research article interval neutrosophic sets and ... - hindawi

Research ArticleInterval Neutrosophic Sets and Their Application inMulticriteria Decision Making Problems

Hong-yu Zhang Jian-qiang Wang and Xiao-hong Chen

School of Business Central South University Changsha 410083 China

Correspondence should be addressed to Jian-qiang Wang jqwangcsueducn

Received 30 August 2013 Accepted 18 December 2013 Published 17 February 2014

Academic Editors A Balbas and P A D Castro

Copyright copy 2014 Hong-yu Zhang et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As a generalization of fuzzy sets and intuitionistic fuzzy sets neutrosophic sets have been developed to represent uncertainimprecise incomplete and inconsistent information existing in the real world And interval neutrosophic sets (INSs) have beenproposed exactly to address issues with a set of numbers in the real unit interval not just a specific number However there are fewerreliable operations for INSs as well as the INS aggregation operators and decisionmakingmethod For this purpose the operationsfor INSs are defined and a comparison approach is put forward based on the related research of interval valued intuitionistic fuzzysets (IVIFSs) in this paper On the basis of the operations and comparison approach two interval neutrosophic number aggregationoperators are developedThen amethod formulticriteria decisionmaking problems is explored applying the aggregation operatorsIn addition an example is provided to illustrate the application of the proposed method

1 Introduction

Zadeh proposed his remarkable theory of fuzzy sets (FSs inshort) in 1965 [1] to encounter different types of uncertaintiesSince then it has been applied successfully in various fields[2] As the traditional fuzzy set uses one single value 120583

119860(119909) isin

[0 1] to represent the grade of membership of the fuzzy setA defined on a universe it cannot handle some cases where120583119860is hard to be defined by a specific value So interval valued

fuzzy sets (IVFSs) were introduced by Turksen [3] And tocope with the lack of knowledge of nonmembership degreesAtanassov introduced intuitionistic fuzzy sets (IFSs in short)[4ndash7] an extension of Zadehrsquos FSs In addition Gau andBuehrer [8] defined vague sets Later on Bustince pointedout that vague sets and Atanassovrsquos IFSs are mathematicallyequivalent objects [9] As for the present IFSs have beenwidely applied in solvingmulticriteria decisionmaking prob-lems [10ndash14] neural networks [15 16] medical diagnosis [17]color region extraction [18 19]market prediction [20] and soforth

IFSs took into account themembership degree nonmem-bership degree and degree of hesitation simultaneously So

IFSs are more flexible and practical in addressing the fuzzi-ness and uncertainty than the traditional FSs Moreover insome actual cases the membership degree nonmembershipdegree and hesitation degree of an element in the IFSmay notbe a specific number Hence it was extended to the intervalvalued intuitionistic fuzzy sets (IVIFSs in brief) [21] Tohandle the situations where people are hesitant in expressingtheir preference over objects in a decision making processhesitant fuzzy sets (HFSs) were introduced by Torra [22] andTorra and Narukawa [23]

Although the FSs theory has been developed and general-ized it can not deal with all sorts of uncertainties in differentreal physical problems Some types of uncertainties such asthe indeterminate information and inconsistent informationcan not be handled For example [24] when we ask about theopinion of an expert about a certain statement he or she maysay that the possibility that the statement is true is 05 thatthe statement is false is 06 and the degree that he or she isnot sure is 02This issue is beyond the scope of FSs and IFSsTherefore some new theories are required

Smarandache coined neutrosophic logic and neutro-sophic sets (NSs) in 1995 [25 26] A NS is a set where each

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 645953 15 pageshttpdxdoiorg1011552014645953

2 The Scientific World Journal

element of the universe has a degree of truth indetermi-nacy and falsity respectively and which lies in ]0minus 1+[ thenonstandard unit interval [27] Obviously it is the extensionto the standard interval [0 1] in IFSs And the uncertaintypresent here that is indeterminacy factor is independent oftruth and falsity values while the incorporated uncertainty isdependent on the degrees of belongingness and nonbelong-ingness in IFSs [28] And for the aforementioned example bymeans of NSs it can be expressed as 119909(05 02 06)

However without being specified it is difficult to apply inthe real applications Hence the single valued neutrosophicset (SVNS)was proposedwhich is an instance ofNSs [24 28]Furthermore the information energy of SVNSs correlationand correlation coefficient of SVNSs and a decision makingmethod by the use of SVNSs were presented [29] In additionYe also introduced the concept of simplified neutrosophic sets(SNSs) which can be described by three real numbers in thereal unit interval [0 1] and proposed a multicriteria decisionmaking method using aggregation operators for SNSs [30]Majumdar and Samant introduced a measure of entropy of aSVNS [28]

In fact sometimes the degree of truth falsity and inde-terminacy of a certain statement can not be defined exactlyin the real situations but denoted by several possible intervalvalues So the interval neutrosophic set (INS) was requiredsimilar to IVIFSWang et al proposed the concept of INS andgave the set-theoretic operators of INS [31] The operationsof INS were discussed in [32] yet the comparison methodswere not seen there Furthermore Ye defined the Hammingand Euclidean distances between INSs and proposed thesimilarity measures between INSs based on the relationshipbetween similarity measures and distances [33] However insome cases the INS operations in [31] might be irrationalFor instance the sum of any element and the maximumvalue should be equal to the maximum one while it does nothold with the operations in [31] In addition to the best ofour knowledge the existing literatures do not put forwardthe INS aggregation operators and decision making methodwhich were vitally important for INSs to be utilized inthe real situations in scientific and engineering applicationsTherefore the operations and comparison approach betweeninterval neutrosophic numbers (INNs) and the aggregationoperators for INSs are defined in this paper to be usedThus a multicriteria decision making method is establishedbased on the proposed operators an illustrative exampleis given to demonstrate the application of the proposedmethod

The rest of the paper is organized as follows Section 2briefly introduces interval numbers properties of t-normand t-conorm and concepts and operations of NSs SNSsand INSs And the operations and comparison approachfor INSs are defined on the basis of the IVIFS theoryin Section 3 The INN aggregation operators are givenand a decision making method is developed for INSsby means of the INN aggregation operators in Section 4In Section 5 an illustrative example is presented to illus-trate the proposed method and the comparison analysisand discussion are given Finally Section 6 concludes thepaper

2 Preliminaries

In this section some basic concepts and definitions relatedto INSs including interval numbers t-norm and t-conormand the definitions and operations of NSs SNSs and INSsare introduced which will be utilized in the rest of the paper

21 Interval Numbers andTheirOperations Interval numbersand their operations are of utmost importance to explore theoperations for INSs So some definitions and operations ofinterval numbers are given below

Definition 1 (see [34ndash37]) Let 119886 = [119886119871 119886119880] = 119909 | 119886119871 le 119909 le119886119880 and then 119886 is called an interval number In particular if

0 le 119886119871le 119909 le 119886

119880 then 119886 is reduced to a positive intervalnumber

Consider any two interval numbers 119886 = [119886119871 119886119880] and =[119887119871 119887119880] and then their operations are defined as follows

(1) 119886 = hArr 119886119871= 119887119871 119886119880= 119887119880

(2) 119886 + = [119886119871 + 119887119871 119886119880 + 119887119880](3) 119886 minus = [119886119871 minus 119887119880 119886119880 minus 119887119871](4) 119886 times = [min119886119871119887119871 119886119871119887119880 119886119880119887119871 119886119880119887119880 max119886119871119887119871

119886119871119887119880 119886119880119887119871 119886119880119887119880]

(5) 119896119886 = [119896119886119871 119896119886119880] 119896 gt 0

Definition 2 (see [37]) Let 119886 = [119886119871 119886119880] and = [119887

119871 119887119880]

119897119886= 119886119880minus119886119871 and 119897

119887= 119880minus119871 and then the degree of possibility

of 119886 ge is formulated by

119901 (119886 ge ) = max1 minusmax(119880minus 119886119871

119897119886+ 119897119887

0) 0 (1)

Suppose that there are 119899 interval numbers 119886119894

=

[119886119871

119894 119886119880

119894] (119894 = 1 2 119899) and each interval number 119886

119894is

compared to all interval numbers 119886119895(119895 = 1 2 119899) by using

(1) namely

119901119894119895= 119901 (119886

119894ge 119886119895) = max1 minusmax(

119886119880

119895minus 119886119871

119894

119897119886119894+ 119897119886119895

0) 0 (2)

Then a complementary matrix can be constructed asfollows

119875 =

[[[[

[

1199011111990112sdot sdot sdot 1199011119899

1199012111990122sdot sdot sdot 1199012119899

11990111989911199011198992sdot sdot sdot 119901119899119899

]]]]

]

(3)

where 119901119894119895ge 0 119901

119894119895+ 119901119895119894= 1 119901

119894119894= 05

22 t-Norm and t-Conorm The t-norm and its dual t-conorm play an important role in the construction of oper-ation rules and averaging operators of INSs Here some basicconcepts are introduced

Definition 3 (see [38 39]) A function 119879 [0 1] times [0 1] rarr[0 1] is called t-norm if it satisfies the following conditions

The Scientific World Journal 3

(1) forall119909 isin [0 1] 119879(1 119909) = 119909

(2) forall119909 119910 isin [0 1] 119879(119909 119910) = 119879(119910 119909)

(3) forall119909 119910 119911 isin [0 1] 119879(119909 119879(119910 119911)) = 119879(119879(119909 119910) 119911)

(4) if 119909 le 1199091015840 119910 le 1199101015840 then 119879(119909 119910) le 119879(1199091015840 1199101015840)

Definition 4 (see [38 39]) A function 119878 [0 1] times [0 1] rarr[0 1] is called t-conorm if it satisfies the following conditions

(1) forall119909 isin [0 1] 119878(0 119909) = 119909

(2) forall119909 119910 isin [0 1] 119878(119909 119910) = 119878(119910 119909)

(3) forall119909 119910 119911 isin [0 1] 119878(119909 119878(119910 119911)) = 119878(119878(119909 119910) 119911)

(4) if 119909 le 1199091015840 119910 le 1199101015840 then 119878(119909 119910) le 119878(1199091015840 1199101015840)

Definition 5 (see [38 39]) A t-norm function 119879(119909 119910) iscalled Archimedean t-norm if it is continuous and 119879(119909 119909) lt119909 for all 119909 isin (0 1) An Archimedean t-norm is calledstrictly Archimedean t-norm if it is strictly increasing in eachvariable for119909 119910 isin (0 1) A t-conorm function 119878(119909 119910) is calledArchimedean t-conorm if it is continuous and 119878(119909 119909) gt 119909 forall 119909 isin (0 1) An Archimedean t-conorm is called strictlyArchimedean t-conorm if it is strictly increasing in eachvariable for 119909 119910 isin (0 1)

It is well known [39 40] that a strict Archimedean t-normcan be expressed via its additive generator 119896 as 119879(119909 119910) =119896minus1(119896(119909) + 119896(119910)) and similarly applied to its dual t-conorm

119878(119909 119910) = 119897minus1(119897(119909) + 119897(119910)) with 119897(119905) = 119896(1 minus 119905) We observe that

an additive generator of a continuous Archimedean t-normis a strictly decreasing function 119896 [0 1] rarr [0infin)

There are some well-knownArchimedean t-conorms andt-norms [41]

(1) Let 119896(119905) = minus log 119905 119897(119905) = minus log(1 minus 119905) 119896minus1(119905) = 119890minus119905 and119897minus1(119905) = 1 minus 119890

minus119905 Then algebraic t-conorm and t-normare obtained

119878 (119909 119910) = 1 minus (1 minus 119909) (1 minus 119910) 119879 (119909 119910) = 119909119910 (4)

(2) Let 119896(119905) = log((2minus119905)119905) 119897(119905) = log((2minus(1minus119905))(1minus119905))119896minus1(119905) = 2(119890

119905+ 1) and 119897minus1(119905) = 1 minus (2(119890119905 + 1)) Then

Einstein t-conorm and t-norm are obtained

119878 (119909 119910) =119909 + 119910

1 + 119909119910 119879 (119909 119910) =

119909119910

1 + (1 minus 119909) (1 minus 119910) (5)

(3) Let 119896(119905) = log((120574 minus (1 minus 120574)119905)119905) 119897(119905) = log((120574 minus (1 minus120574)(1 minus 119905))(1 minus 119905)) 119896minus1(119905) = 120574(119890119905 + 120574 minus 1) and 119897minus1(119905) =1 minus (120574(119890

119905+ 120574 minus 1)) 120574 gt 0 Then Hamacher t-conorm

and t-norm are obtained

119878 (119909 119910) =119909 + 119910 minus 119909119910 minus (1 minus 120574) 119909119910

1 minus (1 minus 120574) 119909119910

119879 (119909 119910) =119909119910

120574 + (1 minus 120574) (119909 + 119910 minus 119909119910) 120574 gt 0

(6)

23 Definitions and Operations of NSs and SNSs

Definition 6 (see [31]) Let 119883 be a space of points (objects)with a generic element in 119883 denoted by 119909 A NS 119860 in 119883is characterized by a truth-membership function 119879

119860(119909) an

indeterminacy-membership function 119868119860(119909) and a falsity-

membership function119865119860(119909) 119879

119860(119909) 119868119860(119909) and119865

119860(119909) are real

standard or nonstandard subsets of ]0minus 1+[ that is 119879119860(119909)

119883 rarr ]0minus 1+[ 119868119860(119909) 119883 rarr ]0

minus 1+[ and 119865

119860(119909) 119883 rarr

]0minus 1+[ There is no restriction on the sum of 119879

119860(119909) 119868119860(119909)

and 119865119860(119909) so 0minus le sup119879

119860(119909) + sup 119868

119860(119909) + sup119865

119860(119909) le 3

+

Definition 7 (see [31]) A NS 119860 is contained in the other NS119861 denoted by 119860 sube 119861 if and only if inf 119879

119860(119909) le inf 119879

119861(119909)

sup119879119860(119909) le sup119879

119861(119909) inf 119868

119860(119909) le inf 119868

119861(119909) sup 119868

119860(119909) le

sup 119868119861(119909) inf 119865

119860(119909) le inf 119865

119861(119909) and sup119865

119860(119909) le sup119865

119861(119909)

for 119909 isin 119883

Since it is difficult to apply NSs to practical problems Yereduced NSs of nonstandard intervals into a kind of SNSs ofstandard intervals that will preserve the operations of NSs[30]

Definition 8 (see [30]) Let 119883 be a space of points (objects)with a generic element in 119883 denoted by 119909 A NS 119860 in 119883 ischaracterized by 119879

119860(119909) 119868119860(119909) and 119865

119860(119909) which are single

subintervalssubsets in the real standard [0 1] that is119879119860(119909)

119883 rarr [0 1] 119868119860(119909) 119883 rarr [0 1] and 119865

119860(119909) 119883 rarr [0 1]

Then a simplification of 119860 is denoted by

119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ | 119909 isin 119883 (7)

which is called a SNS It is a subclass of NSs

The operational relations of SNSs are also defined in [30]

Definition 9 (see [30]) Let119860 and 119861 be two SNSs For any 119909 isin119883

(1) 119860+119861 = ⟨119879119860(119909)+119879

119861(119909)minus119879

119860(119909) sdot119879

119861(119909) 119868119860(119909)+119868

119861(119909)minus

119868119860(119909) sdot 119868

119861(119909) 119865

119860(119909) + 119865

119861(119909) minus 119865

119860(119909) sdot 119865

119861(119909)⟩

(2) 119860 sdot 119861 = ⟨119879119860(119909) sdot 119879

119861(119909) 119868119860(119909) sdot 119868

119861(119909) 119865119860(119909) sdot 119865

119861(119909)⟩

(3) 120582 sdot 119860 = ⟨1 minus (1 minus 119879119860(119909))120582 1 minus (1 minus 119868

119860(119909))120582 1 minus (1 minus

119865119860(119909))120582⟩ 120582 gt 0

(4) 119860120582 = ⟨119879120582119860(119909) 119868120582

119860(119909) 119865

120582

119860(119909)⟩ 120582 gt 0

There are some limitations in Definition 9(1) In some situations the operations such as 119860 + 119861 and

119860 sdot 119861 as given in Definition 9 might be irrational This willbe shown in the example below

For example let two simplified neutrosophic numbers(SNNs) 119886 = ⟨05 05 05⟩ and 119887 = ⟨1 0 0⟩ Obviously 119887 =⟨1 0 0⟩ is the maximum of SNSs It is notable that the sumof any number and the maximum number should be equal tothe maximum one However according to (1) in Definition 9119886+119887 = ⟨1 05 05⟩ = 119887 Hence (1) does not hold and so do theother equations in Definition 9 It shows that the operationsabove are incorrect

(2) In addition the similaritymeasure for SNSs in [30] onthe basis of the operations does not satisfy any cases


For instance let the alternatives 1198861= ⟨01 0 0⟩ 119886

2=

⟨09 0 0⟩ and the ideal alternative 119886lowast = ⟨1 0 0⟩ Accordingto the decisionmakingmethod based on the cosine similaritymeasure for SNSs under the simplified neutrosophic environ-ment in [30] we can obtain that 119878

1(1198861 119886lowast) = 1 119878

2(1198862 119886lowast) =

1 that is the alternative 1198861is equal to the alternative 119886

2

However for 1198791198862(119909) gt 119879

1198861(119909) 119868

1198862(119909) gt 119868

1198861(119909) and 119865

1198862(119909) gt

1198651198861(119909) it is clear that the alternative 119886

2is superior to the

alternative 1198861

24 Definitions and Operations of INSs

Definition 10 (see [31]) Let 119883 be a space of points (objects)with generic elements in 119883 denoted by 119883 An INS 119860

in 119883 is characterized by a truth-membership function119879119860(119909) an indeterminacy-membership function 119868

119860(119909) and

a falsity-membership function 119865119860(119909) For each point 119909 in

119883 we have that 119879119860(119909) = [inf 119879

119860(119909) sup119879

119860(119909)] 119868

119860(119909) =

[inf 119868119860(119909) sup 119868

119860(119909)] 119865

119860(119909) = [inf 119865

119860(119909) sup119865

119860(119909)] sube

[0 1] and 0 le sup119879119860(119909) + sup 119868

119860(119909) + sup119865

119860(119909) le 3 119909 isin 119883

We only consider the subunitary interval of [0 1] It is thesubclass of a NS Therefore all INSs are clearly NSs

Definition 11 (see [31]) An INS 119860 is contained in the otherINS 119861 119860 sube 119861 if and only if

inf 119879119860(119909) le inf 119879

119861(119909) sup119879

119860(119909) le sup119879

119861(119909)

inf 119868119860(119909) ge inf 119868

119861(119909) sup 119868

119860(119909) ge sup 119868

119861(119909)

inf 119865119860(119909) ge inf 119865

119861(119909) and sup119865

119860(119909) ge sup119865

119861(119909)

for any 119909 isin 119883

Definition 12 (see [31]) Two INSs 119860 and 119861 are equal writtenas 119860 = 119861 if and only if 119860 sube 119861 and 119860 supe 119861

Definition 13 (see [31]) The addition of two INSs 119860 and 119861 isan INS 119862 written as 119862 = 119860 + 119861 whose truth-membershipindeterminacy-membership and falsity-membership func-tions are related to those of 119860 and 119861 by

inf 119879119862

= min(inf 119879119860+ inf 119879

119861 1) sup119879

119862=

min(sup119879119860+ sup119879

119861 1)

inf 119868119862= min(inf 119868

119860+inf 119868

119861 1) sup 119868

119862= min(sup 119868

119860+

sup 119868119861 1)

inf 119865119862

= min(inf 119865119860+ inf 119865

119861 1) sup119865

119862=

min(sup119865119860+ sup119865

119861 1)

for all 119909 in119883

As to be known when 119861 = ⟨0 1 1⟩ it should satisfy119860 + 119861 = 119860 and 119860 sdot 119861 = 119861 for B being the minimumvalue of INSs And when 119861 = ⟨1 0 0⟩ as the largest elementof INSs it should satisfy 119860 + 119861 = 119861 and 119860 + 119861 = 119860Let 119861 = ⟨1 0 0⟩ That is inf 119879

119861= sup119879

119861= 1 inf 119868

119861=

sup 119868119861= 0 and inf 119865

119861= sup119865

119861= 0 According to

Definition 13 inf 119879119862= 1 sup119879

119862= 1 inf 119868

119862= inf 119868

119860

sup 119868119862= sup 119868

119860 inf 119865

119862= inf 119865

119860 and sup119865

119862= sup119865

119860 that is

119860 + 119861 = ⟨[1 1] [inf 119868119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ = 119861 so that

Definition 13 does not hold

Definition 14 (see [31]) The Cartesian product of two INSs119860 defined on the universe 119883

1and 119861 defined on the uni-

verse 1198832is an INS 119862 written as 119862 = 119860 sdot 119861 whose

truth-membership indeterminacy-membership and falsity-membership functions are related to those of 119860 and 119861

by

inf 119879119862(119909 119910) = inf 119879

119860(119909) + inf 119879

119861(119910) minus inf 119879

119860(119909) sdot

inf 119879119861(119910)

sup119879119862(119909 119910) = sup119879

119860(119909) + sup119879

119861(119910) minus sup119879

119860(119909) sdot

sup119879119861(119910)

inf 119868119862(119909 119910) = inf 119868

119860(119909) sdot inf 119868

119861(119910) sup 119868

119862(119909 119910) =

sup 119868119860(119909) sdot sup 119868

119861(119910)

inf 119865119862(119909 119910) = inf 119865

119860(119909) sdot inf 119865

119861(119910) sup119865

119862(119909 119910) =

sup119865119860(119909) sdot sup119865

119861(119910)

for all 119909 in1198831 119910 in119883

2

Being similar toDefinition 13 Definition 14 does not holdin some casesTherefore new operation rules for INSs shouldbe explored

3 Operations and Comparison Approachfor INSs

31 Operations for INSs Xu defined some operations of inter-val valued intuitionistic fuzzy numbers [42] Based on theseoperations and preliminaries in Section 2 the operations oftwo INSs can be defined as follows

Definition 15 Let two INNs 119860 = ⟨[inf 119879119860

sup119879119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ 119861 = ⟨[inf 119879

119861

sup119879119861] [inf 119868

119861 sup 119868

119861] [inf 119865

119861 sup119865

119861]⟩ and 120582 gt 0 The

operations for INNs are defined based on the Archimedeant-conorm and t-norm as below

(1)

120582119860 = ⟨[119897minus1(120582119897 (inf 119879

119860)) 119897minus1(120582119897 (sup119879

119860))]

[119896minus1(120582119896 (inf 119868

119860)) 119896minus1(120582119896 (sup 119868

119860))]

[119896minus1(120582119896 (inf 119865

119860)) 119896minus1(120582119896 (sup119865

119860))]⟩

(8)

(2)

119860120582= ⟨[(119896

minus1(120582119896 (inf 119879

119860))) (119896

minus1(120582119896 (sup119879

119860)))]

[119897minus1(120582119897 (inf 119868

119860)) 119897minus1(120582119897 (sup 119868

119860))]

[119897minus1(120582119897 (inf 119865

119860)) 119897minus1(120582119897 (sup119865

119860))]⟩

(9)


(3)

119860 + 119861

= ⟨[119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))

119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))]

[119896minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))

119896minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))]

[119896minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))

119896minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))]⟩

(10)

(4)

119860 sdot 119861

= ⟨[119896minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))

119896minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))]

[119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))

119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))]

[119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))

119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))]⟩

(11)

Let 119860 and 119861 be both INNs If we assign its generator 119896 aspecific form specific operations for INSs will be obtainedWhen 119896(119909) = minus log(119909) we have

(5)

120582119860 = ⟨[1 minus (1 minus inf 119879119860)120582 1 minus (1 minus sup119879

119860)120582]

[(inf 119868119860)120582 (sup 119868

119860)120582]

[(inf 119865119860)120582 (sup119865

119860)120582]⟩

(12)

(6)

119860120582= ⟨[(inf 119879

119860)120582 (sup119879

119860)120582]

[1 minus (1 minus inf 119868119860)120582 1 minus (1 minus sup 119868

119860)120582]

[1 minus (1 minus inf 119865119860)120582 1 minus (1 minus sup119865

119860)120582]⟩

(13)

(7)

119860 + 119861 = ⟨[inf 119879119860+ inf 119879

119861minus inf 119879

119860sdot inf 119879

119861

sup119879119860+ sup119879

119861minus sup119879

119860sdot sup119879

119861]

[inf 119879119860sdot inf 119868119861 sup 119868

119860sdot sup 119868

119861]

[inf 119865119860sdot inf 119865

119861 sup119865

119860sdot sup119865

119861]⟩

(14)

(8)

119860 sdot 119861 = ⟨[inf 119879119860sdot inf 119879

119861 sup119879

119860sdot sup119879

119861]

[inf 119879119860+ inf 119868

119861minus inf 119879

119860sdot inf 119868119861

sup 119868119860+ sup 119868

119861minus sup 119868

119860sdot sup 119868

119861]

[inf 119865119860+ inf 119865

119861minus inf 119865

119860sdot inf 119865

119861

sup119865119860+ sup119865

119861minus sup119865

119860sdot sup119865

119861]⟩

(15)

Theorem 16 Let three INNs 119860 = ⟨[inf 119879119860

sup119879119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ 119861 = ⟨[inf 119879

119861

sup119879119861] [inf 119868

119861 sup 119868

119861] [inf 119865

119861 sup119865

119861]⟩ 119862 = ⟨[inf 119879

119862

sup119879119862] [inf 119868

119862 sup 119868

119862] [inf 119865

119862 sup119865

119862]⟩ and then the

following equations are true

(1) 119860 + 119861 = 119861 + 119860

(2) 119860 sdot 119861 = 119861 sdot 119860

(3) 120582(119860 + 119861) = 120582119860 + 120582119861 120582 gt 0

(4) (119860 sdot 119861)120582 = 119860120582 + 119861120582 120582 gt 0

(5) 1205821119860 + 120582

2119860 = (120582

1+ 1205822)119860 1205821gt 0 120582

2gt 0

(6) 1198601205821 sdot 1198601205822 = 119860(1205821+1205822) 1205821gt 0 120582

2gt 0

(7) (119860 + 119861) + 119862 = 119860 + (119861 + 119862)

(8) (119860 sdot 119861) sdot 119862 = 119860 sdot (119861 sdot 119862)

Proof (1) (2) (7) and (8) are obvious thus we prove theothers

(3)

120582 (119860 + 119861) = 120582 sdot ⟨[119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))

119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))]

[119896minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))

119896minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))]

[119896minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))

119896minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))]⟩

= ⟨[119897minus1(120582119897 (119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))))


119897minus1(120582119897 (119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))))]

[119896minus1(120582119896 (119896

minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))))]

[119896minus1(120582119896 (119896

minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))))]⟩

= ⟨[119897minus1(120582 (119897 (inf 119879

119860) + 119897 (inf 119879

119861)))

119897minus1(120582 (119897 (sup119879

119860) + 119897 (sup119879

119861)))]

[119896minus1(120582 (119896 (inf 119868

119860) + 119896 (inf 119868

119861)))

119896minus1(120582 (119896 (sup 119868

119860) + 119896 (sup 119868

119861)))]

[119896minus1(120582 (119896 (inf 119865

119860) + 119896 (inf 119865

119861)))

119896minus1(120582 (119896 (sup119865

119860) + 119896 (sup119865

119861)))]⟩

= ⟨[119897minus1(120582119897 (inf 119879

119860) + 120582119897 (inf 119879

119861))

119897minus1(120582119897 (sup119879

119860) + 120582119897 (sup119879

119861))]

[119896minus1(120582119896 (inf 119868

119860) + 120582119896 (inf 119868

119861))

119896minus1(120582119896 (sup 119868

119860) + 120582119896 (sup 119868

119861))]

[119896minus1(120582119896 (inf 119865

119860) + 120582119896 (inf 119865

119861))

119896minus1(120582119896 (sup119865

119860) + 120582119896 (sup119865

119861))]⟩

= 120582119860 + 120582119861

(16)

(4)

(119860 sdot 119861)120582= (⟨[119896

minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))

119896minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))]

[119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))

119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))]

[119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))

119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))]⟩)120582

= ⟨[119896minus1(120582119896 (119896

minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))))]

[119897minus1(120582119897 (119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))))

119897minus1(120582119897 (119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))))]

[119897minus1(120582119897 (119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))))

119897minus1(120582119897 (119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))))]⟩

= ⟨[119896minus1(120582 (119896 (inf 119879

119860) + 119896 (inf 119879

119861)))

119896minus1(120582 (119896 (sup119879

119860) + 119896 (sup119879

119861)))]

[119897minus1(120582 (119897 (inf 119868

119860) + 119897 (inf 119868

119861)))

119897minus1(120582 (119897 (sup 119868

119860) + 119897 (sup 119868

119861)))]

[119897minus1(120582 (119897 (inf 119865

119860) + 119897 (inf 119865

119861)))

119897minus1(120582 (119897 (sup119865

119860) + 119897 (sup119865

119861)))]⟩

= ⟨[119896minus1(120582119896 (inf 119879

119860) + 120582119896 (inf 119879

119861))

119896minus1(120582119896 (sup119879

119860) + 120582119896 (sup119879

119861))]

[119897minus1(120582119897 (inf 119868

119860) + 120582119897 (inf 119868

119861))

119897minus1(120582119897 (sup 119868

119860) + 120582119897 (sup 119868

119861))]

[119897minus1(120582119897 (inf 119865

119860) + 120582119897 (inf 119865

119861))

119897minus1(120582119897 (sup119865

119860) + 120582119897 (sup119865

119861))]⟩

= 119860120582sdot 119861120582

(17)

(5)

1205821119860 + 120582

2119860

= ⟨[119897minus1(1205821119897 (inf 119879

119860)) 119897minus1(1205821119897 (sup119879

119860))]

[119896minus1(1205821119896 (inf 119868

119860)) 119896minus1(1205821119896 (sup 119868

119860))]

[119896minus1(1205821119896 (inf 119865

119860)) 119896minus1(1205821119896 (sup119865

119860))]⟩

oplus ⟨[119897minus1(1205822119897 (inf 119879

119860)) 119897minus1(1205822119897 (sup119879

119860))]

[119896minus1(1205822119896 (inf 119868

119860)) 119896minus1(1205822119896 (sup 119868

119860))]

[119896minus1(1205822119896 (inf 119865

119860)) 119896minus1(1205822119896 (sup119865

119860))]⟩

= ⟨[119897minus1(119897 (119897minus1(1205821119897 (inf 119879

119860))) + 119897 (119897

minus1(1205822119897 (inf 119879

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup119879

119860))) + 119897 (119897

minus1(1205822119897 (sup119879

119860))))]

[119896minus1(119896 (119896minus1(1205821119896 (inf 119868

119860)))

+119896 (119896minus1(1205822119896 (inf 119868

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup 119868

119860)))

+119896 (119896minus1(1205822119896 (sup 119868

119860))))]


[119896minus1(119896 (119896minus1(1205821119896 (inf 119865

119860)))

+119896 (119896minus1(1205822119896 (inf 119865

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119865

119860)))

+119896 (119896minus1(1205822119896 (sup119865

119860))))]⟩

= ⟨[119897minus1(1205821119897 (inf 119879

119860) + 1205822119897 (inf 119879

119860))

119897minus1(1205821119897 (sup119879

119860) + 1205822119897 (sup119879

119860))]

[119896minus1(1205821119896 (inf 119868

119860) + 1205822119896 (inf 119868

119860))

119896minus1(1205821119896 (sup 119868

119860) + 1205822119896 (sup 119868

119860))]

[119896minus1(1205821119896 (inf 119865

119860) + 1205822119896 (inf 119865

119860))

119896minus1(1205821119896 (sup119865

119860) + 1205822119896 (sup119865

119860))]⟩

= ⟨[119897minus1((1205821+ 1205822) 119897 (inf 119879

119860))

119897minus1((1205821+ 1205822) 119897 (sup119879

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119868

119860))

119896minus1((1205821+ 1205822) 119896 (sup 119868

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119865

119860))

119896minus1((1205821+ 1205822) 119896 (sup119865

119860))]⟩

= (1205821+ 1205822) 119860

(18)

(6)

1198601205821 sdot 1198601205822

=⟨[119896minus1(1205821119896 (inf 119879

119860)) 119896minus1(1205821119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860)) 119897minus1(1205821119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860)) 119897minus1(1205821119897 (sup119865

119860))]⟩

sdot ⟨[119896minus1(1205822119896 (inf 119879

119860)) 119896minus1(1205822119896 (sup119879

119860))]

[119897minus1(1205822119897 (inf 119868

119860)) 119897minus1(1205822119897 (sup 119868

119860))]

[119897minus1(1205822119897 (inf 119865

119860)) 119897minus1(1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1(119896 (119896minus1(1205821119896 (inf 119879

119860)))

+119896 (119896minus1(1205822119896 (inf 119879

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119879

119860)))

+119896 (119896minus1(1205822119896 (sup119879

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119868

119860)))

+119897 (119897minus1(1205822119897 (inf 119868

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup 119868

119860)))

+119897 (119897minus1(1205822119897 (sup 119868

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119865

119860)))

+ 119897 (119897minus1(1205822119897 (inf 119865

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup119865

119860)))

+119897 (119897minus1(1205822119897 (sup119865

119860))))]⟩

= ⟨[119896minus1(1205821119896 (inf 119879

119860) + 1205822119896 (inf 119879

119860))

119896minus1(1205821119896 (sup119879

119860) + 1205822119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860) + 1205822119897 (inf 119868

119860))

119897minus1(1205821119897 (sup 119868

119860) + 1205822119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860) + 1205822119897 (inf 119865

119860))

119897minus1(1205821119897 (sup119865

119860) + 1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1((1205821+ 1205822) 119896 (inf 119879

119860))

119896minus1((1205821+ 1205822) 119896 (sup119879

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119868

119860))

119897minus1((1205821+ 1205822) 119897 (sup 119868

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119865

119860))

119897minus1((1205821+ 1205822) 119897 (sup119865

119860))]⟩

= 1198601205821+1205822

(19)

Example 17 Assume that119860 = ⟨[07 08] [00 01] [01 02]⟩119861 = ⟨[04 05] [02 03] [03 04]⟩ and 120582 = 2 When 119896(119909) =minus log(119909) then

(1) 2 sdot 119860 = ⟨[091 096] [0 001] [001 004]⟩(2) 1198602 = ⟨[049 064] [0 019] [019 036]⟩(3) 119860 + 119861 = ⟨[082 090] [0 005] [003 008]⟩(4) 119860 sdot 119861 = ⟨[028 040] [020 037] [037 052]⟩

INSs are the extension of SVNSs or SNSs Assume thatinf 119879119860(119909) = sup119879

119860(119909) inf 119868

119860(119909) = sup 119868

119860(119909) inf 119865

119860(119909) =

sup119865119860(119909) inf 119879

119861(119909) = sup119879

119861(119909) inf 119868

119861(119909) = sup 119868

119861(119909)

and inf 119865119861(119909) = sup119865

119861(119909) and then the two INSs 119860 =

⟨119879119860(119909) 119868119860(119909) 119865119860(119909)⟩ and 119861 = ⟨119879

119861(119909) 119868119861(119909) 119865119861(119909)⟩ are


reduced to SNSs and SVNSs According to Definition 15 andTheorem 16 the SNS or SVNS operations can be obtained

IVIFSs are an instance of NSs Let inf 119868119860= sup 119868

119860= 0

inf 119868119861= sup 119868

119861= 0 sup119879

119860+sup119865

119860le 1 and sup119879

119861+sup119865

119861le

1 Then the two INSs 119860 = ⟨119879119860 119868119860 119865119860⟩ and 119861 = ⟨119879

119861 119868119861 119865119861⟩

are reduced to IVIFSs According to Definition 15 when119896(119909) = minus log(119909) the following equations can be obtained

(1) 119860 + 119861 = ⟨[inf 119879119860+ inf 119879

119861minus inf 119879

119860sdot inf 119879

119861 sup119879

119860+

sup119879119861minus sup119879

119860sdot sup119879

119861][inf 119865

119860sdot inf 119865

119861 sup119865

119860sdot

sup119865119861]⟩

(2) 119860 sdot 119861 = ⟨[inf 119879119860sdot inf 119879

119861 sup119879

119860sdot sup119879

119861] [inf 119865

119860+

inf 119865119861minus inf 119865

119860sdot inf 119865

119861 sup119865

119860+ sup119865

119861minus sup119865

119860sdot

sup119865119861]⟩

(3) 120582 sdot 119860 = ⟨[1 minus (1 minus inf 119879119860)120582 1 minus (1 minus sup119879

119860)120582]

[(inf 119865119860)120582 (sup119865

119860)120582]⟩

(4) 119860120582 = ⟨[(inf 119879119860)120582 (sup119879

119860)120582] [1minus(1minusinf 119865

119860)120582 1minus(1minus

sup119865119860)120582]⟩

which coincides with the operations of IVIFSs in [42] Itindicates that the same principles of INSs inDefinition 15 alsoadapt to IVIFSs In fact when the indeterminacy factor i isreplaced by 120587 = 1 minus 119879 minus 119865 the NS is an IFS

32 Comparison Rules Based on the score function andaccuracy function of IVIFSs the score function accuracyfunction and certainty function of an INN 119860 are defined

Definition 18 Let the INN 119860 =

⟨[inf 119879119860 sup119879

119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ and

then

(1) 119904(119860) = [inf 119879119860+ 1 minus sup 119868

119860+ 1 minus sup119865

119860 sup119879

119860+ 1 minus

inf 119868119860+ 1 minus inf 119865

119860]

(2) 119886(119860) = [mininf 119879119860

minus inf 119865119860 sup119879

119860minus

sup119865119860maxinf 119879

119860minus inf 119865

119860 sup119879

119860minus sup119865

119860]

(3) 119888(119860) = [inf 119879119860 sup119879

119860]

where 119904(119860) 119886(119860) and 119888(119860) represent the score functionaccuracy function and certainty function of the INN 119860respectively

The score function is an important index in rankingINNs For an INN A the bigger the truth-membership TAis the greater the INS is And the less the indeterminacy-membership IA is the greater the INS is Similarly the smallerthe false-membershipFA is the greater the INS is At the sametime inf 119879

119860(119909) sup119879

119860(119909) inf 119868

119860(119909) sup 119868

119860(119909) inf 119865

119860(119909)

sup119865119860(119909) sube [0 1] so the score function s(A) is defined as

shown above For the accuracy function if the differencebetween truth and falsity is bigger then the statement issurer That is the larger the values of T I and F are themore the accuracy of the INS is So the accuracy functionis given above As to the certainty function the value oftruth-membership TA is bigger and it means more certaintyof the INS

Example 19 Assume that119860 = ⟨[07 08] [00 01][01 02]⟩and 119861 = ⟨[04 05] [02 03] [03 04]⟩ and then

(1) 119904(119860) = [24 27] 119904(119861) = [17 20]

(2) 119886(119860) = [06 06] 119886(119861) = [01 01]

(3) 119888(119860) = [07 08] 119888(119861) = [04 05]

On the basis of Definition 18 the method to compare INNscan be defined as follows

Definition 20 Let 119860 and 119861 be two INNs The comparisonapproach can be defined as follows

(1) If 119901(119904(119860) ge 119904(119861)) gt 05 then 119860 is greater than 119861 thatis 119860 is superior to 119861 denoted by 119860 ≻ 119861

(2) If 119901(119904(119860) ge 119904(119861)) = 05 and 119901(119886(119860) ge 119886(119861)) gt 05then 119860 is greater than 119861 that is 119860 is superior to 119861denoted by 119860 ≻ 119861

(3) If 119901(119904(119860) ge 119904(119861)) = 05 119901(119886(119860) ge 119886(119861)) = 05 and119901(119888(119860) ge 119888(119861)) gt 05 then119860 is greater than 119861 that is119860 is superior to 119861 denoted by 119860 ≻ 119861

(4) If 119901(119904(119860) ge 119904(119861)) = 05 119901(119886(119860) ge 119886(119861)) = 05 and119901(119888(119860) ge 119888(119861)) = 05 then 119860 is equal to 119861 that is 119860is indifferent to 119861 denoted by 119860 sim 119861

Example 21 Let 119860 and 119861 be two INNs(1) Assume that 119860 = ⟨[07 08] [00 01] [01 02]⟩

and 119861 = ⟨[04 05] [02 03] [03 04]⟩ Referring toDefinition 18 119904(119860) = [24 27] 119904(119861) = [17 20] 119886(119860) =[06 06] 119886(119861) = [01 01] 119888(119860) = [07 08] and 119888(119861) =[04 05] According to Definition 20 119901(119904(119860) ge 119904(119861)) = 1 gt05 Therefore 119860 ≻ 119861

(2) Assuming that 119860 = ⟨[06 07] [03 04] [04 05]⟩

and 119861 = ⟨[04 05] [02 03] [03 04]⟩ referring toDefinition 18 119904(119860) = [17 20] 119904(119861) = [17 20] 119886(119860) =[02 02] 119886(119861) = [01 01] 119888(119860) = [06 07] and 119888(119861) =[04 05] According to Definition 20 119901(119904(119860) ge 119904(119861)) = 05119901(119886(119860) ge 119886(119861)) = 1 gt 05 Therefore 119860 ≻ 119861

(3) For two INNs 119860 = ⟨[06 07] [03 04] [04 05]⟩

and 119861 = ⟨[04 05] [03 04] [02 03]⟩ referring toDefinition 18 119904(119860) = [17 20] 119904(119861) = [17 20] 119886(119860) =[02 02] 119886(119861) = [02 02] 119888(119860) = [06 07] and 119888(119861) =[04 05] According to Definition 20 119901(119904(119860) ge 119904(119861)) = 05119901(119886(119860) ge 119886(119861)) = 05 and 119901(119888(119860) ge 119888(119861)) = 1 gt 05Therefore 119860 ≻ 119861

4 INN Aggregation Operators and TheirApplications to Multicriteria DecisionMaking Problems

In this section applying the INS operations we presentaggregation operators for INNs and propose a method formulticriteria decision making by means of the aggregationoperators


41 INN Aggregation Operators

Definition 22 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a

collection of INNs and let INNWA INN119899 rarr INN

INNWA119908(1198601 1198602 119860

119899) = 11990811198601+ 11990821198602+ sdot sdot sdot + 119908

119899119860119899

=

119899

sum

119895=1

119908119895119860119895

(20)

then INNWA is called the interval neutrosophic numberweighted averaging operator of dimension 119899 where 119882 =

(1199081 1199082 119908

119899) is the weight vector of 119860

119895(119895 = 1 2 119899)

with 119908119895ge 0(119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 23 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a

collection of INNs and 119882 = (1199081 1199082 119908

119899) be the weight

vector of 119860119895(119895 = 1 2 119899) with 119908

119895ge 0 (119895 = 1 2 119899)

andsum119899119895=1119908119895= 1 then their aggregated result using the INNWA

operator is also an INN and

INNWA119908(1198601 1198602 119860

119899)

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602) sdot sdot sdot

+119908119899119897 (inf 119879

119860119899))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879


+119908119899119897 (sup119879

119860119899))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868


+119908119899119896 (inf 119868

119860119899))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868


+119908119899119896 (sup 119868

119860119899))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865


+119908119899119896 (inf 119865

119860119899))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865


+119908119899119896 (sup119865

119860119899))]⟩

(21)

where 119896 is the additive generator of Archimedean t-norm thatis used in the operations of INSs and 119897(119909) = 119896(1 minus 119909)

Let 119896(119909) = minus log(119909) Then 119897(119909) = minus log(1 minus 119909) 119896minus1(119909) =119890minus119909 and 119897minus1(119909) = 1 minus 119890minus119909 And the aggregated result using theINNWA operator in Theorem 23 can be represented by

INNWA119908(1198601 1198602 119860

119899)

= ⟨[1 minus

119899

prod

119894=1

(1 minus inf 119879119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119879119860119894)119908119894]

[

119899

prod

119894=1

inf 119868119908119894119860119894

119899

prod

119894=1

sup 119868119908119894119860119894]

[

119899

prod

119894=1

inf 119865119908119894119860119894

119899

prod

119894=1

sup119865119908119894119860119894]⟩

(22)

where 119882 = (1199081 1199082 119908119899) is the weight vector of 119860

119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1

Proof By using the mathematical induction on 119899we have thefollowing

(1) For 119899 = 2 since

11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601)) 119897minus1(1199081119897 (sup119879

1198601))]

[119896minus1(1199081119896 (inf 119868

1198601)) 119896minus1(1199081119896 (sup 119868

1198601))]

[119896minus1(1199081119896 (inf 119865

1198601)) 119896minus1(1199081119896 (sup119865

1198601))]⟩

oplus ⟨[119897minus1(1199082119897 (inf 119879

1198602)) 119897minus1(1199082119897 (sup119879

1198602))]

[119896minus1(1199082119896 (inf 119868

1198602)) 119896minus1(1199082119896 (sup 119868

1198602))]

[119896minus1(1199082119896 (inf 119865

1198602)) 119896minus1(1199082119896 (sup119865

1198602))]⟩

= ⟨[119897minus1(119897 (119897minus1(1199081119897 (inf 119879

1198601)))

+119897 (119897minus1(1199082119897 (inf 119879

1198602))))

119897minus1(119897 (119897minus1(1199081119897 (sup119879

1198601)))

+119897 (119897minus1(1199082119897 (sup119879

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119868

1198601)))

+119896 (119896minus1(1199082119896 (inf 119868

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup 119868

1198601)))

+119896 (119896minus1(1199082119896 (sup 119868

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119865

1198601)))

+119896 (119896minus1(1199082119896 (inf 119865

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup119865

1198601)))

+119896 (119896minus1(1199082119896 (sup119865

1198602))))]⟩

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))


119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(23)

then

SNNWA119908(1198601 1198602)

= 11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(24)

(2) If (21) holds for 119899 = 119896 that is

SNNWA119908(1198601 1198602 119860

119896)

= ⟨[

[

119897minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(25)

then if 119899 = 119896 + 1 we have

SNNWA119908(1198601 1198602 119860

119896 119860119896+1)

= ⟨[

[

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (inf 119879

119860119896+1)))))

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (sup119879

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (inf 119868

119860119896+1)))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (sup 119868

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (inf 119865

119860119896+1))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (sup119865

119860119896+1))))]

]

⟩

= ⟨[

[

119897minus1(

119896+1

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896+1

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119865

119860119895))


119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(26)

that is (29) holds for 119899 = 119896 + 1 Thus (29) holds for all 119899Then we have

SNNWA119908(1198601 1198602 119860

119899)

= ⟨[

[

119897minus1(

119899

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119899

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(27)

which completes the proof

It is obvious that the INNWG operator has the followingproperties

(1) Idempotency let 119860119895(119895 = 1 2 119899) be a collec-

tion of INNs If all 119860119895(119895 = 1 2 119899) are equal

that is 119860119895= 119860 for all 119895 isin 1 2 119899 then

INNWA119908(1198601 1198602 119860

119899) = 119860

(2) Boundedness assume that 119860119895(119895 = 1 2

119899) is a collection of INNs and 119860minus = ⟨min119895119879119860119895(119909)

max119895119868119860119895(119909)max

119895119865119860119895(119909)⟩ 119860+ = ⟨max

119895119879119860119895(119909)

min119895119868119860119895(119909)min

119895119865119860119895(119909)⟩ for all 119895 isin 1 2 119899 and

then 119860minus sube INNWA119908(1198601 1198602 119860

119899) sube 119860+

(3) Monotonity assuming that 119860119895(119895 = 1 2 119899)

is a collection of INNs if 119860119895

sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWA119908(1198601 1198602 119860

119899) sube

INNWA119908(119860lowast

1 119860lowast

2 119860

lowast

119899)


collection of INNs and let INNWG INN119899 rarr INN

INNWG119908(1198601 1198602 119860

119899) =

119899

prod

119895=1

119860119908119894

119895 (28)

then INNWG is called an interval neutrosophic numberweighted geometric operator of dimension 119899 where 119882 =

(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 25 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a

collection of INNs and one has the following result by usingDefinition 15

119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[119896minus1(1199081119896 (inf 119879

1198601) + 1199082119896 (inf 119879


+119908119899119896 (inf 119879

119860119899))

119896minus1(1199081119896 (sup119879

1198601) + 119908

2119896 (sup119879


+119908119899119896 (sup119879

119860119899))]

[119897minus1(1199081119897 (inf 119868

1198601) + 1199082119897 (inf 119868


+119908119899119897 (inf 119868

119860119899))

119897minus1(1199081119897 (sup 119868

1198601) + 1199082119897 (sup 119868


+119908119899119897 (sup 119868

119860119899))]

[119897minus1(1199081119897 (inf 119865

1198601) + 1199082119897 (inf 119865


+119908119899119897 (inf 119865

119860119899))

119897minus1(1199081119897 (sup119865

1198601) + 1199082119897 (sup119865


+119908119899119897 (sup119865

119860119899))]⟩

(29)

Assume that 119896(119909) = minus log(119909) and then 119897(119909) = minus log(1 minus 119909)119896minus1(119909) = 119890

minus119909 119897minus1(119909) = 1 minus 119890minus119909 The aggregated result using theINNWG operator in Theorem 25 can be represented by

119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[

119899

prod

119894=1

inf 119879119908119894119860119894

119899

prod

119894=1

sup119879119908119894119860119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119868119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup 119868119860119894)119908119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119865119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119865119860119894)119908119894]⟩

(30)

where 119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1

Let 119896(119909) = log((2 minus 119909)119909) and then 119897(119909) = log((1 +119909)(1 minus 119909)) 119896minus1(119909) = 2(119890119909 + 1) and 119897minus1(119909) = 1 minus (2(119890119909 +


1)) The aggregated result using the INNWG operator inTheorem 25 can be denoted by

INNWG119908(1198601 1198602 119860

119899)

= ⟨[

[

2prod119899

119894=1inf 119879119908119894119860119894

prod119899

119894=1(2 minus inf 119879

119860119894)119908119894+prod119899

119894=1inf 119879119908119894119860119894

2prod119899

119894=1sup119879119908119894119860119894

prod119899

119894=1(2 minus sup119879

119860119894)119908119894+prod119899

119894=1sup119879119908119894119860119894

]

]

[

[

prod119899

119894=1(1 + inf 119868

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + inf 119868

119860119894)119908119894+prod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894minusprod119899

119894=1(1 minus sup 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894+prod119899

119894=1(1 minus sup 119868

119860119894)119908119894

]

]

[

[

prod119899

119894=1(1 + inf 119865

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + inf 119865

119860119894)119908119894+prod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894minusprod119899

119894=1(1 minus sup119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894+prod119899

119894=1(1 minus sup119865

119860119894)119908119894

]

]

⟩

(31)

where119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1

Theorem 25 also can be proved by the mathematicalinduction

Similarly it can be proved that the INNWG operator hasthe same properties as the INNWA operator




INNWG119908(1198601 1198602 119860

119899) = 119860

(2) Boundedness assume that 119860119895(119895 =

1 2 119899) is a collection of INNs and119860minus

= ⟨min119895119879119860119895(119909)max

119895119868119860119895(119909)max

119895119865119860119895(119909)⟩

119860+

= ⟨max119895119879119860119895(119909)min

119895119868119860119895(119909)min

119895119865119860119895(119909)⟩

for all 119895 isin 1 2 119899 and then 119860minus

sube

INNWG119908(1198601 1198602 119860

119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWG119908(1198601 1198602 119860

119899) sube

INNWG119908(119860lowast

1 119860lowast

2 119860

lowast

119899)

42 Multicriteria Decision Making Method Based on the INNAggregation Operators Assume that there are m alternatives119860 = 119886

1 1198862 119886119898 and n criteria 119862 = 119888

1 1198882 119888119899 whose

criterion weight vector is 119882 = (1199081 1199082 119908

119899) where

119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Let 119877 =

(119886119894119895)119898times119899

be the interval neutrosophic decision matrix where119886119894119895= ⟨119879119886119894119895 119868119886119894119895 119865119886119894119895⟩ is a criterion value denoted by an INN

where 119879119886119894119895

indicates the truth-membership function wherethe alternative 119886

119894satisfies the criterion 119888

119895 119868119886119894119895

indicates theindeterminacy-membership function where the alternative119886119894satisfies the criterion 119888

119895 and 119865

119886119894119895indicates the falsity-

membership function where the alternative 119886119894satisfies the

criterion 119888119895

In the following a procedure to rank and select the mostdesirable alternative(s) is given

Step 1 Utilize the INNWA operator or the INNWG operatorto obtain the INN 119910

119894for the alternatives 119886

119894(119894 = 1 2 119898)

that is

119910119894= INNWA

119908(1198861198941 1198861198942 119886

119894119899) (32)

or

119910119894= INNWG

119908(1198861198941 1198861198942 119886

119894119899) (33)

Step 2 Calculate the score function value 119904(119910119894) the accuracy

function value 119886(119910119894) and the certainty function value 119888(119910

119894) of

119910119894(119894 = 1 2 119898) by Definition 18 denoted by the function

matrix F

119865 =[[[

[

119904 (1199101) 119886 (119910

1) 119888 (119910

1)

119904 (1199102) 119886 (119910

2) 119888 (119910

2)

sdot sdot sdot sdot sdot sdot sdot sdot sdot

119904 (119910119898) 119886 (119910

119898) 119888 (119910

119898)

]]]

]

(34)

Step 3 Construct the possibility matrix 119875119904of the score

function value 119904(119910119894) as follows according to Definition 2

119875119904=

[[[[

[

11990111990411

11990111990412

sdot sdot sdot 1199011199041119898

11990111990421

11990111990422

sdot sdot sdot 1199011199042119898

1199011199041198981

1199011199041198982

sdot sdot sdot 119901119904119898119898

]]]]

]

(35)

where 119901119904119894119895denotes the degree of possibility of 119904(119910

119894) gt 119904(119910

119895)

and it satisfies 119901119904119894119895ge 0 119901119904

119894119895+ 119901119904119895119894= 1 and 119901119904

119894119894= 05 If

119901119904119894119895= 05 (119894 = 119895) then calculate the degree of possibility of

119886(119910119894) gt 119886(119910

119895) denoted by 119901119886

119894119895 And if 119901119886

119894119895= 05 (119894 = 119895) then

calculate the degree of possibility of 119888(119910119894) gt 119888(119910

119895) denoted by

119901119888119894119895

Step 4 Get the priority of the alternatives 119886119894(119894 = 1 2 119898)

in accordance with 119901119904119894119895 119901119886119894119895 and 119901119888

119894119895 and choose the best

one referring to Definition 20

5 Illustrative Example

In this section an example for the multicriteria decisionmaking problem of alternatives is used as the demonstrationof the application of the proposed decision making methodas well as the effectiveness of the proposed method

Let us consider the decision making problem adaptedfrom [33] There is an investment company which wantsto invest a sum of money in the best option There is apanel with four possible alternatives to invest the money(1) 1198601is a car company (2) 119860

2is a food company (3) 119860

3


is a computer company (4) 1198604is an arms company The

investment company must make a decision according to thefollowing three criteria (1) 119862

1is the risk analysis (2) 119862

2

is the growth analysis (3) 1198623is the environmental impact

analysis where 1198621and 119862

2are benefit criteria and 119862

3is a

cost criterion The weight vector of the criteria is given by119882 = (035 025 04) The four possible alternatives are to beevaluated under the above three criteria by the form of INNsas shown in the following interval neutrosophic decisionmatrix D

119863 =

[[[

[

⟨[04 05] [02 03] [03 04]⟩ ⟨[04 06] [01 03] [02 04]⟩ ⟨[07 09] [02 03] [04 05]⟩

⟨[06 07] [01 02] [02 03]⟩ ⟨[06 07] [01 02] [02 03]⟩ ⟨[03 06] [03 05] [08 09]⟩

⟨[03 06] [02 03] [03 04]⟩ ⟨[05 06] [02 03] [03 04]⟩ ⟨[04 05] [02 04] [07 09]⟩

⟨[07 08] [00 01] [01 02]⟩ ⟨[06 07] [01 02] [01 03]⟩ ⟨[06 07] [03 04] [08 09]⟩

]]]

]

(36)

51 Procedures of Decision Making Based on INSs

Step 1 Utilize the INNWA operator or the INNWG operatorto obtain the INNs The aggregation results based on theINNWAoperator and the INNWGoperator are different andthey are calculated separately Here let 119896(119909) = minus log119909 whichmeans that the operations for INNs are based on algebraic t-conorm and t-norm

By using the INNWA operator the alternatives matrixAWA can be obtained

119860119882119860

=[[[

[

⟨[05453 07516] [01682 03000] [03041 04373]⟩

⟨[04996 06634] [01552 02885] [03482 04656]⟩

⟨[03950 05627] [02000 03366] [04210 05533]⟩

⟨[06383 07397] [00000 02071] [02297 04040]⟩

]]]

]

(37)

With the INNWG operator the alternatives matrix AWG isshown as follows

119860119882119866

=[[[

[

⟨[05004 06620] [01761 03000] [03195 04422]⟩

⟨[04547 06581] [01861 03371] [05405 06786]⟩

⟨[03824 05578] [02000 03419] [05012 07070]⟩

⟨[06333 07335] [01555 02570] [05069 06632]⟩

]]]

]

(38)

Step 2 Calculate the score function value accuracy functionvalue and certainty function value

To the alternativesmatrixAWA by usingDefinition 18 thefunction matrix of AWA can be obtained

119865119882119860

=[[[

[

[18080 22793] [02412 03143] [05453 07516]

[17455 21600] [01514 01978] [04996 06634]

[15051 19417] [minus00260 00094] [03950 05627]

[20272 25100] [03357 04086] [06383 07397]

]]]

]

(39)

To the alternatives matrix AWG by using Definition 20 thefunction matrix of AWG is shown as follows

119865119882119866

=[[[

[

[17582 21664] [01809 02198] [05004 06620]

[14390 19315] [minus00858 minus00205] [04547 06581]

[13335 18566] [minus01492 minus01188] [03824 05578]

[17131 20711] [00703 01264] [06333 07335]

]]]

]

(40)

Step 3 Construct the possibility matrix Each interval num-ber is compared to all interval numbers Referring toDefinition 2 the possibilitymatrix of the score function value119904(119910119894) can be obtained For 119865

119882119860

119875119904 119882119860

=[[[

[

05 06498 08527 02642

03974 05 07695 01480

01473 02305 05 0

07358 08520 1 05

]]]

]

(41)

And for 119865119882119866

119901119904 119882119866

=[[[

[

05 08076 08943 05916

01924 05 05888 02568

01057 04112 05 01629

04084 07432 08371 05

]]]

]

(42)

It is obvious that 119901119904119894119895= 05 (119894 = 119895) so there is no need to

compute 119901119886119894119895and 119901119888

119894119895

Step 4 Get the priority of the alternatives and choose the bestone

According to Definition 20 and results in Step 3 for AWAwe have 119860

1≻ 1198602 1198601≻ 1198603 1198602≻ 1198603 1198604≻ 1198601 and 119860

4≻

1198602 Therefore the ranking of the four alternatives is 119860

4 1198601

1198602 and 119860

3 Obviously 119860

4is the best alternative

Similarly for AWG we have 1198601 ≻ 1198602 1198601 ≻ 1198603 1198601 ≻ 11986041198602≻ 1198603 1198604≻ 1198602 and 119860

4≻ 1198603 Therefore the ranking of

the four alternatives is 1198601 1198604 1198602 and 119860

3 Obviously 119860

1is

the best alternativeWhen 119896(119909) = log((2 minus 119909)119909) for AWA the ranking of the

four alternatives is still 1198604 1198601 1198602 and 119860

3 as well as the

ranking 1198601 1198604 1198602 and 119860

3for AWG


52 Comparison Analysis and Discussion In order to validatethe feasibility of the proposed decisionmakingmethod basedon the INN aggregation operators a comparison analysiswill be conducted In Section 51 the same example adaptedfrom [33] for the multicriteria decision making problem isdemonstrated based on the INN aggregation operators Thisanalysis will be based on the same illustrative example

There is no consensus on the best way to sequence INNsYe proposed the similarity measures between INSs based onthe relationship between similarity measures and distancesand utilized the similarity measures between each alternativeand the ideal alternative to establish a multicriteria decisionmaking method for INSs in [33] By contrast we present theaggregation operators for INNs and put forward a methodformulticriteria decisionmaking bymeans of the aggregationoperators

With the same example [33] gave two rankings of the fouralternatives with different similaritymeasuresThe first one is1198604 1198602 1198603 and 119860

1 The second one is 119860

2 1198604 1198603 and 119860

1

Unlike the results in [33] we obtained the ranking sequencesas 1198604 1198601 1198602 1198603and 119860

1 1198604 1198602 and 119860

3 Obviously

the results in [33] conflict with ours in this paper And thedifference mainly lies in the position of 119860

1

Here for convenience the decision matrixD in Section 5is denoted by

119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)

Certainly the alternatives1198601can be obtained by the decision

vector (11988611 11988612 11988613) with the associated weight vector119882 =

(035 025 04) Firstly consider 1198601and 119860

3 As can be

seen from the decision matrix D the truth-membershipthe indeterminacy-membership and the falsity-membershipsatisfies

11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)

And with Definition 2 119901(11987911988611gt 11987911988631) = 05 so 119879

11988611= 11987911988631

Therefore 11988611= 11988631 Similarly it can obtained that 119886

13= 11988633

significantly And byDefinitions 18 and 20 11988612lt 11988632with a bit

difference that is 11988612is close to 119886

32 so that with the weighted

vector119882 = (035 025 04)1198601≻ 1198603 Thus there is a conflict

of sequences of 1198601and 119860

3in [33]

Similarly it is obvious that 11988621

lt 11988641 11988622

lt 11988642

and 11988623

lt 11988643 so that with the associated weight vector

119882 = (035 025 04) 1198601≺ 1198604 which is not coordinated

with the ranking of 1198602 1198604 1198603 and 119860

1in [33] while the

sequences of 1198601 1198602and 119860

1 1198603obtained by the method

in this paper are consistent with the realities Here are thereasons for this The difference between INSs is distorted Inthe similarity measures in [33] the distances between INSsare calculated firstly and the difference was amplified in theresults because of criteria weights This causes the distortionof similarity between an alternative and the ideal alternativeIn addition the ranking of all alternatives was determinedby the similarity so that the degree of distortion can not bereducedHowever the difference between INSs in themethod

proposed in this paper was reserved to the final calculationCombining the factors above the final result produced by themethod proposed in this paper is more precise and reliablethan the result produced in [33]

6 Conclusion

INSs can be applied in addressing problems with uncertainimprecise incomplete and inconsistent information existingin real scientific and engineering applications However asa new branch of NSs there is no enough research aboutINSs In particular the existing literatures do not put forwardthe aggregation operators and multicriteria decision makingmethod for INSs Based on the related research achievementsin IVIFSs we defined the operations of INSs And theapproach to compare INNs was proposed In addition theaggregation operators of INNWA and INNWG were givenThus a multicriteria decision making method is establishedbased on the proposed operators Utilizing the comparisonapproach the ranking of all alternatives can be determinedand the best one can be easily identified as well The illus-trative example demonstrates the application of the proposeddecision making method Although there is no consensus onthe best way to sequence INNs compared to the multicriteriadecision making method for INSs in [33] the illustrativeexample shows that the final result produced by the methodproposed in this paper is more precise and reliable than theresult produced in [33] In this way the method proposed inthis paper can provide a reliable basis for INSs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the editors and anonymous reviewersfor their helpful comments and suggestions This work wassupported by Humanities and Social Sciences Foundation ofMinistry of Education of China (no 11YJCZH227) and theNationalNatural Science Foundation of China (nos 7127121871221061 and 71210003)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] L A Zadeh ldquoProbability measures of Fuzzy eventsrdquo Journal ofMathematical Analysis and Applications vol 23 no 2 pp 421ndash427 1968

[3] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[4] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[5] K T Atanassov Intuitionistic Fuzzy Sets Springer HeidelbergGermany 1999

[6] K T Atanassov ldquoTwo theorems for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 110 no 2 pp 267ndash269 2000


[7] K T Atanassov ldquoIntuitionistic Fuzzy Sets PastPresent and Futurerdquo httpciteseerxistpsueduviewdocdownloaddoi=10111452484amprep=rep1amptype=pdf

[8] W-L Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

[9] H Bustince and P Burillo ldquoVague sets are intuitionistic fuzzysetsrdquo Fuzzy Sets and Systems vol 79 no 3 pp 403ndash405 1996

[10] H-W Liu and G-J Wang ldquoMulti-criteria decision-makingmethods based on intuitionistic fuzzy setsrdquo European Journalof Operational Research vol 179 no 1 pp 220ndash233 2007

[11] Z Pei and L Zheng ldquoA novel approach to multi-attributedecision making based on intuitionistic fuzzy setsrdquo ExpertSystems with Applications vol 39 no 3 pp 2560ndash2566 2012

[12] T-Y Chen ldquoAn outcome-oriented approach tomulticriteria decision analysis with intuitionistic fuzzyoptimisticpessimistic operatorsrdquo Expert Systems withApplications vol 37 no 12 pp 7762ndash7774 2010

[13] S Zeng and W Su ldquoIntuitionistic fuzzy ordered weighteddistance operatorrdquo Knowledge-Based Systems vol 24 no 8 pp1224ndash1232 2011

[14] Z S Xu ldquoIntuitionistic fuzzymultiattribute decisionmaking aninteractive methodrdquo IEEE Transactions on Fuzzy Systems vol20 no 3 pp 514ndash525 2012

[15] L Li J Yang and W Wu ldquoIntuitionistic fuzzy hopfield neuralnetwork and its stabilityrdquo Expert Systems Applications vol 129pp 589ndash597 2005

[16] S Sotirov E Sotirova and D Orozova ldquoNeural networkfor defining intuitionistic fuzzy sets in e-learningrdquo Notes onIntuitionistic Fuzzy Sets vol 15 no 2 pp 33ndash36 2009

[17] T K Shinoj and J J Sunil ldquoIntuitionistic fuzzy multisets andits application in medical fiagnosisrdquo International Journal ofMathematical and Computational Sciences vol 6 pp 34ndash372012

[18] T Chaira ldquoIntuitionistic fuzzy set approach for color regionextractionrdquo Journal of Scientific and Industrial Research vol 69no 6 pp 426ndash432 2010

[19] T Chaira ldquoA novel intuitionistic fuzzy C means clusteringalgorithm and its application to medical imagesrdquo Applied SoftComputing Journal vol 11 no 2 pp 1711ndash1717 2011

[20] B P Joshi and S Kumar ldquoFuzzy time series model based onintuitionistic fuzzy sets for empirical research in stock marketrdquoInternational Journal of Applied Evolutionary Computation vol3 no 4 pp 71ndash84 2012

[21] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[22] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[23] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Korea August 2009

[24] H Wang F Smarandache Y Q Zhang and R SunderramanldquoSingle valued neutrosophic setsrdquo Multispace and Multistruc-ture vol 4 pp 410ndash413 2010

[25] F Smarandache A Unifying Field in Logics Neutrosophy Neu-trosophic Probability Set and Logic American Research PressRehoboth Mass USA 1999

[26] F Smarandache A Unifying Field in Logics Neutrosophic LogicNeutrosophy Neutrosophic Set Neutrosophic Probability Amer-ican Research Press 2003

[27] U Rivieccio ldquoNeutrosophic logics prospects and problemsrdquoFuzzy Sets and Systems vol 159 no 14 pp 1860ndash1868 2008

[28] P Majumdar and S K Samant ldquoOn similarity and entropy ofneutrosophic setsrdquo Journal of Intelligent and Fuzzy Systems vol26 no 3 pp 1245ndash1252 2014

[29] J Ye ldquoMulticriteria decision-making method using the correla-tion coefficient under single-value neutrosophic environmentrdquoInternational Journal of General Systems vol 42 no 4 pp 386ndash394 2013

[30] J Ye ldquoA multicriteria decision-making method using aggre-gation operators for simplified neutrosophic setsrdquo Journal ofIntelligent and Fuzzy Systems

[31] H Wang F Smarandache Y Q Zhang and R SunderramanInterval Neutrosophic Sets and Logic Theory and Applications inComputing Hexis Phoenix Ariz USA 2005

[32] F G Lupianez ldquoInterval neutrosophic sets and topologyrdquoKybernetes vol 38 no 3-4 pp 621ndash624 2009

[33] J Ye ldquoSimilarity measures between interval neutrosophic setsand their applications in multicriteria decision-makingrdquo Jour-nal of Intelligent and Fuzzy Systems vol 26 no 1 pp 165ndash1722014

[34] A Sengupta and T K Pal ldquoOn comparing interval numbersrdquoEuropean Journal of Operational Research vol 127 no 1 pp 28ndash43 2000

[35] P Chen ldquoAn interval estimation for the number of signalsrdquoSignal Processing vol 85 no 8 pp 1623ndash1633 2005

[36] Z Xu ldquoDependent uncertain ordered weighted aggregationoperatorsrdquo Information Fusion vol 9 no 2 pp 310ndash316 2008

[37] Z Xu ldquoOn method for uncertain multiple attribute decisionmaking problems with uncertain multiplicative preferenceinformation on alternativesrdquo Fuzzy Optimization and DecisionMaking vol 4 no 2 pp 131ndash139 2005

[38] H T Nguyen and R A Walker A First Course in Fuzzy LogicCRC Press Boca Raton Fla USA 1997

[39] Logical Algebraic Analytic and Probabilistic Aspects of Trian-gular Norms Elsevier New York NY USA 2005 edited by EPKlement and R Mesiar

[40] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[41] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer New York NY USA 2007

[42] Z-S Xu ldquoMethods for aggregating interval-valued intuitionis-tic fuzzy information and their application to decisionmakingrdquoControl and Decision vol 22 no 2 pp 215ndash219 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


element of the universe has a degree of truth indetermi-nacy and falsity respectively and which lies in ]0minus 1+[ thenonstandard unit interval [27] Obviously it is the extensionto the standard interval [0 1] in IFSs And the uncertaintypresent here that is indeterminacy factor is independent oftruth and falsity values while the incorporated uncertainty isdependent on the degrees of belongingness and nonbelong-ingness in IFSs [28] And for the aforementioned example bymeans of NSs it can be expressed as 119909(05 02 06)

However without being specified it is difficult to apply inthe real applications Hence the single valued neutrosophicset (SVNS)was proposedwhich is an instance ofNSs [24 28]Furthermore the information energy of SVNSs correlationand correlation coefficient of SVNSs and a decision makingmethod by the use of SVNSs were presented [29] In additionYe also introduced the concept of simplified neutrosophic sets(SNSs) which can be described by three real numbers in thereal unit interval [0 1] and proposed a multicriteria decisionmaking method using aggregation operators for SNSs [30]Majumdar and Samant introduced a measure of entropy of aSVNS [28]

In fact sometimes the degree of truth falsity and inde-terminacy of a certain statement can not be defined exactlyin the real situations but denoted by several possible intervalvalues So the interval neutrosophic set (INS) was requiredsimilar to IVIFSWang et al proposed the concept of INS andgave the set-theoretic operators of INS [31] The operationsof INS were discussed in [32] yet the comparison methodswere not seen there Furthermore Ye defined the Hammingand Euclidean distances between INSs and proposed thesimilarity measures between INSs based on the relationshipbetween similarity measures and distances [33] However insome cases the INS operations in [31] might be irrationalFor instance the sum of any element and the maximumvalue should be equal to the maximum one while it does nothold with the operations in [31] In addition to the best ofour knowledge the existing literatures do not put forwardthe INS aggregation operators and decision making methodwhich were vitally important for INSs to be utilized inthe real situations in scientific and engineering applicationsTherefore the operations and comparison approach betweeninterval neutrosophic numbers (INNs) and the aggregationoperators for INSs are defined in this paper to be usedThus a multicriteria decision making method is establishedbased on the proposed operators an illustrative exampleis given to demonstrate the application of the proposedmethod

The rest of the paper is organized as follows Section 2briefly introduces interval numbers properties of t-normand t-conorm and concepts and operations of NSs SNSsand INSs And the operations and comparison approachfor INSs are defined on the basis of the IVIFS theoryin Section 3 The INN aggregation operators are givenand a decision making method is developed for INSsby means of the INN aggregation operators in Section 4In Section 5 an illustrative example is presented to illus-trate the proposed method and the comparison analysisand discussion are given Finally Section 6 concludes thepaper

2 Preliminaries

In this section some basic concepts and definitions relatedto INSs including interval numbers t-norm and t-conormand the definitions and operations of NSs SNSs and INSsare introduced which will be utilized in the rest of the paper

21 Interval Numbers andTheirOperations Interval numbersand their operations are of utmost importance to explore theoperations for INSs So some definitions and operations ofinterval numbers are given below

Definition 1 (see [34ndash37]) Let 119886 = [119886119871 119886119880] = 119909 | 119886119871 le 119909 le119886119880 and then 119886 is called an interval number In particular if

0 le 119886119871le 119909 le 119886

119880 then 119886 is reduced to a positive intervalnumber

Consider any two interval numbers 119886 = [119886119871 119886119880] and =[119887119871 119887119880] and then their operations are defined as follows

(1) 119886 = hArr 119886119871= 119887119871 119886119880= 119887119880

(2) 119886 + = [119886119871 + 119887119871 119886119880 + 119887119880](3) 119886 minus = [119886119871 minus 119887119880 119886119880 minus 119887119871](4) 119886 times = [min119886119871119887119871 119886119871119887119880 119886119880119887119871 119886119880119887119880 max119886119871119887119871

119886119871119887119880 119886119880119887119871 119886119880119887119880]

(5) 119896119886 = [119896119886119871 119896119886119880] 119896 gt 0

Definition 2 (see [37]) Let 119886 = [119886119871 119886119880] and = [119887

119871 119887119880]

119897119886= 119886119880minus119886119871 and 119897

119887= 119880minus119871 and then the degree of possibility

of 119886 ge is formulated by

119901 (119886 ge ) = max1 minusmax(119880minus 119886119871

119897119886+ 119897119887

0) 0 (1)

Suppose that there are 119899 interval numbers 119886119894

=

[119886119871

119894 119886119880

119894] (119894 = 1 2 119899) and each interval number 119886

119894is

compared to all interval numbers 119886119895(119895 = 1 2 119899) by using

(1) namely

119901119894119895= 119901 (119886

119894ge 119886119895) = max1 minusmax(

119886119880

119895minus 119886119871

119894

119897119886119894+ 119897119886119895

0) 0 (2)

Then a complementary matrix can be constructed asfollows

119875 =

[[[[

[

1199011111990112sdot sdot sdot 1199011119899

1199012111990122sdot sdot sdot 1199012119899

11990111989911199011198992sdot sdot sdot 119901119899119899

]]]]

]

(3)

where 119901119894119895ge 0 119901

119894119895+ 119901119895119894= 1 119901

119894119894= 05

22 t-Norm and t-Conorm The t-norm and its dual t-conorm play an important role in the construction of oper-ation rules and averaging operators of INSs Here some basicconcepts are introduced

Definition 3 (see [38 39]) A function 119879 [0 1] times [0 1] rarr[0 1] is called t-norm if it satisfies the following conditions


(1) forall119909 isin [0 1] 119879(1 119909) = 119909

(2) forall119909 119910 isin [0 1] 119879(119909 119910) = 119879(119910 119909)

(3) forall119909 119910 119911 isin [0 1] 119879(119909 119879(119910 119911)) = 119879(119879(119909 119910) 119911)

(4) if 119909 le 1199091015840 119910 le 1199101015840 then 119879(119909 119910) le 119879(1199091015840 1199101015840)


(1) forall119909 isin [0 1] 119878(0 119909) = 119909

(2) forall119909 119910 isin [0 1] 119878(119909 119910) = 119878(119910 119909)

(3) forall119909 119910 119911 isin [0 1] 119878(119909 119878(119910 119911)) = 119878(119878(119909 119910) 119911)

(4) if 119909 le 1199091015840 119910 le 1199101015840 then 119878(119909 119910) le 119878(1199091015840 1199101015840)












119878 (119909 119910) =119909 + 119910

1 + 119909119910 119879 (119909 119910) =

119909119910

1 + (1 minus 119909) (1 minus 119910) (5)





1 minus (1 minus 120574) 119909119910

119879 (119909 119910) =119909119910

120574 + (1 minus 120574) (119909 + 119910 minus 119909119910) 120574 gt 0

(6)



119860(119909) an



119860(119909) 119868119860(119909) and119865

119860(119909) are real


119883 rarr ]0minus 1+[ 119868119860(119909) 119883 rarr ]0


119860(119909) 119883 rarr


119860(119909) 119868119860(119909)


119860(119909) + sup 119868

119860(119909) + sup119865

119860(119909) le 3

+


119860(119909) le inf 119879

119861(119909)

sup119879119860(119909) le sup119879

119861(119909) inf 119868

119860(119909) le inf 119868

119861(119909) sup 119868

119860(119909) le

sup 119868119861(119909) inf 119865

119860(119909) le inf 119865

119861(119909) and sup119865

119860(119909) le sup119865

119861(119909)

for 119909 isin 119883



119860(119909) 119868119860(119909) and 119865



119883 rarr [0 1] 119868119860(119909) 119883 rarr [0 1] and 119865

119860(119909) 119883 rarr [0 1]


119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ | 119909 isin 119883 (7)




(1) 119860+119861 = ⟨119879119860(119909)+119879

119861(119909)minus119879

119860(119909) sdot119879

119861(119909) 119868119860(119909)+119868

119861(119909)minus

119868119860(119909) sdot 119868

119861(119909) 119865

119860(119909) + 119865

119861(119909) minus 119865

119860(119909) sdot 119865

119861(119909)⟩

(2) 119860 sdot 119861 = ⟨119879119860(119909) sdot 119879

119861(119909) 119868119860(119909) sdot 119868

119861(119909) 119865119860(119909) sdot 119865

119861(119909)⟩


119860(119909))120582 1 minus (1 minus

119865119860(119909))120582⟩ 120582 gt 0

(4) 119860120582 = ⟨119879120582119860(119909) 119868120582

119860(119909) 119865

120582

119860(119909)⟩ 120582 gt 0







2=


1(1198861 119886lowast) = 1 119878

2(1198862 119886lowast) =


2

However for 1198791198862(119909) gt 119879

1198861(119909) 119868

1198862(119909) gt 119868

1198861(119909) and 119865

1198862(119909) gt


2is superior to the

alternative 1198861




119860(119909) and


119883 we have that 119879119860(119909) = [inf 119879

119860(119909) sup119879

119860(119909)] 119868

119860(119909) =

[inf 119868119860(119909) sup 119868

119860(119909)] 119865

119860(119909) = [inf 119865

119860(119909) sup119865

119860(119909)] sube

[0 1] and 0 le sup119879119860(119909) + sup 119868

119860(119909) + sup119865

119860(119909) le 3 119909 isin 119883



inf 119879119860(119909) le inf 119879

119861(119909) sup119879

119860(119909) le sup119879

119861(119909)

inf 119868119860(119909) ge inf 119868

119861(119909) sup 119868

119860(119909) ge sup 119868

119861(119909)

inf 119865119860(119909) ge inf 119865

119861(119909) and sup119865

119860(119909) ge sup119865

119861(119909)

for any 119909 isin 119883



inf 119879119862

= min(inf 119879119860+ inf 119879

119861 1) sup119879

119862=

min(sup119879119860+ sup119879

119861 1)

inf 119868119862= min(inf 119868

119860+inf 119868

119861 1) sup 119868

119862= min(sup 119868

119860+

sup 119868119861 1)

inf 119865119862

= min(inf 119865119860+ inf 119865

119861 1) sup119865

119862=

min(sup119865119860+ sup119865

119861 1)

for all 119909 in119883


119861= sup119879

119861= 1 inf 119868

119861=

sup 119868119861= 0 and inf 119865

119861= sup119865



119862= 1 inf 119868

119862= inf 119868

119860

sup 119868119862= sup 119868

119860 inf 119865

119862= inf 119865

119860 and sup119865

119862= sup119865

119860 that is

119860 + 119861 = ⟨[1 1] [inf 119868119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ = 119861 so that






by

inf 119879119862(119909 119910) = inf 119879

119860(119909) + inf 119879

119861(119910) minus inf 119879

119860(119909) sdot

inf 119879119861(119910)

sup119879119862(119909 119910) = sup119879

119860(119909) + sup119879

119861(119910) minus sup119879

119860(119909) sdot

sup119879119861(119910)

inf 119868119862(119909 119910) = inf 119868

119860(119909) sdot inf 119868

119861(119910) sup 119868

119862(119909 119910) =

sup 119868119860(119909) sdot sup 119868

119861(119910)

inf 119865119862(119909 119910) = inf 119865

119860(119909) sdot inf 119865

119861(119910) sup119865

119862(119909 119910) =

sup119865119860(119909) sdot sup119865

119861(119910)

for all 119909 in1198831 119910 in119883

2





sup119879119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ 119861 = ⟨[inf 119879

119861

sup119879119861] [inf 119868

119861 sup 119868

119861] [inf 119865

119861 sup119865

119861]⟩ and 120582 gt 0 The


(1)

120582119860 = ⟨[119897minus1(120582119897 (inf 119879

119860)) 119897minus1(120582119897 (sup119879

119860))]

[119896minus1(120582119896 (inf 119868

119860)) 119896minus1(120582119896 (sup 119868

119860))]

[119896minus1(120582119896 (inf 119865

119860)) 119896minus1(120582119896 (sup119865

119860))]⟩

(8)

(2)

119860120582= ⟨[(119896

minus1(120582119896 (inf 119879

119860))) (119896

minus1(120582119896 (sup119879

119860)))]

[119897minus1(120582119897 (inf 119868

119860)) 119897minus1(120582119897 (sup 119868

119860))]

[119897minus1(120582119897 (inf 119865

119860)) 119897minus1(120582119897 (sup119865

119860))]⟩

(9)


(3)

119860 + 119861

= ⟨[119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))

119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))]

[119896minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))

119896minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))]

[119896minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))

119896minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))]⟩

(10)

(4)

119860 sdot 119861

= ⟨[119896minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))

119896minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))]

[119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))

119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))]

[119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))

119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))]⟩

(11)


(5)


119860)120582]

[(inf 119868119860)120582 (sup 119868

119860)120582]

[(inf 119865119860)120582 (sup119865

119860)120582]⟩

(12)

(6)

119860120582= ⟨[(inf 119879

119860)120582 (sup119879

119860)120582]


119860)120582]


119860)120582]⟩

(13)

(7)

119860 + 119861 = ⟨[inf 119879119860+ inf 119879

119861minus inf 119879

119860sdot inf 119879

119861

sup119879119860+ sup119879

119861minus sup119879

119860sdot sup119879

119861]

[inf 119879119860sdot inf 119868119861 sup 119868

119860sdot sup 119868

119861]

[inf 119865119860sdot inf 119865

119861 sup119865

119860sdot sup119865

119861]⟩

(14)

(8)

119860 sdot 119861 = ⟨[inf 119879119860sdot inf 119879

119861 sup119879

119860sdot sup119879

119861]

[inf 119879119860+ inf 119868

119861minus inf 119879

119860sdot inf 119868119861

sup 119868119860+ sup 119868

119861minus sup 119868

119860sdot sup 119868

119861]

[inf 119865119860+ inf 119865

119861minus inf 119865

119860sdot inf 119865

119861

sup119865119860+ sup119865

119861minus sup119865

119860sdot sup119865

119861]⟩

(15)


sup119879119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ 119861 = ⟨[inf 119879

119861

sup119879119861] [inf 119868

119861 sup 119868

119861] [inf 119865

119861 sup119865

119861]⟩ 119862 = ⟨[inf 119879

119862

sup119879119862] [inf 119868

119862 sup 119868

119862] [inf 119865

119862 sup119865



(1) 119860 + 119861 = 119861 + 119860

(2) 119860 sdot 119861 = 119861 sdot 119860

(3) 120582(119860 + 119861) = 120582119860 + 120582119861 120582 gt 0

(4) (119860 sdot 119861)120582 = 119860120582 + 119861120582 120582 gt 0

(5) 1205821119860 + 120582

2119860 = (120582

1+ 1205822)119860 1205821gt 0 120582

2gt 0

(6) 1198601205821 sdot 1198601205822 = 119860(1205821+1205822) 1205821gt 0 120582

2gt 0

(7) (119860 + 119861) + 119862 = 119860 + (119861 + 119862)



(3)

120582 (119860 + 119861) = 120582 sdot ⟨[119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))

119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))]

[119896minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))

119896minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))]

[119896minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))

119896minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))]⟩

= ⟨[119897minus1(120582119897 (119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))))


119897minus1(120582119897 (119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))))]

[119896minus1(120582119896 (119896

minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))))]

[119896minus1(120582119896 (119896

minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))))]⟩

= ⟨[119897minus1(120582 (119897 (inf 119879

119860) + 119897 (inf 119879

119861)))

119897minus1(120582 (119897 (sup119879

119860) + 119897 (sup119879

119861)))]

[119896minus1(120582 (119896 (inf 119868

119860) + 119896 (inf 119868

119861)))

119896minus1(120582 (119896 (sup 119868

119860) + 119896 (sup 119868

119861)))]

[119896minus1(120582 (119896 (inf 119865

119860) + 119896 (inf 119865

119861)))

119896minus1(120582 (119896 (sup119865

119860) + 119896 (sup119865

119861)))]⟩

= ⟨[119897minus1(120582119897 (inf 119879

119860) + 120582119897 (inf 119879

119861))

119897minus1(120582119897 (sup119879

119860) + 120582119897 (sup119879

119861))]

[119896minus1(120582119896 (inf 119868

119860) + 120582119896 (inf 119868

119861))

119896minus1(120582119896 (sup 119868

119860) + 120582119896 (sup 119868

119861))]

[119896minus1(120582119896 (inf 119865

119860) + 120582119896 (inf 119865

119861))

119896minus1(120582119896 (sup119865

119860) + 120582119896 (sup119865

119861))]⟩

= 120582119860 + 120582119861

(16)

(4)

(119860 sdot 119861)120582= (⟨[119896

minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))

119896minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))]

[119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))

119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))]

[119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))

119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))]⟩)120582

= ⟨[119896minus1(120582119896 (119896

minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))))]

[119897minus1(120582119897 (119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))))

119897minus1(120582119897 (119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))))]

[119897minus1(120582119897 (119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))))

119897minus1(120582119897 (119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))))]⟩

= ⟨[119896minus1(120582 (119896 (inf 119879

119860) + 119896 (inf 119879

119861)))

119896minus1(120582 (119896 (sup119879

119860) + 119896 (sup119879

119861)))]

[119897minus1(120582 (119897 (inf 119868

119860) + 119897 (inf 119868

119861)))

119897minus1(120582 (119897 (sup 119868

119860) + 119897 (sup 119868

119861)))]

[119897minus1(120582 (119897 (inf 119865

119860) + 119897 (inf 119865

119861)))

119897minus1(120582 (119897 (sup119865

119860) + 119897 (sup119865

119861)))]⟩

= ⟨[119896minus1(120582119896 (inf 119879

119860) + 120582119896 (inf 119879

119861))

119896minus1(120582119896 (sup119879

119860) + 120582119896 (sup119879

119861))]

[119897minus1(120582119897 (inf 119868

119860) + 120582119897 (inf 119868

119861))

119897minus1(120582119897 (sup 119868

119860) + 120582119897 (sup 119868

119861))]

[119897minus1(120582119897 (inf 119865

119860) + 120582119897 (inf 119865

119861))

119897minus1(120582119897 (sup119865

119860) + 120582119897 (sup119865

119861))]⟩

= 119860120582sdot 119861120582

(17)

(5)

1205821119860 + 120582

2119860

= ⟨[119897minus1(1205821119897 (inf 119879

119860)) 119897minus1(1205821119897 (sup119879

119860))]

[119896minus1(1205821119896 (inf 119868

119860)) 119896minus1(1205821119896 (sup 119868

119860))]

[119896minus1(1205821119896 (inf 119865

119860)) 119896minus1(1205821119896 (sup119865

119860))]⟩

oplus ⟨[119897minus1(1205822119897 (inf 119879

119860)) 119897minus1(1205822119897 (sup119879

119860))]

[119896minus1(1205822119896 (inf 119868

119860)) 119896minus1(1205822119896 (sup 119868

119860))]

[119896minus1(1205822119896 (inf 119865

119860)) 119896minus1(1205822119896 (sup119865

119860))]⟩

= ⟨[119897minus1(119897 (119897minus1(1205821119897 (inf 119879

119860))) + 119897 (119897

minus1(1205822119897 (inf 119879

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup119879

119860))) + 119897 (119897

minus1(1205822119897 (sup119879

119860))))]

[119896minus1(119896 (119896minus1(1205821119896 (inf 119868

119860)))

+119896 (119896minus1(1205822119896 (inf 119868

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup 119868

119860)))

+119896 (119896minus1(1205822119896 (sup 119868

119860))))]


[119896minus1(119896 (119896minus1(1205821119896 (inf 119865

119860)))

+119896 (119896minus1(1205822119896 (inf 119865

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119865

119860)))

+119896 (119896minus1(1205822119896 (sup119865

119860))))]⟩

= ⟨[119897minus1(1205821119897 (inf 119879

119860) + 1205822119897 (inf 119879

119860))

119897minus1(1205821119897 (sup119879

119860) + 1205822119897 (sup119879

119860))]

[119896minus1(1205821119896 (inf 119868

119860) + 1205822119896 (inf 119868

119860))

119896minus1(1205821119896 (sup 119868

119860) + 1205822119896 (sup 119868

119860))]

[119896minus1(1205821119896 (inf 119865

119860) + 1205822119896 (inf 119865

119860))

119896minus1(1205821119896 (sup119865

119860) + 1205822119896 (sup119865

119860))]⟩

= ⟨[119897minus1((1205821+ 1205822) 119897 (inf 119879

119860))

119897minus1((1205821+ 1205822) 119897 (sup119879

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119868

119860))

119896minus1((1205821+ 1205822) 119896 (sup 119868

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119865

119860))

119896minus1((1205821+ 1205822) 119896 (sup119865

119860))]⟩

= (1205821+ 1205822) 119860

(18)

(6)

1198601205821 sdot 1198601205822

=⟨[119896minus1(1205821119896 (inf 119879

119860)) 119896minus1(1205821119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860)) 119897minus1(1205821119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860)) 119897minus1(1205821119897 (sup119865

119860))]⟩

sdot ⟨[119896minus1(1205822119896 (inf 119879

119860)) 119896minus1(1205822119896 (sup119879

119860))]

[119897minus1(1205822119897 (inf 119868

119860)) 119897minus1(1205822119897 (sup 119868

119860))]

[119897minus1(1205822119897 (inf 119865

119860)) 119897minus1(1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1(119896 (119896minus1(1205821119896 (inf 119879

119860)))

+119896 (119896minus1(1205822119896 (inf 119879

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119879

119860)))

+119896 (119896minus1(1205822119896 (sup119879

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119868

119860)))

+119897 (119897minus1(1205822119897 (inf 119868

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup 119868

119860)))

+119897 (119897minus1(1205822119897 (sup 119868

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119865

119860)))

+ 119897 (119897minus1(1205822119897 (inf 119865

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup119865

119860)))

+119897 (119897minus1(1205822119897 (sup119865

119860))))]⟩

= ⟨[119896minus1(1205821119896 (inf 119879

119860) + 1205822119896 (inf 119879

119860))

119896minus1(1205821119896 (sup119879

119860) + 1205822119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860) + 1205822119897 (inf 119868

119860))

119897minus1(1205821119897 (sup 119868

119860) + 1205822119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860) + 1205822119897 (inf 119865

119860))

119897minus1(1205821119897 (sup119865

119860) + 1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1((1205821+ 1205822) 119896 (inf 119879

119860))

119896minus1((1205821+ 1205822) 119896 (sup119879

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119868

119860))

119897minus1((1205821+ 1205822) 119897 (sup 119868

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119865

119860))

119897minus1((1205821+ 1205822) 119897 (sup119865

119860))]⟩

= 1198601205821+1205822

(19)


(1) 2 sdot 119860 = ⟨[091 096] [0 001] [001 004]⟩(2) 1198602 = ⟨[049 064] [0 019] [019 036]⟩(3) 119860 + 119861 = ⟨[082 090] [0 005] [003 008]⟩(4) 119860 sdot 119861 = ⟨[028 040] [020 037] [037 052]⟩


119860(119909) inf 119868

119860(119909) = sup 119868

119860(119909) inf 119865

119860(119909) =

sup119865119860(119909) inf 119879

119861(119909) = sup119879

119861(119909) inf 119868

119861(119909) = sup 119868

119861(119909)

and inf 119865119861(119909) = sup119865


⟨119879119860(119909) 119868119860(119909) 119865119860(119909)⟩ and 119861 = ⟨119879

119861(119909) 119868119861(119909) 119865119861(119909)⟩ are




119860= 0

inf 119868119861= sup 119868

119861= 0 sup119879

119860+sup119865

119860le 1 and sup119879

119861+sup119865

119861le


119861 119868119861 119865119861⟩


(1) 119860 + 119861 = ⟨[inf 119879119860+ inf 119879

119861minus inf 119879

119860sdot inf 119879

119861 sup119879

119860+

sup119879119861minus sup119879

119860sdot sup119879

119861][inf 119865

119860sdot inf 119865

119861 sup119865

119860sdot

sup119865119861]⟩

(2) 119860 sdot 119861 = ⟨[inf 119879119860sdot inf 119879

119861 sup119879

119860sdot sup119879

119861] [inf 119865

119860+

inf 119865119861minus inf 119865

119860sdot inf 119865

119861 sup119865

119860+ sup119865

119861minus sup119865

119860sdot

sup119865119861]⟩


119860)120582]

[(inf 119865119860)120582 (sup119865

119860)120582]⟩

(4) 119860120582 = ⟨[(inf 119879119860)120582 (sup119879

119860)120582] [1minus(1minusinf 119865

119860)120582 1minus(1minus

sup119865119860)120582]⟩




⟨[inf 119879119860 sup119879

119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ and

then

(1) 119904(119860) = [inf 119879119860+ 1 minus sup 119868

119860+ 1 minus sup119865

119860 sup119879

119860+ 1 minus

inf 119868119860+ 1 minus inf 119865

119860]

(2) 119886(119860) = [mininf 119879119860

minus inf 119865119860 sup119879

119860minus

sup119865119860maxinf 119879

119860minus inf 119865

119860 sup119879

119860minus sup119865

119860]

(3) 119888(119860) = [inf 119879119860 sup119879

119860]



119860(119909) sup119879

119860(119909) inf 119868

119860(119909) sup 119868

119860(119909) inf 119865

119860(119909)




(1) 119904(119860) = [24 27] 119904(119861) = [17 20]

(2) 119886(119860) = [06 06] 119886(119861) = [01 01]

(3) 119888(119860) = [07 08] 119888(119861) = [04 05]









(2) Assuming that 119860 = ⟨[06 07] [03 04] [04 05]⟩


(3) For two INNs 119860 = ⟨[06 07] [03 04] [04 05]⟩








INNWA119908(1198601 1198602 119860

119899) = 11990811198601+ 11990821198602+ sdot sdot sdot + 119908

119899119860119899

=

119899

sum

119895=1

119908119895119860119895

(20)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0(119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 23 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a



vector of 119860119895(119895 = 1 2 119899) with 119908

119895ge 0 (119895 = 1 2 119899)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879


+119908119899119897 (inf 119879

119860119899))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879


+119908119899119897 (sup119879

119860119899))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868


+119908119899119896 (inf 119868

119860119899))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868


+119908119899119896 (sup 119868

119860119899))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865


+119908119899119896 (inf 119865

119860119899))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865


+119908119899119896 (sup119865

119860119899))]⟩

(21)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[1 minus

119899

prod

119894=1

(1 minus inf 119879119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119879119860119894)119908119894]

[

119899

prod

119894=1

inf 119868119908119894119860119894

119899

prod

119894=1

sup 119868119908119894119860119894]

[

119899

prod

119894=1

inf 119865119908119894119860119894

119899

prod

119894=1

sup119865119908119894119860119894]⟩

(22)


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1


(1) For 119899 = 2 since

11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601)) 119897minus1(1199081119897 (sup119879

1198601))]

[119896minus1(1199081119896 (inf 119868

1198601)) 119896minus1(1199081119896 (sup 119868

1198601))]

[119896minus1(1199081119896 (inf 119865

1198601)) 119896minus1(1199081119896 (sup119865

1198601))]⟩

oplus ⟨[119897minus1(1199082119897 (inf 119879

1198602)) 119897minus1(1199082119897 (sup119879

1198602))]

[119896minus1(1199082119896 (inf 119868

1198602)) 119896minus1(1199082119896 (sup 119868

1198602))]

[119896minus1(1199082119896 (inf 119865

1198602)) 119896minus1(1199082119896 (sup119865

1198602))]⟩

= ⟨[119897minus1(119897 (119897minus1(1199081119897 (inf 119879

1198601)))

+119897 (119897minus1(1199082119897 (inf 119879

1198602))))

119897minus1(119897 (119897minus1(1199081119897 (sup119879

1198601)))

+119897 (119897minus1(1199082119897 (sup119879

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119868

1198601)))

+119896 (119896minus1(1199082119896 (inf 119868

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup 119868

1198601)))

+119896 (119896minus1(1199082119896 (sup 119868

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119865

1198601)))

+119896 (119896minus1(1199082119896 (inf 119865

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup119865

1198601)))

+119896 (119896minus1(1199082119896 (sup119865

1198602))))]⟩

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))


119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(23)

then

SNNWA119908(1198601 1198602)

= 11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(24)


SNNWA119908(1198601 1198602 119860

119896)

= ⟨[

[

119897minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(25)

then if 119899 = 119896 + 1 we have

SNNWA119908(1198601 1198602 119860

119896 119860119896+1)

= ⟨[

[

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (inf 119879

119860119896+1)))))

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (sup119879

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (inf 119868

119860119896+1)))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (sup 119868

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (inf 119865

119860119896+1))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (sup119865

119860119896+1))))]

]

⟩

= ⟨[

[

119897minus1(

119896+1

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896+1

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119865

119860119895))


119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(26)


SNNWA119908(1198601 1198602 119860

119899)

= ⟨[

[

119897minus1(

119899

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119899

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(27)






INNWA119908(1198601 1198602 119860

119899) = 119860



max119895119868119860119895(119909)max

119895119865119860119895(119909)⟩ 119860+ = ⟨max

119895119879119860119895(119909)

min119895119868119860119895(119909)min

119895119865119860119895(119909)⟩ for all 119895 isin 1 2 119899 and


119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWA119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)



INNWG119908(1198601 1198602 119860

119899) =

119899

prod

119895=1

119860119908119894

119895 (28)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 25 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a


119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[119896minus1(1199081119896 (inf 119879

1198601) + 1199082119896 (inf 119879


+119908119899119896 (inf 119879

119860119899))

119896minus1(1199081119896 (sup119879

1198601) + 119908

2119896 (sup119879


+119908119899119896 (sup119879

119860119899))]

[119897minus1(1199081119897 (inf 119868

1198601) + 1199082119897 (inf 119868


+119908119899119897 (inf 119868

119860119899))

119897minus1(1199081119897 (sup 119868

1198601) + 1199082119897 (sup 119868


+119908119899119897 (sup 119868

119860119899))]

[119897minus1(1199081119897 (inf 119865

1198601) + 1199082119897 (inf 119865


+119908119899119897 (inf 119865

119860119899))

119897minus1(1199081119897 (sup119865

1198601) + 1199082119897 (sup119865


+119908119899119897 (sup119865

119860119899))]⟩

(29)



119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[

119899

prod

119894=1

inf 119879119908119894119860119894

119899

prod

119894=1

sup119879119908119894119860119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119868119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup 119868119860119894)119908119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119865119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119865119860119894)119908119894]⟩

(30)

where 119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1




INNWG119908(1198601 1198602 119860

119899)

= ⟨[

[

2prod119899

119894=1inf 119879119908119894119860119894

prod119899

119894=1(2 minus inf 119879

119860119894)119908119894+prod119899

119894=1inf 119879119908119894119860119894

2prod119899

119894=1sup119879119908119894119860119894

prod119899

119894=1(2 minus sup119879

119860119894)119908119894+prod119899

119894=1sup119879119908119894119860119894

]

]

[

[

prod119899

119894=1(1 + inf 119868

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + inf 119868

119860119894)119908119894+prod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894minusprod119899

119894=1(1 minus sup 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894+prod119899

119894=1(1 minus sup 119868

119860119894)119908119894

]

]

[

[

prod119899

119894=1(1 + inf 119865

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + inf 119865

119860119894)119908119894+prod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894minusprod119899

119894=1(1 minus sup119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894+prod119899

119894=1(1 minus sup119865

119860119894)119908119894

]

]

⟩

(31)

where119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1






INNWG119908(1198601 1198602 119860

119899) = 119860



= ⟨min119895119879119860119895(119909)max

119895119868119860119895(119909)max

119895119865119860119895(119909)⟩

119860+

= ⟨max119895119879119860119895(119909)min

119895119868119860119895(119909)min

119895119865119860119895(119909)⟩


sube

INNWG119908(1198601 1198602 119860

119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWG119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)


1 1198862 119886119898 and n criteria 119862 = 119888

1 1198882 119888119899 whose


119899) where

119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Let 119877 =

(119886119894119895)119898times119899


where 119879119886119894119895



119895 119868119886119894119895


119895 and 119865







119894(119894 = 1 2 119898)

that is

119910119894= INNWA

119908(1198861198941 1198861198942 119886

119894119899) (32)

or

119910119894= INNWG

119908(1198861198941 1198861198942 119886

119894119899) (33)



119894) of


matrix F

119865 =[[[

[

119904 (1199101) 119886 (119910

1) 119888 (119910

1)

119904 (1199102) 119886 (119910

2) 119888 (119910

2)


119904 (119910119898) 119886 (119910

119898) 119888 (119910

119898)

]]]

]

(34)



119875119904=

[[[[

[

11990111990411

11990111990412

sdot sdot sdot 1199011199041119898

11990111990421

11990111990422

sdot sdot sdot 1199011199042119898

1199011199041198981

1199011199041198982

sdot sdot sdot 119901119904119898119898

]]]]

]

(35)


119894) gt 119904(119910

119895)


119894119895+ 119901119904119895119894= 1 and 119901119904

119894119894= 05 If


119886(119910119894) gt 119886(119910

119895) denoted by 119901119886

119894119895 And if 119901119886

119894119895= 05 (119894 = 119895) then


119895) denoted by

119901119888119894119895









3





2




3is a


119863 =

[[[

[

⟨[04 05] [02 03] [03 04]⟩ ⟨[04 06] [01 03] [02 04]⟩ ⟨[07 09] [02 03] [04 05]⟩

⟨[06 07] [01 02] [02 03]⟩ ⟨[06 07] [01 02] [02 03]⟩ ⟨[03 06] [03 05] [08 09]⟩

⟨[03 06] [02 03] [03 04]⟩ ⟨[05 06] [02 03] [03 04]⟩ ⟨[04 05] [02 04] [07 09]⟩

⟨[07 08] [00 01] [01 02]⟩ ⟨[06 07] [01 02] [01 03]⟩ ⟨[06 07] [03 04] [08 09]⟩

]]]

]

(36)




119860119882119860

=[[[

[

⟨[05453 07516] [01682 03000] [03041 04373]⟩

⟨[04996 06634] [01552 02885] [03482 04656]⟩

⟨[03950 05627] [02000 03366] [04210 05533]⟩

⟨[06383 07397] [00000 02071] [02297 04040]⟩

]]]

]

(37)


119860119882119866

=[[[

[

⟨[05004 06620] [01761 03000] [03195 04422]⟩

⟨[04547 06581] [01861 03371] [05405 06786]⟩

⟨[03824 05578] [02000 03419] [05012 07070]⟩

⟨[06333 07335] [01555 02570] [05069 06632]⟩

]]]

]

(38)



119865119882119860

=[[[

[

[18080 22793] [02412 03143] [05453 07516]

[17455 21600] [01514 01978] [04996 06634]

[15051 19417] [minus00260 00094] [03950 05627]

[20272 25100] [03357 04086] [06383 07397]

]]]

]

(39)


119865119882119866

=[[[

[

[17582 21664] [01809 02198] [05004 06620]

[14390 19315] [minus00858 minus00205] [04547 06581]

[13335 18566] [minus01492 minus01188] [03824 05578]

[17131 20711] [00703 01264] [06333 07335]

]]]

]

(40)


119882119860

119875119904 119882119860

=[[[

[

05 06498 08527 02642

03974 05 07695 01480

01473 02305 05 0

07358 08520 1 05

]]]

]

(41)

And for 119865119882119866

119901119904 119882119866

=[[[

[

05 08076 08943 05916

01924 05 05888 02568

01057 04112 05 01629

04084 07432 08371 05

]]]

]

(42)


compute 119901119886119894119895and 119901119888

119894119895



1≻ 1198602 1198601≻ 1198603 1198602≻ 1198603 1198604≻ 1198601 and 119860

4≻


4 1198601

1198602 and 119860

3 Obviously 119860





3 Obviously 119860

1is



3 as well as the

ranking 1198601 1198604 1198602 and 119860

3for AWG






2 1198604 1198603 and 119860

1


1 1198604 1198602 and 119860

3 Obviously


1


119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)




3 As can be


11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)


11988611= 11987911988631


13= 11988633






3in [33]


lt 11988641 11988622

lt 11988642

and 11988623




1in [33] while the

sequences of 1198601 1198602and 119860




6 Conclusion




Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(1) forall119909 isin [0 1] 119879(1 119909) = 119909

(2) forall119909 119910 isin [0 1] 119879(119909 119910) = 119879(119910 119909)

(3) forall119909 119910 119911 isin [0 1] 119879(119909 119879(119910 119911)) = 119879(119879(119909 119910) 119911)

(4) if 119909 le 1199091015840 119910 le 1199101015840 then 119879(119909 119910) le 119879(1199091015840 1199101015840)


(1) forall119909 isin [0 1] 119878(0 119909) = 119909

(2) forall119909 119910 isin [0 1] 119878(119909 119910) = 119878(119910 119909)

(3) forall119909 119910 119911 isin [0 1] 119878(119909 119878(119910 119911)) = 119878(119878(119909 119910) 119911)

(4) if 119909 le 1199091015840 119910 le 1199101015840 then 119878(119909 119910) le 119878(1199091015840 1199101015840)












119878 (119909 119910) =119909 + 119910

1 + 119909119910 119879 (119909 119910) =

119909119910

1 + (1 minus 119909) (1 minus 119910) (5)





1 minus (1 minus 120574) 119909119910

119879 (119909 119910) =119909119910

120574 + (1 minus 120574) (119909 + 119910 minus 119909119910) 120574 gt 0

(6)



119860(119909) an



119860(119909) 119868119860(119909) and119865

119860(119909) are real


119883 rarr ]0minus 1+[ 119868119860(119909) 119883 rarr ]0


119860(119909) 119883 rarr


119860(119909) 119868119860(119909)


119860(119909) + sup 119868

119860(119909) + sup119865

119860(119909) le 3

+


119860(119909) le inf 119879

119861(119909)

sup119879119860(119909) le sup119879

119861(119909) inf 119868

119860(119909) le inf 119868

119861(119909) sup 119868

119860(119909) le

sup 119868119861(119909) inf 119865

119860(119909) le inf 119865

119861(119909) and sup119865

119860(119909) le sup119865

119861(119909)

for 119909 isin 119883



119860(119909) 119868119860(119909) and 119865



119883 rarr [0 1] 119868119860(119909) 119883 rarr [0 1] and 119865

119860(119909) 119883 rarr [0 1]


119860 = ⟨119909 119879119860 (119909) 119868119860 (119909) 119865119860 (119909)⟩ | 119909 isin 119883 (7)




(1) 119860+119861 = ⟨119879119860(119909)+119879

119861(119909)minus119879

119860(119909) sdot119879

119861(119909) 119868119860(119909)+119868

119861(119909)minus

119868119860(119909) sdot 119868

119861(119909) 119865

119860(119909) + 119865

119861(119909) minus 119865

119860(119909) sdot 119865

119861(119909)⟩

(2) 119860 sdot 119861 = ⟨119879119860(119909) sdot 119879

119861(119909) 119868119860(119909) sdot 119868

119861(119909) 119865119860(119909) sdot 119865

119861(119909)⟩


119860(119909))120582 1 minus (1 minus

119865119860(119909))120582⟩ 120582 gt 0

(4) 119860120582 = ⟨119879120582119860(119909) 119868120582

119860(119909) 119865

120582

119860(119909)⟩ 120582 gt 0







2=


1(1198861 119886lowast) = 1 119878

2(1198862 119886lowast) =


2

However for 1198791198862(119909) gt 119879

1198861(119909) 119868

1198862(119909) gt 119868

1198861(119909) and 119865

1198862(119909) gt


2is superior to the

alternative 1198861




119860(119909) and


119883 we have that 119879119860(119909) = [inf 119879

119860(119909) sup119879

119860(119909)] 119868

119860(119909) =

[inf 119868119860(119909) sup 119868

119860(119909)] 119865

119860(119909) = [inf 119865

119860(119909) sup119865

119860(119909)] sube

[0 1] and 0 le sup119879119860(119909) + sup 119868

119860(119909) + sup119865

119860(119909) le 3 119909 isin 119883



inf 119879119860(119909) le inf 119879

119861(119909) sup119879

119860(119909) le sup119879

119861(119909)

inf 119868119860(119909) ge inf 119868

119861(119909) sup 119868

119860(119909) ge sup 119868

119861(119909)

inf 119865119860(119909) ge inf 119865

119861(119909) and sup119865

119860(119909) ge sup119865

119861(119909)

for any 119909 isin 119883



inf 119879119862

= min(inf 119879119860+ inf 119879

119861 1) sup119879

119862=

min(sup119879119860+ sup119879

119861 1)

inf 119868119862= min(inf 119868

119860+inf 119868

119861 1) sup 119868

119862= min(sup 119868

119860+

sup 119868119861 1)

inf 119865119862

= min(inf 119865119860+ inf 119865

119861 1) sup119865

119862=

min(sup119865119860+ sup119865

119861 1)

for all 119909 in119883


119861= sup119879

119861= 1 inf 119868

119861=

sup 119868119861= 0 and inf 119865

119861= sup119865



119862= 1 inf 119868

119862= inf 119868

119860

sup 119868119862= sup 119868

119860 inf 119865

119862= inf 119865

119860 and sup119865

119862= sup119865

119860 that is

119860 + 119861 = ⟨[1 1] [inf 119868119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ = 119861 so that






by

inf 119879119862(119909 119910) = inf 119879

119860(119909) + inf 119879

119861(119910) minus inf 119879

119860(119909) sdot

inf 119879119861(119910)

sup119879119862(119909 119910) = sup119879

119860(119909) + sup119879

119861(119910) minus sup119879

119860(119909) sdot

sup119879119861(119910)

inf 119868119862(119909 119910) = inf 119868

119860(119909) sdot inf 119868

119861(119910) sup 119868

119862(119909 119910) =

sup 119868119860(119909) sdot sup 119868

119861(119910)

inf 119865119862(119909 119910) = inf 119865

119860(119909) sdot inf 119865

119861(119910) sup119865

119862(119909 119910) =

sup119865119860(119909) sdot sup119865

119861(119910)

for all 119909 in1198831 119910 in119883

2





sup119879119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ 119861 = ⟨[inf 119879

119861

sup119879119861] [inf 119868

119861 sup 119868

119861] [inf 119865

119861 sup119865

119861]⟩ and 120582 gt 0 The


(1)

120582119860 = ⟨[119897minus1(120582119897 (inf 119879

119860)) 119897minus1(120582119897 (sup119879

119860))]

[119896minus1(120582119896 (inf 119868

119860)) 119896minus1(120582119896 (sup 119868

119860))]

[119896minus1(120582119896 (inf 119865

119860)) 119896minus1(120582119896 (sup119865

119860))]⟩

(8)

(2)

119860120582= ⟨[(119896

minus1(120582119896 (inf 119879

119860))) (119896

minus1(120582119896 (sup119879

119860)))]

[119897minus1(120582119897 (inf 119868

119860)) 119897minus1(120582119897 (sup 119868

119860))]

[119897minus1(120582119897 (inf 119865

119860)) 119897minus1(120582119897 (sup119865

119860))]⟩

(9)


(3)

119860 + 119861

= ⟨[119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))

119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))]

[119896minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))

119896minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))]

[119896minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))

119896minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))]⟩

(10)

(4)

119860 sdot 119861

= ⟨[119896minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))

119896minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))]

[119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))

119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))]

[119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))

119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))]⟩

(11)


(5)


119860)120582]

[(inf 119868119860)120582 (sup 119868

119860)120582]

[(inf 119865119860)120582 (sup119865

119860)120582]⟩

(12)

(6)

119860120582= ⟨[(inf 119879

119860)120582 (sup119879

119860)120582]


119860)120582]


119860)120582]⟩

(13)

(7)

119860 + 119861 = ⟨[inf 119879119860+ inf 119879

119861minus inf 119879

119860sdot inf 119879

119861

sup119879119860+ sup119879

119861minus sup119879

119860sdot sup119879

119861]

[inf 119879119860sdot inf 119868119861 sup 119868

119860sdot sup 119868

119861]

[inf 119865119860sdot inf 119865

119861 sup119865

119860sdot sup119865

119861]⟩

(14)

(8)

119860 sdot 119861 = ⟨[inf 119879119860sdot inf 119879

119861 sup119879

119860sdot sup119879

119861]

[inf 119879119860+ inf 119868

119861minus inf 119879

119860sdot inf 119868119861

sup 119868119860+ sup 119868

119861minus sup 119868

119860sdot sup 119868

119861]

[inf 119865119860+ inf 119865

119861minus inf 119865

119860sdot inf 119865

119861

sup119865119860+ sup119865

119861minus sup119865

119860sdot sup119865

119861]⟩

(15)


sup119879119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ 119861 = ⟨[inf 119879

119861

sup119879119861] [inf 119868

119861 sup 119868

119861] [inf 119865

119861 sup119865

119861]⟩ 119862 = ⟨[inf 119879

119862

sup119879119862] [inf 119868

119862 sup 119868

119862] [inf 119865

119862 sup119865



(1) 119860 + 119861 = 119861 + 119860

(2) 119860 sdot 119861 = 119861 sdot 119860

(3) 120582(119860 + 119861) = 120582119860 + 120582119861 120582 gt 0

(4) (119860 sdot 119861)120582 = 119860120582 + 119861120582 120582 gt 0

(5) 1205821119860 + 120582

2119860 = (120582

1+ 1205822)119860 1205821gt 0 120582

2gt 0

(6) 1198601205821 sdot 1198601205822 = 119860(1205821+1205822) 1205821gt 0 120582

2gt 0

(7) (119860 + 119861) + 119862 = 119860 + (119861 + 119862)



(3)

120582 (119860 + 119861) = 120582 sdot ⟨[119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))

119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))]

[119896minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))

119896minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))]

[119896minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))

119896minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))]⟩

= ⟨[119897minus1(120582119897 (119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))))


119897minus1(120582119897 (119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))))]

[119896minus1(120582119896 (119896

minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))))]

[119896minus1(120582119896 (119896

minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))))]⟩

= ⟨[119897minus1(120582 (119897 (inf 119879

119860) + 119897 (inf 119879

119861)))

119897minus1(120582 (119897 (sup119879

119860) + 119897 (sup119879

119861)))]

[119896minus1(120582 (119896 (inf 119868

119860) + 119896 (inf 119868

119861)))

119896minus1(120582 (119896 (sup 119868

119860) + 119896 (sup 119868

119861)))]

[119896minus1(120582 (119896 (inf 119865

119860) + 119896 (inf 119865

119861)))

119896minus1(120582 (119896 (sup119865

119860) + 119896 (sup119865

119861)))]⟩

= ⟨[119897minus1(120582119897 (inf 119879

119860) + 120582119897 (inf 119879

119861))

119897minus1(120582119897 (sup119879

119860) + 120582119897 (sup119879

119861))]

[119896minus1(120582119896 (inf 119868

119860) + 120582119896 (inf 119868

119861))

119896minus1(120582119896 (sup 119868

119860) + 120582119896 (sup 119868

119861))]

[119896minus1(120582119896 (inf 119865

119860) + 120582119896 (inf 119865

119861))

119896minus1(120582119896 (sup119865

119860) + 120582119896 (sup119865

119861))]⟩

= 120582119860 + 120582119861

(16)

(4)

(119860 sdot 119861)120582= (⟨[119896

minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))

119896minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))]

[119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))

119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))]

[119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))

119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))]⟩)120582

= ⟨[119896minus1(120582119896 (119896

minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))))]

[119897minus1(120582119897 (119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))))

119897minus1(120582119897 (119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))))]

[119897minus1(120582119897 (119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))))

119897minus1(120582119897 (119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))))]⟩

= ⟨[119896minus1(120582 (119896 (inf 119879

119860) + 119896 (inf 119879

119861)))

119896minus1(120582 (119896 (sup119879

119860) + 119896 (sup119879

119861)))]

[119897minus1(120582 (119897 (inf 119868

119860) + 119897 (inf 119868

119861)))

119897minus1(120582 (119897 (sup 119868

119860) + 119897 (sup 119868

119861)))]

[119897minus1(120582 (119897 (inf 119865

119860) + 119897 (inf 119865

119861)))

119897minus1(120582 (119897 (sup119865

119860) + 119897 (sup119865

119861)))]⟩

= ⟨[119896minus1(120582119896 (inf 119879

119860) + 120582119896 (inf 119879

119861))

119896minus1(120582119896 (sup119879

119860) + 120582119896 (sup119879

119861))]

[119897minus1(120582119897 (inf 119868

119860) + 120582119897 (inf 119868

119861))

119897minus1(120582119897 (sup 119868

119860) + 120582119897 (sup 119868

119861))]

[119897minus1(120582119897 (inf 119865

119860) + 120582119897 (inf 119865

119861))

119897minus1(120582119897 (sup119865

119860) + 120582119897 (sup119865

119861))]⟩

= 119860120582sdot 119861120582

(17)

(5)

1205821119860 + 120582

2119860

= ⟨[119897minus1(1205821119897 (inf 119879

119860)) 119897minus1(1205821119897 (sup119879

119860))]

[119896minus1(1205821119896 (inf 119868

119860)) 119896minus1(1205821119896 (sup 119868

119860))]

[119896minus1(1205821119896 (inf 119865

119860)) 119896minus1(1205821119896 (sup119865

119860))]⟩

oplus ⟨[119897minus1(1205822119897 (inf 119879

119860)) 119897minus1(1205822119897 (sup119879

119860))]

[119896minus1(1205822119896 (inf 119868

119860)) 119896minus1(1205822119896 (sup 119868

119860))]

[119896minus1(1205822119896 (inf 119865

119860)) 119896minus1(1205822119896 (sup119865

119860))]⟩

= ⟨[119897minus1(119897 (119897minus1(1205821119897 (inf 119879

119860))) + 119897 (119897

minus1(1205822119897 (inf 119879

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup119879

119860))) + 119897 (119897

minus1(1205822119897 (sup119879

119860))))]

[119896minus1(119896 (119896minus1(1205821119896 (inf 119868

119860)))

+119896 (119896minus1(1205822119896 (inf 119868

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup 119868

119860)))

+119896 (119896minus1(1205822119896 (sup 119868

119860))))]


[119896minus1(119896 (119896minus1(1205821119896 (inf 119865

119860)))

+119896 (119896minus1(1205822119896 (inf 119865

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119865

119860)))

+119896 (119896minus1(1205822119896 (sup119865

119860))))]⟩

= ⟨[119897minus1(1205821119897 (inf 119879

119860) + 1205822119897 (inf 119879

119860))

119897minus1(1205821119897 (sup119879

119860) + 1205822119897 (sup119879

119860))]

[119896minus1(1205821119896 (inf 119868

119860) + 1205822119896 (inf 119868

119860))

119896minus1(1205821119896 (sup 119868

119860) + 1205822119896 (sup 119868

119860))]

[119896minus1(1205821119896 (inf 119865

119860) + 1205822119896 (inf 119865

119860))

119896minus1(1205821119896 (sup119865

119860) + 1205822119896 (sup119865

119860))]⟩

= ⟨[119897minus1((1205821+ 1205822) 119897 (inf 119879

119860))

119897minus1((1205821+ 1205822) 119897 (sup119879

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119868

119860))

119896minus1((1205821+ 1205822) 119896 (sup 119868

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119865

119860))

119896minus1((1205821+ 1205822) 119896 (sup119865

119860))]⟩

= (1205821+ 1205822) 119860

(18)

(6)

1198601205821 sdot 1198601205822

=⟨[119896minus1(1205821119896 (inf 119879

119860)) 119896minus1(1205821119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860)) 119897minus1(1205821119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860)) 119897minus1(1205821119897 (sup119865

119860))]⟩

sdot ⟨[119896minus1(1205822119896 (inf 119879

119860)) 119896minus1(1205822119896 (sup119879

119860))]

[119897minus1(1205822119897 (inf 119868

119860)) 119897minus1(1205822119897 (sup 119868

119860))]

[119897minus1(1205822119897 (inf 119865

119860)) 119897minus1(1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1(119896 (119896minus1(1205821119896 (inf 119879

119860)))

+119896 (119896minus1(1205822119896 (inf 119879

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119879

119860)))

+119896 (119896minus1(1205822119896 (sup119879

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119868

119860)))

+119897 (119897minus1(1205822119897 (inf 119868

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup 119868

119860)))

+119897 (119897minus1(1205822119897 (sup 119868

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119865

119860)))

+ 119897 (119897minus1(1205822119897 (inf 119865

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup119865

119860)))

+119897 (119897minus1(1205822119897 (sup119865

119860))))]⟩

= ⟨[119896minus1(1205821119896 (inf 119879

119860) + 1205822119896 (inf 119879

119860))

119896minus1(1205821119896 (sup119879

119860) + 1205822119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860) + 1205822119897 (inf 119868

119860))

119897minus1(1205821119897 (sup 119868

119860) + 1205822119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860) + 1205822119897 (inf 119865

119860))

119897minus1(1205821119897 (sup119865

119860) + 1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1((1205821+ 1205822) 119896 (inf 119879

119860))

119896minus1((1205821+ 1205822) 119896 (sup119879

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119868

119860))

119897minus1((1205821+ 1205822) 119897 (sup 119868

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119865

119860))

119897minus1((1205821+ 1205822) 119897 (sup119865

119860))]⟩

= 1198601205821+1205822

(19)


(1) 2 sdot 119860 = ⟨[091 096] [0 001] [001 004]⟩(2) 1198602 = ⟨[049 064] [0 019] [019 036]⟩(3) 119860 + 119861 = ⟨[082 090] [0 005] [003 008]⟩(4) 119860 sdot 119861 = ⟨[028 040] [020 037] [037 052]⟩


119860(119909) inf 119868

119860(119909) = sup 119868

119860(119909) inf 119865

119860(119909) =

sup119865119860(119909) inf 119879

119861(119909) = sup119879

119861(119909) inf 119868

119861(119909) = sup 119868

119861(119909)

and inf 119865119861(119909) = sup119865


⟨119879119860(119909) 119868119860(119909) 119865119860(119909)⟩ and 119861 = ⟨119879

119861(119909) 119868119861(119909) 119865119861(119909)⟩ are




119860= 0

inf 119868119861= sup 119868

119861= 0 sup119879

119860+sup119865

119860le 1 and sup119879

119861+sup119865

119861le


119861 119868119861 119865119861⟩


(1) 119860 + 119861 = ⟨[inf 119879119860+ inf 119879

119861minus inf 119879

119860sdot inf 119879

119861 sup119879

119860+

sup119879119861minus sup119879

119860sdot sup119879

119861][inf 119865

119860sdot inf 119865

119861 sup119865

119860sdot

sup119865119861]⟩

(2) 119860 sdot 119861 = ⟨[inf 119879119860sdot inf 119879

119861 sup119879

119860sdot sup119879

119861] [inf 119865

119860+

inf 119865119861minus inf 119865

119860sdot inf 119865

119861 sup119865

119860+ sup119865

119861minus sup119865

119860sdot

sup119865119861]⟩


119860)120582]

[(inf 119865119860)120582 (sup119865

119860)120582]⟩

(4) 119860120582 = ⟨[(inf 119879119860)120582 (sup119879

119860)120582] [1minus(1minusinf 119865

119860)120582 1minus(1minus

sup119865119860)120582]⟩




⟨[inf 119879119860 sup119879

119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ and

then

(1) 119904(119860) = [inf 119879119860+ 1 minus sup 119868

119860+ 1 minus sup119865

119860 sup119879

119860+ 1 minus

inf 119868119860+ 1 minus inf 119865

119860]

(2) 119886(119860) = [mininf 119879119860

minus inf 119865119860 sup119879

119860minus

sup119865119860maxinf 119879

119860minus inf 119865

119860 sup119879

119860minus sup119865

119860]

(3) 119888(119860) = [inf 119879119860 sup119879

119860]



119860(119909) sup119879

119860(119909) inf 119868

119860(119909) sup 119868

119860(119909) inf 119865

119860(119909)




(1) 119904(119860) = [24 27] 119904(119861) = [17 20]

(2) 119886(119860) = [06 06] 119886(119861) = [01 01]

(3) 119888(119860) = [07 08] 119888(119861) = [04 05]









(2) Assuming that 119860 = ⟨[06 07] [03 04] [04 05]⟩


(3) For two INNs 119860 = ⟨[06 07] [03 04] [04 05]⟩








INNWA119908(1198601 1198602 119860

119899) = 11990811198601+ 11990821198602+ sdot sdot sdot + 119908

119899119860119899

=

119899

sum

119895=1

119908119895119860119895

(20)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0(119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 23 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a



vector of 119860119895(119895 = 1 2 119899) with 119908

119895ge 0 (119895 = 1 2 119899)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879


+119908119899119897 (inf 119879

119860119899))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879


+119908119899119897 (sup119879

119860119899))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868


+119908119899119896 (inf 119868

119860119899))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868


+119908119899119896 (sup 119868

119860119899))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865


+119908119899119896 (inf 119865

119860119899))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865


+119908119899119896 (sup119865

119860119899))]⟩

(21)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[1 minus

119899

prod

119894=1

(1 minus inf 119879119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119879119860119894)119908119894]

[

119899

prod

119894=1

inf 119868119908119894119860119894

119899

prod

119894=1

sup 119868119908119894119860119894]

[

119899

prod

119894=1

inf 119865119908119894119860119894

119899

prod

119894=1

sup119865119908119894119860119894]⟩

(22)


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1


(1) For 119899 = 2 since

11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601)) 119897minus1(1199081119897 (sup119879

1198601))]

[119896minus1(1199081119896 (inf 119868

1198601)) 119896minus1(1199081119896 (sup 119868

1198601))]

[119896minus1(1199081119896 (inf 119865

1198601)) 119896minus1(1199081119896 (sup119865

1198601))]⟩

oplus ⟨[119897minus1(1199082119897 (inf 119879

1198602)) 119897minus1(1199082119897 (sup119879

1198602))]

[119896minus1(1199082119896 (inf 119868

1198602)) 119896minus1(1199082119896 (sup 119868

1198602))]

[119896minus1(1199082119896 (inf 119865

1198602)) 119896minus1(1199082119896 (sup119865

1198602))]⟩

= ⟨[119897minus1(119897 (119897minus1(1199081119897 (inf 119879

1198601)))

+119897 (119897minus1(1199082119897 (inf 119879

1198602))))

119897minus1(119897 (119897minus1(1199081119897 (sup119879

1198601)))

+119897 (119897minus1(1199082119897 (sup119879

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119868

1198601)))

+119896 (119896minus1(1199082119896 (inf 119868

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup 119868

1198601)))

+119896 (119896minus1(1199082119896 (sup 119868

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119865

1198601)))

+119896 (119896minus1(1199082119896 (inf 119865

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup119865

1198601)))

+119896 (119896minus1(1199082119896 (sup119865

1198602))))]⟩

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))


119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(23)

then

SNNWA119908(1198601 1198602)

= 11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(24)


SNNWA119908(1198601 1198602 119860

119896)

= ⟨[

[

119897minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(25)

then if 119899 = 119896 + 1 we have

SNNWA119908(1198601 1198602 119860

119896 119860119896+1)

= ⟨[

[

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (inf 119879

119860119896+1)))))

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (sup119879

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (inf 119868

119860119896+1)))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (sup 119868

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (inf 119865

119860119896+1))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (sup119865

119860119896+1))))]

]

⟩

= ⟨[

[

119897minus1(

119896+1

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896+1

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119865

119860119895))


119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(26)


SNNWA119908(1198601 1198602 119860

119899)

= ⟨[

[

119897minus1(

119899

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119899

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(27)






INNWA119908(1198601 1198602 119860

119899) = 119860



max119895119868119860119895(119909)max

119895119865119860119895(119909)⟩ 119860+ = ⟨max

119895119879119860119895(119909)

min119895119868119860119895(119909)min

119895119865119860119895(119909)⟩ for all 119895 isin 1 2 119899 and


119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWA119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)



INNWG119908(1198601 1198602 119860

119899) =

119899

prod

119895=1

119860119908119894

119895 (28)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 25 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a


119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[119896minus1(1199081119896 (inf 119879

1198601) + 1199082119896 (inf 119879


+119908119899119896 (inf 119879

119860119899))

119896minus1(1199081119896 (sup119879

1198601) + 119908

2119896 (sup119879


+119908119899119896 (sup119879

119860119899))]

[119897minus1(1199081119897 (inf 119868

1198601) + 1199082119897 (inf 119868


+119908119899119897 (inf 119868

119860119899))

119897minus1(1199081119897 (sup 119868

1198601) + 1199082119897 (sup 119868


+119908119899119897 (sup 119868

119860119899))]

[119897minus1(1199081119897 (inf 119865

1198601) + 1199082119897 (inf 119865


+119908119899119897 (inf 119865

119860119899))

119897minus1(1199081119897 (sup119865

1198601) + 1199082119897 (sup119865


+119908119899119897 (sup119865

119860119899))]⟩

(29)



119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[

119899

prod

119894=1

inf 119879119908119894119860119894

119899

prod

119894=1

sup119879119908119894119860119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119868119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup 119868119860119894)119908119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119865119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119865119860119894)119908119894]⟩

(30)

where 119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1




INNWG119908(1198601 1198602 119860

119899)

= ⟨[

[

2prod119899

119894=1inf 119879119908119894119860119894

prod119899

119894=1(2 minus inf 119879

119860119894)119908119894+prod119899

119894=1inf 119879119908119894119860119894

2prod119899

119894=1sup119879119908119894119860119894

prod119899

119894=1(2 minus sup119879

119860119894)119908119894+prod119899

119894=1sup119879119908119894119860119894

]

]

[

[

prod119899

119894=1(1 + inf 119868

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + inf 119868

119860119894)119908119894+prod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894minusprod119899

119894=1(1 minus sup 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894+prod119899

119894=1(1 minus sup 119868

119860119894)119908119894

]

]

[

[

prod119899

119894=1(1 + inf 119865

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + inf 119865

119860119894)119908119894+prod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894minusprod119899

119894=1(1 minus sup119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894+prod119899

119894=1(1 minus sup119865

119860119894)119908119894

]

]

⟩

(31)

where119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1






INNWG119908(1198601 1198602 119860

119899) = 119860



= ⟨min119895119879119860119895(119909)max

119895119868119860119895(119909)max

119895119865119860119895(119909)⟩

119860+

= ⟨max119895119879119860119895(119909)min

119895119868119860119895(119909)min

119895119865119860119895(119909)⟩


sube

INNWG119908(1198601 1198602 119860

119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWG119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)


1 1198862 119886119898 and n criteria 119862 = 119888

1 1198882 119888119899 whose


119899) where

119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Let 119877 =

(119886119894119895)119898times119899


where 119879119886119894119895



119895 119868119886119894119895


119895 and 119865







119894(119894 = 1 2 119898)

that is

119910119894= INNWA

119908(1198861198941 1198861198942 119886

119894119899) (32)

or

119910119894= INNWG

119908(1198861198941 1198861198942 119886

119894119899) (33)



119894) of


matrix F

119865 =[[[

[

119904 (1199101) 119886 (119910

1) 119888 (119910

1)

119904 (1199102) 119886 (119910

2) 119888 (119910

2)


119904 (119910119898) 119886 (119910

119898) 119888 (119910

119898)

]]]

]

(34)



119875119904=

[[[[

[

11990111990411

11990111990412

sdot sdot sdot 1199011199041119898

11990111990421

11990111990422

sdot sdot sdot 1199011199042119898

1199011199041198981

1199011199041198982

sdot sdot sdot 119901119904119898119898

]]]]

]

(35)


119894) gt 119904(119910

119895)


119894119895+ 119901119904119895119894= 1 and 119901119904

119894119894= 05 If


119886(119910119894) gt 119886(119910

119895) denoted by 119901119886

119894119895 And if 119901119886

119894119895= 05 (119894 = 119895) then


119895) denoted by

119901119888119894119895









3





2




3is a


119863 =

[[[

[

⟨[04 05] [02 03] [03 04]⟩ ⟨[04 06] [01 03] [02 04]⟩ ⟨[07 09] [02 03] [04 05]⟩

⟨[06 07] [01 02] [02 03]⟩ ⟨[06 07] [01 02] [02 03]⟩ ⟨[03 06] [03 05] [08 09]⟩

⟨[03 06] [02 03] [03 04]⟩ ⟨[05 06] [02 03] [03 04]⟩ ⟨[04 05] [02 04] [07 09]⟩

⟨[07 08] [00 01] [01 02]⟩ ⟨[06 07] [01 02] [01 03]⟩ ⟨[06 07] [03 04] [08 09]⟩

]]]

]

(36)




119860119882119860

=[[[

[

⟨[05453 07516] [01682 03000] [03041 04373]⟩

⟨[04996 06634] [01552 02885] [03482 04656]⟩

⟨[03950 05627] [02000 03366] [04210 05533]⟩

⟨[06383 07397] [00000 02071] [02297 04040]⟩

]]]

]

(37)


119860119882119866

=[[[

[

⟨[05004 06620] [01761 03000] [03195 04422]⟩

⟨[04547 06581] [01861 03371] [05405 06786]⟩

⟨[03824 05578] [02000 03419] [05012 07070]⟩

⟨[06333 07335] [01555 02570] [05069 06632]⟩

]]]

]

(38)



119865119882119860

=[[[

[

[18080 22793] [02412 03143] [05453 07516]

[17455 21600] [01514 01978] [04996 06634]

[15051 19417] [minus00260 00094] [03950 05627]

[20272 25100] [03357 04086] [06383 07397]

]]]

]

(39)


119865119882119866

=[[[

[

[17582 21664] [01809 02198] [05004 06620]

[14390 19315] [minus00858 minus00205] [04547 06581]

[13335 18566] [minus01492 minus01188] [03824 05578]

[17131 20711] [00703 01264] [06333 07335]

]]]

]

(40)


119882119860

119875119904 119882119860

=[[[

[

05 06498 08527 02642

03974 05 07695 01480

01473 02305 05 0

07358 08520 1 05

]]]

]

(41)

And for 119865119882119866

119901119904 119882119866

=[[[

[

05 08076 08943 05916

01924 05 05888 02568

01057 04112 05 01629

04084 07432 08371 05

]]]

]

(42)


compute 119901119886119894119895and 119901119888

119894119895



1≻ 1198602 1198601≻ 1198603 1198602≻ 1198603 1198604≻ 1198601 and 119860

4≻


4 1198601

1198602 and 119860

3 Obviously 119860





3 Obviously 119860

1is



3 as well as the

ranking 1198601 1198604 1198602 and 119860

3for AWG






2 1198604 1198603 and 119860

1


1 1198604 1198602 and 119860

3 Obviously


1


119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)




3 As can be


11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)


11988611= 11987911988631


13= 11988633






3in [33]


lt 11988641 11988622

lt 11988642

and 11988623




1in [33] while the

sequences of 1198601 1198602and 119860




6 Conclusion




Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2=


1(1198861 119886lowast) = 1 119878

2(1198862 119886lowast) =


2

However for 1198791198862(119909) gt 119879

1198861(119909) 119868

1198862(119909) gt 119868

1198861(119909) and 119865

1198862(119909) gt


2is superior to the

alternative 1198861




119860(119909) and


119883 we have that 119879119860(119909) = [inf 119879

119860(119909) sup119879

119860(119909)] 119868

119860(119909) =

[inf 119868119860(119909) sup 119868

119860(119909)] 119865

119860(119909) = [inf 119865

119860(119909) sup119865

119860(119909)] sube

[0 1] and 0 le sup119879119860(119909) + sup 119868

119860(119909) + sup119865

119860(119909) le 3 119909 isin 119883



inf 119879119860(119909) le inf 119879

119861(119909) sup119879

119860(119909) le sup119879

119861(119909)

inf 119868119860(119909) ge inf 119868

119861(119909) sup 119868

119860(119909) ge sup 119868

119861(119909)

inf 119865119860(119909) ge inf 119865

119861(119909) and sup119865

119860(119909) ge sup119865

119861(119909)

for any 119909 isin 119883



inf 119879119862

= min(inf 119879119860+ inf 119879

119861 1) sup119879

119862=

min(sup119879119860+ sup119879

119861 1)

inf 119868119862= min(inf 119868

119860+inf 119868

119861 1) sup 119868

119862= min(sup 119868

119860+

sup 119868119861 1)

inf 119865119862

= min(inf 119865119860+ inf 119865

119861 1) sup119865

119862=

min(sup119865119860+ sup119865

119861 1)

for all 119909 in119883


119861= sup119879

119861= 1 inf 119868

119861=

sup 119868119861= 0 and inf 119865

119861= sup119865



119862= 1 inf 119868

119862= inf 119868

119860

sup 119868119862= sup 119868

119860 inf 119865

119862= inf 119865

119860 and sup119865

119862= sup119865

119860 that is

119860 + 119861 = ⟨[1 1] [inf 119868119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ = 119861 so that






by

inf 119879119862(119909 119910) = inf 119879

119860(119909) + inf 119879

119861(119910) minus inf 119879

119860(119909) sdot

inf 119879119861(119910)

sup119879119862(119909 119910) = sup119879

119860(119909) + sup119879

119861(119910) minus sup119879

119860(119909) sdot

sup119879119861(119910)

inf 119868119862(119909 119910) = inf 119868

119860(119909) sdot inf 119868

119861(119910) sup 119868

119862(119909 119910) =

sup 119868119860(119909) sdot sup 119868

119861(119910)

inf 119865119862(119909 119910) = inf 119865

119860(119909) sdot inf 119865

119861(119910) sup119865

119862(119909 119910) =

sup119865119860(119909) sdot sup119865

119861(119910)

for all 119909 in1198831 119910 in119883

2





sup119879119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ 119861 = ⟨[inf 119879

119861

sup119879119861] [inf 119868

119861 sup 119868

119861] [inf 119865

119861 sup119865

119861]⟩ and 120582 gt 0 The


(1)

120582119860 = ⟨[119897minus1(120582119897 (inf 119879

119860)) 119897minus1(120582119897 (sup119879

119860))]

[119896minus1(120582119896 (inf 119868

119860)) 119896minus1(120582119896 (sup 119868

119860))]

[119896minus1(120582119896 (inf 119865

119860)) 119896minus1(120582119896 (sup119865

119860))]⟩

(8)

(2)

119860120582= ⟨[(119896

minus1(120582119896 (inf 119879

119860))) (119896

minus1(120582119896 (sup119879

119860)))]

[119897minus1(120582119897 (inf 119868

119860)) 119897minus1(120582119897 (sup 119868

119860))]

[119897minus1(120582119897 (inf 119865

119860)) 119897minus1(120582119897 (sup119865

119860))]⟩

(9)


(3)

119860 + 119861

= ⟨[119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))

119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))]

[119896minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))

119896minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))]

[119896minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))

119896minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))]⟩

(10)

(4)

119860 sdot 119861

= ⟨[119896minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))

119896minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))]

[119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))

119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))]

[119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))

119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))]⟩

(11)


(5)


119860)120582]

[(inf 119868119860)120582 (sup 119868

119860)120582]

[(inf 119865119860)120582 (sup119865

119860)120582]⟩

(12)

(6)

119860120582= ⟨[(inf 119879

119860)120582 (sup119879

119860)120582]


119860)120582]


119860)120582]⟩

(13)

(7)

119860 + 119861 = ⟨[inf 119879119860+ inf 119879

119861minus inf 119879

119860sdot inf 119879

119861

sup119879119860+ sup119879

119861minus sup119879

119860sdot sup119879

119861]

[inf 119879119860sdot inf 119868119861 sup 119868

119860sdot sup 119868

119861]

[inf 119865119860sdot inf 119865

119861 sup119865

119860sdot sup119865

119861]⟩

(14)

(8)

119860 sdot 119861 = ⟨[inf 119879119860sdot inf 119879

119861 sup119879

119860sdot sup119879

119861]

[inf 119879119860+ inf 119868

119861minus inf 119879

119860sdot inf 119868119861

sup 119868119860+ sup 119868

119861minus sup 119868

119860sdot sup 119868

119861]

[inf 119865119860+ inf 119865

119861minus inf 119865

119860sdot inf 119865

119861

sup119865119860+ sup119865

119861minus sup119865

119860sdot sup119865

119861]⟩

(15)


sup119879119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ 119861 = ⟨[inf 119879

119861

sup119879119861] [inf 119868

119861 sup 119868

119861] [inf 119865

119861 sup119865

119861]⟩ 119862 = ⟨[inf 119879

119862

sup119879119862] [inf 119868

119862 sup 119868

119862] [inf 119865

119862 sup119865



(1) 119860 + 119861 = 119861 + 119860

(2) 119860 sdot 119861 = 119861 sdot 119860

(3) 120582(119860 + 119861) = 120582119860 + 120582119861 120582 gt 0

(4) (119860 sdot 119861)120582 = 119860120582 + 119861120582 120582 gt 0

(5) 1205821119860 + 120582

2119860 = (120582

1+ 1205822)119860 1205821gt 0 120582

2gt 0

(6) 1198601205821 sdot 1198601205822 = 119860(1205821+1205822) 1205821gt 0 120582

2gt 0

(7) (119860 + 119861) + 119862 = 119860 + (119861 + 119862)



(3)

120582 (119860 + 119861) = 120582 sdot ⟨[119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))

119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))]

[119896minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))

119896minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))]

[119896minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))

119896minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))]⟩

= ⟨[119897minus1(120582119897 (119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))))


119897minus1(120582119897 (119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))))]

[119896minus1(120582119896 (119896

minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))))]

[119896minus1(120582119896 (119896

minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))))]⟩

= ⟨[119897minus1(120582 (119897 (inf 119879

119860) + 119897 (inf 119879

119861)))

119897minus1(120582 (119897 (sup119879

119860) + 119897 (sup119879

119861)))]

[119896minus1(120582 (119896 (inf 119868

119860) + 119896 (inf 119868

119861)))

119896minus1(120582 (119896 (sup 119868

119860) + 119896 (sup 119868

119861)))]

[119896minus1(120582 (119896 (inf 119865

119860) + 119896 (inf 119865

119861)))

119896minus1(120582 (119896 (sup119865

119860) + 119896 (sup119865

119861)))]⟩

= ⟨[119897minus1(120582119897 (inf 119879

119860) + 120582119897 (inf 119879

119861))

119897minus1(120582119897 (sup119879

119860) + 120582119897 (sup119879

119861))]

[119896minus1(120582119896 (inf 119868

119860) + 120582119896 (inf 119868

119861))

119896minus1(120582119896 (sup 119868

119860) + 120582119896 (sup 119868

119861))]

[119896minus1(120582119896 (inf 119865

119860) + 120582119896 (inf 119865

119861))

119896minus1(120582119896 (sup119865

119860) + 120582119896 (sup119865

119861))]⟩

= 120582119860 + 120582119861

(16)

(4)

(119860 sdot 119861)120582= (⟨[119896

minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))

119896minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))]

[119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))

119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))]

[119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))

119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))]⟩)120582

= ⟨[119896minus1(120582119896 (119896

minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))))]

[119897minus1(120582119897 (119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))))

119897minus1(120582119897 (119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))))]

[119897minus1(120582119897 (119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))))

119897minus1(120582119897 (119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))))]⟩

= ⟨[119896minus1(120582 (119896 (inf 119879

119860) + 119896 (inf 119879

119861)))

119896minus1(120582 (119896 (sup119879

119860) + 119896 (sup119879

119861)))]

[119897minus1(120582 (119897 (inf 119868

119860) + 119897 (inf 119868

119861)))

119897minus1(120582 (119897 (sup 119868

119860) + 119897 (sup 119868

119861)))]

[119897minus1(120582 (119897 (inf 119865

119860) + 119897 (inf 119865

119861)))

119897minus1(120582 (119897 (sup119865

119860) + 119897 (sup119865

119861)))]⟩

= ⟨[119896minus1(120582119896 (inf 119879

119860) + 120582119896 (inf 119879

119861))

119896minus1(120582119896 (sup119879

119860) + 120582119896 (sup119879

119861))]

[119897minus1(120582119897 (inf 119868

119860) + 120582119897 (inf 119868

119861))

119897minus1(120582119897 (sup 119868

119860) + 120582119897 (sup 119868

119861))]

[119897minus1(120582119897 (inf 119865

119860) + 120582119897 (inf 119865

119861))

119897minus1(120582119897 (sup119865

119860) + 120582119897 (sup119865

119861))]⟩

= 119860120582sdot 119861120582

(17)

(5)

1205821119860 + 120582

2119860

= ⟨[119897minus1(1205821119897 (inf 119879

119860)) 119897minus1(1205821119897 (sup119879

119860))]

[119896minus1(1205821119896 (inf 119868

119860)) 119896minus1(1205821119896 (sup 119868

119860))]

[119896minus1(1205821119896 (inf 119865

119860)) 119896minus1(1205821119896 (sup119865

119860))]⟩

oplus ⟨[119897minus1(1205822119897 (inf 119879

119860)) 119897minus1(1205822119897 (sup119879

119860))]

[119896minus1(1205822119896 (inf 119868

119860)) 119896minus1(1205822119896 (sup 119868

119860))]

[119896minus1(1205822119896 (inf 119865

119860)) 119896minus1(1205822119896 (sup119865

119860))]⟩

= ⟨[119897minus1(119897 (119897minus1(1205821119897 (inf 119879

119860))) + 119897 (119897

minus1(1205822119897 (inf 119879

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup119879

119860))) + 119897 (119897

minus1(1205822119897 (sup119879

119860))))]

[119896minus1(119896 (119896minus1(1205821119896 (inf 119868

119860)))

+119896 (119896minus1(1205822119896 (inf 119868

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup 119868

119860)))

+119896 (119896minus1(1205822119896 (sup 119868

119860))))]


[119896minus1(119896 (119896minus1(1205821119896 (inf 119865

119860)))

+119896 (119896minus1(1205822119896 (inf 119865

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119865

119860)))

+119896 (119896minus1(1205822119896 (sup119865

119860))))]⟩

= ⟨[119897minus1(1205821119897 (inf 119879

119860) + 1205822119897 (inf 119879

119860))

119897minus1(1205821119897 (sup119879

119860) + 1205822119897 (sup119879

119860))]

[119896minus1(1205821119896 (inf 119868

119860) + 1205822119896 (inf 119868

119860))

119896minus1(1205821119896 (sup 119868

119860) + 1205822119896 (sup 119868

119860))]

[119896minus1(1205821119896 (inf 119865

119860) + 1205822119896 (inf 119865

119860))

119896minus1(1205821119896 (sup119865

119860) + 1205822119896 (sup119865

119860))]⟩

= ⟨[119897minus1((1205821+ 1205822) 119897 (inf 119879

119860))

119897minus1((1205821+ 1205822) 119897 (sup119879

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119868

119860))

119896minus1((1205821+ 1205822) 119896 (sup 119868

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119865

119860))

119896minus1((1205821+ 1205822) 119896 (sup119865

119860))]⟩

= (1205821+ 1205822) 119860

(18)

(6)

1198601205821 sdot 1198601205822

=⟨[119896minus1(1205821119896 (inf 119879

119860)) 119896minus1(1205821119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860)) 119897minus1(1205821119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860)) 119897minus1(1205821119897 (sup119865

119860))]⟩

sdot ⟨[119896minus1(1205822119896 (inf 119879

119860)) 119896minus1(1205822119896 (sup119879

119860))]

[119897minus1(1205822119897 (inf 119868

119860)) 119897minus1(1205822119897 (sup 119868

119860))]

[119897minus1(1205822119897 (inf 119865

119860)) 119897minus1(1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1(119896 (119896minus1(1205821119896 (inf 119879

119860)))

+119896 (119896minus1(1205822119896 (inf 119879

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119879

119860)))

+119896 (119896minus1(1205822119896 (sup119879

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119868

119860)))

+119897 (119897minus1(1205822119897 (inf 119868

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup 119868

119860)))

+119897 (119897minus1(1205822119897 (sup 119868

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119865

119860)))

+ 119897 (119897minus1(1205822119897 (inf 119865

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup119865

119860)))

+119897 (119897minus1(1205822119897 (sup119865

119860))))]⟩

= ⟨[119896minus1(1205821119896 (inf 119879

119860) + 1205822119896 (inf 119879

119860))

119896minus1(1205821119896 (sup119879

119860) + 1205822119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860) + 1205822119897 (inf 119868

119860))

119897minus1(1205821119897 (sup 119868

119860) + 1205822119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860) + 1205822119897 (inf 119865

119860))

119897minus1(1205821119897 (sup119865

119860) + 1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1((1205821+ 1205822) 119896 (inf 119879

119860))

119896minus1((1205821+ 1205822) 119896 (sup119879

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119868

119860))

119897minus1((1205821+ 1205822) 119897 (sup 119868

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119865

119860))

119897minus1((1205821+ 1205822) 119897 (sup119865

119860))]⟩

= 1198601205821+1205822

(19)


(1) 2 sdot 119860 = ⟨[091 096] [0 001] [001 004]⟩(2) 1198602 = ⟨[049 064] [0 019] [019 036]⟩(3) 119860 + 119861 = ⟨[082 090] [0 005] [003 008]⟩(4) 119860 sdot 119861 = ⟨[028 040] [020 037] [037 052]⟩


119860(119909) inf 119868

119860(119909) = sup 119868

119860(119909) inf 119865

119860(119909) =

sup119865119860(119909) inf 119879

119861(119909) = sup119879

119861(119909) inf 119868

119861(119909) = sup 119868

119861(119909)

and inf 119865119861(119909) = sup119865


⟨119879119860(119909) 119868119860(119909) 119865119860(119909)⟩ and 119861 = ⟨119879

119861(119909) 119868119861(119909) 119865119861(119909)⟩ are




119860= 0

inf 119868119861= sup 119868

119861= 0 sup119879

119860+sup119865

119860le 1 and sup119879

119861+sup119865

119861le


119861 119868119861 119865119861⟩


(1) 119860 + 119861 = ⟨[inf 119879119860+ inf 119879

119861minus inf 119879

119860sdot inf 119879

119861 sup119879

119860+

sup119879119861minus sup119879

119860sdot sup119879

119861][inf 119865

119860sdot inf 119865

119861 sup119865

119860sdot

sup119865119861]⟩

(2) 119860 sdot 119861 = ⟨[inf 119879119860sdot inf 119879

119861 sup119879

119860sdot sup119879

119861] [inf 119865

119860+

inf 119865119861minus inf 119865

119860sdot inf 119865

119861 sup119865

119860+ sup119865

119861minus sup119865

119860sdot

sup119865119861]⟩


119860)120582]

[(inf 119865119860)120582 (sup119865

119860)120582]⟩

(4) 119860120582 = ⟨[(inf 119879119860)120582 (sup119879

119860)120582] [1minus(1minusinf 119865

119860)120582 1minus(1minus

sup119865119860)120582]⟩




⟨[inf 119879119860 sup119879

119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ and

then

(1) 119904(119860) = [inf 119879119860+ 1 minus sup 119868

119860+ 1 minus sup119865

119860 sup119879

119860+ 1 minus

inf 119868119860+ 1 minus inf 119865

119860]

(2) 119886(119860) = [mininf 119879119860

minus inf 119865119860 sup119879

119860minus

sup119865119860maxinf 119879

119860minus inf 119865

119860 sup119879

119860minus sup119865

119860]

(3) 119888(119860) = [inf 119879119860 sup119879

119860]



119860(119909) sup119879

119860(119909) inf 119868

119860(119909) sup 119868

119860(119909) inf 119865

119860(119909)




(1) 119904(119860) = [24 27] 119904(119861) = [17 20]

(2) 119886(119860) = [06 06] 119886(119861) = [01 01]

(3) 119888(119860) = [07 08] 119888(119861) = [04 05]









(2) Assuming that 119860 = ⟨[06 07] [03 04] [04 05]⟩


(3) For two INNs 119860 = ⟨[06 07] [03 04] [04 05]⟩








INNWA119908(1198601 1198602 119860

119899) = 11990811198601+ 11990821198602+ sdot sdot sdot + 119908

119899119860119899

=

119899

sum

119895=1

119908119895119860119895

(20)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0(119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 23 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a



vector of 119860119895(119895 = 1 2 119899) with 119908

119895ge 0 (119895 = 1 2 119899)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879


+119908119899119897 (inf 119879

119860119899))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879


+119908119899119897 (sup119879

119860119899))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868


+119908119899119896 (inf 119868

119860119899))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868


+119908119899119896 (sup 119868

119860119899))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865


+119908119899119896 (inf 119865

119860119899))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865


+119908119899119896 (sup119865

119860119899))]⟩

(21)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[1 minus

119899

prod

119894=1

(1 minus inf 119879119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119879119860119894)119908119894]

[

119899

prod

119894=1

inf 119868119908119894119860119894

119899

prod

119894=1

sup 119868119908119894119860119894]

[

119899

prod

119894=1

inf 119865119908119894119860119894

119899

prod

119894=1

sup119865119908119894119860119894]⟩

(22)


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1


(1) For 119899 = 2 since

11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601)) 119897minus1(1199081119897 (sup119879

1198601))]

[119896minus1(1199081119896 (inf 119868

1198601)) 119896minus1(1199081119896 (sup 119868

1198601))]

[119896minus1(1199081119896 (inf 119865

1198601)) 119896minus1(1199081119896 (sup119865

1198601))]⟩

oplus ⟨[119897minus1(1199082119897 (inf 119879

1198602)) 119897minus1(1199082119897 (sup119879

1198602))]

[119896minus1(1199082119896 (inf 119868

1198602)) 119896minus1(1199082119896 (sup 119868

1198602))]

[119896minus1(1199082119896 (inf 119865

1198602)) 119896minus1(1199082119896 (sup119865

1198602))]⟩

= ⟨[119897minus1(119897 (119897minus1(1199081119897 (inf 119879

1198601)))

+119897 (119897minus1(1199082119897 (inf 119879

1198602))))

119897minus1(119897 (119897minus1(1199081119897 (sup119879

1198601)))

+119897 (119897minus1(1199082119897 (sup119879

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119868

1198601)))

+119896 (119896minus1(1199082119896 (inf 119868

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup 119868

1198601)))

+119896 (119896minus1(1199082119896 (sup 119868

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119865

1198601)))

+119896 (119896minus1(1199082119896 (inf 119865

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup119865

1198601)))

+119896 (119896minus1(1199082119896 (sup119865

1198602))))]⟩

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))


119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(23)

then

SNNWA119908(1198601 1198602)

= 11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(24)


SNNWA119908(1198601 1198602 119860

119896)

= ⟨[

[

119897minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(25)

then if 119899 = 119896 + 1 we have

SNNWA119908(1198601 1198602 119860

119896 119860119896+1)

= ⟨[

[

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (inf 119879

119860119896+1)))))

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (sup119879

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (inf 119868

119860119896+1)))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (sup 119868

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (inf 119865

119860119896+1))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (sup119865

119860119896+1))))]

]

⟩

= ⟨[

[

119897minus1(

119896+1

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896+1

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119865

119860119895))


119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(26)


SNNWA119908(1198601 1198602 119860

119899)

= ⟨[

[

119897minus1(

119899

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119899

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(27)






INNWA119908(1198601 1198602 119860

119899) = 119860



max119895119868119860119895(119909)max

119895119865119860119895(119909)⟩ 119860+ = ⟨max

119895119879119860119895(119909)

min119895119868119860119895(119909)min

119895119865119860119895(119909)⟩ for all 119895 isin 1 2 119899 and


119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWA119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)



INNWG119908(1198601 1198602 119860

119899) =

119899

prod

119895=1

119860119908119894

119895 (28)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 25 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a


119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[119896minus1(1199081119896 (inf 119879

1198601) + 1199082119896 (inf 119879


+119908119899119896 (inf 119879

119860119899))

119896minus1(1199081119896 (sup119879

1198601) + 119908

2119896 (sup119879


+119908119899119896 (sup119879

119860119899))]

[119897minus1(1199081119897 (inf 119868

1198601) + 1199082119897 (inf 119868


+119908119899119897 (inf 119868

119860119899))

119897minus1(1199081119897 (sup 119868

1198601) + 1199082119897 (sup 119868


+119908119899119897 (sup 119868

119860119899))]

[119897minus1(1199081119897 (inf 119865

1198601) + 1199082119897 (inf 119865


+119908119899119897 (inf 119865

119860119899))

119897minus1(1199081119897 (sup119865

1198601) + 1199082119897 (sup119865


+119908119899119897 (sup119865

119860119899))]⟩

(29)



119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[

119899

prod

119894=1

inf 119879119908119894119860119894

119899

prod

119894=1

sup119879119908119894119860119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119868119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup 119868119860119894)119908119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119865119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119865119860119894)119908119894]⟩

(30)

where 119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1




INNWG119908(1198601 1198602 119860

119899)

= ⟨[

[

2prod119899

119894=1inf 119879119908119894119860119894

prod119899

119894=1(2 minus inf 119879

119860119894)119908119894+prod119899

119894=1inf 119879119908119894119860119894

2prod119899

119894=1sup119879119908119894119860119894

prod119899

119894=1(2 minus sup119879

119860119894)119908119894+prod119899

119894=1sup119879119908119894119860119894

]

]

[

[

prod119899

119894=1(1 + inf 119868

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + inf 119868

119860119894)119908119894+prod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894minusprod119899

119894=1(1 minus sup 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894+prod119899

119894=1(1 minus sup 119868

119860119894)119908119894

]

]

[

[

prod119899

119894=1(1 + inf 119865

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + inf 119865

119860119894)119908119894+prod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894minusprod119899

119894=1(1 minus sup119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894+prod119899

119894=1(1 minus sup119865

119860119894)119908119894

]

]

⟩

(31)

where119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1






INNWG119908(1198601 1198602 119860

119899) = 119860



= ⟨min119895119879119860119895(119909)max

119895119868119860119895(119909)max

119895119865119860119895(119909)⟩

119860+

= ⟨max119895119879119860119895(119909)min

119895119868119860119895(119909)min

119895119865119860119895(119909)⟩


sube

INNWG119908(1198601 1198602 119860

119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWG119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)


1 1198862 119886119898 and n criteria 119862 = 119888

1 1198882 119888119899 whose


119899) where

119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Let 119877 =

(119886119894119895)119898times119899


where 119879119886119894119895



119895 119868119886119894119895


119895 and 119865







119894(119894 = 1 2 119898)

that is

119910119894= INNWA

119908(1198861198941 1198861198942 119886

119894119899) (32)

or

119910119894= INNWG

119908(1198861198941 1198861198942 119886

119894119899) (33)



119894) of


matrix F

119865 =[[[

[

119904 (1199101) 119886 (119910

1) 119888 (119910

1)

119904 (1199102) 119886 (119910

2) 119888 (119910

2)


119904 (119910119898) 119886 (119910

119898) 119888 (119910

119898)

]]]

]

(34)



119875119904=

[[[[

[

11990111990411

11990111990412

sdot sdot sdot 1199011199041119898

11990111990421

11990111990422

sdot sdot sdot 1199011199042119898

1199011199041198981

1199011199041198982

sdot sdot sdot 119901119904119898119898

]]]]

]

(35)


119894) gt 119904(119910

119895)


119894119895+ 119901119904119895119894= 1 and 119901119904

119894119894= 05 If


119886(119910119894) gt 119886(119910

119895) denoted by 119901119886

119894119895 And if 119901119886

119894119895= 05 (119894 = 119895) then


119895) denoted by

119901119888119894119895









3





2




3is a


119863 =

[[[

[

⟨[04 05] [02 03] [03 04]⟩ ⟨[04 06] [01 03] [02 04]⟩ ⟨[07 09] [02 03] [04 05]⟩

⟨[06 07] [01 02] [02 03]⟩ ⟨[06 07] [01 02] [02 03]⟩ ⟨[03 06] [03 05] [08 09]⟩

⟨[03 06] [02 03] [03 04]⟩ ⟨[05 06] [02 03] [03 04]⟩ ⟨[04 05] [02 04] [07 09]⟩

⟨[07 08] [00 01] [01 02]⟩ ⟨[06 07] [01 02] [01 03]⟩ ⟨[06 07] [03 04] [08 09]⟩

]]]

]

(36)




119860119882119860

=[[[

[

⟨[05453 07516] [01682 03000] [03041 04373]⟩

⟨[04996 06634] [01552 02885] [03482 04656]⟩

⟨[03950 05627] [02000 03366] [04210 05533]⟩

⟨[06383 07397] [00000 02071] [02297 04040]⟩

]]]

]

(37)


119860119882119866

=[[[

[

⟨[05004 06620] [01761 03000] [03195 04422]⟩

⟨[04547 06581] [01861 03371] [05405 06786]⟩

⟨[03824 05578] [02000 03419] [05012 07070]⟩

⟨[06333 07335] [01555 02570] [05069 06632]⟩

]]]

]

(38)



119865119882119860

=[[[

[

[18080 22793] [02412 03143] [05453 07516]

[17455 21600] [01514 01978] [04996 06634]

[15051 19417] [minus00260 00094] [03950 05627]

[20272 25100] [03357 04086] [06383 07397]

]]]

]

(39)


119865119882119866

=[[[

[

[17582 21664] [01809 02198] [05004 06620]

[14390 19315] [minus00858 minus00205] [04547 06581]

[13335 18566] [minus01492 minus01188] [03824 05578]

[17131 20711] [00703 01264] [06333 07335]

]]]

]

(40)


119882119860

119875119904 119882119860

=[[[

[

05 06498 08527 02642

03974 05 07695 01480

01473 02305 05 0

07358 08520 1 05

]]]

]

(41)

And for 119865119882119866

119901119904 119882119866

=[[[

[

05 08076 08943 05916

01924 05 05888 02568

01057 04112 05 01629

04084 07432 08371 05

]]]

]

(42)


compute 119901119886119894119895and 119901119888

119894119895



1≻ 1198602 1198601≻ 1198603 1198602≻ 1198603 1198604≻ 1198601 and 119860

4≻


4 1198601

1198602 and 119860

3 Obviously 119860





3 Obviously 119860

1is



3 as well as the

ranking 1198601 1198604 1198602 and 119860

3for AWG






2 1198604 1198603 and 119860

1


1 1198604 1198602 and 119860

3 Obviously


1


119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)




3 As can be


11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)


11988611= 11987911988631


13= 11988633






3in [33]


lt 11988641 11988622

lt 11988642

and 11988623




1in [33] while the

sequences of 1198601 1198602and 119860




6 Conclusion




Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(3)

119860 + 119861

= ⟨[119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))

119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))]

[119896minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))

119896minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))]

[119896minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))

119896minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))]⟩

(10)

(4)

119860 sdot 119861

= ⟨[119896minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))

119896minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))]

[119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))

119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))]

[119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))

119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))]⟩

(11)


(5)


119860)120582]

[(inf 119868119860)120582 (sup 119868

119860)120582]

[(inf 119865119860)120582 (sup119865

119860)120582]⟩

(12)

(6)

119860120582= ⟨[(inf 119879

119860)120582 (sup119879

119860)120582]


119860)120582]


119860)120582]⟩

(13)

(7)

119860 + 119861 = ⟨[inf 119879119860+ inf 119879

119861minus inf 119879

119860sdot inf 119879

119861

sup119879119860+ sup119879

119861minus sup119879

119860sdot sup119879

119861]

[inf 119879119860sdot inf 119868119861 sup 119868

119860sdot sup 119868

119861]

[inf 119865119860sdot inf 119865

119861 sup119865

119860sdot sup119865

119861]⟩

(14)

(8)

119860 sdot 119861 = ⟨[inf 119879119860sdot inf 119879

119861 sup119879

119860sdot sup119879

119861]

[inf 119879119860+ inf 119868

119861minus inf 119879

119860sdot inf 119868119861

sup 119868119860+ sup 119868

119861minus sup 119868

119860sdot sup 119868

119861]

[inf 119865119860+ inf 119865

119861minus inf 119865

119860sdot inf 119865

119861

sup119865119860+ sup119865

119861minus sup119865

119860sdot sup119865

119861]⟩

(15)


sup119879119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ 119861 = ⟨[inf 119879

119861

sup119879119861] [inf 119868

119861 sup 119868

119861] [inf 119865

119861 sup119865

119861]⟩ 119862 = ⟨[inf 119879

119862

sup119879119862] [inf 119868

119862 sup 119868

119862] [inf 119865

119862 sup119865



(1) 119860 + 119861 = 119861 + 119860

(2) 119860 sdot 119861 = 119861 sdot 119860

(3) 120582(119860 + 119861) = 120582119860 + 120582119861 120582 gt 0

(4) (119860 sdot 119861)120582 = 119860120582 + 119861120582 120582 gt 0

(5) 1205821119860 + 120582

2119860 = (120582

1+ 1205822)119860 1205821gt 0 120582

2gt 0

(6) 1198601205821 sdot 1198601205822 = 119860(1205821+1205822) 1205821gt 0 120582

2gt 0

(7) (119860 + 119861) + 119862 = 119860 + (119861 + 119862)



(3)

120582 (119860 + 119861) = 120582 sdot ⟨[119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))

119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))]

[119896minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))

119896minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))]

[119896minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))

119896minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))]⟩

= ⟨[119897minus1(120582119897 (119897minus1(119897 (inf 119879

119860) + 119897 (inf 119879

119861))))


119897minus1(120582119897 (119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))))]

[119896minus1(120582119896 (119896

minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))))]

[119896minus1(120582119896 (119896

minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))))]⟩

= ⟨[119897minus1(120582 (119897 (inf 119879

119860) + 119897 (inf 119879

119861)))

119897minus1(120582 (119897 (sup119879

119860) + 119897 (sup119879

119861)))]

[119896minus1(120582 (119896 (inf 119868

119860) + 119896 (inf 119868

119861)))

119896minus1(120582 (119896 (sup 119868

119860) + 119896 (sup 119868

119861)))]

[119896minus1(120582 (119896 (inf 119865

119860) + 119896 (inf 119865

119861)))

119896minus1(120582 (119896 (sup119865

119860) + 119896 (sup119865

119861)))]⟩

= ⟨[119897minus1(120582119897 (inf 119879

119860) + 120582119897 (inf 119879

119861))

119897minus1(120582119897 (sup119879

119860) + 120582119897 (sup119879

119861))]

[119896minus1(120582119896 (inf 119868

119860) + 120582119896 (inf 119868

119861))

119896minus1(120582119896 (sup 119868

119860) + 120582119896 (sup 119868

119861))]

[119896minus1(120582119896 (inf 119865

119860) + 120582119896 (inf 119865

119861))

119896minus1(120582119896 (sup119865

119860) + 120582119896 (sup119865

119861))]⟩

= 120582119860 + 120582119861

(16)

(4)

(119860 sdot 119861)120582= (⟨[119896

minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))

119896minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))]

[119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))

119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))]

[119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))

119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))]⟩)120582

= ⟨[119896minus1(120582119896 (119896

minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))))]

[119897minus1(120582119897 (119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))))

119897minus1(120582119897 (119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))))]

[119897minus1(120582119897 (119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))))

119897minus1(120582119897 (119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))))]⟩

= ⟨[119896minus1(120582 (119896 (inf 119879

119860) + 119896 (inf 119879

119861)))

119896minus1(120582 (119896 (sup119879

119860) + 119896 (sup119879

119861)))]

[119897minus1(120582 (119897 (inf 119868

119860) + 119897 (inf 119868

119861)))

119897minus1(120582 (119897 (sup 119868

119860) + 119897 (sup 119868

119861)))]

[119897minus1(120582 (119897 (inf 119865

119860) + 119897 (inf 119865

119861)))

119897minus1(120582 (119897 (sup119865

119860) + 119897 (sup119865

119861)))]⟩

= ⟨[119896minus1(120582119896 (inf 119879

119860) + 120582119896 (inf 119879

119861))

119896minus1(120582119896 (sup119879

119860) + 120582119896 (sup119879

119861))]

[119897minus1(120582119897 (inf 119868

119860) + 120582119897 (inf 119868

119861))

119897minus1(120582119897 (sup 119868

119860) + 120582119897 (sup 119868

119861))]

[119897minus1(120582119897 (inf 119865

119860) + 120582119897 (inf 119865

119861))

119897minus1(120582119897 (sup119865

119860) + 120582119897 (sup119865

119861))]⟩

= 119860120582sdot 119861120582

(17)

(5)

1205821119860 + 120582

2119860

= ⟨[119897minus1(1205821119897 (inf 119879

119860)) 119897minus1(1205821119897 (sup119879

119860))]

[119896minus1(1205821119896 (inf 119868

119860)) 119896minus1(1205821119896 (sup 119868

119860))]

[119896minus1(1205821119896 (inf 119865

119860)) 119896minus1(1205821119896 (sup119865

119860))]⟩

oplus ⟨[119897minus1(1205822119897 (inf 119879

119860)) 119897minus1(1205822119897 (sup119879

119860))]

[119896minus1(1205822119896 (inf 119868

119860)) 119896minus1(1205822119896 (sup 119868

119860))]

[119896minus1(1205822119896 (inf 119865

119860)) 119896minus1(1205822119896 (sup119865

119860))]⟩

= ⟨[119897minus1(119897 (119897minus1(1205821119897 (inf 119879

119860))) + 119897 (119897

minus1(1205822119897 (inf 119879

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup119879

119860))) + 119897 (119897

minus1(1205822119897 (sup119879

119860))))]

[119896minus1(119896 (119896minus1(1205821119896 (inf 119868

119860)))

+119896 (119896minus1(1205822119896 (inf 119868

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup 119868

119860)))

+119896 (119896minus1(1205822119896 (sup 119868

119860))))]


[119896minus1(119896 (119896minus1(1205821119896 (inf 119865

119860)))

+119896 (119896minus1(1205822119896 (inf 119865

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119865

119860)))

+119896 (119896minus1(1205822119896 (sup119865

119860))))]⟩

= ⟨[119897minus1(1205821119897 (inf 119879

119860) + 1205822119897 (inf 119879

119860))

119897minus1(1205821119897 (sup119879

119860) + 1205822119897 (sup119879

119860))]

[119896minus1(1205821119896 (inf 119868

119860) + 1205822119896 (inf 119868

119860))

119896minus1(1205821119896 (sup 119868

119860) + 1205822119896 (sup 119868

119860))]

[119896minus1(1205821119896 (inf 119865

119860) + 1205822119896 (inf 119865

119860))

119896minus1(1205821119896 (sup119865

119860) + 1205822119896 (sup119865

119860))]⟩

= ⟨[119897minus1((1205821+ 1205822) 119897 (inf 119879

119860))

119897minus1((1205821+ 1205822) 119897 (sup119879

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119868

119860))

119896minus1((1205821+ 1205822) 119896 (sup 119868

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119865

119860))

119896minus1((1205821+ 1205822) 119896 (sup119865

119860))]⟩

= (1205821+ 1205822) 119860

(18)

(6)

1198601205821 sdot 1198601205822

=⟨[119896minus1(1205821119896 (inf 119879

119860)) 119896minus1(1205821119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860)) 119897minus1(1205821119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860)) 119897minus1(1205821119897 (sup119865

119860))]⟩

sdot ⟨[119896minus1(1205822119896 (inf 119879

119860)) 119896minus1(1205822119896 (sup119879

119860))]

[119897minus1(1205822119897 (inf 119868

119860)) 119897minus1(1205822119897 (sup 119868

119860))]

[119897minus1(1205822119897 (inf 119865

119860)) 119897minus1(1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1(119896 (119896minus1(1205821119896 (inf 119879

119860)))

+119896 (119896minus1(1205822119896 (inf 119879

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119879

119860)))

+119896 (119896minus1(1205822119896 (sup119879

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119868

119860)))

+119897 (119897minus1(1205822119897 (inf 119868

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup 119868

119860)))

+119897 (119897minus1(1205822119897 (sup 119868

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119865

119860)))

+ 119897 (119897minus1(1205822119897 (inf 119865

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup119865

119860)))

+119897 (119897minus1(1205822119897 (sup119865

119860))))]⟩

= ⟨[119896minus1(1205821119896 (inf 119879

119860) + 1205822119896 (inf 119879

119860))

119896minus1(1205821119896 (sup119879

119860) + 1205822119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860) + 1205822119897 (inf 119868

119860))

119897minus1(1205821119897 (sup 119868

119860) + 1205822119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860) + 1205822119897 (inf 119865

119860))

119897minus1(1205821119897 (sup119865

119860) + 1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1((1205821+ 1205822) 119896 (inf 119879

119860))

119896minus1((1205821+ 1205822) 119896 (sup119879

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119868

119860))

119897minus1((1205821+ 1205822) 119897 (sup 119868

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119865

119860))

119897minus1((1205821+ 1205822) 119897 (sup119865

119860))]⟩

= 1198601205821+1205822

(19)


(1) 2 sdot 119860 = ⟨[091 096] [0 001] [001 004]⟩(2) 1198602 = ⟨[049 064] [0 019] [019 036]⟩(3) 119860 + 119861 = ⟨[082 090] [0 005] [003 008]⟩(4) 119860 sdot 119861 = ⟨[028 040] [020 037] [037 052]⟩


119860(119909) inf 119868

119860(119909) = sup 119868

119860(119909) inf 119865

119860(119909) =

sup119865119860(119909) inf 119879

119861(119909) = sup119879

119861(119909) inf 119868

119861(119909) = sup 119868

119861(119909)

and inf 119865119861(119909) = sup119865


⟨119879119860(119909) 119868119860(119909) 119865119860(119909)⟩ and 119861 = ⟨119879

119861(119909) 119868119861(119909) 119865119861(119909)⟩ are




119860= 0

inf 119868119861= sup 119868

119861= 0 sup119879

119860+sup119865

119860le 1 and sup119879

119861+sup119865

119861le


119861 119868119861 119865119861⟩


(1) 119860 + 119861 = ⟨[inf 119879119860+ inf 119879

119861minus inf 119879

119860sdot inf 119879

119861 sup119879

119860+

sup119879119861minus sup119879

119860sdot sup119879

119861][inf 119865

119860sdot inf 119865

119861 sup119865

119860sdot

sup119865119861]⟩

(2) 119860 sdot 119861 = ⟨[inf 119879119860sdot inf 119879

119861 sup119879

119860sdot sup119879

119861] [inf 119865

119860+

inf 119865119861minus inf 119865

119860sdot inf 119865

119861 sup119865

119860+ sup119865

119861minus sup119865

119860sdot

sup119865119861]⟩


119860)120582]

[(inf 119865119860)120582 (sup119865

119860)120582]⟩

(4) 119860120582 = ⟨[(inf 119879119860)120582 (sup119879

119860)120582] [1minus(1minusinf 119865

119860)120582 1minus(1minus

sup119865119860)120582]⟩




⟨[inf 119879119860 sup119879

119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ and

then

(1) 119904(119860) = [inf 119879119860+ 1 minus sup 119868

119860+ 1 minus sup119865

119860 sup119879

119860+ 1 minus

inf 119868119860+ 1 minus inf 119865

119860]

(2) 119886(119860) = [mininf 119879119860

minus inf 119865119860 sup119879

119860minus

sup119865119860maxinf 119879

119860minus inf 119865

119860 sup119879

119860minus sup119865

119860]

(3) 119888(119860) = [inf 119879119860 sup119879

119860]



119860(119909) sup119879

119860(119909) inf 119868

119860(119909) sup 119868

119860(119909) inf 119865

119860(119909)




(1) 119904(119860) = [24 27] 119904(119861) = [17 20]

(2) 119886(119860) = [06 06] 119886(119861) = [01 01]

(3) 119888(119860) = [07 08] 119888(119861) = [04 05]









(2) Assuming that 119860 = ⟨[06 07] [03 04] [04 05]⟩


(3) For two INNs 119860 = ⟨[06 07] [03 04] [04 05]⟩








INNWA119908(1198601 1198602 119860

119899) = 11990811198601+ 11990821198602+ sdot sdot sdot + 119908

119899119860119899

=

119899

sum

119895=1

119908119895119860119895

(20)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0(119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 23 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a



vector of 119860119895(119895 = 1 2 119899) with 119908

119895ge 0 (119895 = 1 2 119899)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879


+119908119899119897 (inf 119879

119860119899))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879


+119908119899119897 (sup119879

119860119899))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868


+119908119899119896 (inf 119868

119860119899))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868


+119908119899119896 (sup 119868

119860119899))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865


+119908119899119896 (inf 119865

119860119899))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865


+119908119899119896 (sup119865

119860119899))]⟩

(21)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[1 minus

119899

prod

119894=1

(1 minus inf 119879119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119879119860119894)119908119894]

[

119899

prod

119894=1

inf 119868119908119894119860119894

119899

prod

119894=1

sup 119868119908119894119860119894]

[

119899

prod

119894=1

inf 119865119908119894119860119894

119899

prod

119894=1

sup119865119908119894119860119894]⟩

(22)


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1


(1) For 119899 = 2 since

11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601)) 119897minus1(1199081119897 (sup119879

1198601))]

[119896minus1(1199081119896 (inf 119868

1198601)) 119896minus1(1199081119896 (sup 119868

1198601))]

[119896minus1(1199081119896 (inf 119865

1198601)) 119896minus1(1199081119896 (sup119865

1198601))]⟩

oplus ⟨[119897minus1(1199082119897 (inf 119879

1198602)) 119897minus1(1199082119897 (sup119879

1198602))]

[119896minus1(1199082119896 (inf 119868

1198602)) 119896minus1(1199082119896 (sup 119868

1198602))]

[119896minus1(1199082119896 (inf 119865

1198602)) 119896minus1(1199082119896 (sup119865

1198602))]⟩

= ⟨[119897minus1(119897 (119897minus1(1199081119897 (inf 119879

1198601)))

+119897 (119897minus1(1199082119897 (inf 119879

1198602))))

119897minus1(119897 (119897minus1(1199081119897 (sup119879

1198601)))

+119897 (119897minus1(1199082119897 (sup119879

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119868

1198601)))

+119896 (119896minus1(1199082119896 (inf 119868

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup 119868

1198601)))

+119896 (119896minus1(1199082119896 (sup 119868

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119865

1198601)))

+119896 (119896minus1(1199082119896 (inf 119865

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup119865

1198601)))

+119896 (119896minus1(1199082119896 (sup119865

1198602))))]⟩

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))


119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(23)

then

SNNWA119908(1198601 1198602)

= 11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(24)


SNNWA119908(1198601 1198602 119860

119896)

= ⟨[

[

119897minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(25)

then if 119899 = 119896 + 1 we have

SNNWA119908(1198601 1198602 119860

119896 119860119896+1)

= ⟨[

[

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (inf 119879

119860119896+1)))))

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (sup119879

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (inf 119868

119860119896+1)))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (sup 119868

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (inf 119865

119860119896+1))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (sup119865

119860119896+1))))]

]

⟩

= ⟨[

[

119897minus1(

119896+1

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896+1

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119865

119860119895))


119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(26)


SNNWA119908(1198601 1198602 119860

119899)

= ⟨[

[

119897minus1(

119899

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119899

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(27)






INNWA119908(1198601 1198602 119860

119899) = 119860



max119895119868119860119895(119909)max

119895119865119860119895(119909)⟩ 119860+ = ⟨max

119895119879119860119895(119909)

min119895119868119860119895(119909)min

119895119865119860119895(119909)⟩ for all 119895 isin 1 2 119899 and


119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWA119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)



INNWG119908(1198601 1198602 119860

119899) =

119899

prod

119895=1

119860119908119894

119895 (28)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 25 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a


119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[119896minus1(1199081119896 (inf 119879

1198601) + 1199082119896 (inf 119879


+119908119899119896 (inf 119879

119860119899))

119896minus1(1199081119896 (sup119879

1198601) + 119908

2119896 (sup119879


+119908119899119896 (sup119879

119860119899))]

[119897minus1(1199081119897 (inf 119868

1198601) + 1199082119897 (inf 119868


+119908119899119897 (inf 119868

119860119899))

119897minus1(1199081119897 (sup 119868

1198601) + 1199082119897 (sup 119868


+119908119899119897 (sup 119868

119860119899))]

[119897minus1(1199081119897 (inf 119865

1198601) + 1199082119897 (inf 119865


+119908119899119897 (inf 119865

119860119899))

119897minus1(1199081119897 (sup119865

1198601) + 1199082119897 (sup119865


+119908119899119897 (sup119865

119860119899))]⟩

(29)



119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[

119899

prod

119894=1

inf 119879119908119894119860119894

119899

prod

119894=1

sup119879119908119894119860119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119868119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup 119868119860119894)119908119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119865119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119865119860119894)119908119894]⟩

(30)

where 119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1




INNWG119908(1198601 1198602 119860

119899)

= ⟨[

[

2prod119899

119894=1inf 119879119908119894119860119894

prod119899

119894=1(2 minus inf 119879

119860119894)119908119894+prod119899

119894=1inf 119879119908119894119860119894

2prod119899

119894=1sup119879119908119894119860119894

prod119899

119894=1(2 minus sup119879

119860119894)119908119894+prod119899

119894=1sup119879119908119894119860119894

]

]

[

[

prod119899

119894=1(1 + inf 119868

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + inf 119868

119860119894)119908119894+prod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894minusprod119899

119894=1(1 minus sup 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894+prod119899

119894=1(1 minus sup 119868

119860119894)119908119894

]

]

[

[

prod119899

119894=1(1 + inf 119865

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + inf 119865

119860119894)119908119894+prod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894minusprod119899

119894=1(1 minus sup119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894+prod119899

119894=1(1 minus sup119865

119860119894)119908119894

]

]

⟩

(31)

where119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1






INNWG119908(1198601 1198602 119860

119899) = 119860



= ⟨min119895119879119860119895(119909)max

119895119868119860119895(119909)max

119895119865119860119895(119909)⟩

119860+

= ⟨max119895119879119860119895(119909)min

119895119868119860119895(119909)min

119895119865119860119895(119909)⟩


sube

INNWG119908(1198601 1198602 119860

119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWG119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)


1 1198862 119886119898 and n criteria 119862 = 119888

1 1198882 119888119899 whose


119899) where

119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Let 119877 =

(119886119894119895)119898times119899


where 119879119886119894119895



119895 119868119886119894119895


119895 and 119865







119894(119894 = 1 2 119898)

that is

119910119894= INNWA

119908(1198861198941 1198861198942 119886

119894119899) (32)

or

119910119894= INNWG

119908(1198861198941 1198861198942 119886

119894119899) (33)



119894) of


matrix F

119865 =[[[

[

119904 (1199101) 119886 (119910

1) 119888 (119910

1)

119904 (1199102) 119886 (119910

2) 119888 (119910

2)


119904 (119910119898) 119886 (119910

119898) 119888 (119910

119898)

]]]

]

(34)



119875119904=

[[[[

[

11990111990411

11990111990412

sdot sdot sdot 1199011199041119898

11990111990421

11990111990422

sdot sdot sdot 1199011199042119898

1199011199041198981

1199011199041198982

sdot sdot sdot 119901119904119898119898

]]]]

]

(35)


119894) gt 119904(119910

119895)


119894119895+ 119901119904119895119894= 1 and 119901119904

119894119894= 05 If


119886(119910119894) gt 119886(119910

119895) denoted by 119901119886

119894119895 And if 119901119886

119894119895= 05 (119894 = 119895) then


119895) denoted by

119901119888119894119895









3





2




3is a


119863 =

[[[

[

⟨[04 05] [02 03] [03 04]⟩ ⟨[04 06] [01 03] [02 04]⟩ ⟨[07 09] [02 03] [04 05]⟩

⟨[06 07] [01 02] [02 03]⟩ ⟨[06 07] [01 02] [02 03]⟩ ⟨[03 06] [03 05] [08 09]⟩

⟨[03 06] [02 03] [03 04]⟩ ⟨[05 06] [02 03] [03 04]⟩ ⟨[04 05] [02 04] [07 09]⟩

⟨[07 08] [00 01] [01 02]⟩ ⟨[06 07] [01 02] [01 03]⟩ ⟨[06 07] [03 04] [08 09]⟩

]]]

]

(36)




119860119882119860

=[[[

[

⟨[05453 07516] [01682 03000] [03041 04373]⟩

⟨[04996 06634] [01552 02885] [03482 04656]⟩

⟨[03950 05627] [02000 03366] [04210 05533]⟩

⟨[06383 07397] [00000 02071] [02297 04040]⟩

]]]

]

(37)


119860119882119866

=[[[

[

⟨[05004 06620] [01761 03000] [03195 04422]⟩

⟨[04547 06581] [01861 03371] [05405 06786]⟩

⟨[03824 05578] [02000 03419] [05012 07070]⟩

⟨[06333 07335] [01555 02570] [05069 06632]⟩

]]]

]

(38)



119865119882119860

=[[[

[

[18080 22793] [02412 03143] [05453 07516]

[17455 21600] [01514 01978] [04996 06634]

[15051 19417] [minus00260 00094] [03950 05627]

[20272 25100] [03357 04086] [06383 07397]

]]]

]

(39)


119865119882119866

=[[[

[

[17582 21664] [01809 02198] [05004 06620]

[14390 19315] [minus00858 minus00205] [04547 06581]

[13335 18566] [minus01492 minus01188] [03824 05578]

[17131 20711] [00703 01264] [06333 07335]

]]]

]

(40)


119882119860

119875119904 119882119860

=[[[

[

05 06498 08527 02642

03974 05 07695 01480

01473 02305 05 0

07358 08520 1 05

]]]

]

(41)

And for 119865119882119866

119901119904 119882119866

=[[[

[

05 08076 08943 05916

01924 05 05888 02568

01057 04112 05 01629

04084 07432 08371 05

]]]

]

(42)


compute 119901119886119894119895and 119901119888

119894119895



1≻ 1198602 1198601≻ 1198603 1198602≻ 1198603 1198604≻ 1198601 and 119860

4≻


4 1198601

1198602 and 119860

3 Obviously 119860





3 Obviously 119860

1is



3 as well as the

ranking 1198601 1198604 1198602 and 119860

3for AWG






2 1198604 1198603 and 119860

1


1 1198604 1198602 and 119860

3 Obviously


1


119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)




3 As can be


11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)


11988611= 11987911988631


13= 11988633






3in [33]


lt 11988641 11988622

lt 11988642

and 11988623




1in [33] while the

sequences of 1198601 1198602and 119860




6 Conclusion




Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119897minus1(120582119897 (119897minus1(119897 (sup119879

119860) + 119897 (sup119879

119861))))]

[119896minus1(120582119896 (119896

minus1(119896 (inf 119868

119860) + 119896 (inf 119868

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup 119868

119860) + 119896 (sup 119868

119861))))]

[119896minus1(120582119896 (119896

minus1(119896 (inf 119865

119860) + 119896 (inf 119865

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup119865

119860) + 119896 (sup119865

119861))))]⟩

= ⟨[119897minus1(120582 (119897 (inf 119879

119860) + 119897 (inf 119879

119861)))

119897minus1(120582 (119897 (sup119879

119860) + 119897 (sup119879

119861)))]

[119896minus1(120582 (119896 (inf 119868

119860) + 119896 (inf 119868

119861)))

119896minus1(120582 (119896 (sup 119868

119860) + 119896 (sup 119868

119861)))]

[119896minus1(120582 (119896 (inf 119865

119860) + 119896 (inf 119865

119861)))

119896minus1(120582 (119896 (sup119865

119860) + 119896 (sup119865

119861)))]⟩

= ⟨[119897minus1(120582119897 (inf 119879

119860) + 120582119897 (inf 119879

119861))

119897minus1(120582119897 (sup119879

119860) + 120582119897 (sup119879

119861))]

[119896minus1(120582119896 (inf 119868

119860) + 120582119896 (inf 119868

119861))

119896minus1(120582119896 (sup 119868

119860) + 120582119896 (sup 119868

119861))]

[119896minus1(120582119896 (inf 119865

119860) + 120582119896 (inf 119865

119861))

119896minus1(120582119896 (sup119865

119860) + 120582119896 (sup119865

119861))]⟩

= 120582119860 + 120582119861

(16)

(4)

(119860 sdot 119861)120582= (⟨[119896

minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))

119896minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))]

[119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))

119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))]

[119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))

119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))]⟩)120582

= ⟨[119896minus1(120582119896 (119896

minus1(119896 (inf 119879

119860) + 119896 (inf 119879

119861))))

119896minus1(120582119896 (119896

minus1(119896 (sup119879

119860) + 119896 (sup119879

119861))))]

[119897minus1(120582119897 (119897minus1(119897 (inf 119868

119860) + 119897 (inf 119868

119861))))

119897minus1(120582119897 (119897minus1(119897 (sup 119868

119860) + 119897 (sup 119868

119861))))]

[119897minus1(120582119897 (119897minus1(119897 (inf 119865

119860) + 119897 (inf 119865

119861))))

119897minus1(120582119897 (119897minus1(119897 (sup119865

119860) + 119897 (sup119865

119861))))]⟩

= ⟨[119896minus1(120582 (119896 (inf 119879

119860) + 119896 (inf 119879

119861)))

119896minus1(120582 (119896 (sup119879

119860) + 119896 (sup119879

119861)))]

[119897minus1(120582 (119897 (inf 119868

119860) + 119897 (inf 119868

119861)))

119897minus1(120582 (119897 (sup 119868

119860) + 119897 (sup 119868

119861)))]

[119897minus1(120582 (119897 (inf 119865

119860) + 119897 (inf 119865

119861)))

119897minus1(120582 (119897 (sup119865

119860) + 119897 (sup119865

119861)))]⟩

= ⟨[119896minus1(120582119896 (inf 119879

119860) + 120582119896 (inf 119879

119861))

119896minus1(120582119896 (sup119879

119860) + 120582119896 (sup119879

119861))]

[119897minus1(120582119897 (inf 119868

119860) + 120582119897 (inf 119868

119861))

119897minus1(120582119897 (sup 119868

119860) + 120582119897 (sup 119868

119861))]

[119897minus1(120582119897 (inf 119865

119860) + 120582119897 (inf 119865

119861))

119897minus1(120582119897 (sup119865

119860) + 120582119897 (sup119865

119861))]⟩

= 119860120582sdot 119861120582

(17)

(5)

1205821119860 + 120582

2119860

= ⟨[119897minus1(1205821119897 (inf 119879

119860)) 119897minus1(1205821119897 (sup119879

119860))]

[119896minus1(1205821119896 (inf 119868

119860)) 119896minus1(1205821119896 (sup 119868

119860))]

[119896minus1(1205821119896 (inf 119865

119860)) 119896minus1(1205821119896 (sup119865

119860))]⟩

oplus ⟨[119897minus1(1205822119897 (inf 119879

119860)) 119897minus1(1205822119897 (sup119879

119860))]

[119896minus1(1205822119896 (inf 119868

119860)) 119896minus1(1205822119896 (sup 119868

119860))]

[119896minus1(1205822119896 (inf 119865

119860)) 119896minus1(1205822119896 (sup119865

119860))]⟩

= ⟨[119897minus1(119897 (119897minus1(1205821119897 (inf 119879

119860))) + 119897 (119897

minus1(1205822119897 (inf 119879

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup119879

119860))) + 119897 (119897

minus1(1205822119897 (sup119879

119860))))]

[119896minus1(119896 (119896minus1(1205821119896 (inf 119868

119860)))

+119896 (119896minus1(1205822119896 (inf 119868

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup 119868

119860)))

+119896 (119896minus1(1205822119896 (sup 119868

119860))))]


[119896minus1(119896 (119896minus1(1205821119896 (inf 119865

119860)))

+119896 (119896minus1(1205822119896 (inf 119865

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119865

119860)))

+119896 (119896minus1(1205822119896 (sup119865

119860))))]⟩

= ⟨[119897minus1(1205821119897 (inf 119879

119860) + 1205822119897 (inf 119879

119860))

119897minus1(1205821119897 (sup119879

119860) + 1205822119897 (sup119879

119860))]

[119896minus1(1205821119896 (inf 119868

119860) + 1205822119896 (inf 119868

119860))

119896minus1(1205821119896 (sup 119868

119860) + 1205822119896 (sup 119868

119860))]

[119896minus1(1205821119896 (inf 119865

119860) + 1205822119896 (inf 119865

119860))

119896minus1(1205821119896 (sup119865

119860) + 1205822119896 (sup119865

119860))]⟩

= ⟨[119897minus1((1205821+ 1205822) 119897 (inf 119879

119860))

119897minus1((1205821+ 1205822) 119897 (sup119879

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119868

119860))

119896minus1((1205821+ 1205822) 119896 (sup 119868

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119865

119860))

119896minus1((1205821+ 1205822) 119896 (sup119865

119860))]⟩

= (1205821+ 1205822) 119860

(18)

(6)

1198601205821 sdot 1198601205822

=⟨[119896minus1(1205821119896 (inf 119879

119860)) 119896minus1(1205821119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860)) 119897minus1(1205821119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860)) 119897minus1(1205821119897 (sup119865

119860))]⟩

sdot ⟨[119896minus1(1205822119896 (inf 119879

119860)) 119896minus1(1205822119896 (sup119879

119860))]

[119897minus1(1205822119897 (inf 119868

119860)) 119897minus1(1205822119897 (sup 119868

119860))]

[119897minus1(1205822119897 (inf 119865

119860)) 119897minus1(1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1(119896 (119896minus1(1205821119896 (inf 119879

119860)))

+119896 (119896minus1(1205822119896 (inf 119879

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119879

119860)))

+119896 (119896minus1(1205822119896 (sup119879

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119868

119860)))

+119897 (119897minus1(1205822119897 (inf 119868

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup 119868

119860)))

+119897 (119897minus1(1205822119897 (sup 119868

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119865

119860)))

+ 119897 (119897minus1(1205822119897 (inf 119865

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup119865

119860)))

+119897 (119897minus1(1205822119897 (sup119865

119860))))]⟩

= ⟨[119896minus1(1205821119896 (inf 119879

119860) + 1205822119896 (inf 119879

119860))

119896minus1(1205821119896 (sup119879

119860) + 1205822119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860) + 1205822119897 (inf 119868

119860))

119897minus1(1205821119897 (sup 119868

119860) + 1205822119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860) + 1205822119897 (inf 119865

119860))

119897minus1(1205821119897 (sup119865

119860) + 1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1((1205821+ 1205822) 119896 (inf 119879

119860))

119896minus1((1205821+ 1205822) 119896 (sup119879

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119868

119860))

119897minus1((1205821+ 1205822) 119897 (sup 119868

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119865

119860))

119897minus1((1205821+ 1205822) 119897 (sup119865

119860))]⟩

= 1198601205821+1205822

(19)


(1) 2 sdot 119860 = ⟨[091 096] [0 001] [001 004]⟩(2) 1198602 = ⟨[049 064] [0 019] [019 036]⟩(3) 119860 + 119861 = ⟨[082 090] [0 005] [003 008]⟩(4) 119860 sdot 119861 = ⟨[028 040] [020 037] [037 052]⟩


119860(119909) inf 119868

119860(119909) = sup 119868

119860(119909) inf 119865

119860(119909) =

sup119865119860(119909) inf 119879

119861(119909) = sup119879

119861(119909) inf 119868

119861(119909) = sup 119868

119861(119909)

and inf 119865119861(119909) = sup119865


⟨119879119860(119909) 119868119860(119909) 119865119860(119909)⟩ and 119861 = ⟨119879

119861(119909) 119868119861(119909) 119865119861(119909)⟩ are




119860= 0

inf 119868119861= sup 119868

119861= 0 sup119879

119860+sup119865

119860le 1 and sup119879

119861+sup119865

119861le


119861 119868119861 119865119861⟩


(1) 119860 + 119861 = ⟨[inf 119879119860+ inf 119879

119861minus inf 119879

119860sdot inf 119879

119861 sup119879

119860+

sup119879119861minus sup119879

119860sdot sup119879

119861][inf 119865

119860sdot inf 119865

119861 sup119865

119860sdot

sup119865119861]⟩

(2) 119860 sdot 119861 = ⟨[inf 119879119860sdot inf 119879

119861 sup119879

119860sdot sup119879

119861] [inf 119865

119860+

inf 119865119861minus inf 119865

119860sdot inf 119865

119861 sup119865

119860+ sup119865

119861minus sup119865

119860sdot

sup119865119861]⟩


119860)120582]

[(inf 119865119860)120582 (sup119865

119860)120582]⟩

(4) 119860120582 = ⟨[(inf 119879119860)120582 (sup119879

119860)120582] [1minus(1minusinf 119865

119860)120582 1minus(1minus

sup119865119860)120582]⟩




⟨[inf 119879119860 sup119879

119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ and

then

(1) 119904(119860) = [inf 119879119860+ 1 minus sup 119868

119860+ 1 minus sup119865

119860 sup119879

119860+ 1 minus

inf 119868119860+ 1 minus inf 119865

119860]

(2) 119886(119860) = [mininf 119879119860

minus inf 119865119860 sup119879

119860minus

sup119865119860maxinf 119879

119860minus inf 119865

119860 sup119879

119860minus sup119865

119860]

(3) 119888(119860) = [inf 119879119860 sup119879

119860]



119860(119909) sup119879

119860(119909) inf 119868

119860(119909) sup 119868

119860(119909) inf 119865

119860(119909)




(1) 119904(119860) = [24 27] 119904(119861) = [17 20]

(2) 119886(119860) = [06 06] 119886(119861) = [01 01]

(3) 119888(119860) = [07 08] 119888(119861) = [04 05]









(2) Assuming that 119860 = ⟨[06 07] [03 04] [04 05]⟩


(3) For two INNs 119860 = ⟨[06 07] [03 04] [04 05]⟩








INNWA119908(1198601 1198602 119860

119899) = 11990811198601+ 11990821198602+ sdot sdot sdot + 119908

119899119860119899

=

119899

sum

119895=1

119908119895119860119895

(20)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0(119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 23 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a



vector of 119860119895(119895 = 1 2 119899) with 119908

119895ge 0 (119895 = 1 2 119899)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879


+119908119899119897 (inf 119879

119860119899))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879


+119908119899119897 (sup119879

119860119899))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868


+119908119899119896 (inf 119868

119860119899))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868


+119908119899119896 (sup 119868

119860119899))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865


+119908119899119896 (inf 119865

119860119899))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865


+119908119899119896 (sup119865

119860119899))]⟩

(21)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[1 minus

119899

prod

119894=1

(1 minus inf 119879119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119879119860119894)119908119894]

[

119899

prod

119894=1

inf 119868119908119894119860119894

119899

prod

119894=1

sup 119868119908119894119860119894]

[

119899

prod

119894=1

inf 119865119908119894119860119894

119899

prod

119894=1

sup119865119908119894119860119894]⟩

(22)


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1


(1) For 119899 = 2 since

11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601)) 119897minus1(1199081119897 (sup119879

1198601))]

[119896minus1(1199081119896 (inf 119868

1198601)) 119896minus1(1199081119896 (sup 119868

1198601))]

[119896minus1(1199081119896 (inf 119865

1198601)) 119896minus1(1199081119896 (sup119865

1198601))]⟩

oplus ⟨[119897minus1(1199082119897 (inf 119879

1198602)) 119897minus1(1199082119897 (sup119879

1198602))]

[119896minus1(1199082119896 (inf 119868

1198602)) 119896minus1(1199082119896 (sup 119868

1198602))]

[119896minus1(1199082119896 (inf 119865

1198602)) 119896minus1(1199082119896 (sup119865

1198602))]⟩

= ⟨[119897minus1(119897 (119897minus1(1199081119897 (inf 119879

1198601)))

+119897 (119897minus1(1199082119897 (inf 119879

1198602))))

119897minus1(119897 (119897minus1(1199081119897 (sup119879

1198601)))

+119897 (119897minus1(1199082119897 (sup119879

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119868

1198601)))

+119896 (119896minus1(1199082119896 (inf 119868

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup 119868

1198601)))

+119896 (119896minus1(1199082119896 (sup 119868

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119865

1198601)))

+119896 (119896minus1(1199082119896 (inf 119865

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup119865

1198601)))

+119896 (119896minus1(1199082119896 (sup119865

1198602))))]⟩

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))


119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(23)

then

SNNWA119908(1198601 1198602)

= 11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(24)


SNNWA119908(1198601 1198602 119860

119896)

= ⟨[

[

119897minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(25)

then if 119899 = 119896 + 1 we have

SNNWA119908(1198601 1198602 119860

119896 119860119896+1)

= ⟨[

[

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (inf 119879

119860119896+1)))))

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (sup119879

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (inf 119868

119860119896+1)))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (sup 119868

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (inf 119865

119860119896+1))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (sup119865

119860119896+1))))]

]

⟩

= ⟨[

[

119897minus1(

119896+1

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896+1

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119865

119860119895))


119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(26)


SNNWA119908(1198601 1198602 119860

119899)

= ⟨[

[

119897minus1(

119899

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119899

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(27)






INNWA119908(1198601 1198602 119860

119899) = 119860



max119895119868119860119895(119909)max

119895119865119860119895(119909)⟩ 119860+ = ⟨max

119895119879119860119895(119909)

min119895119868119860119895(119909)min

119895119865119860119895(119909)⟩ for all 119895 isin 1 2 119899 and


119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWA119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)



INNWG119908(1198601 1198602 119860

119899) =

119899

prod

119895=1

119860119908119894

119895 (28)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 25 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a


119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[119896minus1(1199081119896 (inf 119879

1198601) + 1199082119896 (inf 119879


+119908119899119896 (inf 119879

119860119899))

119896minus1(1199081119896 (sup119879

1198601) + 119908

2119896 (sup119879


+119908119899119896 (sup119879

119860119899))]

[119897minus1(1199081119897 (inf 119868

1198601) + 1199082119897 (inf 119868


+119908119899119897 (inf 119868

119860119899))

119897minus1(1199081119897 (sup 119868

1198601) + 1199082119897 (sup 119868


+119908119899119897 (sup 119868

119860119899))]

[119897minus1(1199081119897 (inf 119865

1198601) + 1199082119897 (inf 119865


+119908119899119897 (inf 119865

119860119899))

119897minus1(1199081119897 (sup119865

1198601) + 1199082119897 (sup119865


+119908119899119897 (sup119865

119860119899))]⟩

(29)



119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[

119899

prod

119894=1

inf 119879119908119894119860119894

119899

prod

119894=1

sup119879119908119894119860119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119868119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup 119868119860119894)119908119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119865119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119865119860119894)119908119894]⟩

(30)

where 119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1




INNWG119908(1198601 1198602 119860

119899)

= ⟨[

[

2prod119899

119894=1inf 119879119908119894119860119894

prod119899

119894=1(2 minus inf 119879

119860119894)119908119894+prod119899

119894=1inf 119879119908119894119860119894

2prod119899

119894=1sup119879119908119894119860119894

prod119899

119894=1(2 minus sup119879

119860119894)119908119894+prod119899

119894=1sup119879119908119894119860119894

]

]

[

[

prod119899

119894=1(1 + inf 119868

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + inf 119868

119860119894)119908119894+prod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894minusprod119899

119894=1(1 minus sup 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894+prod119899

119894=1(1 minus sup 119868

119860119894)119908119894

]

]

[

[

prod119899

119894=1(1 + inf 119865

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + inf 119865

119860119894)119908119894+prod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894minusprod119899

119894=1(1 minus sup119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894+prod119899

119894=1(1 minus sup119865

119860119894)119908119894

]

]

⟩

(31)

where119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1






INNWG119908(1198601 1198602 119860

119899) = 119860



= ⟨min119895119879119860119895(119909)max

119895119868119860119895(119909)max

119895119865119860119895(119909)⟩

119860+

= ⟨max119895119879119860119895(119909)min

119895119868119860119895(119909)min

119895119865119860119895(119909)⟩


sube

INNWG119908(1198601 1198602 119860

119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWG119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)


1 1198862 119886119898 and n criteria 119862 = 119888

1 1198882 119888119899 whose


119899) where

119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Let 119877 =

(119886119894119895)119898times119899


where 119879119886119894119895



119895 119868119886119894119895


119895 and 119865







119894(119894 = 1 2 119898)

that is

119910119894= INNWA

119908(1198861198941 1198861198942 119886

119894119899) (32)

or

119910119894= INNWG

119908(1198861198941 1198861198942 119886

119894119899) (33)



119894) of


matrix F

119865 =[[[

[

119904 (1199101) 119886 (119910

1) 119888 (119910

1)

119904 (1199102) 119886 (119910

2) 119888 (119910

2)


119904 (119910119898) 119886 (119910

119898) 119888 (119910

119898)

]]]

]

(34)



119875119904=

[[[[

[

11990111990411

11990111990412

sdot sdot sdot 1199011199041119898

11990111990421

11990111990422

sdot sdot sdot 1199011199042119898

1199011199041198981

1199011199041198982

sdot sdot sdot 119901119904119898119898

]]]]

]

(35)


119894) gt 119904(119910

119895)


119894119895+ 119901119904119895119894= 1 and 119901119904

119894119894= 05 If


119886(119910119894) gt 119886(119910

119895) denoted by 119901119886

119894119895 And if 119901119886

119894119895= 05 (119894 = 119895) then


119895) denoted by

119901119888119894119895









3





2




3is a


119863 =

[[[

[

⟨[04 05] [02 03] [03 04]⟩ ⟨[04 06] [01 03] [02 04]⟩ ⟨[07 09] [02 03] [04 05]⟩

⟨[06 07] [01 02] [02 03]⟩ ⟨[06 07] [01 02] [02 03]⟩ ⟨[03 06] [03 05] [08 09]⟩

⟨[03 06] [02 03] [03 04]⟩ ⟨[05 06] [02 03] [03 04]⟩ ⟨[04 05] [02 04] [07 09]⟩

⟨[07 08] [00 01] [01 02]⟩ ⟨[06 07] [01 02] [01 03]⟩ ⟨[06 07] [03 04] [08 09]⟩

]]]

]

(36)




119860119882119860

=[[[

[

⟨[05453 07516] [01682 03000] [03041 04373]⟩

⟨[04996 06634] [01552 02885] [03482 04656]⟩

⟨[03950 05627] [02000 03366] [04210 05533]⟩

⟨[06383 07397] [00000 02071] [02297 04040]⟩

]]]

]

(37)


119860119882119866

=[[[

[

⟨[05004 06620] [01761 03000] [03195 04422]⟩

⟨[04547 06581] [01861 03371] [05405 06786]⟩

⟨[03824 05578] [02000 03419] [05012 07070]⟩

⟨[06333 07335] [01555 02570] [05069 06632]⟩

]]]

]

(38)



119865119882119860

=[[[

[

[18080 22793] [02412 03143] [05453 07516]

[17455 21600] [01514 01978] [04996 06634]

[15051 19417] [minus00260 00094] [03950 05627]

[20272 25100] [03357 04086] [06383 07397]

]]]

]

(39)


119865119882119866

=[[[

[

[17582 21664] [01809 02198] [05004 06620]

[14390 19315] [minus00858 minus00205] [04547 06581]

[13335 18566] [minus01492 minus01188] [03824 05578]

[17131 20711] [00703 01264] [06333 07335]

]]]

]

(40)


119882119860

119875119904 119882119860

=[[[

[

05 06498 08527 02642

03974 05 07695 01480

01473 02305 05 0

07358 08520 1 05

]]]

]

(41)

And for 119865119882119866

119901119904 119882119866

=[[[

[

05 08076 08943 05916

01924 05 05888 02568

01057 04112 05 01629

04084 07432 08371 05

]]]

]

(42)


compute 119901119886119894119895and 119901119888

119894119895



1≻ 1198602 1198601≻ 1198603 1198602≻ 1198603 1198604≻ 1198601 and 119860

4≻


4 1198601

1198602 and 119860

3 Obviously 119860





3 Obviously 119860

1is



3 as well as the

ranking 1198601 1198604 1198602 and 119860

3for AWG






2 1198604 1198603 and 119860

1


1 1198604 1198602 and 119860

3 Obviously


1


119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)




3 As can be


11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)


11988611= 11987911988631


13= 11988633






3in [33]


lt 11988641 11988622

lt 11988642

and 11988623




1in [33] while the

sequences of 1198601 1198602and 119860




6 Conclusion




Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[119896minus1(119896 (119896minus1(1205821119896 (inf 119865

119860)))

+119896 (119896minus1(1205822119896 (inf 119865

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119865

119860)))

+119896 (119896minus1(1205822119896 (sup119865

119860))))]⟩

= ⟨[119897minus1(1205821119897 (inf 119879

119860) + 1205822119897 (inf 119879

119860))

119897minus1(1205821119897 (sup119879

119860) + 1205822119897 (sup119879

119860))]

[119896minus1(1205821119896 (inf 119868

119860) + 1205822119896 (inf 119868

119860))

119896minus1(1205821119896 (sup 119868

119860) + 1205822119896 (sup 119868

119860))]

[119896minus1(1205821119896 (inf 119865

119860) + 1205822119896 (inf 119865

119860))

119896minus1(1205821119896 (sup119865

119860) + 1205822119896 (sup119865

119860))]⟩

= ⟨[119897minus1((1205821+ 1205822) 119897 (inf 119879

119860))

119897minus1((1205821+ 1205822) 119897 (sup119879

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119868

119860))

119896minus1((1205821+ 1205822) 119896 (sup 119868

119860))]

[119896minus1((1205821+ 1205822) 119896 (inf 119865

119860))

119896minus1((1205821+ 1205822) 119896 (sup119865

119860))]⟩

= (1205821+ 1205822) 119860

(18)

(6)

1198601205821 sdot 1198601205822

=⟨[119896minus1(1205821119896 (inf 119879

119860)) 119896minus1(1205821119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860)) 119897minus1(1205821119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860)) 119897minus1(1205821119897 (sup119865

119860))]⟩

sdot ⟨[119896minus1(1205822119896 (inf 119879

119860)) 119896minus1(1205822119896 (sup119879

119860))]

[119897minus1(1205822119897 (inf 119868

119860)) 119897minus1(1205822119897 (sup 119868

119860))]

[119897minus1(1205822119897 (inf 119865

119860)) 119897minus1(1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1(119896 (119896minus1(1205821119896 (inf 119879

119860)))

+119896 (119896minus1(1205822119896 (inf 119879

119860))))

119896minus1(119896 (119896minus1(1205821119896 (sup119879

119860)))

+119896 (119896minus1(1205822119896 (sup119879

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119868

119860)))

+119897 (119897minus1(1205822119897 (inf 119868

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup 119868

119860)))

+119897 (119897minus1(1205822119897 (sup 119868

119860))))]

[119897minus1(119897 (119897minus1(1205821119897 (inf 119865

119860)))

+ 119897 (119897minus1(1205822119897 (inf 119865

119860))))

119897minus1(119897 (119897minus1(1205821119897 (sup119865

119860)))

+119897 (119897minus1(1205822119897 (sup119865

119860))))]⟩

= ⟨[119896minus1(1205821119896 (inf 119879

119860) + 1205822119896 (inf 119879

119860))

119896minus1(1205821119896 (sup119879

119860) + 1205822119896 (sup119879

119860))]

[119897minus1(1205821119897 (inf 119868

119860) + 1205822119897 (inf 119868

119860))

119897minus1(1205821119897 (sup 119868

119860) + 1205822119897 (sup 119868

119860))]

[119897minus1(1205821119897 (inf 119865

119860) + 1205822119897 (inf 119865

119860))

119897minus1(1205821119897 (sup119865

119860) + 1205822119897 (sup119865

119860))]⟩

= ⟨[119896minus1((1205821+ 1205822) 119896 (inf 119879

119860))

119896minus1((1205821+ 1205822) 119896 (sup119879

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119868

119860))

119897minus1((1205821+ 1205822) 119897 (sup 119868

119860))]

[119897minus1((1205821+ 1205822) 119897 (inf 119865

119860))

119897minus1((1205821+ 1205822) 119897 (sup119865

119860))]⟩

= 1198601205821+1205822

(19)


(1) 2 sdot 119860 = ⟨[091 096] [0 001] [001 004]⟩(2) 1198602 = ⟨[049 064] [0 019] [019 036]⟩(3) 119860 + 119861 = ⟨[082 090] [0 005] [003 008]⟩(4) 119860 sdot 119861 = ⟨[028 040] [020 037] [037 052]⟩


119860(119909) inf 119868

119860(119909) = sup 119868

119860(119909) inf 119865

119860(119909) =

sup119865119860(119909) inf 119879

119861(119909) = sup119879

119861(119909) inf 119868

119861(119909) = sup 119868

119861(119909)

and inf 119865119861(119909) = sup119865


⟨119879119860(119909) 119868119860(119909) 119865119860(119909)⟩ and 119861 = ⟨119879

119861(119909) 119868119861(119909) 119865119861(119909)⟩ are




119860= 0

inf 119868119861= sup 119868

119861= 0 sup119879

119860+sup119865

119860le 1 and sup119879

119861+sup119865

119861le


119861 119868119861 119865119861⟩


(1) 119860 + 119861 = ⟨[inf 119879119860+ inf 119879

119861minus inf 119879

119860sdot inf 119879

119861 sup119879

119860+

sup119879119861minus sup119879

119860sdot sup119879

119861][inf 119865

119860sdot inf 119865

119861 sup119865

119860sdot

sup119865119861]⟩

(2) 119860 sdot 119861 = ⟨[inf 119879119860sdot inf 119879

119861 sup119879

119860sdot sup119879

119861] [inf 119865

119860+

inf 119865119861minus inf 119865

119860sdot inf 119865

119861 sup119865

119860+ sup119865

119861minus sup119865

119860sdot

sup119865119861]⟩


119860)120582]

[(inf 119865119860)120582 (sup119865

119860)120582]⟩

(4) 119860120582 = ⟨[(inf 119879119860)120582 (sup119879

119860)120582] [1minus(1minusinf 119865

119860)120582 1minus(1minus

sup119865119860)120582]⟩




⟨[inf 119879119860 sup119879

119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ and

then

(1) 119904(119860) = [inf 119879119860+ 1 minus sup 119868

119860+ 1 minus sup119865

119860 sup119879

119860+ 1 minus

inf 119868119860+ 1 minus inf 119865

119860]

(2) 119886(119860) = [mininf 119879119860

minus inf 119865119860 sup119879

119860minus

sup119865119860maxinf 119879

119860minus inf 119865

119860 sup119879

119860minus sup119865

119860]

(3) 119888(119860) = [inf 119879119860 sup119879

119860]



119860(119909) sup119879

119860(119909) inf 119868

119860(119909) sup 119868

119860(119909) inf 119865

119860(119909)




(1) 119904(119860) = [24 27] 119904(119861) = [17 20]

(2) 119886(119860) = [06 06] 119886(119861) = [01 01]

(3) 119888(119860) = [07 08] 119888(119861) = [04 05]









(2) Assuming that 119860 = ⟨[06 07] [03 04] [04 05]⟩


(3) For two INNs 119860 = ⟨[06 07] [03 04] [04 05]⟩








INNWA119908(1198601 1198602 119860

119899) = 11990811198601+ 11990821198602+ sdot sdot sdot + 119908

119899119860119899

=

119899

sum

119895=1

119908119895119860119895

(20)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0(119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 23 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a



vector of 119860119895(119895 = 1 2 119899) with 119908

119895ge 0 (119895 = 1 2 119899)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879


+119908119899119897 (inf 119879

119860119899))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879


+119908119899119897 (sup119879

119860119899))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868


+119908119899119896 (inf 119868

119860119899))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868


+119908119899119896 (sup 119868

119860119899))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865


+119908119899119896 (inf 119865

119860119899))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865


+119908119899119896 (sup119865

119860119899))]⟩

(21)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[1 minus

119899

prod

119894=1

(1 minus inf 119879119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119879119860119894)119908119894]

[

119899

prod

119894=1

inf 119868119908119894119860119894

119899

prod

119894=1

sup 119868119908119894119860119894]

[

119899

prod

119894=1

inf 119865119908119894119860119894

119899

prod

119894=1

sup119865119908119894119860119894]⟩

(22)


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1


(1) For 119899 = 2 since

11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601)) 119897minus1(1199081119897 (sup119879

1198601))]

[119896minus1(1199081119896 (inf 119868

1198601)) 119896minus1(1199081119896 (sup 119868

1198601))]

[119896minus1(1199081119896 (inf 119865

1198601)) 119896minus1(1199081119896 (sup119865

1198601))]⟩

oplus ⟨[119897minus1(1199082119897 (inf 119879

1198602)) 119897minus1(1199082119897 (sup119879

1198602))]

[119896minus1(1199082119896 (inf 119868

1198602)) 119896minus1(1199082119896 (sup 119868

1198602))]

[119896minus1(1199082119896 (inf 119865

1198602)) 119896minus1(1199082119896 (sup119865

1198602))]⟩

= ⟨[119897minus1(119897 (119897minus1(1199081119897 (inf 119879

1198601)))

+119897 (119897minus1(1199082119897 (inf 119879

1198602))))

119897minus1(119897 (119897minus1(1199081119897 (sup119879

1198601)))

+119897 (119897minus1(1199082119897 (sup119879

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119868

1198601)))

+119896 (119896minus1(1199082119896 (inf 119868

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup 119868

1198601)))

+119896 (119896minus1(1199082119896 (sup 119868

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119865

1198601)))

+119896 (119896minus1(1199082119896 (inf 119865

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup119865

1198601)))

+119896 (119896minus1(1199082119896 (sup119865

1198602))))]⟩

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))


119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(23)

then

SNNWA119908(1198601 1198602)

= 11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(24)


SNNWA119908(1198601 1198602 119860

119896)

= ⟨[

[

119897minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(25)

then if 119899 = 119896 + 1 we have

SNNWA119908(1198601 1198602 119860

119896 119860119896+1)

= ⟨[

[

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (inf 119879

119860119896+1)))))

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (sup119879

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (inf 119868

119860119896+1)))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (sup 119868

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (inf 119865

119860119896+1))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (sup119865

119860119896+1))))]

]

⟩

= ⟨[

[

119897minus1(

119896+1

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896+1

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119865

119860119895))


119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(26)


SNNWA119908(1198601 1198602 119860

119899)

= ⟨[

[

119897minus1(

119899

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119899

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(27)






INNWA119908(1198601 1198602 119860

119899) = 119860



max119895119868119860119895(119909)max

119895119865119860119895(119909)⟩ 119860+ = ⟨max

119895119879119860119895(119909)

min119895119868119860119895(119909)min

119895119865119860119895(119909)⟩ for all 119895 isin 1 2 119899 and


119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWA119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)



INNWG119908(1198601 1198602 119860

119899) =

119899

prod

119895=1

119860119908119894

119895 (28)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 25 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a


119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[119896minus1(1199081119896 (inf 119879

1198601) + 1199082119896 (inf 119879


+119908119899119896 (inf 119879

119860119899))

119896minus1(1199081119896 (sup119879

1198601) + 119908

2119896 (sup119879


+119908119899119896 (sup119879

119860119899))]

[119897minus1(1199081119897 (inf 119868

1198601) + 1199082119897 (inf 119868


+119908119899119897 (inf 119868

119860119899))

119897minus1(1199081119897 (sup 119868

1198601) + 1199082119897 (sup 119868


+119908119899119897 (sup 119868

119860119899))]

[119897minus1(1199081119897 (inf 119865

1198601) + 1199082119897 (inf 119865


+119908119899119897 (inf 119865

119860119899))

119897minus1(1199081119897 (sup119865

1198601) + 1199082119897 (sup119865


+119908119899119897 (sup119865

119860119899))]⟩

(29)



119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[

119899

prod

119894=1

inf 119879119908119894119860119894

119899

prod

119894=1

sup119879119908119894119860119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119868119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup 119868119860119894)119908119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119865119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119865119860119894)119908119894]⟩

(30)

where 119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1




INNWG119908(1198601 1198602 119860

119899)

= ⟨[

[

2prod119899

119894=1inf 119879119908119894119860119894

prod119899

119894=1(2 minus inf 119879

119860119894)119908119894+prod119899

119894=1inf 119879119908119894119860119894

2prod119899

119894=1sup119879119908119894119860119894

prod119899

119894=1(2 minus sup119879

119860119894)119908119894+prod119899

119894=1sup119879119908119894119860119894

]

]

[

[

prod119899

119894=1(1 + inf 119868

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + inf 119868

119860119894)119908119894+prod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894minusprod119899

119894=1(1 minus sup 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894+prod119899

119894=1(1 minus sup 119868

119860119894)119908119894

]

]

[

[

prod119899

119894=1(1 + inf 119865

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + inf 119865

119860119894)119908119894+prod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894minusprod119899

119894=1(1 minus sup119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894+prod119899

119894=1(1 minus sup119865

119860119894)119908119894

]

]

⟩

(31)

where119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1






INNWG119908(1198601 1198602 119860

119899) = 119860



= ⟨min119895119879119860119895(119909)max

119895119868119860119895(119909)max

119895119865119860119895(119909)⟩

119860+

= ⟨max119895119879119860119895(119909)min

119895119868119860119895(119909)min

119895119865119860119895(119909)⟩


sube

INNWG119908(1198601 1198602 119860

119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWG119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)


1 1198862 119886119898 and n criteria 119862 = 119888

1 1198882 119888119899 whose


119899) where

119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Let 119877 =

(119886119894119895)119898times119899


where 119879119886119894119895



119895 119868119886119894119895


119895 and 119865







119894(119894 = 1 2 119898)

that is

119910119894= INNWA

119908(1198861198941 1198861198942 119886

119894119899) (32)

or

119910119894= INNWG

119908(1198861198941 1198861198942 119886

119894119899) (33)



119894) of


matrix F

119865 =[[[

[

119904 (1199101) 119886 (119910

1) 119888 (119910

1)

119904 (1199102) 119886 (119910

2) 119888 (119910

2)


119904 (119910119898) 119886 (119910

119898) 119888 (119910

119898)

]]]

]

(34)



119875119904=

[[[[

[

11990111990411

11990111990412

sdot sdot sdot 1199011199041119898

11990111990421

11990111990422

sdot sdot sdot 1199011199042119898

1199011199041198981

1199011199041198982

sdot sdot sdot 119901119904119898119898

]]]]

]

(35)


119894) gt 119904(119910

119895)


119894119895+ 119901119904119895119894= 1 and 119901119904

119894119894= 05 If


119886(119910119894) gt 119886(119910

119895) denoted by 119901119886

119894119895 And if 119901119886

119894119895= 05 (119894 = 119895) then


119895) denoted by

119901119888119894119895









3





2




3is a


119863 =

[[[

[

⟨[04 05] [02 03] [03 04]⟩ ⟨[04 06] [01 03] [02 04]⟩ ⟨[07 09] [02 03] [04 05]⟩

⟨[06 07] [01 02] [02 03]⟩ ⟨[06 07] [01 02] [02 03]⟩ ⟨[03 06] [03 05] [08 09]⟩

⟨[03 06] [02 03] [03 04]⟩ ⟨[05 06] [02 03] [03 04]⟩ ⟨[04 05] [02 04] [07 09]⟩

⟨[07 08] [00 01] [01 02]⟩ ⟨[06 07] [01 02] [01 03]⟩ ⟨[06 07] [03 04] [08 09]⟩

]]]

]

(36)




119860119882119860

=[[[

[

⟨[05453 07516] [01682 03000] [03041 04373]⟩

⟨[04996 06634] [01552 02885] [03482 04656]⟩

⟨[03950 05627] [02000 03366] [04210 05533]⟩

⟨[06383 07397] [00000 02071] [02297 04040]⟩

]]]

]

(37)


119860119882119866

=[[[

[

⟨[05004 06620] [01761 03000] [03195 04422]⟩

⟨[04547 06581] [01861 03371] [05405 06786]⟩

⟨[03824 05578] [02000 03419] [05012 07070]⟩

⟨[06333 07335] [01555 02570] [05069 06632]⟩

]]]

]

(38)



119865119882119860

=[[[

[

[18080 22793] [02412 03143] [05453 07516]

[17455 21600] [01514 01978] [04996 06634]

[15051 19417] [minus00260 00094] [03950 05627]

[20272 25100] [03357 04086] [06383 07397]

]]]

]

(39)


119865119882119866

=[[[

[

[17582 21664] [01809 02198] [05004 06620]

[14390 19315] [minus00858 minus00205] [04547 06581]

[13335 18566] [minus01492 minus01188] [03824 05578]

[17131 20711] [00703 01264] [06333 07335]

]]]

]

(40)


119882119860

119875119904 119882119860

=[[[

[

05 06498 08527 02642

03974 05 07695 01480

01473 02305 05 0

07358 08520 1 05

]]]

]

(41)

And for 119865119882119866

119901119904 119882119866

=[[[

[

05 08076 08943 05916

01924 05 05888 02568

01057 04112 05 01629

04084 07432 08371 05

]]]

]

(42)


compute 119901119886119894119895and 119901119888

119894119895



1≻ 1198602 1198601≻ 1198603 1198602≻ 1198603 1198604≻ 1198601 and 119860

4≻


4 1198601

1198602 and 119860

3 Obviously 119860





3 Obviously 119860

1is



3 as well as the

ranking 1198601 1198604 1198602 and 119860

3for AWG






2 1198604 1198603 and 119860

1


1 1198604 1198602 and 119860

3 Obviously


1


119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)




3 As can be


11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)


11988611= 11987911988631


13= 11988633






3in [33]


lt 11988641 11988622

lt 11988642

and 11988623




1in [33] while the

sequences of 1198601 1198602and 119860




6 Conclusion




Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119860= 0

inf 119868119861= sup 119868

119861= 0 sup119879

119860+sup119865

119860le 1 and sup119879

119861+sup119865

119861le


119861 119868119861 119865119861⟩


(1) 119860 + 119861 = ⟨[inf 119879119860+ inf 119879

119861minus inf 119879

119860sdot inf 119879

119861 sup119879

119860+

sup119879119861minus sup119879

119860sdot sup119879

119861][inf 119865

119860sdot inf 119865

119861 sup119865

119860sdot

sup119865119861]⟩

(2) 119860 sdot 119861 = ⟨[inf 119879119860sdot inf 119879

119861 sup119879

119860sdot sup119879

119861] [inf 119865

119860+

inf 119865119861minus inf 119865

119860sdot inf 119865

119861 sup119865

119860+ sup119865

119861minus sup119865

119860sdot

sup119865119861]⟩


119860)120582]

[(inf 119865119860)120582 (sup119865

119860)120582]⟩

(4) 119860120582 = ⟨[(inf 119879119860)120582 (sup119879

119860)120582] [1minus(1minusinf 119865

119860)120582 1minus(1minus

sup119865119860)120582]⟩




⟨[inf 119879119860 sup119879

119860] [inf 119868

119860 sup 119868

119860] [inf 119865

119860 sup119865

119860]⟩ and

then

(1) 119904(119860) = [inf 119879119860+ 1 minus sup 119868

119860+ 1 minus sup119865

119860 sup119879

119860+ 1 minus

inf 119868119860+ 1 minus inf 119865

119860]

(2) 119886(119860) = [mininf 119879119860

minus inf 119865119860 sup119879

119860minus

sup119865119860maxinf 119879

119860minus inf 119865

119860 sup119879

119860minus sup119865

119860]

(3) 119888(119860) = [inf 119879119860 sup119879

119860]



119860(119909) sup119879

119860(119909) inf 119868

119860(119909) sup 119868

119860(119909) inf 119865

119860(119909)




(1) 119904(119860) = [24 27] 119904(119861) = [17 20]

(2) 119886(119860) = [06 06] 119886(119861) = [01 01]

(3) 119888(119860) = [07 08] 119888(119861) = [04 05]









(2) Assuming that 119860 = ⟨[06 07] [03 04] [04 05]⟩


(3) For two INNs 119860 = ⟨[06 07] [03 04] [04 05]⟩








INNWA119908(1198601 1198602 119860

119899) = 11990811198601+ 11990821198602+ sdot sdot sdot + 119908

119899119860119899

=

119899

sum

119895=1

119908119895119860119895

(20)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0(119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 23 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a



vector of 119860119895(119895 = 1 2 119899) with 119908

119895ge 0 (119895 = 1 2 119899)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879


+119908119899119897 (inf 119879

119860119899))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879


+119908119899119897 (sup119879

119860119899))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868


+119908119899119896 (inf 119868

119860119899))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868


+119908119899119896 (sup 119868

119860119899))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865


+119908119899119896 (inf 119865

119860119899))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865


+119908119899119896 (sup119865

119860119899))]⟩

(21)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[1 minus

119899

prod

119894=1

(1 minus inf 119879119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119879119860119894)119908119894]

[

119899

prod

119894=1

inf 119868119908119894119860119894

119899

prod

119894=1

sup 119868119908119894119860119894]

[

119899

prod

119894=1

inf 119865119908119894119860119894

119899

prod

119894=1

sup119865119908119894119860119894]⟩

(22)


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1


(1) For 119899 = 2 since

11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601)) 119897minus1(1199081119897 (sup119879

1198601))]

[119896minus1(1199081119896 (inf 119868

1198601)) 119896minus1(1199081119896 (sup 119868

1198601))]

[119896minus1(1199081119896 (inf 119865

1198601)) 119896minus1(1199081119896 (sup119865

1198601))]⟩

oplus ⟨[119897minus1(1199082119897 (inf 119879

1198602)) 119897minus1(1199082119897 (sup119879

1198602))]

[119896minus1(1199082119896 (inf 119868

1198602)) 119896minus1(1199082119896 (sup 119868

1198602))]

[119896minus1(1199082119896 (inf 119865

1198602)) 119896minus1(1199082119896 (sup119865

1198602))]⟩

= ⟨[119897minus1(119897 (119897minus1(1199081119897 (inf 119879

1198601)))

+119897 (119897minus1(1199082119897 (inf 119879

1198602))))

119897minus1(119897 (119897minus1(1199081119897 (sup119879

1198601)))

+119897 (119897minus1(1199082119897 (sup119879

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119868

1198601)))

+119896 (119896minus1(1199082119896 (inf 119868

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup 119868

1198601)))

+119896 (119896minus1(1199082119896 (sup 119868

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119865

1198601)))

+119896 (119896minus1(1199082119896 (inf 119865

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup119865

1198601)))

+119896 (119896minus1(1199082119896 (sup119865

1198602))))]⟩

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))


119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(23)

then

SNNWA119908(1198601 1198602)

= 11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(24)


SNNWA119908(1198601 1198602 119860

119896)

= ⟨[

[

119897minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(25)

then if 119899 = 119896 + 1 we have

SNNWA119908(1198601 1198602 119860

119896 119860119896+1)

= ⟨[

[

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (inf 119879

119860119896+1)))))

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (sup119879

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (inf 119868

119860119896+1)))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (sup 119868

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (inf 119865

119860119896+1))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (sup119865

119860119896+1))))]

]

⟩

= ⟨[

[

119897minus1(

119896+1

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896+1

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119865

119860119895))


119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(26)


SNNWA119908(1198601 1198602 119860

119899)

= ⟨[

[

119897minus1(

119899

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119899

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(27)






INNWA119908(1198601 1198602 119860

119899) = 119860



max119895119868119860119895(119909)max

119895119865119860119895(119909)⟩ 119860+ = ⟨max

119895119879119860119895(119909)

min119895119868119860119895(119909)min

119895119865119860119895(119909)⟩ for all 119895 isin 1 2 119899 and


119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWA119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)



INNWG119908(1198601 1198602 119860

119899) =

119899

prod

119895=1

119860119908119894

119895 (28)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 25 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a


119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[119896minus1(1199081119896 (inf 119879

1198601) + 1199082119896 (inf 119879


+119908119899119896 (inf 119879

119860119899))

119896minus1(1199081119896 (sup119879

1198601) + 119908

2119896 (sup119879


+119908119899119896 (sup119879

119860119899))]

[119897minus1(1199081119897 (inf 119868

1198601) + 1199082119897 (inf 119868


+119908119899119897 (inf 119868

119860119899))

119897minus1(1199081119897 (sup 119868

1198601) + 1199082119897 (sup 119868


+119908119899119897 (sup 119868

119860119899))]

[119897minus1(1199081119897 (inf 119865

1198601) + 1199082119897 (inf 119865


+119908119899119897 (inf 119865

119860119899))

119897minus1(1199081119897 (sup119865

1198601) + 1199082119897 (sup119865


+119908119899119897 (sup119865

119860119899))]⟩

(29)



119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[

119899

prod

119894=1

inf 119879119908119894119860119894

119899

prod

119894=1

sup119879119908119894119860119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119868119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup 119868119860119894)119908119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119865119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119865119860119894)119908119894]⟩

(30)

where 119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1




INNWG119908(1198601 1198602 119860

119899)

= ⟨[

[

2prod119899

119894=1inf 119879119908119894119860119894

prod119899

119894=1(2 minus inf 119879

119860119894)119908119894+prod119899

119894=1inf 119879119908119894119860119894

2prod119899

119894=1sup119879119908119894119860119894

prod119899

119894=1(2 minus sup119879

119860119894)119908119894+prod119899

119894=1sup119879119908119894119860119894

]

]

[

[

prod119899

119894=1(1 + inf 119868

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + inf 119868

119860119894)119908119894+prod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894minusprod119899

119894=1(1 minus sup 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894+prod119899

119894=1(1 minus sup 119868

119860119894)119908119894

]

]

[

[

prod119899

119894=1(1 + inf 119865

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + inf 119865

119860119894)119908119894+prod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894minusprod119899

119894=1(1 minus sup119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894+prod119899

119894=1(1 minus sup119865

119860119894)119908119894

]

]

⟩

(31)

where119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1






INNWG119908(1198601 1198602 119860

119899) = 119860



= ⟨min119895119879119860119895(119909)max

119895119868119860119895(119909)max

119895119865119860119895(119909)⟩

119860+

= ⟨max119895119879119860119895(119909)min

119895119868119860119895(119909)min

119895119865119860119895(119909)⟩


sube

INNWG119908(1198601 1198602 119860

119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWG119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)


1 1198862 119886119898 and n criteria 119862 = 119888

1 1198882 119888119899 whose


119899) where

119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Let 119877 =

(119886119894119895)119898times119899


where 119879119886119894119895



119895 119868119886119894119895


119895 and 119865







119894(119894 = 1 2 119898)

that is

119910119894= INNWA

119908(1198861198941 1198861198942 119886

119894119899) (32)

or

119910119894= INNWG

119908(1198861198941 1198861198942 119886

119894119899) (33)



119894) of


matrix F

119865 =[[[

[

119904 (1199101) 119886 (119910

1) 119888 (119910

1)

119904 (1199102) 119886 (119910

2) 119888 (119910

2)


119904 (119910119898) 119886 (119910

119898) 119888 (119910

119898)

]]]

]

(34)



119875119904=

[[[[

[

11990111990411

11990111990412

sdot sdot sdot 1199011199041119898

11990111990421

11990111990422

sdot sdot sdot 1199011199042119898

1199011199041198981

1199011199041198982

sdot sdot sdot 119901119904119898119898

]]]]

]

(35)


119894) gt 119904(119910

119895)


119894119895+ 119901119904119895119894= 1 and 119901119904

119894119894= 05 If


119886(119910119894) gt 119886(119910

119895) denoted by 119901119886

119894119895 And if 119901119886

119894119895= 05 (119894 = 119895) then


119895) denoted by

119901119888119894119895









3





2




3is a


119863 =

[[[

[

⟨[04 05] [02 03] [03 04]⟩ ⟨[04 06] [01 03] [02 04]⟩ ⟨[07 09] [02 03] [04 05]⟩

⟨[06 07] [01 02] [02 03]⟩ ⟨[06 07] [01 02] [02 03]⟩ ⟨[03 06] [03 05] [08 09]⟩

⟨[03 06] [02 03] [03 04]⟩ ⟨[05 06] [02 03] [03 04]⟩ ⟨[04 05] [02 04] [07 09]⟩

⟨[07 08] [00 01] [01 02]⟩ ⟨[06 07] [01 02] [01 03]⟩ ⟨[06 07] [03 04] [08 09]⟩

]]]

]

(36)




119860119882119860

=[[[

[

⟨[05453 07516] [01682 03000] [03041 04373]⟩

⟨[04996 06634] [01552 02885] [03482 04656]⟩

⟨[03950 05627] [02000 03366] [04210 05533]⟩

⟨[06383 07397] [00000 02071] [02297 04040]⟩

]]]

]

(37)


119860119882119866

=[[[

[

⟨[05004 06620] [01761 03000] [03195 04422]⟩

⟨[04547 06581] [01861 03371] [05405 06786]⟩

⟨[03824 05578] [02000 03419] [05012 07070]⟩

⟨[06333 07335] [01555 02570] [05069 06632]⟩

]]]

]

(38)



119865119882119860

=[[[

[

[18080 22793] [02412 03143] [05453 07516]

[17455 21600] [01514 01978] [04996 06634]

[15051 19417] [minus00260 00094] [03950 05627]

[20272 25100] [03357 04086] [06383 07397]

]]]

]

(39)


119865119882119866

=[[[

[

[17582 21664] [01809 02198] [05004 06620]

[14390 19315] [minus00858 minus00205] [04547 06581]

[13335 18566] [minus01492 minus01188] [03824 05578]

[17131 20711] [00703 01264] [06333 07335]

]]]

]

(40)


119882119860

119875119904 119882119860

=[[[

[

05 06498 08527 02642

03974 05 07695 01480

01473 02305 05 0

07358 08520 1 05

]]]

]

(41)

And for 119865119882119866

119901119904 119882119866

=[[[

[

05 08076 08943 05916

01924 05 05888 02568

01057 04112 05 01629

04084 07432 08371 05

]]]

]

(42)


compute 119901119886119894119895and 119901119888

119894119895



1≻ 1198602 1198601≻ 1198603 1198602≻ 1198603 1198604≻ 1198601 and 119860

4≻


4 1198601

1198602 and 119860

3 Obviously 119860





3 Obviously 119860

1is



3 as well as the

ranking 1198601 1198604 1198602 and 119860

3for AWG






2 1198604 1198603 and 119860

1


1 1198604 1198602 and 119860

3 Obviously


1


119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)




3 As can be


11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)


11988611= 11987911988631


13= 11988633






3in [33]


lt 11988641 11988622

lt 11988642

and 11988623




1in [33] while the

sequences of 1198601 1198602and 119860




6 Conclusion




Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













INNWA119908(1198601 1198602 119860

119899) = 11990811198601+ 11990821198602+ sdot sdot sdot + 119908

119899119860119899

=

119899

sum

119895=1

119908119895119860119895

(20)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0(119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 23 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a



vector of 119860119895(119895 = 1 2 119899) with 119908

119895ge 0 (119895 = 1 2 119899)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879


+119908119899119897 (inf 119879

119860119899))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879


+119908119899119897 (sup119879

119860119899))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868


+119908119899119896 (inf 119868

119860119899))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868


+119908119899119896 (sup 119868

119860119899))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865


+119908119899119896 (inf 119865

119860119899))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865


+119908119899119896 (sup119865

119860119899))]⟩

(21)



INNWA119908(1198601 1198602 119860

119899)

= ⟨[1 minus

119899

prod

119894=1

(1 minus inf 119879119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119879119860119894)119908119894]

[

119899

prod

119894=1

inf 119868119908119894119860119894

119899

prod

119894=1

sup 119868119908119894119860119894]

[

119899

prod

119894=1

inf 119865119908119894119860119894

119899

prod

119894=1

sup119865119908119894119860119894]⟩

(22)


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1


(1) For 119899 = 2 since

11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601)) 119897minus1(1199081119897 (sup119879

1198601))]

[119896minus1(1199081119896 (inf 119868

1198601)) 119896minus1(1199081119896 (sup 119868

1198601))]

[119896minus1(1199081119896 (inf 119865

1198601)) 119896minus1(1199081119896 (sup119865

1198601))]⟩

oplus ⟨[119897minus1(1199082119897 (inf 119879

1198602)) 119897minus1(1199082119897 (sup119879

1198602))]

[119896minus1(1199082119896 (inf 119868

1198602)) 119896minus1(1199082119896 (sup 119868

1198602))]

[119896minus1(1199082119896 (inf 119865

1198602)) 119896minus1(1199082119896 (sup119865

1198602))]⟩

= ⟨[119897minus1(119897 (119897minus1(1199081119897 (inf 119879

1198601)))

+119897 (119897minus1(1199082119897 (inf 119879

1198602))))

119897minus1(119897 (119897minus1(1199081119897 (sup119879

1198601)))

+119897 (119897minus1(1199082119897 (sup119879

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119868

1198601)))

+119896 (119896minus1(1199082119896 (inf 119868

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup 119868

1198601)))

+119896 (119896minus1(1199082119896 (sup 119868

1198602))))]

[119896minus1(119896 (119896minus1(1199081119896 (inf 119865

1198601)))

+119896 (119896minus1(1199082119896 (inf 119865

1198602))))

119896minus1(119896 (119896minus1(1199081119896 (sup119865

1198601)))

+119896 (119896minus1(1199082119896 (sup119865

1198602))))]⟩

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))


119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(23)

then

SNNWA119908(1198601 1198602)

= 11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(24)


SNNWA119908(1198601 1198602 119860

119896)

= ⟨[

[

119897minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(25)

then if 119899 = 119896 + 1 we have

SNNWA119908(1198601 1198602 119860

119896 119860119896+1)

= ⟨[

[

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (inf 119879

119860119896+1)))))

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (sup119879

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (inf 119868

119860119896+1)))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (sup 119868

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (inf 119865

119860119896+1))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (sup119865

119860119896+1))))]

]

⟩

= ⟨[

[

119897minus1(

119896+1

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896+1

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119865

119860119895))


119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(26)


SNNWA119908(1198601 1198602 119860

119899)

= ⟨[

[

119897minus1(

119899

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119899

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(27)






INNWA119908(1198601 1198602 119860

119899) = 119860



max119895119868119860119895(119909)max

119895119865119860119895(119909)⟩ 119860+ = ⟨max

119895119879119860119895(119909)

min119895119868119860119895(119909)min

119895119865119860119895(119909)⟩ for all 119895 isin 1 2 119899 and


119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWA119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)



INNWG119908(1198601 1198602 119860

119899) =

119899

prod

119895=1

119860119908119894

119895 (28)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 25 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a


119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[119896minus1(1199081119896 (inf 119879

1198601) + 1199082119896 (inf 119879


+119908119899119896 (inf 119879

119860119899))

119896minus1(1199081119896 (sup119879

1198601) + 119908

2119896 (sup119879


+119908119899119896 (sup119879

119860119899))]

[119897minus1(1199081119897 (inf 119868

1198601) + 1199082119897 (inf 119868


+119908119899119897 (inf 119868

119860119899))

119897minus1(1199081119897 (sup 119868

1198601) + 1199082119897 (sup 119868


+119908119899119897 (sup 119868

119860119899))]

[119897minus1(1199081119897 (inf 119865

1198601) + 1199082119897 (inf 119865


+119908119899119897 (inf 119865

119860119899))

119897minus1(1199081119897 (sup119865

1198601) + 1199082119897 (sup119865


+119908119899119897 (sup119865

119860119899))]⟩

(29)



119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[

119899

prod

119894=1

inf 119879119908119894119860119894

119899

prod

119894=1

sup119879119908119894119860119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119868119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup 119868119860119894)119908119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119865119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119865119860119894)119908119894]⟩

(30)

where 119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1




INNWG119908(1198601 1198602 119860

119899)

= ⟨[

[

2prod119899

119894=1inf 119879119908119894119860119894

prod119899

119894=1(2 minus inf 119879

119860119894)119908119894+prod119899

119894=1inf 119879119908119894119860119894

2prod119899

119894=1sup119879119908119894119860119894

prod119899

119894=1(2 minus sup119879

119860119894)119908119894+prod119899

119894=1sup119879119908119894119860119894

]

]

[

[

prod119899

119894=1(1 + inf 119868

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + inf 119868

119860119894)119908119894+prod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894minusprod119899

119894=1(1 minus sup 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894+prod119899

119894=1(1 minus sup 119868

119860119894)119908119894

]

]

[

[

prod119899

119894=1(1 + inf 119865

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + inf 119865

119860119894)119908119894+prod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894minusprod119899

119894=1(1 minus sup119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894+prod119899

119894=1(1 minus sup119865

119860119894)119908119894

]

]

⟩

(31)

where119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1






INNWG119908(1198601 1198602 119860

119899) = 119860



= ⟨min119895119879119860119895(119909)max

119895119868119860119895(119909)max

119895119865119860119895(119909)⟩

119860+

= ⟨max119895119879119860119895(119909)min

119895119868119860119895(119909)min

119895119865119860119895(119909)⟩


sube

INNWG119908(1198601 1198602 119860

119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWG119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)


1 1198862 119886119898 and n criteria 119862 = 119888

1 1198882 119888119899 whose


119899) where

119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Let 119877 =

(119886119894119895)119898times119899


where 119879119886119894119895



119895 119868119886119894119895


119895 and 119865







119894(119894 = 1 2 119898)

that is

119910119894= INNWA

119908(1198861198941 1198861198942 119886

119894119899) (32)

or

119910119894= INNWG

119908(1198861198941 1198861198942 119886

119894119899) (33)



119894) of


matrix F

119865 =[[[

[

119904 (1199101) 119886 (119910

1) 119888 (119910

1)

119904 (1199102) 119886 (119910

2) 119888 (119910

2)


119904 (119910119898) 119886 (119910

119898) 119888 (119910

119898)

]]]

]

(34)



119875119904=

[[[[

[

11990111990411

11990111990412

sdot sdot sdot 1199011199041119898

11990111990421

11990111990422

sdot sdot sdot 1199011199042119898

1199011199041198981

1199011199041198982

sdot sdot sdot 119901119904119898119898

]]]]

]

(35)


119894) gt 119904(119910

119895)


119894119895+ 119901119904119895119894= 1 and 119901119904

119894119894= 05 If


119886(119910119894) gt 119886(119910

119895) denoted by 119901119886

119894119895 And if 119901119886

119894119895= 05 (119894 = 119895) then


119895) denoted by

119901119888119894119895









3





2




3is a


119863 =

[[[

[

⟨[04 05] [02 03] [03 04]⟩ ⟨[04 06] [01 03] [02 04]⟩ ⟨[07 09] [02 03] [04 05]⟩

⟨[06 07] [01 02] [02 03]⟩ ⟨[06 07] [01 02] [02 03]⟩ ⟨[03 06] [03 05] [08 09]⟩

⟨[03 06] [02 03] [03 04]⟩ ⟨[05 06] [02 03] [03 04]⟩ ⟨[04 05] [02 04] [07 09]⟩

⟨[07 08] [00 01] [01 02]⟩ ⟨[06 07] [01 02] [01 03]⟩ ⟨[06 07] [03 04] [08 09]⟩

]]]

]

(36)




119860119882119860

=[[[

[

⟨[05453 07516] [01682 03000] [03041 04373]⟩

⟨[04996 06634] [01552 02885] [03482 04656]⟩

⟨[03950 05627] [02000 03366] [04210 05533]⟩

⟨[06383 07397] [00000 02071] [02297 04040]⟩

]]]

]

(37)


119860119882119866

=[[[

[

⟨[05004 06620] [01761 03000] [03195 04422]⟩

⟨[04547 06581] [01861 03371] [05405 06786]⟩

⟨[03824 05578] [02000 03419] [05012 07070]⟩

⟨[06333 07335] [01555 02570] [05069 06632]⟩

]]]

]

(38)



119865119882119860

=[[[

[

[18080 22793] [02412 03143] [05453 07516]

[17455 21600] [01514 01978] [04996 06634]

[15051 19417] [minus00260 00094] [03950 05627]

[20272 25100] [03357 04086] [06383 07397]

]]]

]

(39)


119865119882119866

=[[[

[

[17582 21664] [01809 02198] [05004 06620]

[14390 19315] [minus00858 minus00205] [04547 06581]

[13335 18566] [minus01492 minus01188] [03824 05578]

[17131 20711] [00703 01264] [06333 07335]

]]]

]

(40)


119882119860

119875119904 119882119860

=[[[

[

05 06498 08527 02642

03974 05 07695 01480

01473 02305 05 0

07358 08520 1 05

]]]

]

(41)

And for 119865119882119866

119901119904 119882119866

=[[[

[

05 08076 08943 05916

01924 05 05888 02568

01057 04112 05 01629

04084 07432 08371 05

]]]

]

(42)


compute 119901119886119894119895and 119901119888

119894119895



1≻ 1198602 1198601≻ 1198603 1198602≻ 1198603 1198604≻ 1198601 and 119860

4≻


4 1198601

1198602 and 119860

3 Obviously 119860





3 Obviously 119860

1is



3 as well as the

ranking 1198601 1198604 1198602 and 119860

3for AWG






2 1198604 1198603 and 119860

1


1 1198604 1198602 and 119860

3 Obviously


1


119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)




3 As can be


11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)


11988611= 11987911988631


13= 11988633






3in [33]


lt 11988641 11988622

lt 11988642

and 11988623




1in [33] while the

sequences of 1198601 1198602and 119860




6 Conclusion




Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(23)

then

SNNWA119908(1198601 1198602)

= 11990811198601+ 11990821198602

= ⟨[119897minus1(1199081119897 (inf 119879

1198601) + 1199082119897 (inf 119879

1198602))

119897minus1(1199081119897 (sup119879

1198601) + 1199082119897 (sup119879

1198602))]

[119896minus1(1199081119896 (inf 119868

1198601) + 1199082119896 (inf 119868

1198602))

119896minus1(1199081119896 (sup 119868

1198601) + 1199082119896 (sup 119868

1198602))]

[119896minus1(1199081119896 (inf 119865

1198601) + 1199082119896 (inf 119865

1198602))

119896minus1(1199081119896 (sup119865

1198601) + 1199082119896 (sup119865

1198602))]⟩

(24)


SNNWA119908(1198601 1198602 119860

119896)

= ⟨[

[

119897minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(25)

then if 119899 = 119896 + 1 we have

SNNWA119908(1198601 1198602 119860

119896 119860119896+1)

= ⟨[

[

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (inf 119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (inf 119879

119860119896+1)))))

119897minus1(119897(119897

minus1(

119896

sum

119895=1

119908119895119897 (sup119879

119860119895))

+ 119897 (119897minus1(119908119896+1119897 (sup119879

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (inf 119868

119860119896+1)))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup 119868

119860119895))

+ 119896 (119896minus1(119908119896+1119896 (sup 119868

119860119896+1)))))]

]

[

[

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (inf 119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (inf 119865

119860119896+1))))

119896minus1(119896(119896

minus1(

119896

sum

119895=1

119908119895119896 (sup119865

119860119895)))

+ 119896 (119896minus1(119908119896+1119896 (sup119865

119860119896+1))))]

]

⟩

= ⟨[

[

119897minus1(

119896+1

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119896+1

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119896+1

sum

119895=1

119908119895119896 (inf 119865

119860119895))


119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(26)


SNNWA119908(1198601 1198602 119860

119899)

= ⟨[

[

119897minus1(

119899

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119899

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(27)






INNWA119908(1198601 1198602 119860

119899) = 119860



max119895119868119860119895(119909)max

119895119865119860119895(119909)⟩ 119860+ = ⟨max

119895119879119860119895(119909)

min119895119868119860119895(119909)min

119895119865119860119895(119909)⟩ for all 119895 isin 1 2 119899 and


119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWA119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)



INNWG119908(1198601 1198602 119860

119899) =

119899

prod

119895=1

119860119908119894

119895 (28)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 25 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a


119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[119896minus1(1199081119896 (inf 119879

1198601) + 1199082119896 (inf 119879


+119908119899119896 (inf 119879

119860119899))

119896minus1(1199081119896 (sup119879

1198601) + 119908

2119896 (sup119879


+119908119899119896 (sup119879

119860119899))]

[119897minus1(1199081119897 (inf 119868

1198601) + 1199082119897 (inf 119868


+119908119899119897 (inf 119868

119860119899))

119897minus1(1199081119897 (sup 119868

1198601) + 1199082119897 (sup 119868


+119908119899119897 (sup 119868

119860119899))]

[119897minus1(1199081119897 (inf 119865

1198601) + 1199082119897 (inf 119865


+119908119899119897 (inf 119865

119860119899))

119897minus1(1199081119897 (sup119865

1198601) + 1199082119897 (sup119865


+119908119899119897 (sup119865

119860119899))]⟩

(29)



119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[

119899

prod

119894=1

inf 119879119908119894119860119894

119899

prod

119894=1

sup119879119908119894119860119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119868119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup 119868119860119894)119908119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119865119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119865119860119894)119908119894]⟩

(30)

where 119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1




INNWG119908(1198601 1198602 119860

119899)

= ⟨[

[

2prod119899

119894=1inf 119879119908119894119860119894

prod119899

119894=1(2 minus inf 119879

119860119894)119908119894+prod119899

119894=1inf 119879119908119894119860119894

2prod119899

119894=1sup119879119908119894119860119894

prod119899

119894=1(2 minus sup119879

119860119894)119908119894+prod119899

119894=1sup119879119908119894119860119894

]

]

[

[

prod119899

119894=1(1 + inf 119868

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + inf 119868

119860119894)119908119894+prod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894minusprod119899

119894=1(1 minus sup 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894+prod119899

119894=1(1 minus sup 119868

119860119894)119908119894

]

]

[

[

prod119899

119894=1(1 + inf 119865

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + inf 119865

119860119894)119908119894+prod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894minusprod119899

119894=1(1 minus sup119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894+prod119899

119894=1(1 minus sup119865

119860119894)119908119894

]

]

⟩

(31)

where119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1






INNWG119908(1198601 1198602 119860

119899) = 119860



= ⟨min119895119879119860119895(119909)max

119895119868119860119895(119909)max

119895119865119860119895(119909)⟩

119860+

= ⟨max119895119879119860119895(119909)min

119895119868119860119895(119909)min

119895119865119860119895(119909)⟩


sube

INNWG119908(1198601 1198602 119860

119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWG119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)


1 1198862 119886119898 and n criteria 119862 = 119888

1 1198882 119888119899 whose


119899) where

119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Let 119877 =

(119886119894119895)119898times119899


where 119879119886119894119895



119895 119868119886119894119895


119895 and 119865







119894(119894 = 1 2 119898)

that is

119910119894= INNWA

119908(1198861198941 1198861198942 119886

119894119899) (32)

or

119910119894= INNWG

119908(1198861198941 1198861198942 119886

119894119899) (33)



119894) of


matrix F

119865 =[[[

[

119904 (1199101) 119886 (119910

1) 119888 (119910

1)

119904 (1199102) 119886 (119910

2) 119888 (119910

2)


119904 (119910119898) 119886 (119910

119898) 119888 (119910

119898)

]]]

]

(34)



119875119904=

[[[[

[

11990111990411

11990111990412

sdot sdot sdot 1199011199041119898

11990111990421

11990111990422

sdot sdot sdot 1199011199042119898

1199011199041198981

1199011199041198982

sdot sdot sdot 119901119904119898119898

]]]]

]

(35)


119894) gt 119904(119910

119895)


119894119895+ 119901119904119895119894= 1 and 119901119904

119894119894= 05 If


119886(119910119894) gt 119886(119910

119895) denoted by 119901119886

119894119895 And if 119901119886

119894119895= 05 (119894 = 119895) then


119895) denoted by

119901119888119894119895









3





2




3is a


119863 =

[[[

[

⟨[04 05] [02 03] [03 04]⟩ ⟨[04 06] [01 03] [02 04]⟩ ⟨[07 09] [02 03] [04 05]⟩

⟨[06 07] [01 02] [02 03]⟩ ⟨[06 07] [01 02] [02 03]⟩ ⟨[03 06] [03 05] [08 09]⟩

⟨[03 06] [02 03] [03 04]⟩ ⟨[05 06] [02 03] [03 04]⟩ ⟨[04 05] [02 04] [07 09]⟩

⟨[07 08] [00 01] [01 02]⟩ ⟨[06 07] [01 02] [01 03]⟩ ⟨[06 07] [03 04] [08 09]⟩

]]]

]

(36)




119860119882119860

=[[[

[

⟨[05453 07516] [01682 03000] [03041 04373]⟩

⟨[04996 06634] [01552 02885] [03482 04656]⟩

⟨[03950 05627] [02000 03366] [04210 05533]⟩

⟨[06383 07397] [00000 02071] [02297 04040]⟩

]]]

]

(37)


119860119882119866

=[[[

[

⟨[05004 06620] [01761 03000] [03195 04422]⟩

⟨[04547 06581] [01861 03371] [05405 06786]⟩

⟨[03824 05578] [02000 03419] [05012 07070]⟩

⟨[06333 07335] [01555 02570] [05069 06632]⟩

]]]

]

(38)



119865119882119860

=[[[

[

[18080 22793] [02412 03143] [05453 07516]

[17455 21600] [01514 01978] [04996 06634]

[15051 19417] [minus00260 00094] [03950 05627]

[20272 25100] [03357 04086] [06383 07397]

]]]

]

(39)


119865119882119866

=[[[

[

[17582 21664] [01809 02198] [05004 06620]

[14390 19315] [minus00858 minus00205] [04547 06581]

[13335 18566] [minus01492 minus01188] [03824 05578]

[17131 20711] [00703 01264] [06333 07335]

]]]

]

(40)


119882119860

119875119904 119882119860

=[[[

[

05 06498 08527 02642

03974 05 07695 01480

01473 02305 05 0

07358 08520 1 05

]]]

]

(41)

And for 119865119882119866

119901119904 119882119866

=[[[

[

05 08076 08943 05916

01924 05 05888 02568

01057 04112 05 01629

04084 07432 08371 05

]]]

]

(42)


compute 119901119886119894119895and 119901119888

119894119895



1≻ 1198602 1198601≻ 1198603 1198602≻ 1198603 1198604≻ 1198601 and 119860

4≻


4 1198601

1198602 and 119860

3 Obviously 119860





3 Obviously 119860

1is



3 as well as the

ranking 1198601 1198604 1198602 and 119860

3for AWG






2 1198604 1198603 and 119860

1


1 1198604 1198602 and 119860

3 Obviously


1


119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)




3 As can be


11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)


11988611= 11987911988631


13= 11988633






3in [33]


lt 11988641 11988622

lt 11988642

and 11988623




1in [33] while the

sequences of 1198601 1198602and 119860




6 Conclusion




Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119896minus1(

119896+1

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(26)


SNNWA119908(1198601 1198602 119860

119899)

= ⟨[

[

119897minus1(

119899

sum

119895=1

119908119895119897 (inf 119879

119860119895))

119897minus1(

119899

sum

119895=1

119908119895119897 (sup119879

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119868

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup 119868

119860119895))]

]

[

[

119896minus1(

119899

sum

119895=1

119908119895119896 (inf 119865

119860119895))

119896minus1(

119899

sum

119895=1

119908119895119896 (sup119865

119860119895))]

]

⟩

(27)






INNWA119908(1198601 1198602 119860

119899) = 119860



max119895119868119860119895(119909)max

119895119865119860119895(119909)⟩ 119860+ = ⟨max

119895119879119860119895(119909)

min119895119868119860119895(119909)min

119895119865119860119895(119909)⟩ for all 119895 isin 1 2 119899 and


119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWA119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)



INNWG119908(1198601 1198602 119860

119899) =

119899

prod

119895=1

119860119908119894

119895 (28)


(1199081 1199082 119908


119895(119895 = 1 2 119899)

with 119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1

Theorem 25 Let 119860119895= ⟨119879119860119895 119868119860119895 119865119860119895⟩ (119895 = 1 2 119899) be a


119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[119896minus1(1199081119896 (inf 119879

1198601) + 1199082119896 (inf 119879


+119908119899119896 (inf 119879

119860119899))

119896minus1(1199081119896 (sup119879

1198601) + 119908

2119896 (sup119879


+119908119899119896 (sup119879

119860119899))]

[119897minus1(1199081119897 (inf 119868

1198601) + 1199082119897 (inf 119868


+119908119899119897 (inf 119868

119860119899))

119897minus1(1199081119897 (sup 119868

1198601) + 1199082119897 (sup 119868


+119908119899119897 (sup 119868

119860119899))]

[119897minus1(1199081119897 (inf 119865

1198601) + 1199082119897 (inf 119865


+119908119899119897 (inf 119865

119860119899))

119897minus1(1199081119897 (sup119865

1198601) + 1199082119897 (sup119865


+119908119899119897 (sup119865

119860119899))]⟩

(29)



119868119873119873119882119866119908(1198601 1198602 119860

119899)

= ⟨[

119899

prod

119894=1

inf 119879119908119894119860119894

119899

prod

119894=1

sup119879119908119894119860119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119868119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup 119868119860119894)119908119894]

[1 minus

119899

prod

119894=1

(1 minus inf 119865119860119894)119908119894 1 minus

119899

prod

119894=1

(1 minus sup119865119860119894)119908119894]⟩

(30)

where 119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1




INNWG119908(1198601 1198602 119860

119899)

= ⟨[

[

2prod119899

119894=1inf 119879119908119894119860119894

prod119899

119894=1(2 minus inf 119879

119860119894)119908119894+prod119899

119894=1inf 119879119908119894119860119894

2prod119899

119894=1sup119879119908119894119860119894

prod119899

119894=1(2 minus sup119879

119860119894)119908119894+prod119899

119894=1sup119879119908119894119860119894

]

]

[

[

prod119899

119894=1(1 + inf 119868

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + inf 119868

119860119894)119908119894+prod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894minusprod119899

119894=1(1 minus sup 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894+prod119899

119894=1(1 minus sup 119868

119860119894)119908119894

]

]

[

[

prod119899

119894=1(1 + inf 119865

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + inf 119865

119860119894)119908119894+prod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894minusprod119899

119894=1(1 minus sup119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894+prod119899

119894=1(1 minus sup119865

119860119894)119908119894

]

]

⟩

(31)

where119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1






INNWG119908(1198601 1198602 119860

119899) = 119860



= ⟨min119895119879119860119895(119909)max

119895119868119860119895(119909)max

119895119865119860119895(119909)⟩

119860+

= ⟨max119895119879119860119895(119909)min

119895119868119860119895(119909)min

119895119865119860119895(119909)⟩


sube

INNWG119908(1198601 1198602 119860

119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWG119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)


1 1198862 119886119898 and n criteria 119862 = 119888

1 1198882 119888119899 whose


119899) where

119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Let 119877 =

(119886119894119895)119898times119899


where 119879119886119894119895



119895 119868119886119894119895


119895 and 119865







119894(119894 = 1 2 119898)

that is

119910119894= INNWA

119908(1198861198941 1198861198942 119886

119894119899) (32)

or

119910119894= INNWG

119908(1198861198941 1198861198942 119886

119894119899) (33)



119894) of


matrix F

119865 =[[[

[

119904 (1199101) 119886 (119910

1) 119888 (119910

1)

119904 (1199102) 119886 (119910

2) 119888 (119910

2)


119904 (119910119898) 119886 (119910

119898) 119888 (119910

119898)

]]]

]

(34)



119875119904=

[[[[

[

11990111990411

11990111990412

sdot sdot sdot 1199011199041119898

11990111990421

11990111990422

sdot sdot sdot 1199011199042119898

1199011199041198981

1199011199041198982

sdot sdot sdot 119901119904119898119898

]]]]

]

(35)


119894) gt 119904(119910

119895)


119894119895+ 119901119904119895119894= 1 and 119901119904

119894119894= 05 If


119886(119910119894) gt 119886(119910

119895) denoted by 119901119886

119894119895 And if 119901119886

119894119895= 05 (119894 = 119895) then


119895) denoted by

119901119888119894119895









3





2




3is a


119863 =

[[[

[

⟨[04 05] [02 03] [03 04]⟩ ⟨[04 06] [01 03] [02 04]⟩ ⟨[07 09] [02 03] [04 05]⟩

⟨[06 07] [01 02] [02 03]⟩ ⟨[06 07] [01 02] [02 03]⟩ ⟨[03 06] [03 05] [08 09]⟩

⟨[03 06] [02 03] [03 04]⟩ ⟨[05 06] [02 03] [03 04]⟩ ⟨[04 05] [02 04] [07 09]⟩

⟨[07 08] [00 01] [01 02]⟩ ⟨[06 07] [01 02] [01 03]⟩ ⟨[06 07] [03 04] [08 09]⟩

]]]

]

(36)




119860119882119860

=[[[

[

⟨[05453 07516] [01682 03000] [03041 04373]⟩

⟨[04996 06634] [01552 02885] [03482 04656]⟩

⟨[03950 05627] [02000 03366] [04210 05533]⟩

⟨[06383 07397] [00000 02071] [02297 04040]⟩

]]]

]

(37)


119860119882119866

=[[[

[

⟨[05004 06620] [01761 03000] [03195 04422]⟩

⟨[04547 06581] [01861 03371] [05405 06786]⟩

⟨[03824 05578] [02000 03419] [05012 07070]⟩

⟨[06333 07335] [01555 02570] [05069 06632]⟩

]]]

]

(38)



119865119882119860

=[[[

[

[18080 22793] [02412 03143] [05453 07516]

[17455 21600] [01514 01978] [04996 06634]

[15051 19417] [minus00260 00094] [03950 05627]

[20272 25100] [03357 04086] [06383 07397]

]]]

]

(39)


119865119882119866

=[[[

[

[17582 21664] [01809 02198] [05004 06620]

[14390 19315] [minus00858 minus00205] [04547 06581]

[13335 18566] [minus01492 minus01188] [03824 05578]

[17131 20711] [00703 01264] [06333 07335]

]]]

]

(40)


119882119860

119875119904 119882119860

=[[[

[

05 06498 08527 02642

03974 05 07695 01480

01473 02305 05 0

07358 08520 1 05

]]]

]

(41)

And for 119865119882119866

119901119904 119882119866

=[[[

[

05 08076 08943 05916

01924 05 05888 02568

01057 04112 05 01629

04084 07432 08371 05

]]]

]

(42)


compute 119901119886119894119895and 119901119888

119894119895



1≻ 1198602 1198601≻ 1198603 1198602≻ 1198603 1198604≻ 1198601 and 119860

4≻


4 1198601

1198602 and 119860

3 Obviously 119860





3 Obviously 119860

1is



3 as well as the

ranking 1198601 1198604 1198602 and 119860

3for AWG






2 1198604 1198603 and 119860

1


1 1198604 1198602 and 119860

3 Obviously


1


119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)




3 As can be


11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)


11988611= 11987911988631


13= 11988633






3in [33]


lt 11988641 11988622

lt 11988642

and 11988623




1in [33] while the

sequences of 1198601 1198602and 119860




6 Conclusion




Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











INNWG119908(1198601 1198602 119860

119899)

= ⟨[

[

2prod119899

119894=1inf 119879119908119894119860119894

prod119899

119894=1(2 minus inf 119879

119860119894)119908119894+prod119899

119894=1inf 119879119908119894119860119894

2prod119899

119894=1sup119879119908119894119860119894

prod119899

119894=1(2 minus sup119879

119860119894)119908119894+prod119899

119894=1sup119879119908119894119860119894

]

]

[

[

prod119899

119894=1(1 + inf 119868

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + inf 119868

119860119894)119908119894+prod119899

119894=1(1 minus inf 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894minusprod119899

119894=1(1 minus sup 119868

119860119894)119908119894

prod119899

119894=1(1 + sup 119868

119860119894)119908119894+prod119899

119894=1(1 minus sup 119868

119860119894)119908119894

]

]

[

[

prod119899

119894=1(1 + inf 119865

119860119894)119908119894minusprod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + inf 119865

119860119894)119908119894+prod119899

119894=1(1 minus inf 119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894minusprod119899

119894=1(1 minus sup119865

119860119894)119908119894

prod119899

119894=1(1 + sup119865

119860119894)119908119894+prod119899

119894=1(1 minus sup119865

119860119894)119908119894

]

]

⟩

(31)

where119882 = (1199081 1199082 119908


119895(119895 =

1 2 119899) with 119908119895isin [0 1] and sum119899

119895=1119908119895= 1






INNWG119908(1198601 1198602 119860

119899) = 119860



= ⟨min119895119879119860119895(119909)max

119895119868119860119895(119909)max

119895119865119860119895(119909)⟩

119860+

= ⟨max119895119879119860119895(119909)min

119895119868119860119895(119909)min

119895119865119860119895(119909)⟩


sube

INNWG119908(1198601 1198602 119860

119899) sube 119860+



sube 119860lowast

119895 for

119895 isin 1 2 119899 then INNWG119908(1198601 1198602 119860

119899) sube


1 119860lowast

2 119860

lowast

119899)


1 1198862 119886119898 and n criteria 119862 = 119888

1 1198882 119888119899 whose


119899) where

119908119895ge 0 (119895 = 1 2 119899) and sum119899

119895=1119908119895= 1 Let 119877 =

(119886119894119895)119898times119899


where 119879119886119894119895



119895 119868119886119894119895


119895 and 119865







119894(119894 = 1 2 119898)

that is

119910119894= INNWA

119908(1198861198941 1198861198942 119886

119894119899) (32)

or

119910119894= INNWG

119908(1198861198941 1198861198942 119886

119894119899) (33)



119894) of


matrix F

119865 =[[[

[

119904 (1199101) 119886 (119910

1) 119888 (119910

1)

119904 (1199102) 119886 (119910

2) 119888 (119910

2)


119904 (119910119898) 119886 (119910

119898) 119888 (119910

119898)

]]]

]

(34)



119875119904=

[[[[

[

11990111990411

11990111990412

sdot sdot sdot 1199011199041119898

11990111990421

11990111990422

sdot sdot sdot 1199011199042119898

1199011199041198981

1199011199041198982

sdot sdot sdot 119901119904119898119898

]]]]

]

(35)


119894) gt 119904(119910

119895)


119894119895+ 119901119904119895119894= 1 and 119901119904

119894119894= 05 If


119886(119910119894) gt 119886(119910

119895) denoted by 119901119886

119894119895 And if 119901119886

119894119895= 05 (119894 = 119895) then


119895) denoted by

119901119888119894119895









3





2




3is a


119863 =

[[[

[

⟨[04 05] [02 03] [03 04]⟩ ⟨[04 06] [01 03] [02 04]⟩ ⟨[07 09] [02 03] [04 05]⟩

⟨[06 07] [01 02] [02 03]⟩ ⟨[06 07] [01 02] [02 03]⟩ ⟨[03 06] [03 05] [08 09]⟩

⟨[03 06] [02 03] [03 04]⟩ ⟨[05 06] [02 03] [03 04]⟩ ⟨[04 05] [02 04] [07 09]⟩

⟨[07 08] [00 01] [01 02]⟩ ⟨[06 07] [01 02] [01 03]⟩ ⟨[06 07] [03 04] [08 09]⟩

]]]

]

(36)




119860119882119860

=[[[

[

⟨[05453 07516] [01682 03000] [03041 04373]⟩

⟨[04996 06634] [01552 02885] [03482 04656]⟩

⟨[03950 05627] [02000 03366] [04210 05533]⟩

⟨[06383 07397] [00000 02071] [02297 04040]⟩

]]]

]

(37)


119860119882119866

=[[[

[

⟨[05004 06620] [01761 03000] [03195 04422]⟩

⟨[04547 06581] [01861 03371] [05405 06786]⟩

⟨[03824 05578] [02000 03419] [05012 07070]⟩

⟨[06333 07335] [01555 02570] [05069 06632]⟩

]]]

]

(38)



119865119882119860

=[[[

[

[18080 22793] [02412 03143] [05453 07516]

[17455 21600] [01514 01978] [04996 06634]

[15051 19417] [minus00260 00094] [03950 05627]

[20272 25100] [03357 04086] [06383 07397]

]]]

]

(39)


119865119882119866

=[[[

[

[17582 21664] [01809 02198] [05004 06620]

[14390 19315] [minus00858 minus00205] [04547 06581]

[13335 18566] [minus01492 minus01188] [03824 05578]

[17131 20711] [00703 01264] [06333 07335]

]]]

]

(40)


119882119860

119875119904 119882119860

=[[[

[

05 06498 08527 02642

03974 05 07695 01480

01473 02305 05 0

07358 08520 1 05

]]]

]

(41)

And for 119865119882119866

119901119904 119882119866

=[[[

[

05 08076 08943 05916

01924 05 05888 02568

01057 04112 05 01629

04084 07432 08371 05

]]]

]

(42)


compute 119901119886119894119895and 119901119888

119894119895



1≻ 1198602 1198601≻ 1198603 1198602≻ 1198603 1198604≻ 1198601 and 119860

4≻


4 1198601

1198602 and 119860

3 Obviously 119860





3 Obviously 119860

1is



3 as well as the

ranking 1198601 1198604 1198602 and 119860

3for AWG






2 1198604 1198603 and 119860

1


1 1198604 1198602 and 119860

3 Obviously


1


119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)




3 As can be


11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)


11988611= 11987911988631


13= 11988633






3in [33]


lt 11988641 11988622

lt 11988642

and 11988623




1in [33] while the

sequences of 1198601 1198602and 119860




6 Conclusion




Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













2




3is a


119863 =

[[[

[

⟨[04 05] [02 03] [03 04]⟩ ⟨[04 06] [01 03] [02 04]⟩ ⟨[07 09] [02 03] [04 05]⟩

⟨[06 07] [01 02] [02 03]⟩ ⟨[06 07] [01 02] [02 03]⟩ ⟨[03 06] [03 05] [08 09]⟩

⟨[03 06] [02 03] [03 04]⟩ ⟨[05 06] [02 03] [03 04]⟩ ⟨[04 05] [02 04] [07 09]⟩

⟨[07 08] [00 01] [01 02]⟩ ⟨[06 07] [01 02] [01 03]⟩ ⟨[06 07] [03 04] [08 09]⟩

]]]

]

(36)




119860119882119860

=[[[

[

⟨[05453 07516] [01682 03000] [03041 04373]⟩

⟨[04996 06634] [01552 02885] [03482 04656]⟩

⟨[03950 05627] [02000 03366] [04210 05533]⟩

⟨[06383 07397] [00000 02071] [02297 04040]⟩

]]]

]

(37)


119860119882119866

=[[[

[

⟨[05004 06620] [01761 03000] [03195 04422]⟩

⟨[04547 06581] [01861 03371] [05405 06786]⟩

⟨[03824 05578] [02000 03419] [05012 07070]⟩

⟨[06333 07335] [01555 02570] [05069 06632]⟩

]]]

]

(38)



119865119882119860

=[[[

[

[18080 22793] [02412 03143] [05453 07516]

[17455 21600] [01514 01978] [04996 06634]

[15051 19417] [minus00260 00094] [03950 05627]

[20272 25100] [03357 04086] [06383 07397]

]]]

]

(39)


119865119882119866

=[[[

[

[17582 21664] [01809 02198] [05004 06620]

[14390 19315] [minus00858 minus00205] [04547 06581]

[13335 18566] [minus01492 minus01188] [03824 05578]

[17131 20711] [00703 01264] [06333 07335]

]]]

]

(40)


119882119860

119875119904 119882119860

=[[[

[

05 06498 08527 02642

03974 05 07695 01480

01473 02305 05 0

07358 08520 1 05

]]]

]

(41)

And for 119865119882119866

119901119904 119882119866

=[[[

[

05 08076 08943 05916

01924 05 05888 02568

01057 04112 05 01629

04084 07432 08371 05

]]]

]

(42)


compute 119901119886119894119895and 119901119888

119894119895



1≻ 1198602 1198601≻ 1198603 1198602≻ 1198603 1198604≻ 1198601 and 119860

4≻


4 1198601

1198602 and 119860

3 Obviously 119860





3 Obviously 119860

1is



3 as well as the

ranking 1198601 1198604 1198602 and 119860

3for AWG






2 1198604 1198603 and 119860

1


1 1198604 1198602 and 119860

3 Obviously


1


119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)




3 As can be


11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)


11988611= 11987911988631


13= 11988633






3in [33]


lt 11988641 11988622

lt 11988642

and 11988623




1in [33] while the

sequences of 1198601 1198602and 119860




6 Conclusion




Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














2 1198604 1198603 and 119860

1


1 1198604 1198602 and 119860

3 Obviously


1


119863 =[[[

[

119886111198861211988613

119886211198862211988623

119886311198863211988633

119886411198864211988643

]]]

]

(43)




3 As can be


11986811988611= 11986811988631 119865

11988611= 11986511988631 (44)


11988611= 11987911988631


13= 11988633






3in [33]


lt 11988641 11988622

lt 11988642

and 11988623




1in [33] while the

sequences of 1198601 1198602and 119860




6 Conclusion




Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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