The Interval-Valued Intuitionistic Fuzzy Optimized Weighted
Transcript of The Interval-Valued Intuitionistic Fuzzy Optimized Weighted
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 981762, 10 pageshttp://dx.doi.org/10.1155/2013/981762
Research ArticleThe Interval-Valued Intuitionistic Fuzzy Optimized WeightedBonferroni Means and Their Application
Ya-ming Shi and Jian-min He
School of Economics and Management, Southeast University, Nanjing 211189, China
Correspondence should be addressed to Ya-ming Shi; [email protected]
Received 3 May 2013; Accepted 18 June 2013
Academic Editor: Guangchen Wang
Copyright ยฉ 2013 Y.-m. Shi and J.-m. He. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
We investigate and propose two new Bonferroni means, that is, the optimized weighted BM (OWBM) and the generalizedoptimized weighted BM (GOWBM), whose characteristics are to reflect the preference and interrelationship of the aggregatedarguments and can satisfy the basic properties of the aggregation techniques simultaneously. Further, we propose the interval-valued intuitionistic fuzzy optimized weighted Bonferroni mean (IIFOWBM) and the generalized interval-valued intuitionisticfuzzy optimized weighted Bonferroni mean (GIIFOWBM) and detailed study of their desirable properties such as idempotency,monotonicity, transformation, and boundary. Finally, based on IIFOWBMandGIIFOWBM,we give an approach to group decisionmaking under the interval-valued intuitionistic fuzzy environment and utilize a practical case involving the assessment of a set ofagroecological regions in Hubei Province, China, to illustrate the developed methods.
1. Introduction
As a useful aggregation technique, the Bonferronimean (BM)can capture the interrelationship between input argumentsand has been a hot research topic recently. Bonferroni [1]originally introducedamean-typeaggregationoperator, calledthe Bonferroni mean, whose prominent characteristic is thatit cannot only consider the importance of each criterion butalso reflect the interrelationship of the individual criterion.Recently, Yager [2] extended the BM by two mean-type op-erators, such as the Choquet integral operator [3] and theordered weighted averaging operator [4], as well as associatesdiffering importance with the arguments. Mordelova andRuckschlossova [5] also investigated the generalizations ofBM referred to as ABC-aggregation functions. Beliakov et al.[6] further extended the BM by considering the correlationsof any three aggregated arguments instead of any two andproposed the generalized Bonferroni mean (GBM).
Nevertheless, the arguments suitable to be aggregated bythe BM and GBM can only take the forms of crisp numbers.In the real world, due to the increasing complexity of thesocioeconomic environment and the lack of knowledge anddata, crisp data are sometimes unavailable. Thus, the input
arguments may be more suitable with representation of fuzzyformats, such as fuzzy number [7], interval-valued fuzzynumber [8], intuitionistic fuzzy value [9], interval-valuedintuitionistic fuzzy value [10], and hesitant fuzzy element[11]. Thus, Xu and Yager [12] introduced the intuitionisticfuzzy Bonferroni mean (IFBM) and the intuitionistic fuzzyweighted Bonferroni mean (IFWBM). Xu and Chen [13]further proposed the interval-valued intuitionistic fuzzyBonferroni mean (IIFBM) and the interval-valued intuition-istic fuzzy weighted Bonferroni mean (IIFWBM). Xia et al.[14] proposed the generalized intuitionistic fuzzy Bonferronimeans. Zhou and He [15] developed some geometric Bonfer-roni Means. Furthermore, Beliakov and James [16] definedBonferroni means over lattices which is a new viewpoint.Recently, Zhou andHe [17] constructed an intuitionistic fuzzyweighted Bonferroni mean, Xia et al. [18] further investi-gated the generalized geometric Bonferroni means, Beliakovand James [19] extend the generalized Bonferroni means toAtanassov orthopairs, and so on.
The desirable characteristic of the BM is its capabilityto capture the interrelationship between input arguments.However, the classical BM andGBM, even the extended BMs,cannot reflect the interrelationship between the individual
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criterion and other criteria. To deal with these issues, in thispaper, we propose the optimized weighted Bonferroni mean(OWBM) and the generalized optimized weighted Bonfer-roni mean (GOWBM), whose characteristics are to reflectthe preference and interrelationship of the aggregated argu-ments and can satisfy the basic properties of the aggrega-tion techniques simultaneously. Further, we have proposedthe interval-valued intuitionistic fuzzy optimized weightedBonferroni mean (IIFOWBM) and the generalized interval-valued intuitionistic fuzzy optimized weighted Bonferronimean (GIIFOWBM) and detailed study of their desirableproperties such as idempotency, monotonicity, transforma-tion, and boundary.
The remainder of this paper is organized as follows.We briefly review some basic definitions of the interval-valued intuitionistic fuzzy values and Bonferroni mean.Then, in Section 3, we propose two BMs including OWBMand GOWBM and study their properties. Furthermore, inSection 4, we develop the IIFOWBM and GIIFOWBM oper-ators, and their idempotency, monotonicity, transformation,and boundary are also investigated. A practical example isprovided in Section 5 to demonstrate the application of theOWBM, GOWBM, and two interval-valued intuitionisticfuzzy BMs. The paper ends in Section 6 with concludingremarks.
2. Some Basic Concepts
2.1. Interval-Valued Intuitionistic Fuzzy Values. In the follow-ing, we introduce the basic concepts and operations relatedto interval-valued intuitionistic fuzzy value [20], which isan extended definition of the intuitionistic fuzzy value andinterval-valued intuitionistic fuzzy set [21, 22].
Definition 1 (see [10]). Let ๐ = (๐ฅ1, ๐ฅ
2, . . . , ๐ฅ
๐) be fixed. An
interval-valued intuitionistic fuzzy set (IIFS) ๐ด in ๐ can bedefined as
๐ด = {(๐ฅ, ๐๐ด(๐ฅ) , ]
๐ด(๐ฅ)) | ๐ฅ โ ๐} , (1)
where ๐๐ด(๐ฅ) โ [0, 1] and ]
๐ด(๐ฅ) โ [0, 1] satisfy sup ๐
๐ด(๐ฅ) +
sup ]๐ด(๐ฅ) โค 1 for all ๐ฅ
๐โ ๐ and ๐
๐ด(๐ฅ) and ]
๐ด(๐ฅ) are,
respectively, called the degree of membership and the degreeof nonmembership of the element ๐ฅ
๐โ ๐ to ๐ด.
Definition 2 (see [20]). Let ๐ด = {(๐ฅ, ๐๐ด(๐ฅ), ]
๐ด(๐ฅ)) | ๐ฅ โ ๐}
be an IIFS; the pair (๐๐ด(๐ฅ), ]
๐ด(๐ฅ)) is called an interval-valued
intuitionistic fuzzy value (IIFV).For computational convenience, an IIFV can be denoted
by ([๐, ๐], [๐, ๐]), with the condition that [๐, ๐] โ [0, 1],[๐, ๐] โ [0, 1], and ๐ + ๐ โค 1. Furthermore, based onthe operations of interval-valued intuitionistic fuzzy values[21, 22], Xu [23] defined some operations of IIFNs asfollows.
Definition 3 (see [23]). Letting ๏ฟฝ๏ฟฝ = ([๐, ๐], [๐, ๐]), ๏ฟฝ๏ฟฝ1=
([๐1, ๐
1], [๐
1, ๐
1]), and ๏ฟฝ๏ฟฝ
2= ([๐
2, ๐
2], [๐
2, ๐
2]) be three IIFVs,
then the following operational laws are valid:
(1) ๏ฟฝ๏ฟฝ1โ ๏ฟฝ๏ฟฝ
2= ([๐
1+ ๐
2โ ๐
1๐2, ๐
1+ ๐
2โ ๐
1๐2], [๐
1๐2, ๐
1๐2]),
(2) ๏ฟฝ๏ฟฝ1โ ๏ฟฝ๏ฟฝ
2= ([๐
1๐2, ๐
1๐2], [๐
1+ ๐
2โ ๐
1๐2, ๐
1+ ๐
2โ ๐
1๐2]),
(3) ๐๏ฟฝ๏ฟฝ = ([1 โ (1 โ ๐)๐, 1 โ (1 โ ๐)๐], [๐๐, ๐๐]), ๐ > 0,
(4) ๏ฟฝ๏ฟฝ๐ = ([๐๐, ๐๐], [1 โ (1 โ ๐)๐, 1 โ (1 โ ๐)๐]), ๐ > 0.
Then, Xu [23] introduced the score function ๐ (๏ฟฝ๏ฟฝ) = (๐ โ๐+๐โ๐)/2 and the accuracy function โ(๏ฟฝ๏ฟฝ) = (๐+๐+๐+๐)/2to calculate the score value and accuracy degree of IIFV ๏ฟฝ๏ฟฝ =([๐, ๐], [๐, ๐]) and gave an order relation between two IIFNs๏ฟฝ๏ฟฝ1and ๏ฟฝ๏ฟฝ
2as follows:
(1) if ๐ (๏ฟฝ๏ฟฝ1) < ๐ (๏ฟฝ๏ฟฝ
2), then ๏ฟฝ๏ฟฝ
1< ๏ฟฝ๏ฟฝ
2;
(2) if ๐ (๏ฟฝ๏ฟฝ1) = ๐ (๏ฟฝ๏ฟฝ
2), then
(i) if โ(๏ฟฝ๏ฟฝ1) < โ(๏ฟฝ๏ฟฝ
2), then ๏ฟฝ๏ฟฝ
1< ๏ฟฝ๏ฟฝ
2;
(ii) if โ(๏ฟฝ๏ฟฝ1) = โ(๏ฟฝ๏ฟฝ
2), then ๏ฟฝ๏ฟฝ
1โผ ๏ฟฝ๏ฟฝ
2.
2.2. Bonferroni Means
Definition 4 (see [1]). Let ๐, ๐ โฅ 0 and ๐๐(๐ = 1, 2, . . . , ๐) a
collection of nonnegative numbers. If
BM๐,๐(๐
1, ๐
2, . . . , ๐
๐) = (
1
๐(๐ โ 1)
๐
โ
๐,๐=1
๐ = ๐
๐๐
๐๐๐
๐)
1/(๐+๐)
,
(2)
then BM๐,๐ is called the Bonferroni mean (BM).
Definition 5 (see [6]). Let ๐, ๐, ๐ โฅ 0 and ๐๐(๐ = 1, 2, . . . , ๐)
be a collection of nonnegative numbers. If
GBM๐,๐,๐(๐
1, ๐
2, . . . , ๐
๐)
= (1
๐(๐ โ 1)(๐ โ 2)
๐
โ
๐,๐,๐=1
๐ = ๐ = ๐
๐๐
๐๐๐
๐๐๐
๐)
1/(๐+๐+๐)
,
(3)
then GBM๐,๐,๐ is called the generalized Bonferroni mean(GBM).
To deal with intuitionistic fuzzy value and hesitant fuzzyvalue, Xu and Yager [12] extended these BMs to fuzzy envi-ronment and gave the following concepts.
Definition 6 (see [12]). Let ๐ผ๐(๐ = 1, 2, . . . , ๐) be a set of
intuitionistic fuzzy values.The intuitionistic fuzzy Bonferronimean (IFBM), the intuitionistic fuzzy weighted Bonferronimean (IFWBM), and the generalized intuitionistic fuzzy
Journal of Applied Mathematics 3
weighted Bonferroni mean (GIFWBM) are, respectively,defined as
IFBM๐,๐(๐ผ
1, ๐ผ
2, . . . , ๐ผ
๐)
= (1
๐ (๐ โ 1)
๐
โจ
๐,๐=1,
๐ = ๐
(๐ผ๐
๐โ ๐ผ
๐
๐))
1/(๐+๐)
,
IFWBM๐,๐(๐ผ
1, ๐ผ
2, . . . , ๐ผ
๐)
= (1
๐ (๐ โ 1)
๐
โจ
๐,๐=1,
๐ = ๐
((๐ค๐๐ผ๐
๐) โ (๐ค
๐๐ผ๐
๐)))
1/(๐+๐)
,
GIFBM๐,๐,๐(๐ผ
1, ๐ผ
2, . . . , ๐ผ
๐)
= (1
๐ (๐ โ 1) (๐ โ 2)
๐
โจ
๐,๐,๐=1,
๐ = ๐ = ๐
((๐ค๐๐ผ๐
๐) โ (๐ค
๐๐ผ๐
๐) โ (๐ค
๐๐ผ๐
๐)))
1/(๐+๐+๐)
.
(4)
However, it is noted that the above BMs could noteffectively aggregate the general fuzzy value, that is, interval-valued intuitionistic fuzzy value. On the other hand, theabove BMs just consider the whole correlationship betweenthe criterion ๐
๐and all criteria โ๐
๐=1๐๐
๐โ โ
๐
๐=1๐๐
๐and cannot
reflect the interrelationship between the individual criterion๐๐and other criteria V๐,๐
๐which is the main advantage of the
BM. To overcome this drawback, we propose the followingOWBM, GOWBM, IIFOWBM, and GIIFOWBM operators.
3. The Optimized Weighted BM andIts Generalized Form
The BM and GBM, including IFWBM and GIFWBM, justconsider the whole correlationship between the criterion๐๐and all criteria and cannot reflect the interrelationship
between the individual criterion ๐๐and other criteria V๐,๐
๐
which is the main advantage of the BM. To deal with theseissues, in the following subsections, we propose the optimizedweighted versions of BM and its generalized form, that is,the optimized weighted BM (OWBM) and the generalizedoptimized weighted BM (GOWBM). Based on the Bonfer-roni mean, we can define the following optimized weight-ed BM (OWBM) and generalized optimized weighted BM(GOWBM).
Definition 7. Let ๐, ๐ โฅ 0 and ๐๐(๐ = 1, 2, . . . , ๐) be a
collection of nonnegative numbers with the weight vector๐ค = (๐ค
1, ๐ค
2, . . . , ๐ค
๐) such that ๐ค
๐โฅ 0 and โ๐
๐=1๐ค๐= 1. Then
the optimized weighted Bonferroni mean (OWBM) can bedefined as follows:
OWBM๐,๐(๐
1, ๐
2, . . . , ๐
๐) = (
๐
โ
๐,๐=1
๐ = ๐
๐ค๐๐ค๐
1 โ ๐ค๐
๐๐
๐๐๐
๐)
1/(๐+๐)
.
(5)
Definition 8. Let ๐, ๐, ๐ โฅ 0 and ๐๐(๐ = 1, 2, . . . , ๐) a
collection of nonnegative numbers with the weight vector๐ค = (๐ค
1, ๐ค
2, . . . , ๐ค
๐) such that ๐ค
๐โฅ 0 and โ๐
๐=1๐ค๐= 1.
Then the generalized optimized weighted Bonferroni mean(GOWBM) can be defined as follows:
GOWBM๐,๐,๐(๐
1, ๐
2, . . . , ๐
๐)
= (
๐
โ
๐,๐,๐=1
๐ = ๐ = ๐
๐ค๐๐ค๐๐ค๐
(1 โ ๐ค๐) (1 โ ๐ค
๐โ ๐ค
๐)
๐๐
๐๐๐
๐๐๐
๐)
1/(๐+๐+๐)
.
(6)
Furthermore,wecan transformtheOWBMandGOWBMinto the interrelationship between OWBM and GOWBMforms as follows:
OWBM๐,๐(๐
1, ๐
2, . . . , ๐
๐)= (
๐
โ
๐=1
๐ค๐๐๐
๐
๐
โ
๐=1
๐ = ๐
๐ค๐
1 โ ๐ค๐
๐๐
๐)
1/(๐+๐)
,
(7)
GOWBM๐,๐,๐(๐
1, ๐
2, . . . , ๐
๐)
= (
๐
โ
๐=1
๐ค๐๐๐
๐
๐
โ
๐=1
๐ = ๐
๐ค๐
1 โ ๐ค๐
๐๐
๐
๐
โ
๐=1
๐ = ๐ = ๐
๐ค๐
(1 โ ๐ค๐โ ๐ค
๐)
๐๐
๐)
1/(๐+๐+๐)
.
(8)
According to (6)-(7), we see that the terms โ๐
๐=1,๐ = ๐(๐ค
๐/
(1 โ ๐ค๐))๐
๐
๐and โ๐
๐,๐,๐=1,๐ = ๐ = ๐(๐ค
๐๐ค๐๐ค๐/(1 โ ๐ค
๐)(1 โ ๐ค
๐โ
๐ค๐))๐
๐
๐๐๐
๐are the weighted power average satisfaction of all
criteria except ๐ด๐, and โ๐
๐=1,๐ = ๐(๐ค
๐/(1 โ ๐ค
๐)) = 1 and
โ๐
๐,๐,๐=1,๐ = ๐ = ๐(๐ค
๐๐ค๐๐ค๐/(1 โ ๐ค
๐)(1 โ ๐ค
๐โ ๐ค
๐)) = 1. If we,
respectively, denote the above terms as ๐ข๐๐and ๐ข๐
๐V๐๐, thus
OWBM๐,๐(๐
1, ๐
2, . . . , ๐
๐) = (
๐
โ
๐=1
๐ค๐๐๐
๐๐ข๐
๐)
1/(๐+๐)
,
GOWBM๐,๐,๐(๐
1, ๐
2, . . . , ๐
๐) = (
๐
โ
๐=1
๐ค๐๐๐
๐๐ข๐
๐V๐๐)
1/(๐+๐+๐)
.
(9)
Here then ๐ข๐๐and ๐ข๐
๐V๐๐are the weighted power average
satisfaction to all criteria except ๐ด๐, which represents the
interrelationship between the individual criterion ๐๐and
other criteria ๐๐(๐ = ๐). This is similar to the BM.
4 Journal of Applied Mathematics
4. Two Interval-Valued IntuitionisticFuzzy BM Operators Based on OWBMand GOWBM
To aggregate the fuzzy information, Xu and Yager [12] pro-posed the IFBM and IFWBM. However, it is noted that theabove BMs could not effectively aggregate the general fuzzyvalue, that is, interval-valued intuitionistic fuzzy value. Thus,we further propose two interval-valued intuitionistic fuzzyBM operators to aggregate the interval-valued intuitionisticfuzzy correlated information based on the optimized weight-ed and generalized optimized weighted BMs, respectively,that is, IIFOWBM and GIIFOWBM.
Definition 9. Let ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐]) be a collection of IIFVs
with the weight vector ๐ค = (๐ค1, ๐ค
2, . . . , ๐ค
๐) such that ๐, ๐ โฅ
0, ๐ค๐โฅ 0, and โ๐
๐=1๐ค๐= 1. If
IIFOWBM๐,๐(๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐)
= (
๐
โจ
๐,๐=1,
๐ = ๐
๐ค๐๐ค๐
1 โ ๐ค๐
(๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ
๐
๐))
1/(๐+๐)
,
(10)
then IIFOWBM๐,๐ is called the interval-valued intuitionisticfuzzy optimized weighted Bonferroni mean (IIFOWBM).
On the basis of the operational laws of IIFVs, we have thefollowing.
Theorem 10. Letting ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐]) (๐ = 1, 2, . . . , ๐) be
a collection of IIFVs with the weight vector (๐ค1, ๐ค
2, . . . , ๐ค
๐),
such that๐, ๐ โฅ 0,๐ค๐โฅ 0, andโ๐
๐=1๐ค๐= 1, then the aggregated
value by using the IIFOWBM is also an IIFV, and
IIFOWBM๐,๐(๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐)
= (
๐
โจ
๐,๐=1,
๐ = ๐
๐ค๐๐ค๐
1 โ ๐ค๐
(๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ
๐
๐))
1/(๐+๐)
= (
[[[
[
(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ ๐๐
๐๐๐
๐)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
,
(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ ๐๐
๐๐๐
๐)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
]]]
]
,
[[[
[
1 โ(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ (1 โ ๐๐)๐
ร (1 โ ๐๐)๐
)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
,
1 โ(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ (1 โ ๐๐)๐
ร (1 โ๐๐)๐
)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
]]]
]
) .
(11)
Proof. By the operational laws for IIFVs, we get
๏ฟฝ๏ฟฝ๐
๐= ([๐
๐
๐, ๐
๐
๐] , [1 โ (1 โ ๐
๐)๐
, 1 โ (1 โ ๐๐)๐
]) ,
๏ฟฝ๏ฟฝ๐
๐= ([๐
๐
๐, ๐
๐
๐] , [1 โ (1 โ ๐
๐)๐
, 1 โ (1 โ ๐๐)๐
]) ,
๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ
๐
๐
= ([๐๐
๐๐๐
๐, ๐
๐
๐๐๐
๐] ,
[1 โ (1 โ ๐๐)๐
(1 โ ๐๐)๐
, 1 โ (1 โ ๐๐)๐
(1 โ ๐๐)๐
]) ,
๐ค๐๐ค๐
1 โ ๐ค๐
๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ
๐
๐
= ([1 โ (1 โ ๐๐
๐๐๐
๐)๐ค๐๐ค๐/(1โ๐ค๐)
, 1 โ (1 โ ๐๐
๐๐๐
๐)๐ค๐๐ค๐/(1โ๐ค๐)
] ,
[(1 โ (1 โ ๐๐)๐
(1 โ ๐๐)๐
)๐ค๐๐ค๐/(1โ๐ค๐)
,
(1 โ (1 โ ๐๐)๐
(1 โ ๐๐)๐
)๐ค๐๐ค๐/(1โ๐ค๐)
]) ,
๐
โจ
๐,๐=1,
๐ = ๐
๐ค๐๐ค๐
1 โ ๐ค๐
๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ
๐
๐
= (
[[[
[
1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ ๐๐
๐๐๐
๐)๐ค๐๐ค๐/(1โ๐ค๐)
,
1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ ๐๐
๐๐๐
๐)๐ค๐๐ค๐/(1โ๐ค๐)
]]]
]
,
[[[
[
๐
โ
๐,๐=1,
๐ = ๐
(1 โ (1 โ ๐๐)๐
(1 โ ๐๐)๐
)๐ค๐๐ค๐/(1โ๐ค๐)
,
๐
โ
๐,๐=1,
๐ = ๐
(1 โ (1 โ ๐๐)๐
(1 โ ๐๐)๐
)๐ค๐๐ค๐/(1โ๐ค๐)]
]]
]
) .
(12)
Journal of Applied Mathematics 5
Therefore,
IIFOWBM๐,๐(๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐)
= (
๐
โจ
๐,๐=1,
๐ = ๐
๐ค๐๐ค๐
1 โ ๐ค๐
(๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ
๐
๐))
1/(๐+๐)
= (
[[[
[
(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ ๐๐
๐๐๐
๐)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
,
(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ ๐๐
๐๐๐
๐)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
]]]
]
,
[[[
[
1โ(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ(1 โ ๐๐)๐
(1 โ ๐๐)๐
)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
,
1 โ(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ (1 โ ๐๐)๐
ร (1 โ ๐๐)๐
)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
]]]
]
),
0 โค(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ๐๐
๐๐๐
๐)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
โค 1,
0 โค (1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ ๐๐
๐๐๐
๐)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
โค 1.
(13)
We have
0โค1 โ(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ (1 โ ๐๐)๐
(1 โ ๐๐)๐
)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
โค 1,
0โค1 โ(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ(1 โ ๐๐)๐
(1 โ ๐๐)๐
)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
โค 1,
(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ ๐๐
๐๐๐
๐)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
+ 1
โ(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ (1 โ ๐๐)๐
(1 โ ๐๐)๐
)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
โค 1
+(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ (1 โ ๐๐)๐
(1 โ ๐๐)๐
)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
โ(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ(1 โ ๐๐)๐
(1 โ ๐๐)๐
)๐ค๐๐ค๐/(1โ๐ค๐)
)
1/(๐+๐)
= 1
(14)
which completes the proof.
Property 1 (idempotency). If all IIFVs ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐])
(๐ = 1, 2, . . . , ๐) are equal, that is, ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐]) = ๏ฟฝ๏ฟฝ =
([๐, ๐], [๐, ๐]), for all ๐, we have
IIFOWBM๐,๐(๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐) = ๏ฟฝ๏ฟฝ = ([๐, ๐] , [๐, ๐]) . (15)
Proof. Since ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐]) = ๏ฟฝ๏ฟฝ = ([๐, ๐], [๐, ๐]), then
IIFOWBM๐,๐(๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐)
= (
๐
โจ
๐,๐=1,
๐ = ๐
๐ค๐๐ค๐
1 โ ๐ค๐
(๏ฟฝ๏ฟฝ๐โ ๏ฟฝ๏ฟฝ
๐))
1/(๐+๐)
= (๏ฟฝ๏ฟฝ๐โ ๏ฟฝ๏ฟฝ
๐)1/(๐+๐)
= ๏ฟฝ๏ฟฝ๐
(16)
which completes the proof of Property 1.
Corollary 11. If ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐]) (๐ = 1, 2, . . . , ๐) is a
collection of the largest IIFVs, that is, ๏ฟฝ๏ฟฝ๐= ๏ฟฝ๏ฟฝ = ([1, 1], [0, 0]),
6 Journal of Applied Mathematics
for all ๐, and (๐ค๐(1), ๐ค
๐(2), . . . , ๐ค
๐(๐)) is precise weight vector of
๏ฟฝ๏ฟฝ๐, then
IIFOWBM๐,๐(๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐)
= (
๐
โจ
๐,๐=1,
๐ = ๐
๐ค๐๐ค๐
1 โ ๐ค๐
(๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ
๐
๐))
1/(๐+๐)
= (
[[[
[
(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ 1))
1/(๐+๐)
,
(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ 1))
1/(๐+๐)
]]]
]
,
[1 โ (1)1/(๐+๐)
, 1 โ (1)1/(๐+๐)
])
= ([1, 1] , [0, 0])
(17)
which is also the largest IIFV.
Property 2 (monotonicity). Let ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐]) and
๐ฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐]) (๐, ๐ = 1, 2, . . . , ๐) be two collections
of IIFVs; if ๐๐โค ๐
๐, ๐
๐โค ๐
๐, ๐
๐โค ๐
๐and ๐
๐โค ๐
๐, for all ๐, then
IIFOWBM๐,๐(๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐)
โค IIFOWBM๐,๐(๐ฝ
1, ๐ฝ
2, . . . , ๐ฝ
๐) .
(18)
Proof. Theproof of Property 2 is similar to Property 10 in [15].
Property 3 (transformation). Letting ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐])
(๐ = 1, 2, . . . , ๐) be a set of IIFVs and (๐ผ1, ๐ผ
2, . . . , ๐ผ
๐) the per-
mutation of (๏ฟฝ๏ฟฝ1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐), then
IIFOWBM๐,๐(๐ผ
1, ๐ผ
2, . . . , ๐ผ
๐)
= IIFOWBM๐,๐(๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐) .
(19)
Proof. Since (๐ผ1, ๐ผ
2, . . . , ๐ผ
๐) is a permutation of (๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2,
. . . , ๏ฟฝ๏ฟฝ๐), then
(
๐
โจ
๐,๐=1,
๐ = ๐
๐ค๐๐ค๐
1 โ ๐ค๐
๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ
๐
๐)
1/(๐+๐)
= (
๐
โจ
๐,๐=1,
๐ = ๐
๐ค๐๐ค๐
1 โ ๐ค๐
๐ผ๐
๐โ ๐ผ
๐
๐)
1/(๐+๐)
.
(20)
Thus,
IIFOWBM๐,๐(๐ผ
1, ๐ผ
2, . . . , ๐ผ
๐)
= IIFOWBM๐,๐(๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐) .
(21)
Definition 12. Let ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐]) (๐ = 1, 2, . . . , ๐)
be a collection of IIFVs with the weight vector ๐ค =
(๐ค1, ๐ค
2, . . . , ๐ค
๐) such that ๐, ๐, ๐ โฅ 0 and โ๐
๐=1๐ค๐= 1. If
GIIFOWBM๐,๐,๐(๏ฟฝ๏ฟฝ
1, . . . , ๏ฟฝ๏ฟฝ
๐)
=(
๐
โจ
๐,๐,๐=1
๐ = ๐ = ๐
๐ค๐๐ค๐๐ค๐
(1 โ ๐ค๐) (1 โ ๐ค
๐โ ๐ค
๐)
(๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ
๐
๐โ ๏ฟฝ๏ฟฝ
๐
๐))
1/(๐+๐+๐)
,
(22)
then GIIFOWBM๐,๐,๐ is called the generalized interval-valued intuitionistic fuzzy optimized weighted Bonferronimean (GIIFOWBM).
Theorem 13. Letting ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐]) (๐ = 1, 2, . . . , ๐) be
a set of IIFVs with the weight vector (๐ค1, ๐ค
2, . . . , ๐ค
๐), such that
๐, ๐, ๐ โฅ 0 andโ๐
๐=1๐ค๐= 1, then the aggregated value by using
the GIIFOWBM is also an IIFV and
GIIFOWBM๐,๐,๐ (๏ฟฝ๏ฟฝ1, ๏ฟฝ๏ฟฝ2, . . . , ๏ฟฝ๏ฟฝ๐)
= (
๐
โจ
๐,๐,๐=1
๐ = ๐ = ๐
๐ค๐๐ค๐๐ค๐
(1 โ ๐ค๐) (1 โ ๐ค๐ โ ๐ค๐)
(๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ๐
๐))
1/(๐+๐+๐)
= (
[[[[
[
(1 โ
๐
โ
๐,๐,๐=1
๐ = ๐ = ๐
(1 โ ๐๐
๐๐๐
๐๐๐
๐)
๐ค๐๐ค๐๐ค๐/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
)
1/(๐+๐+๐)
,
(1 โ
๐
โ
๐,๐,๐=1
๐ = ๐ = ๐
(1 โ ๐๐
๐๐๐
๐๐๐
๐)
๐ค๐๐ค๐๐ค๐/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
)
1/(๐+๐+๐)
]]]]
]
ร
[[[[
[
1 โ(1 โ
๐
โ
๐,๐,๐=1
๐ = ๐ = ๐
(1 โ (1 โ ๐๐)๐(1 โ ๐๐)
๐
ร(1 โ ๐๐)๐)๐ค๐๐ค๐๐ค๐
/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
)
1/(๐+๐+๐)
,
1 โ(1 โ
๐
โ
๐,๐,๐=1
๐ = ๐ = ๐
(1 โ (1 โ ๐๐)๐(1 โ ๐๐)
๐
ร(1 โ ๐๐)๐)๐ค๐๐ค๐๐ค๐
/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
)
1/(๐+๐+๐)
]]]]
]
).
(23)
Journal of Applied Mathematics 7
Proof. By the operational laws for IIFVs, we have
๏ฟฝ๏ฟฝ๐
๐= ([๐
๐
๐, ๐
๐
๐] , [1 โ (1 โ ๐
๐)๐
, 1 โ (1 โ ๐๐)๐
]) ,
๏ฟฝ๏ฟฝ๐
๐= ([๐
๐
๐, ๐
๐
๐] , [1 โ (1 โ ๐
๐)๐
, 1 โ (1 โ ๐๐)๐
]) ,
๏ฟฝ๏ฟฝ๐
๐= ([๐
๐
๐, ๐
๐
๐] , [1 โ (1 โ ๐
๐)๐
, 1 โ (1 โ ๐๐)๐
]) ,
๐ค๐๐ค๐๐ค๐
(1 โ ๐ค๐) (1 โ ๐ค
๐โ ๐ค
๐)
๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ
๐
๐โ ๏ฟฝ๏ฟฝ
๐
๐
= ([1 โ (1 โ ๐๐
๐๐๐
๐๐๐
๐)๐ค๐๐ค๐๐ค๐/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
,
1 โ (1 โ ๐๐
๐๐๐
๐๐๐
๐)๐ค๐๐ค๐๐ค๐/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
] ,
[(1 โ(1 โ ๐๐)๐
(1โ ๐๐)๐
(1 โ ๐๐)๐
)๐ค๐๐ค๐๐ค๐/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
,
(1 โ (1 โ ๐๐)๐
(1 โ ๐๐)๐
ร(1 โ ๐๐)๐
)๐ค๐๐ค๐๐ค๐/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
]) .
(24)
Therefore,
GIIFOWBM๐,๐,๐ (๏ฟฝ๏ฟฝ1, ๏ฟฝ๏ฟฝ2, . . . , ๏ฟฝ๏ฟฝ๐)
= (
๐
โจ
๐,๐,๐=1
๐ = ๐ = ๐
๐ค๐๐ค๐๐ค๐
(1 โ ๐ค๐) (1 โ ๐ค๐ โ ๐ค๐)
(๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ๐
๐โ ๏ฟฝ๏ฟฝ๐
๐))
1/(๐+๐+๐)
= (
[[[[
[
(1 โ
๐
โ
๐,๐,๐=1
๐ = ๐ = ๐
(1 โ ๐๐
๐๐๐
๐๐๐
๐)
๐ค๐๐ค๐๐ค๐/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
)
1/(๐+๐+๐)
,
(1 โ
๐
โ
๐,๐,๐=1
๐ = ๐ = ๐
(1 โ ๐๐
๐๐๐
๐๐๐
๐)
๐ค๐๐ค๐๐ค๐/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
)
1/(๐+๐+๐)
]]]]
]
,
[[[
[
1 โ(1 โ
๐
โ
๐,๐,๐=1
๐ = ๐ = ๐
(1 โ (1 โ ๐๐)๐(1 โ ๐๐)
๐
ร(1 โ ๐๐)๐)๐ค๐๐ค๐๐ค๐
/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
)
1/(๐+๐+๐)
,
1 โ(1 โ
๐
โ
๐,๐,๐=1
๐ = ๐ = ๐
(1 โ (1 โ ๐๐)๐(1 โ ๐๐)
๐
ร(1 โ ๐๐)๐)๐ค๐๐ค๐๐ค๐
/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
)
1/(๐+๐+๐)
]]]]
]
).
(25)
In addition, since
0โค(1โ
๐
โ
๐,๐,๐=1,
๐ = ๐ = ๐
(1โ๐๐
๐๐๐
๐๐๐
๐)๐ค๐๐ค๐๐ค๐/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
)
1/(๐+๐+๐)
โค 1,
0โค(1 โ
๐
โ
๐,๐,๐=1,
๐ = ๐ = ๐
(1 โ ๐๐
๐๐๐
๐๐๐
๐)๐ค๐๐ค๐๐ค๐/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
)
1/(๐+๐+๐)
โค 1,
(26)
then
0 โค 1 โ(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ (1 โ ๐๐)๐
(1 โ ๐๐)๐
ร(1 โ ๐๐)๐
)๐ค๐๐ค๐๐ค๐/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
)
1/(๐+๐+๐)
โค 1,
0 โค 1 โ(1 โ
๐
โ
๐,๐=1,
๐ = ๐
(1 โ (1 โ ๐๐)๐
(1 โ ๐๐)๐
ร(1 โ ๐๐)๐
)๐ค๐๐ค๐๐ค๐/(1โ๐ค๐)(1โ๐ค๐โ๐ค๐)
)
1/(๐+๐+๐)
โค 1
(27)
which completes the proof of Theorem 13.
Property 4 (idempotency). If all IIFVs ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐],
[๐๐, ๐
๐]) (๐ = 1, 2, . . . , ๐) are equal, that is, ๏ฟฝ๏ฟฝ
๐=
([๐๐, ๐
๐], [๐
๐, ๐
๐]) = ๏ฟฝ๏ฟฝ = ([๐, ๐], [๐, ๐]), for all ๐, then
GIIFOWBM๐,๐,๐(๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐) = ๏ฟฝ๏ฟฝ = ([๐, ๐] , [๐, ๐]) .
(28)
Proof. The proof of Property 4 is similar to Property 1.
Corollary 14. If ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐]) (๐ = 1, 2, . . . , ๐) is a set
of the largest IIFVs, that is, ๏ฟฝ๏ฟฝ๐= ๏ฟฝ๏ฟฝ = ([1, 1], [0, 0]), for all ๐,
and (๐ค๐(1), ๐ค
๐(2), . . . , ๐ค
๐(๐)) is precise weight vector of ๏ฟฝ๏ฟฝ
๐, then
GIIFOWBM๐,๐(๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐) = ([1, 1] , [0, 0]) (29)
which is also the largest IIFV.
8 Journal of Applied Mathematics
Property 5 (monotonicity). Let ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐]) and ๐ฝ
๐=
([๐๐, ๐
๐], [๐
๐, ๐
๐]) (๐, ๐ = 1, 2, . . . , ๐) be two sets of IIFVs; if ๐
๐โค
๐๐, ๐
๐โค ๐
๐, ๐
๐โค ๐
๐and ๐
๐โค ๐
๐, for all ๐, then
GIIFOWBM๐,๐,๐(๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐)
โค GIIFOWBM๐,๐,๐(๐ฝ
1, ๐ฝ
2, . . . , ๐ฝ
๐) .
(30)
Proof. The proof of Property 5 is similar to Property 2.
Property 6 (transformation). Letting ๏ฟฝ๏ฟฝ๐= ([๐
๐, ๐
๐], [๐
๐, ๐
๐])
(๐ = 1, 2, . . . , ๐) be a collection of IIFVs and (๐ผ1, ๐ผ
2, . . . , ๐ผ
๐)
is any permutation of (๏ฟฝ๏ฟฝ1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐), then
GIIFOWBM๐,๐,๐(๐ผ
1, ๐ผ
2, . . . , ๐ผ
๐)
= GIIFOWBM๐,๐,๐(๏ฟฝ๏ฟฝ
1, ๏ฟฝ๏ฟฝ
2, . . . , ๏ฟฝ๏ฟฝ
๐) .
(31)
Proof. The proof of Property 6 is similar to Property 3.
5. Case Illustration
In the following,wewill apply the IIFOWBMandGIIFOWBMoperators to group decision making and utilize a practicalcase (adapted from Xu, 2007) [23] involving the assessmentof a set of agroecological regions in Hubei Province, China,to illustrate the developed methods.
Located in Central China and the middle reaches of theChangjiang (Yangtze) River, Hubei Province is distributed ina transitional belt where physical conditions and landscapesare on the transition from north to south and from eastto west. Thus, Hubei Province is well known as โa land ofrice and fishโ since the region enjoys some of the favorablephysical conditions, with diversity of natural resources andthe suitability for growing various crops. At the same time,however, there are also some restrictive factors for developingagriculture such as a tight manland relation between a con-stant degradation of natural resources and a growing popu-lation pressure on land resource reserve. Despite cherishinga burning desire to promote their standard of living, peopleliving in the area are frustrated because they have no abilityto enhance the power to accelerate economic developmentbecause of a dramatic decline in quantity of natural resourcesand a deteriorating environment. Based on the distinctnessand differences in environment and natural resources, HubeiProvince can be roughly divided into four agroecologi-cal regions: ๐
1-Wuhan-Ezhou-Huanggang; ๐
2-Northeast of
Hubei; ๐3-Southeast of Hubei; and ๐
4-West of Hubei. In order
to prioritize these agroecological regions ๐๐(๐ = 1, 2, 3, 4)
with respect to their comprehensive functions, a committeecomprised of four experts ๐
๐(๐ = 1, 2, 3, 4) (whose weight
vector is ๐ค = (0.35, 0.20, 0.15, 0.30)) has been set up toprovide assessment information on ๐
๐(๐ = 1, 2, 3, 4). The
expert ๐๐compared these four agroecological regions with
respect to their comprehensive functions and construct andrepresented the IIFVs ๐ผ
๐๐= ([๐
๐๐, ๐
๐๐], [๐
๐๐, ๐
๐๐]), where [๐
๐๐, ๐
๐๐]
indicates the agreement degree and [๐๐๐, ๐
๐๐] indicates the
unagreement degree. To get the optimal alternative by thenew IIOWBM and GIIFOWBM operators, the followingsteps are given.
Table 1: Interval-valued intuitionistic fuzzy decision matrix ๐ .
๐1
๐2
๐1
([0.1595, 0.6264],[0.1707, 0.2472])
([0.2034, 0.6718],[0.1539, 0.2335])
๐2
([0.2617, 0.6424],[0.1424, 0.2658])
([0.1776, 0.7239],[0.0000, 0.1971])
๐3
([0.2789, 0.6039],[0.1667, 0.2643])
([0.3000, 0.6120],[0.1438, 0.2877])
๐4
([0.1971, 0.7480],[0.0000, 0.0000])
([0.2827, 0.6000],[0.1668, 0.2678])
๐3
๐4
๐1
([0.2768, 0.6693],[0.1725, 0.2672])
([0.3068, 0.6319],[0.1819, 0.2677)
๐2
([0.2001, 0.6744],[0.1175, 0.2161])
([0.2959, 0.6343],[0.1879, 0.2493])
๐3
([0.3492, 0.5811],[0.2005, 0.3271])
([0.1941, 0.6091],[0.2127, 0.2947])
๐4
([0.2562, 0.5386],[0.2474, 0.2973])
([0.2728, 0.5636],[0.2576, 0.3826])
Step 1. We normalize ๐๐๐to ๐
๐๐and construct the normaliza-
tion interval-valued intuitionistic fuzzy decision matrix ๐ =(๐๐๐)4ร5
(see Table 1).
Step 2. Aggregate all the preference values ๐๐๐(๐, ๐ = 1, 2, 3, 4)
of the ๐th line and get the overall performance value ๐๐corre-
sponding to the expert ๐๐by the IIFOWBMoperator (here we
let ๐ = ๐ = 1):
๐1= ([0.2326, 0.6441] , [0.1707, 0.2532]) ,
๐2= ([0.2472, 0.6629] , [0.0000, 0.2381]) ,
๐3= ([0.2702, 0.6038] , [0.1790, 0.2868]) ,
๐4= ([0.2467, 0.6432] , [0.0000, 0.0000]) .
(32)
Step 3. Calculating the score ๐ 2(๐๐) of ๐
๐(๐ = 1, 2, 3, 4),
respectively, we can get
๐ 2(๐1) = 0.6132, ๐
2(๐2) = 0.6680,
๐ 2(๐3) = 0.6021, ๐
2(๐4) = 0.7225.
(33)
Therefore,
๐ 2(๐4) > ๐
2(๐2) > ๐
2(๐1) > ๐
2(๐3) (34)
and ๐4> ๐
2> ๐
1> ๐
3, and ๐ฆ
4is still the optimal alternative.
Based on GOWBM and GIIFOWBMoperators, the mainsteps are as follows.
Step 1โ. See Step 1.
Step 2โ. See Step 2.
Step 3โ. Utilize the GOWBM and GIIFOWBM operatorsto aggregate all the interval-valued individual intuitionistic
Journal of Applied Mathematics 9
Table 2: Interval-valued intuitionistic fuzzy decision matrix of ๐ท.
๐1
๐2
๐1
([0.1348, 0.5504],[0.2320, 0.3113])
([0.1705, 0.6424],[0.1761, 0.2825])
๐2
([0.2279, 0.6064],[0.1524, 0.2832])
([0.1555, 0.6917],[0.1006, 0.2293])
๐3
([0.2742, 0.5941],[0.1749, 0.2806])
([0.3000, 0.6020],[0.1539, 0.3120])
๐4
([0.1769, 0.6868],[0.1530, 0.2183])
([0.2657, 0.6000],[0.2109, 0.2846])
๐3
๐4
๐1
([0.2112, 0.6360],[0.2134, 0.2834])
([0.2924, 0.6154],[0.1998, 0.2733])
๐2
([0.1832, 0.6459],[0.1243, 0.2528])
([0.2469, 0.5951],[0.2188, 0.2995])
๐3
([0.3437, 0.5778],[0.2259, 0.3477])
([0.1770, 0.4885],[0.2534, 0.3349])
๐4
([0.2196, 0.5028],[0.3157, 0.3367])
([0.2427, 0.5387],[0.3145, 0.3915])
fuzzy decisionmatrices๐ท(๐)= (๐
๐๐
๐)4ร4(๐ = 1, 2, 3, 4) into the
collective interval-valued intuitionistic fuzzy decision matrix๐ท = (๐
๐๐)4ร4
(see Table 2).
Step 4. Aggregate all the preference values๐๐๐(๐, ๐ = 1, 2, 3, 4)
of the ๐th line and get the overall performance value ๐๐
corresponding to the expert ๐๐by the GIIFOWBM operator
(here we let ๐ = ๐ = ๐ = 1):
๐1= ([0.1907, 0.5999] , [0.2086, 0.2901]) ,
๐2= ([0.2093, 0.6250] , [0.1589, 0.2732]) ,
๐3= ([0.2532, 0.5593] , [0.2029, 0.3137]) ,
๐4= ([0.2179, 0.5931] , [0.2409, 0.3050]) .
(35)
Step 5. Calculating the score ๐ 2(๐๐) of ๐
๐(๐ = 1, 2, 3, 4),
respectively, we have
๐ 2(๐1) = 0.5729, ๐
2(๐2) = 0.6005,
๐ 2(๐3) = 0.5740, ๐
2(๐4) = 0.5663.
(36)
Therefore,
๐ 2(๐2) > ๐
2(๐3) > ๐
2(๐1) > ๐
2(๐4) (37)
and ๐2> ๐
3> ๐
1> ๐
4, and ๐ฆ
4is still the optimal alternative.
Based on the previous analysis, it could be found that themost comprehensive function is the West of Hubei.
It should be noted out that the whole ranking of the alter-natives has changed. The IIFOWBM1,1 produces the rankingof all the alternatives as ๐
4> ๐
2> ๐
1> ๐
3, which is slightly
different from the ranking of alternatives ๐2> ๐
3> ๐
1>
๐4, derived by the GIIFOWBM1,1,1.Therefore, we can see that
the value derived by the IIFOWBMorGIIFOWBMoperatorsdepends on the choice of the parameters ๐, ๐, and ๐.
6. Concluding Remarks
To further develop the BM, we have proposed the optimizedweighted Bonferroni mean (OWBM) and the generalizedoptimized weighted Bonferroni mean (GOWBM) in thispaper. Then, we proposed two new BM operators underthe interval-valued intuitionistic fuzzy environment, that is,the interval-valued intuitionistic fuzzy optimized weightedBonferroni mean (IIFOWBM) and the generalized interval-valued intuitionistic fuzzy optimized weighted Bonferronimean (GIIFOWBM).The new BMs can reflect the preferenceand interrelationship of the aggregated arguments and cansatisfy the basic properties of the aggregation techniquessimultaneously. Furthermore, some desirable properties ofthe IFIOWBM andGIIFOWBMoperators are investigated indetail, including idempotency,monotonicity, transformation,and boundary. Finally, based on IIFOWBMandGIIFOWBM,we give a utilized practical case involving the assessment of aset of agroecological regions in China to illustrate these newBMs.
Acknowledgment
This work was supported by National Basic Research Pro-gram of China (973 Program, no. 2010 CB328104-02).
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