this paper, the fuzzy part of grey fuzzy numbers takes the form of the
linguistic variables, and the grey part takes the form of the interval
numbers. So in order to use the method proposed by Meng et al. (2007),
we change the linguistic variables of fuzzy part of grey fuzzy numbers
to interval numbers according to Liu and Meng (2009) firstly. The
ranking result is shown as follows:
[A.sub.3] > [A.sub.2] > [A.sub.1] > [A.sub.4].
Obviously, two methods have the same ranking results; this verifies
the validity of the method in this paper.
In order to further illustrate the validity of the proposed method,
we use the evaluation data presented by Meng et al. (2007). Firstly, we
convert the interval numbers to the linguistic variables by the methods
proposed by Liu (2009), and then we can use the methods in this paper.
By calculating, we get the ranking of 4 alternatives. It is shown as
follows:
[y.sub.2] > [y.sub.4] > [y.sub.1] > [y.sub.3].
It is the same as the ranking result produced in Meng et al.
(2007).
Conclusions
The traditional grey fuzzy decision making methods are generally
suitable for decision making the information taking the form of crisp
numbers, or interval numbers in the fuzzy part and the grey part of grey
fuzzy numbers, and yet they will fail in dealing with the linguistic
information of grey fuzzy numbers. In this paper, with respect to
multiple attribute group decision making (MAGDM) problems in which the
attribute values and the attribute weights take the form of the interval
grey linguistic variables, some new group decision making analysis
methods are developed. Firstly, the concept of interval grey linguistic
variables is proposed, in which the fuzzy part takes the form of the
linguistic variables, and the grey part takes the form of the interval
numbers. Then, the operation rules of interval grey linguistic variables
are defined, and some operators (such as interval grey linguistic
weighted harmonic aggregation (IGLWHA) operator, interval grey
linguistic ordered weighted harmonic aggregation (IGLOWHA) operator, and
interval grey linguistic hybrid weighted harmonic aggregation (IGLHWHA)
operator) are proposed to solve the group decision making problems.
Finally, the computational results from an illustrative example have
shown that the proposed approach is feasible and effective for the
group-decision making problems, and it is easier to understand and
implement. This study promotes the development of the theory and method
of grey fuzzy multiple attribute decision making and provides a new idea
to solve the grey fuzzy multiple attribute decision making. Because
interval grey linguistic variables take into account the greyness and
fuzziness of the same problem, they can get more rational
decision-making results. However, in order to do this, we must get more
data, and sometimes, it is very difficult to obtain these data. In the
future, we will research data acquisition methods, and continue working
in the extension and application of the developed operators and methods
to other domains.
Caption: Fig. 1. The flowchart of the proposed method
doi:10.3846/20294913.2013.821685
Acknowledgement
This paper is supported by the National Natural Science Foundation
of China (No. 71271124), the Humanities and Social Sciences Research
Project of Ministry of Education of China (No. 10YJA630073, No.
09YJA630088), the Natural Science Foundation of Shandong Province (No.
ZR2011FM036), National High Technology Research and Development Program
(863 Program) Foundation (No. 2011AA100700) and Technology innovation
fund project of high technology-based SMEs(2012-34-2). The author also
would like to express appreciation to the anonymous reviewers and
Associate Editor for their very helpful comments that improved the
paper.
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Fang JIN (a), Peide LIU (b), Xin ZHANG (c)
(a) The Mathematical and Economic Research Institute, Shandong
University of Finance and Economics, Jinan Shandong 250014, China
(b,c) School of Management Science and Engineering, Shandong
University of Finance and Economics, Jinan Shandong 250014, China
Corresponding author Peide Liu
E-mail: peide.liu@gmail.com
Received 18 August 2011; accepted 28 January 2012
Fang JIN has a PhD and is a chief research worker at The
Mathematical and Economic Research Institute in Shandong University of
Finance and Economics, China. She is the author of more than 30 research
papers. Her research interests include decision-making theory, expert
systems, and their applications.
Peide LIU has a PhD and is a chief research worker at School of
Management Science and Engineering, Shandong University of Finance and
Economics, China. He has authored or coauthored more than 90
publications. His research interests include aggregation operators,
fuzzy logics, fuzzy decision making, and their applications.
Xin ZHANG has a PhD and is a Director of the School of Management
Science and Engineering at The Mathematical and Economic Research
Institute at Shandong University of Finance and Economics, China. He is
the author of more than 50 research papers. His research interests
include decision-making theory, expert systems, and their applications.
Table 1. The attribute values of each attribute with respect
to four enterprises given by expert [e.sub.1]
Enterprises Attribute ([C.sub.1]) Attribute ([C.sub.2])
A1 ([s.sub.5], [0.2,0.3]) ([s.sub.2], [0.4,0.4])
A2 ([s.sub.4], [0.4,0.4]) ([s.sub.5], [0.4,0.5])
A3 ([s.sub.3], [0.2,0.3]) ([s.sub.4], [0.2,0.3])
A4 ([s.sub.5], [0.5,0.6]) ([s.sub.2], [0.2,0.2])
Enterprises Attribute ([C.sub.3]) Attribute ([C.sub.4])
A1 ([s.sub.5], [0.5,0.5]) ([s.sub.3], [0.2,0.4])
A2 ([s.sub.5], [0.1,0.2]) ([s.sub.4], [0.5,0.5])
A3 ([s.sub.5], [0.3,0.3]) ([s.sub.5], [0.2,0.3])
A4 ([s.sub.5], [0.2,0.4]) ([s.sub.3], [0.3,0.4])
Table 2. The attribute values of each attribute with respect
to four enterprises given by expert [e.sub.2]
Enterprises Attribute ([C.sub.1]) Attribute ([C.sub.2])
A1 ([s.sub.4], [0.1,0.3]) ([s.sub.3], [0.2,0.3])
A2 ([s.sub.5], [0.4,0.5]) ([s.sub.3], [0.3,0.4])
A3 ([s.sub.4], [0.2,0.4]) ([s.sub.4], [0.2,0.3])
A4 ([s.sub.5], [0.3,0.4]) ([s.sub.4], [0.4,0.5])
Enterprises Attribute ([C.sub.3]) Attribute ([C.sub.4])
A1 ([s.sub.3], [0.2,0.2]) ([s.sub.6], [0.4,0.5])
A2 ([s.sub.4], [0.2,0.4]) ([s.sub.3], [0.2,0.3])
A3 ([s.sub.2], [0.4,0.4]) ([s.sub.4], [0.3,0.3])
A4 ([s.sub.4], [0.3,0.4]) ([s.sub.4], [0.2,0.4])
Table 3. The attribute values of each attribute with respect
to four enterprises given by expert [e.sub.3]
Enterprises Attribute ([C.sub.1]) Attribute ([C.sub.2])
A1 ([s.sub.5], [0.2,0.4]) ([s.sub.3], [0.3,0.3])
A2 ([s.sub.4], [0.3,0.3]) ([s.sub.5], [0.3,0.4])
A3 ([s.sub.4], [0.2,0.3]) ([s.sub.5], [0.3,0.4])
A4 ([s.sub.3], [0.2,0.3]) ([s.sub.3], [0.1,0.3])
Enterprises Attribute ([C.sub.3]) Attribute ([C.sub.4])
A1 ([s.sub.4], [0.4,0.5]) ([s.sub.4], [0.2,0.3])
A2 ([s.sub.2], [0.1,0.2]) ([s.sub.3], [0.1,0.2])
A3 ([s.sub.1], [0.1,0.2]) ([s.sub.4], [0.2,0.3])
A4 ([s.sub.4], [0.3,0.4]) ([s.sub.5], [0.4,0.5])
Table 4. The attribute weight value given by experts
Experts Attribute ([C.sub.1]) Attribute ([C.sub.2])
[e.sub.1] ([s.sub.5], [0.2,0.3]) ([s.sub.3], [0.1,0.2])
[e.sub.2] ([s.sub.3], [0.2,0.3]) ([s.sub.4], [0.2,0.4])
[e.sub.3] ([s.sub.4], [0.2,0.2]) ([s.sub.3], [0.1,0.2])
Experts Attribute ([C.sub.3]) Attribute ([C.sub.4])
[e.sub.1] ([s.sub.2], [0.3,0.4]) ([s.sub.3], [0.2,0.3])
[e.sub.2] ([s.sub.3], [0.1,0.2]) ([s.sub.3], [0.1,0.2])
[e.sub.3] ([s.sub.2], [0.1,0.2]) ([s.sub.3], [0.2,0.3])
