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 Β© 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
2 Journal of Applied Mathematics
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 π = (π₯1, π₯
2, . . . , π₯
π) be fixed. An
interval-valued intuitionistic fuzzy set (IIFS) π΄ in π can bedefined as
π΄ = {(π₯, ππ΄(π₯) , ]
π΄(π₯)) | π₯ β π} , (1)
where ππ΄(π₯) β [0, 1] and ]
π΄(π₯) β [0, 1] satisfy sup π
π΄(π₯) +
sup ]π΄(π₯) β€ 1 for all π₯
πβ π and π
π΄(π₯) and ]
π΄(π₯) are,
respectively, called the degree of membership and the degreeof nonmembership of the element π₯
πβ π to π΄.
Definition 2 (see [20]). Let π΄ = {(π₯, ππ΄(π₯), ]
π΄(π₯)) | π₯ β π}
be an IIFS; the pair (ππ΄(π₯), ]
π΄(π₯)) is called an interval-valued
intuitionistic fuzzy value (IIFV).For computational convenience, an IIFV can be denoted
by ([π, π], [π, π]), with the condition that [π, π] β [0, 1],[π, π] β [0, 1], and π + π β€ 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 οΏ½οΏ½ = ([π, π], [π, π]), οΏ½οΏ½1=
([π1, π
1], [π
1, π
1]), and οΏ½οΏ½
2= ([π
2, π
2], [π
2, π
2]) be three IIFVs,
then the following operational laws are valid:
(1) οΏ½οΏ½1β οΏ½οΏ½
2= ([π
1+ π
2β π
1π2, π
1+ π
2β π
1π2], [π
1π2, π
1π2]),
(2) οΏ½οΏ½1β οΏ½οΏ½
2= ([π
1π2, π
1π2], [π
1+ π
2β π
1π2, π
1+ π
2β π
1π2]),
(3) ποΏ½οΏ½ = ([1 β (1 β π)π, 1 β (1 β π)π], [ππ, ππ]), π > 0,
(4) οΏ½οΏ½π = ([ππ, ππ], [1 β (1 β π)π, 1 β (1 β π)π]), π > 0.
Then, Xu [23] introduced the score function π (οΏ½οΏ½) = (π βπ+πβπ)/2 and the accuracy function β(οΏ½οΏ½) = (π+π+π+π)/2to calculate the score value and accuracy degree of IIFV οΏ½οΏ½ =([π, π], [π, π]) and gave an order relation between two IIFNsοΏ½οΏ½1and οΏ½οΏ½
2as follows:
(1) if π (οΏ½οΏ½1) < π (οΏ½οΏ½
2), then οΏ½οΏ½
1< οΏ½οΏ½
2;
(2) if π (οΏ½οΏ½1) = π (οΏ½οΏ½
2), then
(i) if β(οΏ½οΏ½1) < β(οΏ½οΏ½
2), then οΏ½οΏ½
1< οΏ½οΏ½
2;
(ii) if β(οΏ½οΏ½1) = β(οΏ½οΏ½
2), then οΏ½οΏ½
1βΌ οΏ½οΏ½
2.
2.2. Bonferroni Means
Definition 4 (see [1]). Let π, π β₯ 0 and ππ(π = 1, 2, . . . , π) a
collection of nonnegative numbers. If
BMπ,π(π
1, π
2, . . . , π
π) = (
1
π(π β 1)
π
β
π,π=1
π = π
ππ
πππ
π)
1/(π+π)
,
(2)
then BMπ,π is called the Bonferroni mean (BM).
Definition 5 (see [6]). Let π, π, π β₯ 0 and ππ(π = 1, 2, . . . , π)
be a collection of nonnegative numbers. If
GBMπ,π,π(π
1, π
2, . . . , π
π)
= (1
π(π β 1)(π β 2)
π
β
π,π,π=1
π = π = π
ππ
πππ
πππ
π)
1/(π+π+π)
,
(3)
then GBMπ,π,π 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 πΌπ(π = 1, 2, . . . , π) 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
IFBMπ,π(πΌ
1, πΌ
2, . . . , πΌ
π)
= (1
π (π β 1)
π
β¨
π,π=1,
π = π
(πΌπ
πβ πΌ
π
π))
1/(π+π)
,
IFWBMπ,π(πΌ
1, πΌ
2, . . . , πΌ
π)
= (1
π (π β 1)
π
β¨
π,π=1,
π = π
((π€ππΌπ
π) β (π€
ππΌπ
π)))
1/(π+π)
,
GIFBMπ,π,π(πΌ
1, πΌ
2, . . . , πΌ
π)
= (1
π (π β 1) (π β 2)
π
β¨
π,π,π=1,
π = π = π
((π€ππΌπ
π) β (π€
ππΌπ
π) β (π€
ππΌπ
π)))
1/(π+π+π)
.
(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 π
πand all criteria βπ
π=1ππ
πβ β
π
π=1ππ
πand cannot
reflect the interrelationship between the individual criterionππand other criteria Vπ,π
πwhich 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 criterionππand all criteria and cannot reflect the interrelationship
between the individual criterion ππand other criteria Vπ,π
π
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 π, π β₯ 0 and ππ(π = 1, 2, . . . , π) be a
collection of nonnegative numbers with the weight vectorπ€ = (π€
1, π€
2, . . . , π€
π) such that π€
πβ₯ 0 and βπ
π=1π€π= 1. Then
the optimized weighted Bonferroni mean (OWBM) can bedefined as follows:
OWBMπ,π(π
1, π
2, . . . , π
π) = (
π
β
π,π=1
π = π
π€ππ€π
1 β π€π
ππ
πππ
π)
1/(π+π)
.
(5)
Definition 8. Let π, π, π β₯ 0 and ππ(π = 1, 2, . . . , π) a
collection of nonnegative numbers with the weight vectorπ€ = (π€
1, π€
2, . . . , π€
π) such that π€
πβ₯ 0 and βπ
π=1π€π= 1.
Then the generalized optimized weighted Bonferroni mean(GOWBM) can be defined as follows:
GOWBMπ,π,π(π
1, π
2, . . . , π
π)
= (
π
β
π,π,π=1
π = π = π
π€ππ€ππ€π
(1 β π€π) (1 β π€
πβ π€
π)
ππ
πππ
πππ
π)
1/(π+π+π)
.
(6)
Furthermore,wecan transformtheOWBMandGOWBMinto the interrelationship between OWBM and GOWBMforms as follows:
OWBMπ,π(π
1, π
2, . . . , π
π)= (
π
β
π=1
π€πππ
π
π
β
π=1
π = π
π€π
1 β π€π
ππ
π)
1/(π+π)
,
(7)
GOWBMπ,π,π(π
1, π
2, . . . , π
π)
= (
π
β
π=1
π€πππ
π
π
β
π=1
π = π
π€π
1 β π€π
ππ
π
π
β
π=1
π = π = π
π€π
(1 β π€πβ π€
π)
ππ
π)
1/(π+π+π)
.
(8)
According to (6)-(7), we see that the terms βπ
π=1,π = π(π€
π/
(1 β π€π))π
π
πand βπ
π,π,π=1,π = π = π(π€
ππ€ππ€π/(1 β π€
π)(1 β π€
πβ
π€π))π
π
πππ
πare the weighted power average satisfaction of all
criteria except π΄π, and βπ
π=1,π = π(π€
π/(1 β π€
π)) = 1 and
βπ
π,π,π=1,π = π = π(π€
ππ€ππ€π/(1 β π€
π)(1 β π€
πβ π€
π)) = 1. If we,
respectively, denote the above terms as π’ππand π’π
πVππ, thus
OWBMπ,π(π
1, π
2, . . . , π
π) = (
π
β
π=1
π€πππ
ππ’π
π)
1/(π+π)
,
GOWBMπ,π,π(π
1, π
2, . . . , π
π) = (
π
β
π=1
π€πππ
ππ’π
πVππ)
1/(π+π+π)
.
(9)
Here then π’ππand π’π
πVππare the weighted power average
satisfaction to all criteria except π΄π, which represents the
interrelationship between the individual criterion ππand
other criteria ππ(π = π). 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 οΏ½οΏ½π= ([π
π, π
π], [π
π, π
π]) be a collection of IIFVs
with the weight vector π€ = (π€1, π€
2, . . . , π€
π) such that π, π β₯
0, π€πβ₯ 0, and βπ
π=1π€π= 1. If
IIFOWBMπ,π(οΏ½οΏ½
1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π)
= (
π
β¨
π,π=1,
π = π
π€ππ€π
1 β π€π
(οΏ½οΏ½π
πβ οΏ½οΏ½
π
π))
1/(π+π)
,
(10)
then IIFOWBMπ,π 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 οΏ½οΏ½π= ([π
π, π
π], [π
π, π
π]) (π = 1, 2, . . . , π) be
a collection of IIFVs with the weight vector (π€1, π€
2, . . . , π€
π),
such thatπ, π β₯ 0,π€πβ₯ 0, andβπ
π=1π€π= 1, then the aggregated
value by using the IIFOWBM is also an IIFV, and
IIFOWBMπ,π(οΏ½οΏ½
1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π)
= (
π
β¨
π,π=1,
π = π
π€ππ€π
1 β π€π
(οΏ½οΏ½π
πβ οΏ½οΏ½
π
π))
1/(π+π)
= (
[[[
[
(1 β
π
β
π,π=1,
π = π
(1 β ππ
πππ
π)π€ππ€π/(1βπ€π)
)
1/(π+π)
,
(1 β
π
β
π,π=1,
π = π
(1 β ππ
πππ
π)π€ππ€π/(1βπ€π)
)
1/(π+π)
]]]
]
,
[[[
[
1 β(1 β
π
β
π,π=1,
π = π
(1 β (1 β ππ)π
Γ (1 β ππ)π
)π€ππ€π/(1βπ€π)
)
1/(π+π)
,
1 β(1 β
π
β
π,π=1,
π = π
(1 β (1 β ππ)π
Γ (1 βππ)π
)π€ππ€π/(1βπ€π)
)
1/(π+π)
]]]
]
) .
(11)
Proof. By the operational laws for IIFVs, we get
οΏ½οΏ½π
π= ([π
π
π, π
π
π] , [1 β (1 β π
π)π
, 1 β (1 β ππ)π
]) ,
οΏ½οΏ½π
π= ([π
π
π, π
π
π] , [1 β (1 β π
π)π
, 1 β (1 β ππ)π
]) ,
οΏ½οΏ½π
πβ οΏ½οΏ½
π
π
= ([ππ
πππ
π, π
π
πππ
π] ,
[1 β (1 β ππ)π
(1 β ππ)π
, 1 β (1 β ππ)π
(1 β ππ)π
]) ,
π€ππ€π
1 β π€π
οΏ½οΏ½π
πβ οΏ½οΏ½
π
π
= ([1 β (1 β ππ
πππ
π)π€ππ€π/(1βπ€π)
, 1 β (1 β ππ
πππ
π)π€ππ€π/(1βπ€π)
] ,
[(1 β (1 β ππ)π
(1 β ππ)π
)π€ππ€π/(1βπ€π)
,
(1 β (1 β ππ)π
(1 β ππ)π
)π€ππ€π/(1βπ€π)
]) ,
π
β¨
π,π=1,
π = π
π€ππ€π
1 β π€π
οΏ½οΏ½π
πβ οΏ½οΏ½
π
π
= (
[[[
[
1 β
π
β
π,π=1,
π = π
(1 β ππ
πππ
π)π€ππ€π/(1βπ€π)
,
1 β
π
β
π,π=1,
π = π
(1 β ππ
πππ
π)π€ππ€π/(1βπ€π)
]]]
]
,
[[[
[
π
β
π,π=1,
π = π
(1 β (1 β ππ)π
(1 β ππ)π
)π€ππ€π/(1βπ€π)
,
π
β
π,π=1,
π = π
(1 β (1 β ππ)π
(1 β ππ)π
)π€ππ€π/(1βπ€π)]
]]
]
) .
(12)
Journal of Applied Mathematics 5
Therefore,
IIFOWBMπ,π(οΏ½οΏ½
1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π)
= (
π
β¨
π,π=1,
π = π
π€ππ€π
1 β π€π
(οΏ½οΏ½π
πβ οΏ½οΏ½
π
π))
1/(π+π)
= (
[[[
[
(1 β
π
β
π,π=1,
π = π
(1 β ππ
πππ
π)π€ππ€π/(1βπ€π)
)
1/(π+π)
,
(1 β
π
β
π,π=1,
π = π
(1 β ππ
πππ
π)π€ππ€π/(1βπ€π)
)
1/(π+π)
]]]
]
,
[[[
[
1β(1 β
π
β
π,π=1,
π = π
(1 β(1 β ππ)π
(1 β ππ)π
)π€ππ€π/(1βπ€π)
)
1/(π+π)
,
1 β(1 β
π
β
π,π=1,
π = π
(1 β (1 β ππ)π
Γ (1 β ππ)π
)π€ππ€π/(1βπ€π)
)
1/(π+π)
]]]
]
),
0 β€(1 β
π
β
π,π=1,
π = π
(1 βππ
πππ
π)π€ππ€π/(1βπ€π)
)
1/(π+π)
β€ 1,
0 β€ (1 β
π
β
π,π=1,
π = π
(1 β ππ
πππ
π)π€ππ€π/(1βπ€π)
)
1/(π+π)
β€ 1.
(13)
We have
0β€1 β(1 β
π
β
π,π=1,
π = π
(1 β (1 β ππ)π
(1 β ππ)π
)π€ππ€π/(1βπ€π)
)
1/(π+π)
β€ 1,
0β€1 β(1 β
π
β
π,π=1,
π = π
(1 β(1 β ππ)π
(1 β ππ)π
)π€ππ€π/(1βπ€π)
)
1/(π+π)
β€ 1,
(1 β
π
β
π,π=1,
π = π
(1 β ππ
πππ
π)π€ππ€π/(1βπ€π)
)
1/(π+π)
+ 1
β(1 β
π
β
π,π=1,
π = π
(1 β (1 β ππ)π
(1 β ππ)π
)π€ππ€π/(1βπ€π)
)
1/(π+π)
β€ 1
+(1 β
π
β
π,π=1,
π = π
(1 β (1 β ππ)π
(1 β ππ)π
)π€ππ€π/(1βπ€π)
)
1/(π+π)
β(1 β
π
β
π,π=1,
π = π
(1 β(1 β ππ)π
(1 β ππ)π
)π€ππ€π/(1βπ€π)
)
1/(π+π)
= 1
(14)
which completes the proof.
Property 1 (idempotency). If all IIFVs οΏ½οΏ½π= ([π
π, π
π], [π
π, π
π])
(π = 1, 2, . . . , π) are equal, that is, οΏ½οΏ½π= ([π
π, π
π], [π
π, π
π]) = οΏ½οΏ½ =
([π, π], [π, π]), for all π, we have
IIFOWBMπ,π(οΏ½οΏ½
1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π) = οΏ½οΏ½ = ([π, π] , [π, π]) . (15)
Proof. Since οΏ½οΏ½π= ([π
π, π
π], [π
π, π
π]) = οΏ½οΏ½ = ([π, π], [π, π]), then
IIFOWBMπ,π(οΏ½οΏ½
1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π)
= (
π
β¨
π,π=1,
π = π
π€ππ€π
1 β π€π
(οΏ½οΏ½πβ οΏ½οΏ½
π))
1/(π+π)
= (οΏ½οΏ½πβ οΏ½οΏ½
π)1/(π+π)
= οΏ½οΏ½π
(16)
which completes the proof of Property 1.
Corollary 11. If οΏ½οΏ½π= ([π
π, π
π], [π
π, π
π]) (π = 1, 2, . . . , π) is a
collection of the largest IIFVs, that is, οΏ½οΏ½π= οΏ½οΏ½ = ([1, 1], [0, 0]),
6 Journal of Applied Mathematics
for all π, and (π€π(1), π€
π(2), . . . , π€
π(π)) is precise weight vector of
οΏ½οΏ½π, then
IIFOWBMπ,π(οΏ½οΏ½
1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π)
= (
π
β¨
π,π=1,
π = π
π€ππ€π
1 β π€π
(οΏ½οΏ½π
πβ οΏ½οΏ½
π
π))
1/(π+π)
= (
[[[
[
(1 β
π
β
π,π=1,
π = π
(1 β 1))
1/(π+π)
,
(1 β
π
β
π,π=1,
π = π
(1 β 1))
1/(π+π)
]]]
]
,
[1 β (1)1/(π+π)
, 1 β (1)1/(π+π)
])
= ([1, 1] , [0, 0])
(17)
which is also the largest IIFV.
Property 2 (monotonicity). Let οΏ½οΏ½π= ([π
π, π
π], [π
π, π
π]) and
π½π= ([π
π, π
π], [π
π, π
π]) (π, π = 1, 2, . . . , π) be two collections
of IIFVs; if ππβ€ π
π, π
πβ€ π
π, π
πβ€ π
πand π
πβ€ π
π, for all π, then
IIFOWBMπ,π(οΏ½οΏ½
1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π)
β€ IIFOWBMπ,π(π½
1, π½
2, . . . , π½
π) .
(18)
Proof. Theproof of Property 2 is similar to Property 10 in [15].
Property 3 (transformation). Letting οΏ½οΏ½π= ([π
π, π
π], [π
π, π
π])
(π = 1, 2, . . . , π) be a set of IIFVs and (πΌ1, πΌ
2, . . . , πΌ
π) the per-
mutation of (οΏ½οΏ½1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π), then
IIFOWBMπ,π(πΌ
1, πΌ
2, . . . , πΌ
π)
= IIFOWBMπ,π(οΏ½οΏ½
1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π) .
(19)
Proof. Since (πΌ1, πΌ
2, . . . , πΌ
π) is a permutation of (οΏ½οΏ½
1, οΏ½οΏ½
2,
. . . , οΏ½οΏ½π), then
(
π
β¨
π,π=1,
π = π
π€ππ€π
1 β π€π
οΏ½οΏ½π
πβ οΏ½οΏ½
π
π)
1/(π+π)
= (
π
β¨
π,π=1,
π = π
π€ππ€π
1 β π€π
πΌπ
πβ πΌ
π
π)
1/(π+π)
.
(20)
Thus,
IIFOWBMπ,π(πΌ
1, πΌ
2, . . . , πΌ
π)
= IIFOWBMπ,π(οΏ½οΏ½
1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π) .
(21)
Definition 12. Let οΏ½οΏ½π= ([π
π, π
π], [π
π, π
π]) (π = 1, 2, . . . , π)
be a collection of IIFVs with the weight vector π€ =
(π€1, π€
2, . . . , π€
π) such that π, π, π β₯ 0 and βπ
π=1π€π= 1. If
GIIFOWBMπ,π,π(οΏ½οΏ½
1, . . . , οΏ½οΏ½
π)
=(
π
β¨
π,π,π=1
π = π = π
π€ππ€ππ€π
(1 β π€π) (1 β π€
πβ π€
π)
(οΏ½οΏ½π
πβ οΏ½οΏ½
π
πβ οΏ½οΏ½
π
π))
1/(π+π+π)
,
(22)
then GIIFOWBMπ,π,π is called the generalized interval-valued intuitionistic fuzzy optimized weighted Bonferronimean (GIIFOWBM).
Theorem 13. Letting οΏ½οΏ½π= ([π
π, π
π], [π
π, π
π]) (π = 1, 2, . . . , π) be
a set of IIFVs with the weight vector (π€1, π€
2, . . . , π€
π), such that
π, π, π β₯ 0 andβπ
π=1π€π= 1, then the aggregated value by using
the GIIFOWBM is also an IIFV and
GIIFOWBMπ,π,π (οΏ½οΏ½1, οΏ½οΏ½2, . . . , οΏ½οΏ½π)
= (
π
β¨
π,π,π=1
π = π = π
π€ππ€ππ€π
(1 β π€π) (1 β π€π β π€π)
(οΏ½οΏ½π
πβ οΏ½οΏ½π
πβ οΏ½οΏ½π
π))
1/(π+π+π)
= (
[[[[
[
(1 β
π
β
π,π,π=1
π = π = π
(1 β ππ
πππ
πππ
π)
π€ππ€ππ€π/(1βπ€π)(1βπ€πβπ€π)
)
1/(π+π+π)
,
(1 β
π
β
π,π,π=1
π = π = π
(1 β ππ
πππ
πππ
π)
π€ππ€ππ€π/(1βπ€π)(1βπ€πβπ€π)
)
1/(π+π+π)
]]]]
]
Γ
[[[[
[
1 β(1 β
π
β
π,π,π=1
π = π = π
(1 β (1 β ππ)π(1 β ππ)
π
Γ(1 β ππ)π)π€ππ€ππ€π
/(1βπ€π)(1βπ€πβπ€π)
)
1/(π+π+π)
,
1 β(1 β
π
β
π,π,π=1
π = π = π
(1 β (1 β ππ)π(1 β ππ)
π
Γ(1 β ππ)π)π€ππ€ππ€π
/(1βπ€π)(1βπ€πβπ€π)
)
1/(π+π+π)
]]]]
]
).
(23)
Journal of Applied Mathematics 7
Proof. By the operational laws for IIFVs, we have
οΏ½οΏ½π
π= ([π
π
π, π
π
π] , [1 β (1 β π
π)π
, 1 β (1 β ππ)π
]) ,
οΏ½οΏ½π
π= ([π
π
π, π
π
π] , [1 β (1 β π
π)π
, 1 β (1 β ππ)π
]) ,
οΏ½οΏ½π
π= ([π
π
π, π
π
π] , [1 β (1 β π
π)π
, 1 β (1 β ππ)π
]) ,
π€ππ€ππ€π
(1 β π€π) (1 β π€
πβ π€
π)
οΏ½οΏ½π
πβ οΏ½οΏ½
π
πβ οΏ½οΏ½
π
π
= ([1 β (1 β ππ
πππ
πππ
π)π€ππ€ππ€π/(1βπ€π)(1βπ€πβπ€π)
,
1 β (1 β ππ
πππ
πππ
π)π€ππ€ππ€π/(1βπ€π)(1βπ€πβπ€π)
] ,
[(1 β(1 β ππ)π
(1β ππ)π
(1 β ππ)π
)π€ππ€ππ€π/(1βπ€π)(1βπ€πβπ€π)
,
(1 β (1 β ππ)π
(1 β ππ)π
Γ(1 β ππ)π
)π€ππ€ππ€π/(1βπ€π)(1βπ€πβπ€π)
]) .
(24)
Therefore,
GIIFOWBMπ,π,π (οΏ½οΏ½1, οΏ½οΏ½2, . . . , οΏ½οΏ½π)
= (
π
β¨
π,π,π=1
π = π = π
π€ππ€ππ€π
(1 β π€π) (1 β π€π β π€π)
(οΏ½οΏ½π
πβ οΏ½οΏ½π
πβ οΏ½οΏ½π
π))
1/(π+π+π)
= (
[[[[
[
(1 β
π
β
π,π,π=1
π = π = π
(1 β ππ
πππ
πππ
π)
π€ππ€ππ€π/(1βπ€π)(1βπ€πβπ€π)
)
1/(π+π+π)
,
(1 β
π
β
π,π,π=1
π = π = π
(1 β ππ
πππ
πππ
π)
π€ππ€ππ€π/(1βπ€π)(1βπ€πβπ€π)
)
1/(π+π+π)
]]]]
]
,
[[[
[
1 β(1 β
π
β
π,π,π=1
π = π = π
(1 β (1 β ππ)π(1 β ππ)
π
Γ(1 β ππ)π)π€ππ€ππ€π
/(1βπ€π)(1βπ€πβπ€π)
)
1/(π+π+π)
,
1 β(1 β
π
β
π,π,π=1
π = π = π
(1 β (1 β ππ)π(1 β ππ)
π
Γ(1 β ππ)π)π€ππ€ππ€π
/(1βπ€π)(1βπ€πβπ€π)
)
1/(π+π+π)
]]]]
]
).
(25)
In addition, since
0β€(1β
π
β
π,π,π=1,
π = π = π
(1βππ
πππ
πππ
π)π€ππ€ππ€π/(1βπ€π)(1βπ€πβπ€π)
)
1/(π+π+π)
β€ 1,
0β€(1 β
π
β
π,π,π=1,
π = π = π
(1 β ππ
πππ
πππ
π)π€ππ€ππ€π/(1βπ€π)(1βπ€πβπ€π)
)
1/(π+π+π)
β€ 1,
(26)
then
0 β€ 1 β(1 β
π
β
π,π=1,
π = π
(1 β (1 β ππ)π
(1 β ππ)π
Γ(1 β ππ)π
)π€ππ€ππ€π/(1βπ€π)(1βπ€πβπ€π)
)
1/(π+π+π)
β€ 1,
0 β€ 1 β(1 β
π
β
π,π=1,
π = π
(1 β (1 β ππ)π
(1 β ππ)π
Γ(1 β ππ)π
)π€ππ€ππ€π/(1βπ€π)(1βπ€πβπ€π)
)
1/(π+π+π)
β€ 1
(27)
which completes the proof of Theorem 13.
Property 4 (idempotency). If all IIFVs οΏ½οΏ½π= ([π
π, π
π],
[ππ, π
π]) (π = 1, 2, . . . , π) are equal, that is, οΏ½οΏ½
π=
([ππ, π
π], [π
π, π
π]) = οΏ½οΏ½ = ([π, π], [π, π]), for all π, then
GIIFOWBMπ,π,π(οΏ½οΏ½
1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π) = οΏ½οΏ½ = ([π, π] , [π, π]) .
(28)
Proof. The proof of Property 4 is similar to Property 1.
Corollary 14. If οΏ½οΏ½π= ([π
π, π
π], [π
π, π
π]) (π = 1, 2, . . . , π) is a set
of the largest IIFVs, that is, οΏ½οΏ½π= οΏ½οΏ½ = ([1, 1], [0, 0]), for all π,
and (π€π(1), π€
π(2), . . . , π€
π(π)) is precise weight vector of οΏ½οΏ½
π, then
GIIFOWBMπ,π(οΏ½οΏ½
1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π) = ([1, 1] , [0, 0]) (29)
which is also the largest IIFV.
8 Journal of Applied Mathematics
Property 5 (monotonicity). Let οΏ½οΏ½π= ([π
π, π
π], [π
π, π
π]) and π½
π=
([ππ, π
π], [π
π, π
π]) (π, π = 1, 2, . . . , π) be two sets of IIFVs; if π
πβ€
ππ, π
πβ€ π
π, π
πβ€ π
πand π
πβ€ π
π, for all π, then
GIIFOWBMπ,π,π(οΏ½οΏ½
1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π)
β€ GIIFOWBMπ,π,π(π½
1, π½
2, . . . , π½
π) .
(30)
Proof. The proof of Property 5 is similar to Property 2.
Property 6 (transformation). Letting οΏ½οΏ½π= ([π
π, π
π], [π
π, π
π])
(π = 1, 2, . . . , π) be a collection of IIFVs and (πΌ1, πΌ
2, . . . , πΌ
π)
is any permutation of (οΏ½οΏ½1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π), then
GIIFOWBMπ,π,π(πΌ
1, πΌ
2, . . . , πΌ
π)
= GIIFOWBMπ,π,π(οΏ½οΏ½
1, οΏ½οΏ½
2, . . . , οΏ½οΏ½
π) .
(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 βa land ofrice and fishβ 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: π
1-Wuhan-Ezhou-Huanggang; π
2-Northeast of
Hubei; π3-Southeast of Hubei; and π
4-West of Hubei. In order
to prioritize these agroecological regions ππ(π = 1, 2, 3, 4)
with respect to their comprehensive functions, a committeecomprised of four experts π
π(π = 1, 2, 3, 4) (whose weight
vector is π€ = (0.35, 0.20, 0.15, 0.30)) has been set up toprovide assessment information on π
π(π = 1, 2, 3, 4). The
expert ππcompared these four agroecological regions with
respect to their comprehensive functions and construct andrepresented the IIFVs πΌ
ππ= ([π
ππ, π
ππ], [π
ππ, π
ππ]), where [π
ππ, π
ππ]
indicates the agreement degree and [πππ, π
ππ] 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 π .
π1
π2
π1
([0.1595, 0.6264],[0.1707, 0.2472])
([0.2034, 0.6718],[0.1539, 0.2335])
π2
([0.2617, 0.6424],[0.1424, 0.2658])
([0.1776, 0.7239],[0.0000, 0.1971])
π3
([0.2789, 0.6039],[0.1667, 0.2643])
([0.3000, 0.6120],[0.1438, 0.2877])
π4
([0.1971, 0.7480],[0.0000, 0.0000])
([0.2827, 0.6000],[0.1668, 0.2678])
π3
π4
π1
([0.2768, 0.6693],[0.1725, 0.2672])
([0.3068, 0.6319],[0.1819, 0.2677)
π2
([0.2001, 0.6744],[0.1175, 0.2161])
([0.2959, 0.6343],[0.1879, 0.2493])
π3
([0.3492, 0.5811],[0.2005, 0.3271])
([0.1941, 0.6091],[0.2127, 0.2947])
π4
([0.2562, 0.5386],[0.2474, 0.2973])
([0.2728, 0.5636],[0.2576, 0.3826])
Step 1. We normalize πππto π
ππand construct the normaliza-
tion interval-valued intuitionistic fuzzy decision matrix π =(πππ)4Γ5
(see Table 1).
Step 2. Aggregate all the preference values πππ(π, π = 1, 2, 3, 4)
of the πth line and get the overall performance value ππcorre-
sponding to the expert ππby the IIFOWBMoperator (here we
let π = π = 1):
π1= ([0.2326, 0.6441] , [0.1707, 0.2532]) ,
π2= ([0.2472, 0.6629] , [0.0000, 0.2381]) ,
π3= ([0.2702, 0.6038] , [0.1790, 0.2868]) ,
π4= ([0.2467, 0.6432] , [0.0000, 0.0000]) .
(32)
Step 3. Calculating the score π 2(ππ) of π
π(π = 1, 2, 3, 4),
respectively, we can get
π 2(π1) = 0.6132, π
2(π2) = 0.6680,
π 2(π3) = 0.6021, π
2(π4) = 0.7225.
(33)
Therefore,
π 2(π4) > π
2(π2) > π
2(π1) > π
2(π3) (34)
and π4> π
2> π
1> π
3, and π¦
4is still the optimal alternative.
Based on GOWBM and GIIFOWBMoperators, the mainsteps are as follows.
Step 1β. See Step 1.
Step 2β. See Step 2.
Step 3β. 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 π·.
π1
π2
π1
([0.1348, 0.5504],[0.2320, 0.3113])
([0.1705, 0.6424],[0.1761, 0.2825])
π2
([0.2279, 0.6064],[0.1524, 0.2832])
([0.1555, 0.6917],[0.1006, 0.2293])
π3
([0.2742, 0.5941],[0.1749, 0.2806])
([0.3000, 0.6020],[0.1539, 0.3120])
π4
([0.1769, 0.6868],[0.1530, 0.2183])
([0.2657, 0.6000],[0.2109, 0.2846])
π3
π4
π1
([0.2112, 0.6360],[0.2134, 0.2834])
([0.2924, 0.6154],[0.1998, 0.2733])
π2
([0.1832, 0.6459],[0.1243, 0.2528])
([0.2469, 0.5951],[0.2188, 0.2995])
π3
([0.3437, 0.5778],[0.2259, 0.3477])
([0.1770, 0.4885],[0.2534, 0.3349])
π4
([0.2196, 0.5028],[0.3157, 0.3367])
([0.2427, 0.5387],[0.3145, 0.3915])
fuzzy decisionmatricesπ·(π)= (π
ππ
π)4Γ4(π = 1, 2, 3, 4) into the
collective interval-valued intuitionistic fuzzy decision matrixπ· = (π
ππ)4Γ4
(see Table 2).
Step 4. Aggregate all the preference valuesπππ(π, π = 1, 2, 3, 4)
of the πth line and get the overall performance value ππ
corresponding to the expert ππby the GIIFOWBM operator
(here we let π = π = π = 1):
π1= ([0.1907, 0.5999] , [0.2086, 0.2901]) ,
π2= ([0.2093, 0.6250] , [0.1589, 0.2732]) ,
π3= ([0.2532, 0.5593] , [0.2029, 0.3137]) ,
π4= ([0.2179, 0.5931] , [0.2409, 0.3050]) .
(35)
Step 5. Calculating the score π 2(ππ) of π
π(π = 1, 2, 3, 4),
respectively, we have
π 2(π1) = 0.5729, π
2(π2) = 0.6005,
π 2(π3) = 0.5740, π
2(π4) = 0.5663.
(36)
Therefore,
π 2(π2) > π
2(π3) > π
2(π1) > π
2(π4) (37)
and π2> π
3> π
1> π
4, and π¦
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 π
4> π
2> π
1> π
3, which is slightly
different from the ranking of alternatives π2> π
3> π
1>
π4, derived by the GIIFOWBM1,1,1.Therefore, we can see that
the value derived by the IIFOWBMorGIIFOWBMoperatorsdepends on the choice of the parameters π, π, and π.
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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