Decision-MakingFrameworkforanEffectiveSanitizertoReduce...

Research ArticleDecision-Making Framework for an Effective Sanitizer to ReduceCOVID-19 under Fermatean Fuzzy Environment

Muhammad Akram 1 Gulfam Shahzadi1 and Abdullah Ali H Ahmadini2

1Department of Mathematics University of the Punjab New Campus Lahore Pakistan2Department of Mathematics Faculty of Science Jazan University Jazan Saudi Arabia

Correspondence should be addressed to Muhammad Akram makrampucitedupk

Received 13 July 2020 Accepted 29 August 2020 Published 30 October 2020

Academic Editor Tahir Mahmood

Copyright copy 2020 Muhammad Akram et al ampis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

ampe purpose of this article is to develop some general aggregation operators (AOs) based on Einsteinrsquos norm operations tocumulate the Fermatean fuzzy data in decision-making environments A Fermatean fuzzy set (FFS) possessing the more flexiblestructure than the intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS) is a competent tool to handle vague informationin the decision-making process by the means of membership degree (MD) and nonmembership degree (NMD) Our target is toempower the AOs using the theoretical basis of Einstein norms for the FFS to establish some advantageous operators namelyFermatean fuzzy Einstein weighted averaging (FFEWA) Fermatean fuzzy Einstein ordered weighted averaging (FFEOWA)generalized Fermatean fuzzy Einstein weighted averaging (GFFEWA) and generalized Fermatean fuzzy Einstein orderedweighted averaging (GFFEOWA) operators Some properties and important results of the proposed operators are highlighted Asan addition to the MADM strategies an approach based on the proposed operators is presented to deal with Fermatean fuzzydata in MADM problems Moreover multiattribute decision-making (MADM) problem for the selection of an effective sanitizerto reduce coronavirus is presented to show the capability and proficiency of this new idea ampe results are compared with theFermatean fuzzy TOPSIS method to exhibit the potency of the proposed model

1 Introduction

In decision sciences it is an important aspect to find theranking order of the alternatives corresponding to differentattributes according to the preferences of the decision-making experts amperefore selection of various attributes ofthe alternatives is a very complex task ampese decisionscannot be interpreted by the exact data so the need of apowerful model was raised to handle the ambiguous dataFor that issue Zadeh [1] initiated the innovative idea offuzzy set (FS) which served as the backbone of the FS theoryFS permits the experts to describe their satisfaction level(membership degree) regarding performance of a memberwithin the unit interval Although the FSs provide thegrounds to the uncertain assessments but they were notadequate enough to describe the NMD To overcome thelimitations of FS Atanassov [2] introduced a more

dominant model namely IFS which has both MD μ andNMD ]with condition μ + ]le 1ampe theory of IFS was felt tobe inept and insufficient to represent the inexact data asthere are a lot of problems where the sum of MD and NMDis exceeded by 1 To reduce such type of complicationsYager [3] delivered the idea of PFS with conditionμ2 + ]2 le 1 However PFS has also some limitations if MD ofan element is 08 and NMD is 07 then sum of square ofthese values is greater than 1 ampen Yager [4] developed thetheory of q-rung orthopair fuzzy set (q-ROFS) with con-dition μq + ]q le 1 Recently Senapati and Yager [5] gave theconcept of FFS as a generalization of IFS and PFS

ampeworthwhile theory of AOs is widely applied to decision-making scenarios for the sake of data aggregation and to identifythe best alternative from the possible choices Xu [6] gave theidea of intuitionistic fuzzy (IF) AOsampe concept of generalizedAOs for IFS was developed by Zhao et al [7] Rahman et al [8]

HindawiJournal of MathematicsVolume 2020 Article ID 3263407 19 pageshttpsdoiorg10115520203263407

introduced Pythagorean fuzzy (PF) AOs Zhao and Wei [9]studied the Einstein hybrid AOs under IF environment ampeidea of IF aggregation using Einstein operations was discussedby Wang and Liu [10] ampe induced interval-valued IF EinsteinAOs were developed by Cai andHan [11] Garg [12] studied thegeneralized PFEinsteinweighted arithmetic AOsGarg [13] alsoproposed the generalized PF Einstein weighted geometric AOsampe Pythagorean Dombi fuzzy AOs with applications werediscussed by Akram et al [14] Shahzadi et al [15] proposed thedecision-making approach using PF Yager AOs Liu andWang[16] expressed q-rung orthopair fuzzy (q-ROF) weighted AOsWei et al [17] studied weighted Heronian mean AOs underq-ROF information q-ROF power Maclaurin AOs were de-veloped by Liu et al [18] Jana et al [19] studiedDombi AOs forq-ROFS Liu and Liu [20] proposed q-ROF Bonferroni meanoperators Joshi and Gegov [21] studied the confidence levelsq-ROF aggregation operators Akram and Shahzadi [22] de-veloped the hybrid decision-makingmodel under q-ROF YagerAOs Liu et al [23] extended the concept of prioritized weightedAOs for complex q-ROFS Senapati and Yager [24] studiedsubtraction division and Fermatean arithmetic mean opera-tions over FFS ampe idea of Fermatean fuzzy (FF) weightedaveraginggeometric operators was also given by Senapati andYager [25] For more information and applications the readerscan refer to [26ndash59]

ampe motivations of this article are described as follows

(1) ampe judgement of a perfect alternative in an FFenvironment is a laborious MADM problem ampeprevalent model possessing the more space than theIF model and PF model vigorously elaborates theimprecise decisions for the selection of bestalternative

(2) As Einstein AOs are the simplest and quite creativeapproach for dealing with DM affairs basically thisarticle directs Einstein AOs in FF surroundings toface complex issues

(3) ampe outcomes based on conclusion are quite accurateunder Einstein AOs when it is put on to the reality-based MADM problems in FF data

(4) ampe proposed operators are keen to provide theoptimal solution not only for FF environment butalso to work efficiently for IF and PF environment

ampe contributions of this article are described as follows

(1) ampe feasibility of FFNs is merged with the aggre-gation skills of Einstein norms to establish morepowerful multiskilled and practical AOs which canbe deployed to aggregate FF data and to get moreaccuracy in decision-making scenarios

(2) ampe dominant properties as well as the notable re-sults of the proposed operators are highlighted

(3) An algorithm is studied to handle complex realisticproblems with FF data

(4) A MADM problem for the selection of an effectivesanitizer to reduce coronavirus is discussed by usingproposed operators

(5) A validity test is discussed for the approval andauthenticity of proposed theory

(6) At the end the benefits and characteristics of theproposed work are discussed by comparison analysis

ampe remaining paper is as follows In Section 2 we recallthe concept of FFS and related score functions Section 3provides Einstein operational laws for FFNs In Sections 4and 5 we study the FFEWA and FFEOWA operators re-spectively and related properties to them In Sections 6 and7 we present the idea of GFFEWA and GFFEOWA oper-ators respectively In Section 8 we propose an algorithm forour new model and discuss a MADM problem for the se-lection of a good sanitizer to reduce the coronavirus Section9 provides the validity criteria to prove the consistency of theproposed work Section 10 gives the comparison analysis ofproposed theory with the FF TOPSIS method In Section 11we have concluded the results related to the proposed model

ampe list of acronyms in research paper is given in Table 1

2 Preliminaries

In this section we recall some basic definitions includingIFS PFS FFS and score functions related to FFS

Definition 1 (see [2]) An IFS I on nonempty setV is givenby

I x μI(x) ]I(x)1113866 11138671113864 1113865 (1)

where μI V⟶ [0 1] and ]I V⟶ [0 1] specify MDand NMD of an element x isin V respectively ϖI(x) 1 minus

μI(x) minus ]I(x) is indeterminacy degree (InD) of an elementx isin V

Definition 2 (see [3]) A PFS P on nonempty setV is givenby

P x μP(x) ]P(x)1113866 11138671113864 1113865 (2)

where μP V⟶ [0 1] and ]P V⟶ [0 1] specify MDand NMD of an element respectively ϖP(x)

1 minus (μP(x))2 minus (]P(x))21113969

is InD

Definition 3 (see [5]) An FFS R on nonempty set V isgiven by

R x μR(x) ]R(x)1113866 11138671113864 1113865 (3)

where μR V⟶ [0 1] ]R V⟶ [0 1] and ϖR(x)

1 minus (μR(x))3 minus (]R(x))33

1113969

specify MD NMD and InDrespectively FFNs are components of the FFS

Definition 4 (see [5]) ampe score function and accuracyfunction for FFN R (μR ]R) are represented by

S(R) μ3R minus ]3R where S(R) isin [minus 1 1]

A(R) μ3R + ]3R whereA(R) isin [0 1](4)

2 Journal of Mathematics

Definition 5 (see [5]) Consider two FFNsR1 μR1 ]R1

1113924 1113925

and R2 μR2 ]R2

1113924 1113925 ampen

(1) If S(R1)lt S(R2) then R1≺R2(2) If S(R1)gt S(R2) then R1≻R2(3) If S(R1) S(R2) then

(a) If A(R1)ltA(R2) then R1≺R2(b) If A(R1)gtA(R2) then R1≻R2(c) If A(R1) A(R2) then R1 sim R2

3 Einstein Operational Law of FFNs

In this section we present concepts of the Einstein t-normand t-conorm operations for FFNs and some of theirproperties ampe Einstein operations on FFNs are defined asfollows

Definition 6 Let R μ ]1113866 1113867 R1 μ1 ]11113866 1113867 and R2

μ2 ]21113866 1113867 be FFNs and λgt 0 then

(i) R ]R μR1113866 1113867

(ii) R1andεR2 min μ1 μ21113864 1113865 max ]1 ]21113864 11138651113866 1113867

(iii) R1orεR2 max μ1 μ21113864 1113865 min ]1 ]21113864 11138651113866 1113867

(iv) R1oplusεR2 (μ31 + μ32)(1 + μ31 middot εμ32)

31113969

(]1 middot ε]21113884

1 + (1 minus ]31) middot ε(1 minus ]32)3

1113969

)rang

(v) R1oplusεR2 (μ1 middot εμ2

1 + (1 minus μ31) middot ε(1 minus μ32)3

1113969

)1113884(]31 + ]32)(1 + ]31 middot ε]32)

31113969

rang

(vi) λmiddotεR

((1+μ3)λminus (1minus μ3)λ)((1+μ3)λ+(1minus μ3)λ)31113969

1113884

(23

radic]λ

(2minus ]3)λ+(]3)λ31113969

)rang

(vii) Rλ (23

radicμλ

(2 minus μ3)λ + (μ3)λ31113969

)1113884

((1 + ]3)λ minus (1 minus ]3)λ)((1 + ]3)λ + (1 minus ]3)λ)31113969

rang

Theorem 1 Let R μR ]R1113866 1113867 R1 μ1 ]11113866 1113867 and R2

μ2 ]21113866 1113867 be three FFNs thenR3 R1oplusεR2 andR4 λ middot εR

are also FFNs

Proof Since λgt 0 andR is an FFN therefore 0le μR(x)le 10le ]R(x)le 1 and 0le (μR(x))3 + (]R(x))3 le 1 then1 minus (μR(x))3 ge (]R(x))3 ge 0 1 minus (]R(x))3 ge (μR(x))3 ge 0and (1 minus (μR(x))3)λ ge (]R(x))3 then

1 + μR(x)( 11138573

1113872 1113873λ

minus 1 minus μR(x)( 11138573

1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ 1 minus μR(x)( 11138573

1113872 1113873λ

3

11139741113972

le

1 + μR(x)( 11138573

1113872 1113873λ

minus ]R(x)( 11138573

1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ ]R(x)( 11138573

1113872 1113873λ

3

11139741113972

23

radic]R(x)( 1113857

λ

2 minus ]R(x)( 11138573

1113872 1113873λ

+ ]R(x)3

1113872 1113873λ3

1113970

le23

radic]R(x)( 1113857

λ

1 + μR(x)( 11138573

1113872 1113873λ

+ ]R(x)( 11138573

1113872 1113873λ3

1113970

(5)

ampus

1 + μR(x)( 11138573

1113872 1113873λ


1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ 1 minus μR(x)( 11138573

1113872 1113873λ

3

11139741113972

⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠

3

+

23

radic]R(x)( 1113857

λ

2 minus ]R(x)( 11138573

1113872 1113873λ

+ ]R(x)( 11138573

1113872 1113873λ3

1113970⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

3

le 1

(6)

Furthermore

1 + μR(x)( 11138573

1113872 1113873λ


1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ 1 minus μR(x)( 11138573

1113872 1113873λ

3

11139741113972

⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠

3

+

23

radic]R(x)( 1113857

λ

2 minus ]R(x)( 11138573

1113872 1113873λ

+ ]R(x)31113872 1113873λ3

1113970⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

3

0

(7)

iff μR(x) ]R(x) 0 and

1 + μR(x)( 11138573

1113872 1113873λ


1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ 1 minus μR(x)( 11138573

1113872 1113873λ

3

11139741113972

⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠

3

+

23

radic]R(x)( 1113857

λ

2 minus ]R(x)( 11138573

1113872 1113873λ

+ ]R(x)31113872 1113873λ3

1113970⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

3

1

(8)

iff (μR(x))3 + (]R(x))3 1ampus R4 λ middot εR is an FFN for λgt 0

Theorem 2 Let λ λ1 λ2 ge 0 then

(i) R1oplusεR2 R2oplusεR1

(ii) R1 otimes εR2 R1 otimes εR2

(iii) λ middot ε(R1oplusεR2) λεR1oplusελ middot εR2

Table 1 List of acronyms

Acronyms DescriptionFS Fuzzy setIFS Intuitionistic fuzzy setPFS Pythagorean fuzzy setq-ROFS q-rung orthopair fuzzy setFFS Fermatean fuzzy setFFN Fermatean fuzzy numberAOs Aggregation operatorsMADM Multiattribute decision-makingFF TOPSIS Fermatean fuzzy TOPSIS

Journal of Mathematics 3

(iv) (R1 otimes εR2)λ Rλ

1 otimes εRλ2

(v) λ1 middot εRoplusελ2 middot εR (λ1 + λ2) middot εR

(vi) Rλ1 otimes εRλ2 R(λ1+λ2)

Proof(i)

R1oplusεR2

μ31 + μ321 + μ31 middot εμ

32

3

11139741113972

]1 middot ε]2

1 + 1 minus ]311113872 1113873 middot ε 1 minus ]321113872 11138733

11139691113898 1113899


31

3

11139741113972

]2 middot ε]1


11139691113898 1113899

R2oplusεR1

(9)

(ii)

R1oplusεR2


32

3

11139741113972

]1 middot ε]2


11139691113898 1113899

(10)

is equivalent to

R1oplusεR2

1 + μ311113872 1113873 middot ε 1 + μ321113872 1113873 minus 1 minus μ311113872 1113873 middot ε 1 minus μ321113872 1113873

1 + μ311113872 1113873 middot ε 1 + μ321113872 1113873 + 1 minus μ311113872 1113873 middot ε 1 minus μ321113872 1113873

3

11139741113972

1113898

23

radic]1 middot ε]2

2 minus ]311113872 1113873 middot ε 2 minus ]321113872 1113873 + ]31 middot ε]32

31113969 1113899

(11)

Take a (1 + μ31) middot ε(1 + μ32) b (1 minus μ31) middot ε(1 minus μ32)c ]31 middot ε]32 and d (2 minus ]31) middot ε(2 minus ]32) then

R1oplusεR2

a minus b

a + b

31113971

2c3

radic

d + c

3radic1113898 1113899 (12)

By the Einstein FF law

λ middot ε R1oplusεR2( 1113857 λ middot ε

a minus b

a + b

31113971

2c3

radic

d + c

3radic1113898 1113899

(1 +((a minus b)(a + b)))λ

minus (1 minus ((a minus b)(a + b)))λ

(1 +((a minus b)(a + b)))λ

+(1 minus ((a minus b)(a + b)))λ

3

11139741113972

23

radicmiddot

2c

3radic

d + c

3radic

1113872 1113873λ

(2 minus (2c(d + c)))λ

+(2c(d + c))λ3

11139691113898 1113899

aλ

minus bλ

aλ

+ bλ

3

1113971

2c

λ31113968

dλ

+ cλ3

11139681113898 1113899

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

minus 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

+ 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

3

11139741113972

23

radic]λ1 middot ε]

λ2

2 minus ]311113872 1113873λ

middot ε 2 minus ]321113872 1113873λ

+ ]311113872 1113873λ

middot ε ]321113872 1113873λ3

11139691113898 1113899

(13)

On the other hand

λ middot εR1

1 + μ311113872 1113873λ

minus 1 minus μ311113872 1113873λ

1 + μ311113872 1113873λ

+ 1 minus μ311113872 1113873λ

3

11139741113972

23

radic]λ1

2 minus ]311113872 1113873λ

+ ]311113872 1113873λ3

11139691113898 1113899

a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899

λ middot εR2

1 + μ321113872 1113873λ

minus 1 minus μ321113872 1113873λ

1 + μ321113872 1113873λ

+ 1 minus μ321113872 1113873λ

3

11139741113972

23

radic]λ2

2 minus ]321113872 1113873λ

+ ]321113872 1113873λ3

11139691113898 1113899

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899

(14)


where a1 (1 + μ31)λ b1 (1 minus μ31)

λ c1 (]31)λ

d1 (2 minus ]31)λ a2 (1 + μ32)

λ b2 (1 minus μ32)λ c2 (]32)

λand d2 (2 minus ]32)

λ therefore

λ middot εR1( 1113857oplusε λ middot εR2( 1113857

a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899oplusε

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899

a1 minus b1( 1113857 a1 + b1( 1113857( 1113857 + a2 minus b2( 1113857 a2 + b2( 1113857( 1113857

1 + a1 minus b1( 1113857 a1 + b1( 1113857( 1113857 middot ε a2 minus b2( 1113857 a2 + b2( 1113857( 1113857

31113971

223

c1 middot εc2 d1 + c1( 1113857 middot ε d2 + c2( 11138573

1113969

1 + 1 minus 2c1 d1 + c1( 1113857( 1113857( 1113857ε 1 minus 2c2 d2 + c2( 1113857( 1113857( 1113857

311139691113898 1113899

a1 middot εa2 minus b1 middot εb2a1 middot εa2 + b1 middot εb2

31113971

2c1 middot εc2

31113968

d1 middot εd2 + c1 middot εc2

311139681113898 1113899

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

minus 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

+ 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

3

11139741113972

23


λ2

2 minus ]311113872 1113873λ

middot ε 2 minus ]321113872 1113873λ

+ ]311113872 1113873λ

middot ε ]321113872 1113873λ3

11139691113898 1113899

(15)

Hence λ middot ε(R1oplusεR2) λ middot εR1oplusελ middot εR2 (v) For λ1 λ2 gt 0

λ1 middot εR

1 + μ31113872 1113873λ1

minus 1 minus μ31113872 1113873λ1

1 + μ31113872 1113873λ1

+ 1 minus μ31113872 1113873λ1

3

11139741113972

23

radic]λ1

2 minus ]31113872 1113873λ1

+ ]31113872 1113873λ13

11139691113898 1113899

a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899

λ2 middot εR

1 + μ31113872 1113873λ2

minus 1 minus μ31113872 1113873λ2

1 + μ31113872 1113873λ2

+ 1 minus μ31113872 1113873λ2

3

11139741113972

23

radic]λ2

2 minus ]31113872 1113873λ2

+ ]31113872 1113873λ23

11139691113898 1113899

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899

(16)

where aj (1 + μ3)λj bj (1 minus μ3)λj cj (]3)λj anddj (2 minus ]3)λj for j 1 2

λ1 middot εR( 1113857oplusε λ2 middot εR( 1113857

a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899oplusε

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899



31113971

223


1113969

1 + 1 minus 2c1 d1 + c1( 1113857( 1113857( 1113857 middot ε 1 minus 2c2 d2 + c2( 1113857( 1113857( 1113857

311139691113898 1113899


31113971

2c1 middot εc2

31113968


311139681113898 1113899

1 + μ31113872 1113873λ1+λ2

minus 1 minus μ31113872 1113873λ1+λ2

1 + μ31113872 1113873λ1+λ2

+ 1 minus μ31113872 1113873λ1+λ2

3

11139741113972

23

radic]λ1+λ2

2 minus ]31113872 1113873λ1+λ2

+ ]311113872 1113873λ1+λ23

11139691113898 1113899

λ1 + λ2( 1113857 middot εR

(17)


Hence λ1 middot εRoplusελ2 middot εR (λ1 + λ2) middot εRSimilarly others can be verified

Theorem 3 Let R1 μ1 ]11113866 1113867 and R2 μ2 ]21113866 1113867 be FFNsthen

(i) Rc1andεR

c2 (R1orεR2)

c

(ii) Rc1orεR

c2 (R1andεR2)

c

(iii) Rc1oplusεR

c2 (R1 otimes εR2)

c

(iv) Rc1 otimes εR

c2 (R1oplusεR2)

c

(v) (R1orεR2)oplusε(R1andεR2) R1oplusεR2

(vi) (R1orεR2)oplusε(R1andεR2) R1oplusεR2

Proof It is obvious

Theorem 4 Let R1 μ1 ]11113866 1113867 R2 μ2 ]21113866 1113867 and R3

μ3 ]31113866 1113867 be three FFNs then

(i) (R1orεR2)andεR3 (R1andεR3)orε(R2andεR3)

(ii) (R1andεR2)orεR3 (R1orεR3)andε(R2orεR3)

(iii) (R1orεR2)oplusεR3 (R1oplusεR3)orε(R2oplusεR3)

(iv) (R1andεR2)oplusεR3 (R1oplusεR3)andε(R2oplusεR3)

(v) (R1orεR2)otimes εR3 (R1 otimes εR3)orε(R2 otimes εR3)

(vi) (R1andεR2)otimes εR3 (R1 otimes εR3)andε(R2 otimes εR3)

Proof ampe proof is trivial so we omit it

4 Fermatean Fuzzy Einstein WeightedAveraging Operators

ampe Einstein weighted averaging operators under FF envi-ronment are defined here

Definition 7 Let Rj μj ]j1113924 1113925(j 1 2 s) be a col-lection of FFNs and wj be the weight vector (WV) of Rj

with wj gt 0 and 1113936sj1 wj 1 then FFEWA operator is a

mapping Qs⟶ Q such that

FFEWA R1R2 Rs( 1113857 w1 middot εR1oplusεw2 middot εR2oplusε middot middot middotoplusεws middot εRs

(18)

If wj (1s) forallj then FFEWA operator becomesFFWA operator

FFA R1R2 Rs( 1113857 1s

R1oplusεR2oplusε middot middot middotoplusεRs( 1113857 (19)

Theorem 5 Let Rj (μj ]j) be FFNs then the aggre-gated value by using equation (18) is

FFEWA R1R2 Rs( 1113857

1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 + μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 + μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj311139691113898 1113899 (20)

Proof Use the mathematical induction to prove equation(20)

When s 2

FFEWA R1R2( 1113857 w1 middot εR1oplusεw2 middot εR2 (21)

Byampeorem 1 both w1 middot εR1 and w2 middot εR2 are FFNs andvalue of w1 middot εR1oplusεw2 middot εR2 is an FFN By using (vi) inDefinition 6

w1 middot εR1

1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

3

11139741113972

2]w11

31113969

2 minus ]311113872 1113873w1

+ ]311113872 1113873w13

11139691113898 1113899

w2 middot εR2

1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

3

11139741113972

23

radic]w22

2 minus ]321113872 1113873w2

+ ]321113872 1113873w23

11139691113898 1113899

(22)


ampen

FFEWA R1R2( 1113857 w1 middot εR1oplusεw2 middot εR2

1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 + 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

1 + 1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 middot ε 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

3

11139741113972

1113898

23

radic]w11

2 minus ]311113872 1113873w1

+ ]311113872 1113873w13

1113969

1113874 1113875 middot ε23

radic]w21

2 minus ]321113872 1113873w2

+ ]321113872 1113873w23

1113969

1113874 1113875

1 + 1 minus 2]3w11 2 minus ]311113872 1113873

w1+ ]311113872 1113873

w11113872 11138731113872 11138731113872 1113873 middot ε 1 minus 2]3w2

2 2 minus ]321113872 1113873w2

+ ]321113872 1113873w2

1113872 11138731113872 11138731113872 11138733

1113969 1113899

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

minus 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

+ 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

3

11139741113972

23

radic]w11 middot ε]

w22

2 minus ]311113872 1113873w1

middot ε 2 minus ]321113872 1113873w2

+ ]311113872 1113873w1

middot ε ]321113872 1113873w23

11139691113898 1113899

(23)

ampus equation (20) is true when s 2 Suppose result is true for s k

FFEWA R1R2 Rk( 1113857

1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899 (24)

Now for s k + 1

FFEWA R1R2 Rk+1( 1113857

1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899

oplusε

1 + μ3k+11113872 1113873wk+1

minus 1 minus μ3k+11113872 1113873wk+1

1 + μ3k+11113872 1113873wk+1

+ 1 minus μ3k+11113872 1113873wk+1

3

11139741113972

23

radic]wk+1

k+1

2 minus ]3k+11113872 1113873wk+1

+ ]3k+11113872 1113873wk+13

11139691113898 1113899

1113937k+1j1 1 + μ3j1113872 1113873

wjminus 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

1113937k+1j1 1 + μ3j1113872 1113873

wj+ 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

k+1j1]

wj

j

1113937k+1j1 2 minus ]3j1113872 1113873

wj+ 1113937

k+1j1 ]3j1113872 1113873

wj311139691113898 1113899

(25)

ampus the result is true for s k + 1 Hence equation(20) holds foralls

Lemma 1 Let Rj μj ]j1113924 1113925 wj gt 0 and 1113936sj1 wj 1

then

1113945

s

j1R

wj

j le 1113944s

j1wjRj (26)

where equality holds iff R1 R2 middot middot middot Rs

Theorem 6 If Rj μj ]j1113924 1113925 are FFNs then FFEWA(R1R2 Rs) is also an FFN

Proof Since Rj μj ]j1113924 1113925 are FFNs so 0le μj ]j le 1 and0le μ3j + ]3j le 1 amperefore


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj 1

minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le 1 minus 1113945s

j11 minus μ3j1113872 1113873

wj le 1

(27)

Also (1 + μ3j)ge (1 minus μ3j)rArr1113937sj1 (1 + μ3j) minus 1113937

sj1

(1 minus μ3j)ge 0 amperefore

1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjge 0 (28)

ampus 0le μFFEWA le 1Moreover

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le1113945s

j11 minus μ3j1113872 1113873

wj le 1

(29)

Also

1113945

s

j1]3j1113872 1113873

wj ge 0⟺21113937

sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge 0

(30)

ampus 0le ]FFEWA le 1 Moreover

μ3FFEWA + ]3FFEWA 1113937

sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le 1 minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

1

(31)

Hence FFEWA isin [0 1] amperefore FFEWA(R1R2 Rs) isin FFN

Corollary 1 lte FFEWA and FFWA operators have therelationship

FFEWA R1R2 Rs( 1113857le FFWA R1R2 Rs( 1113857

(32)

Proof Let FFEWA (R1R2 Rs) (μβR ]βR) Rβ

and FFWA (R1R2 Rs) (μR ]R) R Since1113937

sj1 (1 + μ3j)wj + 1113937

sj1 (1 minus μ3j)wj le 1113936

sj1 (1 + μ3j)wj+

1113936sj1 (1 minus μ3j)wj 2 then from equation (27) we obtain

1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le

1 minus 1113945s

j11 minus μ3j1113872 1113873

wj⟺ μβR le μR3

11139741113972

(33)

equality holds iff μ1 μ2 middot middot middot μsAlso

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge

21113937sj1 ]3j1113872 1113873

wj

1113936sj1 wj 2 minus ]3j1113872 1113873 + 1113936

sj1 wj]

3j

ge1113945s

j1]3j1113872 1113873

wjrArr

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

3

11139741113972

ge1113945s

j1]wj

j rArr]βR le ]R

(34)


equality holds iff ]1 ]2 middot middot middot ]sampus

S Rβ

1113872 1113873 μβR1113872 11138733

minus ]βR1113872 11138733le μR( 1113857

3minus ]R( 1113857

3 S(R) (35)

If S(Rβ)ltS(R) then

FFEWA R1R2 Rs( 1113857lt FFWA R1R2 Rs( 1113857

(36)

If S(Rβ) S(R) that is (μβR)3 minus (]βR)3

(μR)3 minus (]R)3 then by condition μβR le μR and ]βR ge ]Rthus the accuracy function A(Rβ) (μβR)3 minus (]βR)3

(μR)3 minus (]R)3 A(R) ampus

FFEWA R1R2 Rs( 1113857 FFWA R1R2 Rs( 1113857

(37)

Hence


(38)

equality holds iff R1 R2 middot middot middot Rs

Example 1 Let R1 (08 05) R2 (09 04)R3 (06 07) and R4 (08 07) be four FFNs andw (04 02 02 02)T then

FFEWA R1R2R3R4( 1113857

11139374j1 1 + μ3j1113872 1113873

wjminus 1113937

4j1 1 minus μ3j1113872 1113873

wj

11139374j1 1 + μ3j1113872 1113873

wj+ 1113937

4j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

j

11139374j1 2 minus ]3j1113872 1113873

wj+ 1113937

4j1 ]3j1113872 1113873

wj311139691113898 1113899

149 minus 048149 + 048

31113970

23

radictimes 055

180 + 0163

radic1113898 1113899 080 055lang rang

(39)

Now

FFWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3j1113872 1113873

wj 1113945

4

j1]j1113872 1113873

wjrang 080 055lang rangrArrFFEWA R1R2R3R4( 1113857le FFWA R1R2R3R4( 1113857

3

11139741113972

1113898(40)

Proposition 1 Let Rj μj ]j1113924 1113925 be FFNs and wj be theWV of Rj such that wj isin [0 1] and 1113936

sj1 wj 1

(i) Idempotency if Rj Ro μo ]o1113866 1113867 for all j then

FFEWA R1R2 Rs( 1113857 Ro (41)

(ii) Boundedness let Rminus (minj(μj)maxj(]j)) andR+ (maxj(μj)minj(]j)) then

Rminus le FFEWA R1R2 Rs( 1113857leR+

(42)

(iii) Monotonicity when Rj lePj forallj then

FFEWA R1R2 Rs( 1113857le FFEWA P1P2 Ps( 1113857

(43)

Proof (i) As Rj μo ]o1113866 1113867 are FFNs forallj then


1113937sj1 1 + μ3o1113872 1113873

wjminus 1113937

sj1 1 minus μ3o1113872 1113873

wj

1113937sj1 1 + μ3o1113872 1113873

wj+ 1113937

sj1 1 minus μ3o1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

o

1113937sj1 2 minus ]3o1113872 1113873

wj+ 1113937

sj1 ]3o1113872 1113873

wj311139691113898 1113899

1 + μ3o1113872 11138731113936

sj1 wj minus 1 minus μ3o1113872 1113873

1113936sj1 wj

1 + μ3o1113872 11138731113936

sj1 wj + 1 minus μ3o1113872 1113873

1113936sj1 wj

3

111397411139731113972

23

radic]1113936

sj1 wj

o

2 minus ]3o1113872 11138731113936

sj1 wj + ]3o1113872 1113873

1113936sj1 wj

311139701113898 1113899 μo ]o1113866 1113867

(44)


(ii) Consider f(x) ((1 minus x)(1 + x)) x isin [0 1] thenfprime(x) minus (2(1 + x)2)lt 0 so f(x) is a decreasing function(DF) As μ3jmin le μ3j le μ3jmax forallj 1 2 s thenf(μ3jmax)lef(μ3j)lef(μ3jmin) forallj that is ((1 minus μ3jmax)

(1 + μ3jmax))le ((1 minus μ3j) (1 + μ3j))le ((1 minus μ3jmin)(1+

μ3jmin)) for all j Let wj isin [0 1] and 1113936sj1 wj 1 and we

have

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj

le1 minus μ3j1 + μ3j

1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

1113945

s

j1

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj

le1113945s

j1

1 minus μ3j1 + μ3j

1113888 1113889

wj

le1113945s

j1

1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

⟺1 minus μ3jmax

1 + μ3jmax1113888 1113889

1113944s

j1wj

le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

1113944s

j1wj

⟺1 minus μ3jmax

1 + μ3jmax

⎛⎝ ⎞⎠le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin

⎛⎝ ⎞⎠

⟺2

1 + μ3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1


1113888 1113889

wj

le2

1 + μ3jmin

⎛⎝ ⎞⎠

⟺1 + μ3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle

1 + μ3jmax

2⎛⎝ ⎞⎠

⟺ 1 + μ3jmin1113872 1113873le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle 1 + μ3jmax1113872 1113873

⟺ μ3jmin le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjminus 1le μ3jmax

⟺ μ3jmin le1113937

sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjle μ3jmax

(45)

ampus

μjmin le

1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le μjmax (46)

Consider g(y) ((2 minus y)y) y isin (0 1] then gprime(y) minus

(2y2) ie g(y) is a DF on (0 1] Since ]3

jmin le ]3

j le ]3

jmax

forallj then g(]3

jmax)leg(]3

j)le g(]3

jmin) forallj that is ((2 minus

]3

jmax)]3

jmax)le ((2minus ]3

j)]3

j)le ((2 minus ]3

jmin)]3

jmin) ampen


2 minus ]3jmax

]3jmax1113888 1113889

wj

le2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

wj

1113945

s

j1

2 minus ]3jmax

]3jmax1113888 1113889

wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le1113945s

j1

2 minus ]3jmin

]3jmin1113888 1113889

wj

rArr2 minus ]3jmax

]3jmax1113888 1113889

1113944

s

j1wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

1113944

s

j1wj

rArr2 minus ]3jmax

]3jmax

⎛⎝ ⎞⎠le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin

⎛⎝ ⎞⎠

rArr2

]3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2

]3jmin

⎛⎝ ⎞⎠

rArr]3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle

]3jmax

2⎛⎝ ⎞⎠

rArr ]3jmin1113872 1113873le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax1113872 1113873

rArr]3jmin le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax

rArr]3jmin le21113937

sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjle ]3jmax

rArr]jmin le

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj31113969 le ]jmax

(47)

Let FFEWA (R1R2 Rs) R μR ]R1113866 1113867 thenfrom equations (46) and (47)

μmin le μR le μmax

]min le ]R le ]max(48)

where μmin minj μj1113966 1113967 μmax maxj μj1113966 1113967 ]min minj]j1113966 1113967 and ]max maxj ]j1113966 1113967 So S(R) μ3R minus ]3R le μ3max minus

]3min S(R+) and S(R) μ3R minus ]3R ge μ3minminus

]3max S(Rminus ) As S(R)lt S(R+) and S(R)gt S(Rminus ) so


(49)

(iii) It is similar to (ii) so we omit it

5 Fermatean Fuzzy Einstein Ordered WeightedAveraging Operators

Definition 8 LetRj μj ]j1113924 1113925 be a family of FFNs andwj

be the WV of Rj with wj gt 0 and 1113936sj1 wj 1 then

FFEOWA operator is a mapping Qs⟶ Q such that

FFEOWA R1R2 Rs( 1113857

w1 middot εRϱ(1)oplusεw2 middot εRϱ(2)oplusε middot middot middotoplusεws middot εRϱ(s)(50)

where (ϱ(1) ϱ(2) ϱ(s)) is the permutation of(j 1 2 s) such that Rϱ(jminus 1) geRϱ(j) forallj

1 2 s

Theorem 7 Let Rj (μj ]j) be FFNs then the aggre-gated value by using FFEOWA is an FFN and


1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

1113898

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj31113969 1113899

(51)


Proof It is similar to ampeorem 4We give some properties without their proofs

Corollary 2 lte FFEOWA and FFOWA operators have therelation

FFEOWA R1R2 Rs( 1113857le FFOWA R1R2 Rs( 1113857

(52)

Example 2 Let R1 (06 07) R2 (08 07)R3 (06 09) and R4 (09 04) be four FFNs and w

(03 03 02 02)T as

S R1( 1113857 (06)3

minus (07)3

minus 013

S R2( 1113857 (08)3

minus (07)3

017

S R3( 1113857 (06)3

minus (09)3

minus 051

S R4( 1113857 (09)3

minus (04)3

067

(53)

Since S(R4)gtS(R2)gtS(R1)gtS(R3) therefore

Rϱ(1) R4 (09 04)

Rϱ(2) R2 (08 07)

Rϱ(3) R1 (06 07)

Rϱ(4) R3 (06 09)

(54)

ampus by applying the FFEOWA operator we obtain

FFEOWA R1R2R3R4( 1113857

11139374j1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

11139374j1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

ϱ(j)

11139374j1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

4j1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

144 minus 049144 + 049

31113970

23

radictimes 066

162 + 0663

radic1113898 1113899 079 055lang rang

(55)

Now

FFOWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3ϱ(j)1113872 1113873

wj3

11139741113972

11139454

j1]ϱ(j)1113872 1113873

wj1113898 1113899 080 066lang rang

rArrFFEOWA R1R2R3R4( 1113857lt FFOWA R1R2R3R4( 1113857

(56)


sj1 wj 1

(i) Idempotency if Rj Ro μo ]o1113866 1113867 forallj then

FFEOWA R1R2 Rs( 1113857 Ro (57)


Rminus le FFEOWA R1R2 Rs( 1113857leR+

(58)


FFEOWA R1R2 Rs( 1113857le FFEOWA P1P2 Ps( 1113857

(59)

6 Generalized Fermatean Fuzzy EinsteinWeighted Averaging Operators

Definition 9 LetRj μj ]j1113924 1113925 be a collection of FFNs andwj be the WV of Rj with wj gt 0 and 1113936

sj1 wj 1 then

GFFEWA operator is a mapping Qs⟶ Q such that

GFFEWA R1R2 Rs( 1113857 oplussj1 wj middot εRλj1113872 11138731113872 1113873

1λ

(60)

where λgt 0Particularly

(i) If λ 1 then GFFEWA becomes FFEWA(ii) If w ((1s) (1s) (1s))T then GFFEWA

(R1R2 Rs) ((1s) middot εoplussj1Rλj)1λ

Theorem 8 LetRj μj ]j1113924 1113925 be FFNs and wj be the WVof Rj with wj gt 0 and 1113936

sj1 wj 1 then the aggregated

value by applying the GFFEWA operator is an FFN and


GFFEWA R1R2 Rs( 1113857

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

middot

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(61)

Proof Since

Rλj

23

radicμλj

2 minus μ3j1113872 1113873λ

+ μ3j1113872 1113873λ3

1113969

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

3

11139741113972

1113898 1113899

rArroplussj1wj middot εRλj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

minus 1113937sj1 1 minus 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 minus 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1113874 1113875 1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

1113874 11138753

1113970

1113888 1113889

wj

1113937sj1 2 minus 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 1113875

wj3

1113970 1113899

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ3

1113970

1113888 1113889

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj3

1113970 1113899

(62)

amperefore

oplussj1wj middot εRλj1113872 1113873

1λ

2 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

1113971

2 minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

11139711113898

1 + 21113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

minus 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

1 + 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

3

1113974111397311139731113972

1113899

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

11139711113898

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(63)


When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)

7 Generalized Fermatean Fuzzy EinsteinOrdered Weighted Averaging Operators

Definition 10 Let Rj μj ]j1113924 1113925 be a collection of FFNsand wj be the WV of Rj with wj gt 0 and 1113936

sj1 wj 1

then the GFFEOWA operator is a mapping Qs⟶ Q suchthat

GFFEOWA R1R2 Rs( 1113857 oplussj1 wj middot εRλϱ(j)1113872 11138731113872 1113873

1λ

(65)

where λgt 0



value by applying the GFFEOWA operator is an FFN and

GFFEOWA R1R2 Rs( 1113857

23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)

Proof It is similar to ampeorem 6 and we can prove it When λ 1 then


1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)

8 MADM Problem Using FF Information

To handle an MADM problem under FF environment letK K1K2 Km1113864 1113865 be a set of possible alternatives andJ J1J2 Js1113864 1113865 be a set of possible attributes chosenby the decision maker Let w (w1 w2 ws)T be theWV

with wj gt 0 and 1113936sj1 wj 1 Suppose that

1113957E (μlj ]lj)mtimess is the FF decision matrix (FFDM) whereμlj and ]lj are the MD and NMD of the alternative Kl forthe attribute Jj respectively where 0le μ3lj + ]3lj le 1

ampe following Algorithm 1 is used to solve the MADMproblem with FFN based on using the GFFEWA operator


Bl GFFEWA Kl1Kl2 Kls( 1113857

23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)

for overall preference values Bl(l 1 2 m) ofthe alternatives Kl

(3) Use the score function S(Bl)(l 1 2 m) forthe ranking of alternatives If score values are equalthen compute the accuracy functions A(Bl) andrank according to these valuesOutput the alternative containing maximum scorevalue will be the decision

81 Selection of an Effective Sanitizer to Reduce CoronavirusHand sanitizer is a liquid or gel mostly used to reduce in-fectious agents on the hands Alcohol-based hand sanitizers arepreferred for hand washing in most healthcare settings ampeCenters for Disease Control and Prevention (CDC) advise thepeople to wash hands with soap and water to restrain thespread of infections and decrease the endanger of getting sickIn shortage of soap and water CDC suggests people to use analcohol-based (at least 60 percent) hand sanitizer According tothe World Health Organization (WHO) in this pandemicsituation of coronavirus good hygiene and physical distancingare the best ways to protect ourself and everyone around usfrom coronavirus ampis virus spreads by a person who has thedisease and also spread by touching a sick person We cannotisolate ourselves entirely to prudent from coronavirus So goodhand hygiene can be the final barrier between us and thedisease WHO recommends alcohol-based hand sanitizers toremove the novel coronavirus Alcohol-based hand sanitizerworks to prevent the proteins of microbesmdashincluding bacteriaand some virusesmdashfrom functioning normally Hand sanitizersmust contain ethanol isopropanol n-propanol or a combi-nation of these alcohols All are effective against viruses such asthe novel coronavirus

Demand of a hand sanitizer is increased in such criticalsituation of COVID-19 Due to increasing demand it isdifficult to get good and effective hand sanitizers in local

markets Increasing demand has also led to low quality handsanitizers entering the market ampe main motive of thisapplication is to select an effective sanitizer to mitigatetransmission of coronavirus by applying the GFFYWAoperator Let K K1K2 K3K41113864 1113865 be a set of sanitizersLet J J1J2J31113864 1113865 be a set of three attributes for theevaluation of an effective sanitizer where

J1 represents quantity of ethanol

J2 represents quantity of glycerol

J3 represents quantity of hydrogenperoxide

(69)

(1) ampe FFDM is shown in Table 2(2) ampe weights assigned by the decision maker are

w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)

We use the GFFEWA operator for the selection of aneffective sanitizer

Step 1 For performance values Bl of sanitizers use theGFFEWA operator for λ 1

B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)

Step 2 Calculate the scoresS(Bl) of FFNsBl and rank thesanitizers

(1) Input selection of suitable alternatives and attributes(2) Use the FFDM and GFFEWA operator

ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)

ampe ranking of sanitizers is

K4 gtK3 gtK1 gtK2 (73)

Step 3 amperefore K4 is the best sanitizer

9 Validity Test

For the validity and authenticity of MADM methods Wangand Triantaphyllou [53] developed testing criteria given asfollows

(i) Criterion 1 a MADM technique is valid if the mostdesirable alternative remains same on changes anonoptimal alternative with some other poor orweak alternative without changing the respectivedecision criteria

(ii) Criterion 2 the transitive property should be fol-lowed by a valid MADM technique

(iii) Criterion 3 the ranking result of alternatives shouldnot change on splitting the problem into the smallersubproblems and by applying the same MADMtechnique on subproblems

Now we discuss the validity of our proposed MADMtechnique by testing the above criteria

(1) Validity test by criterion 1 if we replace the decisionvalues of a nonoptimal alternative K2 by 1113957K2 thenthe new DM is given in Table 3By applying the GFFEWA operator for λ 1 andscore function the score values of alternatives are

S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)

ampe ranking of sanitizers is K4 gtK3 gtK1 gt 1113957K2which is the same as the original ranking order andthe best sanitizer isK4 ampus our presented MADMmodel fulfills the test criterion 1

(2) Validity test by criteria 2 and 3 for the validity ofproposed algorithm using criteria 2 and 3 we splitthe problem into the smaller subproblemsK1K2K41113864 1113865 K1K3K41113864 1113865 and K2K3K41113864 1113865

By utilizing the proposed technique the rankingorders of alternatives in these subproblems areK4 gtK1 gtK2K4 gtK3 gtK1 and K4 gtK3 gtK2respectively ampe combined ranking of alternatives isK4 gtK3 gtK1 gtK2 which is the same as that of theoriginal ranking Hence the proposed MADMtechnique is authentic and proficient under criteria 2and 3

10 Comparison Analysis

Here we discuss the comparison of proposed theory with theFF TOPSIS method [5] ampe steps to find out the best al-ternative by the FF TOPSIS method are

(1) Table 2 represents the FF decision matrix in whicheach entry corresponds to an FFN

(2) ampe FF positive ideal solution (FFPIS) S+ and FFnegative ideal solution (FFNIS) Sminus are

S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)

(3) ampe distance between the alternative Kl and FFPISS+ together with the FFNIS Sminus are given in Table 4

(4) ampe revised closeness degree of each alternative isgiven as

ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)

(5) We get the following ranking list by arranging thealternatives in the decreasing order with respect toξ(Kl)


(6) K4 is the best alternative

Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)

Table 3 Reconstructed FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)

Table 4 Distance of alternatives from FFPIS and FFNIS

D(KlS+) D(KlS

minus )

030 008046 029018 021009 032


From the outcomes of proposed operators and FFTOPSIS method as shown in Figure 1 we conclude thatranking lists obtained from both compared methods are thesame and the best alternative from both approaches is K4ampe FF TOPSIS method is a good approach to solve DMproblems but there are many hindrances which can besolved by using our proposed theory ampe FF Einstein AOsare more flexible and easy approach A best alternative canbe obtained by a short process ampe results from proposedtheory are more accurate and closest to original results

ampe steps to solve any MADM problem by FF EinsteinAOs and FF TOPSIS method are shown in Figure 2

101 Advantages and Limitations of Proposed Model ampeproposed model is superior than the IF and PF modelsbecause it contains the space of IF and PF models ampe cubicsum of membership and nonmembership degrees isbounded by 1 in the proposed model ampe MADM

approaches discussed in [10 12 13 15] failed to handle theproposed application because 09 + 06gt 1 and092 + 062 gt 1 but proposed approach covers all such situ-ationsampe results are more precise and accurate by using theproposed model However there are some limitations of thismodel It cannot be applied in the situations where we takethe parameters for the evaluation of anything It means thistheory lacks parametrization property

11 Conclusions

An FFS is an extension of IFS and PFS which has moreflexible structure to solve decision-making problems owingto the condition μ3 + ]3 le 1 Moreover Einsteinrsquos t-normand t-conorm have more generalized structure that operatesefficiently to integrate the intricate information ampe limi-tations of existing operators and beneficial characteristics ofEinstein AOs motivated us to endeavor for the developmentof a fruitful combination of Einstein AOs with FFNs

Amajor contribution of the study is the development newtremendous AOs called FFEWA FFEOWA GFFEWA andGFFEOWA operators Some captivating properties of theseoperators have been discussed Another achievement of thisstudy is the establishment of a MADM technique on the basisof the proposed operators to manifest the application of theproposed operators AMADMproblem for the selection of aneffective sanitizer to reduce the coronavirus has been pre-sented to demonstrate the potency of the proposed strategy

ampe validity test has been discussed to unfold theconsistency of proposed work A comparison analysis ofour proposed theory with the FF TOPSIS method has beenpresented to exhibit the dominance of our proposed op-erators over the FF TOPSIS method In short this articlebuilds up a tool that has the rich properties of Einstein AOsand flexibility of the FF model In future our aim is todevelop some worthwhile AOs using the theoreticalfoundations of Einstein norms for the Fermatean fuzzy softset

Data Availability

No data were used to support this study

Conflicts of Interest

ampe authors declare that they have no conflicts of interestregarding the publication of the research article

References

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

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

[3] R R Yager ldquoPythagorean fuzzy subsetsrdquo in Proceedings of the2013 Joint IFSA World Congress and NAFIPS Annual Meeting(IFSANAFIPS) pp 57ndash61 Edmonton Canada June 2013

[4] R R Yager ldquoGeneralized orthopair fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 25 no 5 pp 1222ndash12302016

ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4

GFFEWA operatorFF TOPSIS method

Figure 1 Comparison of the GFFEWA operator and FF TOPSISmethod

FF decision matrix

FF decision matrix

Revised closeness degree

FFPIS and FFNIS

Distances of alternativesfrom FFPIS and FFNIS

Aggregated value of alternatives

Score values of alternatives

Ranking of alternatives


FF Einstein AOs

FF TOPSISmethod

Figure 2 Flow chart for MADM problem using FF Einstein AOsand FF TOPSIS method


[5] T Senapati and R R Yager ldquoFermatean fuzzy setsrdquo Journal ofAmbient Intelligence and Humanized Computing vol 11no 2 pp 663ndash674 2020

[6] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEETransactions on Fuzzy Systems vol 15 no 6 pp 1179ndash11872007

[7] H Zhao Z Xu M Ni and S Liu ldquoGeneralized aggregationoperators for intuitionistic fuzzy setsrdquo International Journal ofIntelligent Systems vol 25 no 1 pp 1ndash30 2010

[8] K Rahman A Ali M Shakeel M S A Khan and M UllahldquoPythagorean fuzzy weighted averaging aggregation operatorand its application to decision-making theoryrdquo lte Nucleusvol 54 no 3 pp 190ndash196 2017

[9] X Zhao and G Wei ldquoSome intuitionistic fuzzy Einsteinhybrid aggregation operators and their application to multipleattribute decision makingrdquo Knowledge-Based Systems vol 37pp 472ndash479 2013

[10] W Wang and X Liu ldquoIntuitionistic fuzzy information ag-gregation using Einstein operationsrdquo IEEE Transactions onFuzzy Systems vol 20 no 5 pp 923ndash938 2012

[11] X Cai and L Han ldquoSome induced Einstein aggregationoperators based on the data mining with interval-valuedintuitionistic fuzzy information and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 27 no 1 pp 331ndash338 2014

[12] H Garg ldquoA new generalized Pythagorean fuzzy informationaggregation using Einstein operations and its application todecision makingrdquo International Journal of Intelligent Systemsvol 31 no 9 pp 886ndash920 2016

[13] H Garg ldquoGeneralized Pythagorean fuzzy geometric aggre-gation operators using Einsteint-norm andt-conorm formulticriteria decision-making processrdquo International Journalof Intelligent Systems vol 32 no 6 pp 597ndash630 2017

[14] M Akram W A Dudek and J M Dar ldquoPythagorean Dombifuzzy aggregation operators with application in multicriteriadecision-makingrdquo International Journal of Intelligent Systemsvol 34 no 11 pp 3000ndash3019 2019

[15] G Shahzadi M Akram and A N Al-Kenani ldquoDecision-making approach under Pythagorean fuzzy Yager weightedoperatorsrdquo Mathematics vol 8 no 1 p 70 2020

[16] P Liu and P Wang ldquoSome q-rung orthopair fuzzy aggre-gation operators and their applications to multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 33 no 2 pp 259ndash280 2018

[17] G Wei H Gao and Y Wei ldquoSome q-rung orthopair fuzzyHeronian mean operators in multiple attribute decisionmakingrdquo International Journal of Intelligent Systems vol 33no 7 pp 1426ndash1458 2018

[18] P Liu S-M Chen and P Wang ldquoMultiple-attribute groupdecision-making based on q-rung orthopair fuzzy powerMaclaurin symmetric mean operatorsrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 50 no 10pp 3741ndash3756 2020

[19] C Jana GMuhiuddin andM Pal ldquoSomeDombi aggregationof q-rung orthopair fuzzy numbers in multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 34 no 12 pp 3220ndash3240 2019

[20] P Liu and J Liu ldquoSome q-rung orthopair fuzzy Bonferronimean operators and their application to multi-attribute groupdecision-makingrdquo International Journal of Intelligent Systemsvol 33 no 2 pp 315ndash347 2018

[21] B P Joshi and A Gegov ldquoConfidence levels q-rung orthopairfuzzy aggregation operators and its applications to MCDM

problemsrdquo International Journal of Intelligent Systems vol 35no 1 pp 125ndash149 2019

[22] M Akram and G Shahzadi ldquoA hybrid decision-makingmodel under q-rung orthopair fuzzy Yager aggregation op-eratorsrdquo Granular Computing pp 1ndash15 2020

[23] P Liu M Akram and A Sattar ldquoExtensions of prioritizedweighted aggregation operators for decision-making undercomplex q-rung orthopair fuzzy informationrdquo Journal ofIntelligent amp Fuzzy Systems pp 1ndash25 2020

[24] T Senapati and R R Yager ldquoSome new operations overfermatean fuzzy numbers and application of fermatean fuzzyWPM in multiple criteria decision makingrdquo Informaticavol 30 no 2 pp 391ndash412 2019

[25] T Senapati and R R Yager ldquoFermatean fuzzy weightedaveraginggeometric operators and its application in multi-criteria decision-making methodsrdquo Engineering Applicationsof Artificial Intelligence vol 85 pp 112ndash121 2019

[26] M Akram W A Dudek and F Ilyas ldquoGroup decision-making based on pythagorean fuzzy TOPSIS methodrdquo In-ternational Journal of Intelligent Systems vol 34 no 7pp 1455ndash1475 2019

[27] M Akram H Garg and K Zahid ldquoExtensions of ELECTRE-Iand TOPSIS methods for group decision-making undercomplex Pythagorean fuzzy environmentrdquo Iranian Journal ofFuzzy Systems vol 17 no 5 pp 147ndash164 2020

[28] M Akram N Yaqoob G Ali and W Chammam ldquoExten-sions of Dombi aggregation operators for decision-makingunder m-polar fuzzy informationrdquo Journal of Mathematicsvol 2020 Article ID 4739567 20 pages 2020

[29] M Akram F Ilyas and H Garg ldquoMulti-criteria group de-cision making based on ELECTRE I method in Pythagoreanfuzzy informationrdquo Soft Computing vol 24 no 5pp 3425ndash3453 2020

[30] M Akram A Bashir and H Garg ldquoDecision-making modelunder complex picture fuzzy Hamacher aggregation opera-torsrdquo Computational amp Applied Mathematics vol 39 p 2262020

[31] M Akram X Peng A N Al-Kenani and A Sattar ldquoPri-oritized weighted aggregation operators under complex Py-thagorean fuzzy informationrdquo Journal of Intelligent amp FuzzySystems vol 39 no 3 pp 4763ndash4783 2020

[32] M Akram G Shahzadi and X Peng ldquoExtension of Einsteingeometric operators to multi-attribute decision-making un-der q-rung orthopair fuzzy informationrdquo Granular Com-puting 2020

[33] J C R Alcantud ldquoCharacterization of the existence ofmaximal elements of acyclic relationsrdquo Economic lteoryvol 19 no 2 pp 407ndash416 2002

[34] F Feng M Liang H Fujita R Yager and X Liu ldquoLexico-graphic orders of intuitionistic fuzzy values and their rela-tionshipsrdquo Mathematics vol 7 no 2 p 166 20190

[35] F Feng H Fujita M I Ali R R Yager and X Liu ldquoAnotherview on generalized intuitionistic fuzzy soft sets and relatedmultiattribute decision making methodsrdquo IEEE Transactionson Fuzzy Systems vol 27 no 3 pp 474ndash488 2019

[36] H Garg J Gwak T Mahmood and Z Ali ldquoPower aggre-gation operators and VIKOR methods for complex q-rungorthopair fuzzy sets and their applicationsrdquo Mathematicsvol 8 no 4 p 538 2020

[37] H Garg G Shahzadi and M Akram ldquoDecision-makinganalysis based on Fermatean fuzzy Yager aggregation oper-ators with application in COVID-19 testing facilityrdquo Math-ematical Problems in Engineering vol 2020 Article ID7279027 16 pages 2020


[38] P Liu G Shahzadi and M Akram ldquoSpecific types of q-rungpicture fuzzy Yager aggregation operators for decision-makingrdquo International Journal of Computational IntelligenceSystems vol 13 no 1 pp 1072ndash1091 2020

[39] P Liu Q Khan and T Mahmood ldquoGroup decision makingbased on power Heronian aggregation operators underneutrosophic cubic environmentrdquo Soft Computing vol 24no 3 pp 1971ndash1997 2020

[40] X Ma M Akram K Zahid and J C R Alcantud ldquoGroupdecision-making framework using complex Pythagoreanfuzzy informationrdquo Neural Computing and Applicationspp 1ndash21 2020

[41] X Peng J Dai and H Garg ldquoExponential operation andaggregation operator for q-rung orthopair fuzzy set and theirdecision-making method with a new score functionrdquo Inter-national Journal of Intelligent Systems vol 33 no 11pp 2255ndash2282 2018

[42] X Peng and Y Yang ldquoFundamental properties of interval-valued Pythagorean fuzzy aggregation operatorsrdquo Interna-tional Journal of Intelligent Systems vol 31 no 5 pp 444ndash4872016

[43] D Rani and H Garg ldquoComplex intuitionistic fuzzy poweraggregation operators and their applications in multi-criteriadecision-makingrdquo Expert Systems vol 35 no 6 Article IDe12325 2018

[44] K Ullah H Garg T Mahmood N Jan and Z Ali ldquoCor-relation coefficients for T-spherical fuzzy sets and their ap-plications in clustering and multi-attribute decision makingrdquoSoft Computing vol 24 no 3 pp 1647ndash1659 2020

[45] N Waseem M Akram and J C R Alcantud ldquoMulti-attri-bute decision-making based on m-polar fuzzy Hamacheraggregation operatorsrdquo Symmetry vol 11 no 12 p 14982019

[46] G Wei and M Lu ldquoPythagorean fuzzy Maclaurin symmetricmean operators in multiple attribute decision makingrdquo In-ternational Journal of Intelligent Systems vol 33 no 5pp 1043ndash1070 2018

[47] G Wei and M Lu ldquoPythagorean fuzzy power aggregationoperators in multiple attribute decision makingrdquo Interna-tional Journal of Intelligent Systems vol 33 no 1 pp 169ndash1862018

[48] R R Yager ldquoAggregation operators and fuzzy systemsmodelingrdquo Fuzzy Sets and Systems vol 67 no 2 pp 129ndash1451994

[49] R R Yager ldquoPythagorean membership grades in multi-cri-teria decision-makingrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 4 pp 958ndash965 2013

[50] Z Yang X Li H Garg and M Qi ldquoDecision support al-gorithm for selecting an antivirus mask over COVID-19pandemic under spherical normal fuzzy environmentrdquo In-ternational Journal of Environmental Research and PublicHealth vol 17 no 10 p 3407 2020

[51] S Zeng J Chen and X Li ldquoA hybrid method for Pythagoreanfuzzy multiple-criteria decision makingrdquo InternationalJournal of Information Technology amp Decision Making vol 15no 2 pp 403ndash422 2016

[52] J Zhan B Sun and X Zhang ldquoPF-TOPSIS method based onCPFRS models an application to unconventional emergencyeventsrdquo Computers amp Industrial Engineering vol 139 ArticleID 106192 2020

[53] X Wang and E Triantaphyllou ldquoRanking irregularities whenevaluating alternatives by using some ELECTRE methodsrdquoOmega vol 36 no 1 pp 45ndash63 2008

[54] K Bai X Zhu J Wang and R Zhang ldquoSome partitionedMaclaurin symmetric mean based on q-rung orthopair fuzzyinformation for dealing with multi-attribute group decisionmakingrdquo Symmetry vol 10 no 9 p 383 2018

[55] A Fahmi F Amin S Abdullah and A Ali ldquoCubic fuzzyEinstein aggregation operators and its application to decision-makingrdquo International Journal of Systems Science vol 49no 11 pp 2385ndash2397 2018

[56] A Khan S Ashraf S Abdullah M Qiyas J Luo and S KhanldquoPythagorean fuzzy Dombi aggregation operators and theirapplication in decision support systemrdquo Symmetry vol 11no 3 p 383 2019

[57] X Peng and Y Yang ldquoSome results for Pythagorean fuzzysetsrdquo International Journal of Intelligent Systems vol 30no 11 pp 1133ndash1160 2015

[58] X Peng and H Yuan ldquoFundamental properties of Pythag-orean fuzzy aggregation operatorsrdquo Fundamenta Informati-cae vol 147 no 4 pp 415ndash446 2016

[59] X Peng and G Selvachandran ldquoPythagorean fuzzy set state ofthe art and future directionsrdquo Artificial Intelligence Reviewvol 52 no 3 pp 1873ndash1927 2019


introduced Pythagorean fuzzy (PF) AOs Zhao and Wei [9]studied the Einstein hybrid AOs under IF environment ampeidea of IF aggregation using Einstein operations was discussedby Wang and Liu [10] ampe induced interval-valued IF EinsteinAOs were developed by Cai andHan [11] Garg [12] studied thegeneralized PFEinsteinweighted arithmetic AOsGarg [13] alsoproposed the generalized PF Einstein weighted geometric AOsampe Pythagorean Dombi fuzzy AOs with applications werediscussed by Akram et al [14] Shahzadi et al [15] proposed thedecision-making approach using PF Yager AOs Liu andWang[16] expressed q-rung orthopair fuzzy (q-ROF) weighted AOsWei et al [17] studied weighted Heronian mean AOs underq-ROF information q-ROF power Maclaurin AOs were de-veloped by Liu et al [18] Jana et al [19] studiedDombi AOs forq-ROFS Liu and Liu [20] proposed q-ROF Bonferroni meanoperators Joshi and Gegov [21] studied the confidence levelsq-ROF aggregation operators Akram and Shahzadi [22] de-veloped the hybrid decision-makingmodel under q-ROF YagerAOs Liu et al [23] extended the concept of prioritized weightedAOs for complex q-ROFS Senapati and Yager [24] studiedsubtraction division and Fermatean arithmetic mean opera-tions over FFS ampe idea of Fermatean fuzzy (FF) weightedaveraginggeometric operators was also given by Senapati andYager [25] For more information and applications the readerscan refer to [26ndash59]

ampe motivations of this article are described as follows

(1) ampe judgement of a perfect alternative in an FFenvironment is a laborious MADM problem ampeprevalent model possessing the more space than theIF model and PF model vigorously elaborates theimprecise decisions for the selection of bestalternative

(2) As Einstein AOs are the simplest and quite creativeapproach for dealing with DM affairs basically thisarticle directs Einstein AOs in FF surroundings toface complex issues

(3) ampe outcomes based on conclusion are quite accurateunder Einstein AOs when it is put on to the reality-based MADM problems in FF data

(4) ampe proposed operators are keen to provide theoptimal solution not only for FF environment butalso to work efficiently for IF and PF environment

ampe contributions of this article are described as follows

(1) ampe feasibility of FFNs is merged with the aggre-gation skills of Einstein norms to establish morepowerful multiskilled and practical AOs which canbe deployed to aggregate FF data and to get moreaccuracy in decision-making scenarios

(2) ampe dominant properties as well as the notable re-sults of the proposed operators are highlighted

(3) An algorithm is studied to handle complex realisticproblems with FF data

(4) A MADM problem for the selection of an effectivesanitizer to reduce coronavirus is discussed by usingproposed operators

(5) A validity test is discussed for the approval andauthenticity of proposed theory

(6) At the end the benefits and characteristics of theproposed work are discussed by comparison analysis

ampe remaining paper is as follows In Section 2 we recallthe concept of FFS and related score functions Section 3provides Einstein operational laws for FFNs In Sections 4and 5 we study the FFEWA and FFEOWA operators re-spectively and related properties to them In Sections 6 and7 we present the idea of GFFEWA and GFFEOWA oper-ators respectively In Section 8 we propose an algorithm forour new model and discuss a MADM problem for the se-lection of a good sanitizer to reduce the coronavirus Section9 provides the validity criteria to prove the consistency of theproposed work Section 10 gives the comparison analysis ofproposed theory with the FF TOPSIS method In Section 11we have concluded the results related to the proposed model

ampe list of acronyms in research paper is given in Table 1

2 Preliminaries

In this section we recall some basic definitions includingIFS PFS FFS and score functions related to FFS

Definition 1 (see [2]) An IFS I on nonempty setV is givenby

I x μI(x) ]I(x)1113866 11138671113864 1113865 (1)

where μI V⟶ [0 1] and ]I V⟶ [0 1] specify MDand NMD of an element x isin V respectively ϖI(x) 1 minus

μI(x) minus ]I(x) is indeterminacy degree (InD) of an elementx isin V

Definition 2 (see [3]) A PFS P on nonempty setV is givenby

P x μP(x) ]P(x)1113866 11138671113864 1113865 (2)

where μP V⟶ [0 1] and ]P V⟶ [0 1] specify MDand NMD of an element respectively ϖP(x)

1 minus (μP(x))2 minus (]P(x))21113969

is InD

Definition 3 (see [5]) An FFS R on nonempty set V isgiven by

R x μR(x) ]R(x)1113866 11138671113864 1113865 (3)

where μR V⟶ [0 1] ]R V⟶ [0 1] and ϖR(x)

1 minus (μR(x))3 minus (]R(x))33

1113969

specify MD NMD and InDrespectively FFNs are components of the FFS

Definition 4 (see [5]) ampe score function and accuracyfunction for FFN R (μR ]R) are represented by

S(R) μ3R minus ]3R where S(R) isin [minus 1 1]

A(R) μ3R + ]3R whereA(R) isin [0 1](4)



1113924 1113925

and R2 μR2 ]R2

1113924 1113925 ampen







(i) R ]R μR1113866 1113867




31113969

(]1 middot ε]21113884


1113969

)rang



1113969

)1113884(]31 + ]32)(1 + ]31 middot ε]32)

31113969

rang

(vi) λmiddotεR


1113884

(23

radic]λ

(2minus ]3)λ+(]3)λ31113969

)rang

(vii) Rλ (23

radicμλ

(2 minus μ3)λ + (μ3)λ31113969

)1113884


rang



are also FFNs


1 + μR(x)( 11138573

1113872 1113873λ


1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ 1 minus μR(x)( 11138573

1113872 1113873λ

3

11139741113972

le

1 + μR(x)( 11138573

1113872 1113873λ

minus ]R(x)( 11138573

1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ ]R(x)( 11138573

1113872 1113873λ

3

11139741113972

23

radic]R(x)( 1113857

λ

2 minus ]R(x)( 11138573

1113872 1113873λ

+ ]R(x)3

1113872 1113873λ3

1113970

le23

radic]R(x)( 1113857

λ

1 + μR(x)( 11138573

1113872 1113873λ

+ ]R(x)( 11138573

1113872 1113873λ3

1113970

(5)

ampus

1 + μR(x)( 11138573

1113872 1113873λ


1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ 1 minus μR(x)( 11138573

1113872 1113873λ

3

11139741113972

⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠

3

+

23

radic]R(x)( 1113857

λ

2 minus ]R(x)( 11138573

1113872 1113873λ

+ ]R(x)( 11138573

1113872 1113873λ3

1113970⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

3

le 1

(6)

Furthermore

1 + μR(x)( 11138573

1113872 1113873λ


1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ 1 minus μR(x)( 11138573

1113872 1113873λ

3

11139741113972

⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠

3

+

23

radic]R(x)( 1113857

λ

2 minus ]R(x)( 11138573

1113872 1113873λ

+ ]R(x)31113872 1113873λ3

1113970⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

3

0

(7)


1 + μR(x)( 11138573

1113872 1113873λ


1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ 1 minus μR(x)( 11138573

1113872 1113873λ

3

11139741113972

⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠

3

+

23

radic]R(x)( 1113857

λ

2 minus ]R(x)( 11138573

1113872 1113873λ

+ ]R(x)31113872 1113873λ3

1113970⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

3

1

(8)










1 otimes εRλ2



Proof(i)

R1oplusεR2


32

3

11139741113972

]1 middot ε]2


11139691113898 1113899


31

3

11139741113972

]2 middot ε]1


11139691113898 1113899

R2oplusεR1

(9)

(ii)

R1oplusεR2


32

3

11139741113972

]1 middot ε]2


11139691113898 1113899

(10)

is equivalent to

R1oplusεR2



3

11139741113972

1113898

23

radic]1 middot ε]2


31113969 1113899

(11)


R1oplusεR2

a minus b

a + b

31113971

2c3

radic

d + c

3radic1113898 1113899 (12)



a minus b

a + b

31113971

2c3

radic

d + c

3radic1113898 1113899

(1 +((a minus b)(a + b)))λ


(1 +((a minus b)(a + b)))λ


3

11139741113972

23

radicmiddot

2c

3radic

d + c

3radic

1113872 1113873λ


+(2c(d + c))λ3

11139691113898 1113899

aλ

minus bλ

aλ

+ bλ

3

1113971

2c

λ31113968

dλ

+ cλ3

11139681113898 1113899

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

minus 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

+ 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

3

11139741113972

23


λ2

2 minus ]311113872 1113873λ

middot ε 2 minus ]321113872 1113873λ

+ ]311113872 1113873λ

middot ε ]321113872 1113873λ3

11139691113898 1113899

(13)

On the other hand

λ middot εR1

1 + μ311113872 1113873λ

minus 1 minus μ311113872 1113873λ

1 + μ311113872 1113873λ

+ 1 minus μ311113872 1113873λ

3

11139741113972

23

radic]λ1

2 minus ]311113872 1113873λ

+ ]311113872 1113873λ3

11139691113898 1113899

a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899

λ middot εR2

1 + μ321113872 1113873λ

minus 1 minus μ321113872 1113873λ

1 + μ321113872 1113873λ

+ 1 minus μ321113872 1113873λ

3

11139741113972

23

radic]λ2

2 minus ]321113872 1113873λ

+ ]321113872 1113873λ3

11139691113898 1113899

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899

(14)



λ c1 (]31)λ

d1 (2 minus ]31)λ a2 (1 + μ32)

λ b2 (1 minus μ32)λ c2 (]32)


λ therefore


a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899oplusε

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899



31113971

223


1113969


311139691113898 1113899


31113971

2c1 middot εc2

31113968


311139681113898 1113899

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

minus 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

+ 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

3

11139741113972

23


λ2

2 minus ]311113872 1113873λ

middot ε 2 minus ]321113872 1113873λ

+ ]311113872 1113873λ

middot ε ]321113872 1113873λ3

11139691113898 1113899

(15)


λ1 middot εR

1 + μ31113872 1113873λ1

minus 1 minus μ31113872 1113873λ1

1 + μ31113872 1113873λ1

+ 1 minus μ31113872 1113873λ1

3

11139741113972

23

radic]λ1

2 minus ]31113872 1113873λ1

+ ]31113872 1113873λ13

11139691113898 1113899

a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899

λ2 middot εR

1 + μ31113872 1113873λ2

minus 1 minus μ31113872 1113873λ2

1 + μ31113872 1113873λ2

+ 1 minus μ31113872 1113873λ2

3

11139741113972

23

radic]λ2

2 minus ]31113872 1113873λ2

+ ]31113872 1113873λ23

11139691113898 1113899

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899

(16)



a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899oplusε

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899



31113971

223


1113969


311139691113898 1113899


31113971

2c1 middot εc2

31113968


311139681113898 1113899

1 + μ31113872 1113873λ1+λ2

minus 1 minus μ31113872 1113873λ1+λ2

1 + μ31113872 1113873λ1+λ2

+ 1 minus μ31113872 1113873λ1+λ2

3

11139741113972

23

radic]λ1+λ2

2 minus ]31113872 1113873λ1+λ2

+ ]311113872 1113873λ1+λ23

11139691113898 1113899

λ1 + λ2( 1113857 middot εR

(17)




(i) Rc1andεR

c2 (R1orεR2)

c

(ii) Rc1orεR

c2 (R1andεR2)

c

(iii) Rc1oplusεR

c2 (R1 otimes εR2)

c

(iv) Rc1 otimes εR

c2 (R1oplusεR2)

c



Proof It is obvious
















(18)


FFA R1R2 Rs( 1113857 1s




1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 + μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 + μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj311139691113898 1113899 (20)


When s 2



w1 middot εR1

1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

3

11139741113972

2]w11

31113969

2 minus ]311113872 1113873w1

+ ]311113872 1113873w13

11139691113898 1113899

w2 middot εR2

1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

3

11139741113972

23

radic]w22

2 minus ]321113872 1113873w2

+ ]321113872 1113873w23

11139691113898 1113899

(22)


ampen


1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 + 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

1 + 1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 middot ε 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

3

11139741113972

1113898

23

radic]w11

2 minus ]311113872 1113873w1

+ ]311113872 1113873w13

1113969

1113874 1113875 middot ε23

radic]w21

2 minus ]321113872 1113873w2

+ ]321113872 1113873w23

1113969

1113874 1113875

1 + 1 minus 2]3w11 2 minus ]311113872 1113873

w1+ ]311113872 1113873

w11113872 11138731113872 11138731113872 1113873 middot ε 1 minus 2]3w2

2 2 minus ]321113872 1113873w2

+ ]321113872 1113873w2

1113872 11138731113872 11138731113872 11138733

1113969 1113899

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

minus 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

+ 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

3

11139741113972

23


w22

2 minus ]311113872 1113873w1

middot ε 2 minus ]321113872 1113873w2

+ ]311113872 1113873w1

middot ε ]321113872 1113873w23

11139691113898 1113899

(23)



1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899 (24)

Now for s k + 1

FFEWA R1R2 Rk+1( 1113857

1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899

oplusε

1 + μ3k+11113872 1113873wk+1

minus 1 minus μ3k+11113872 1113873wk+1

1 + μ3k+11113872 1113873wk+1

+ 1 minus μ3k+11113872 1113873wk+1

3

11139741113972

23

radic]wk+1

k+1

2 minus ]3k+11113872 1113873wk+1

+ ]3k+11113872 1113873wk+13

11139691113898 1113899

1113937k+1j1 1 + μ3j1113872 1113873

wjminus 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

1113937k+1j1 1 + μ3j1113872 1113873

wj+ 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

k+1j1]

wj

j

1113937k+1j1 2 minus ]3j1113872 1113873

wj+ 1113937

k+1j1 ]3j1113872 1113873

wj311139691113898 1113899

(25)



then

1113945

s

j1R

wj

j le 1113944s

j1wjRj (26)





1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj 1

minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le 1 minus 1113945s

j11 minus μ3j1113872 1113873

wj le 1

(27)


sj1


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjge 0 (28)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le1113945s

j11 minus μ3j1113872 1113873

wj le 1

(29)

Also

1113945

s

j1]3j1113872 1113873

wj ge 0⟺21113937

sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge 0

(30)



sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le 1 minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

1

(31)




(32)



sj1 (1 + μ3j)wj + 1113937


sj1 (1 + μ3j)wj+


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le

1 minus 1113945s

j11 minus μ3j1113872 1113873

wj⟺ μβR le μR3

11139741113972

(33)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge

21113937sj1 ]3j1113872 1113873

wj

1113936sj1 wj 2 minus ]3j1113872 1113873 + 1113936

sj1 wj]

3j

ge1113945s

j1]3j1113872 1113873

wjrArr

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

3

11139741113972

ge1113945s

j1]wj

j rArr]βR le ]R

(34)



S Rβ

1113872 1113873 μβR1113872 11138733

minus ]βR1113872 11138733le μR( 1113857

3minus ]R( 1113857

3 S(R) (35)



(36)





(37)

Hence


(38)



FFEWA R1R2R3R4( 1113857

11139374j1 1 + μ3j1113872 1113873

wjminus 1113937

4j1 1 minus μ3j1113872 1113873

wj

11139374j1 1 + μ3j1113872 1113873

wj+ 1113937

4j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

j

11139374j1 2 minus ]3j1113872 1113873

wj+ 1113937

4j1 ]3j1113872 1113873

wj311139691113898 1113899

149 minus 048149 + 048

31113970

23

radictimes 055

180 + 0163

radic1113898 1113899 080 055lang rang

(39)

Now

FFWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3j1113872 1113873

wj 1113945

4

j1]j1113872 1113873


3

11139741113972

1113898(40)


sj1 wj 1


FFEWA R1R2 Rs( 1113857 Ro (41)



(42)



(43)



1113937sj1 1 + μ3o1113872 1113873

wjminus 1113937

sj1 1 minus μ3o1113872 1113873

wj

1113937sj1 1 + μ3o1113872 1113873

wj+ 1113937

sj1 1 minus μ3o1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

o

1113937sj1 2 minus ]3o1113872 1113873

wj+ 1113937

sj1 ]3o1113872 1113873

wj311139691113898 1113899

1 + μ3o1113872 11138731113936


1113936sj1 wj

1 + μ3o1113872 11138731113936

sj1 wj + 1 minus μ3o1113872 1113873

1113936sj1 wj

3

111397411139731113972

23

radic]1113936

sj1 wj

o

2 minus ]3o1113872 11138731113936

sj1 wj + ]3o1113872 1113873

1113936sj1 wj

311139701113898 1113899 μo ]o1113866 1113867

(44)





have

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

1113945

s

j1

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj

le1113945s

j1


1113888 1113889

wj

le1113945s

j1

1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

⟺1 minus μ3jmax

1 + μ3jmax1113888 1113889

1113944s

j1wj

le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

1113944s

j1wj

⟺1 minus μ3jmax

1 + μ3jmax

⎛⎝ ⎞⎠le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin

⎛⎝ ⎞⎠

⟺2

1 + μ3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1


1113888 1113889

wj

le2

1 + μ3jmin

⎛⎝ ⎞⎠

⟺1 + μ3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle

1 + μ3jmax

2⎛⎝ ⎞⎠

⟺ 1 + μ3jmin1113872 1113873le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle 1 + μ3jmax1113872 1113873

⟺ μ3jmin le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjminus 1le μ3jmax

⟺ μ3jmin le1113937

sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjle μ3jmax

(45)

ampus

μjmin le

1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le μjmax (46)



jmin le ]3

j le ]3

jmax

forallj then g(]3

jmax)leg(]3

j)le g(]3


]3

jmax)]3

jmax)le ((2minus ]3

j)]3

j)le ((2 minus ]3

jmin)]3

jmin) ampen


2 minus ]3jmax

]3jmax1113888 1113889

wj

le2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

wj

1113945

s

j1

2 minus ]3jmax

]3jmax1113888 1113889

wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le1113945s

j1

2 minus ]3jmin

]3jmin1113888 1113889

wj

rArr2 minus ]3jmax

]3jmax1113888 1113889

1113944

s

j1wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

1113944

s

j1wj

rArr2 minus ]3jmax

]3jmax

⎛⎝ ⎞⎠le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin

⎛⎝ ⎞⎠

rArr2

]3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2

]3jmin

⎛⎝ ⎞⎠

rArr]3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle

]3jmax

2⎛⎝ ⎞⎠

rArr ]3jmin1113872 1113873le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax1113872 1113873

rArr]3jmin le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax


sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjle ]3jmax

rArr]jmin le

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj31113969 le ]jmax

(47)








(49)









1 2 s



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

1113898

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj31113969 1113899

(51)





(52)


(03 03 02 02)T as

S R1( 1113857 (06)3

minus (07)3

minus 013

S R2( 1113857 (08)3

minus (07)3

017

S R3( 1113857 (06)3

minus (09)3

minus 051

S R4( 1113857 (09)3

minus (04)3

067

(53)


Rϱ(1) R4 (09 04)

Rϱ(2) R2 (08 07)

Rϱ(3) R1 (06 07)

Rϱ(4) R3 (06 09)

(54)



11139374j1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

11139374j1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

ϱ(j)

11139374j1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

4j1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

144 minus 049144 + 049

31113970

23

radictimes 066

162 + 0663

radic1113898 1113899 079 055lang rang

(55)

Now

FFOWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3ϱ(j)1113872 1113873

wj3

11139741113972

11139454

j1]ϱ(j)1113872 1113873

wj1113898 1113899 080 066lang rang


(56)


sj1 wj 1


FFEOWA R1R2 Rs( 1113857 Ro (57)



(58)



(59)



sj1 wj 1 then



1λ

(60)









23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

middot

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(61)

Proof Since

Rλj

23

radicμλj

2 minus μ3j1113872 1113873λ

+ μ3j1113872 1113873λ3

1113969

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

3

11139741113972

1113898 1113899


1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1113874 1113875 1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

1113874 11138753

1113970

1113888 1113889

wj

1113937sj1 2 minus 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 1113875

wj3

1113970 1113899

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ3

1113970

1113888 1113889

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj3

1113970 1113899

(62)

amperefore


1λ

2 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

1113971

2 minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

11139711113898

1 + 21113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

minus 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

1 + 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

3

1113974111397311139731113972

1113899

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

11139711113898

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(63)


When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)



sj1 wj 1



1λ

(65)

where λgt 0





23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)








23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod






























































1113924 1113925

and R2 μR2 ]R2

1113924 1113925 ampen







(i) R ]R μR1113866 1113867




31113969

(]1 middot ε]21113884


1113969

)rang



1113969

)1113884(]31 + ]32)(1 + ]31 middot ε]32)

31113969

rang

(vi) λmiddotεR


1113884

(23

radic]λ

(2minus ]3)λ+(]3)λ31113969

)rang

(vii) Rλ (23

radicμλ

(2 minus μ3)λ + (μ3)λ31113969

)1113884


rang



are also FFNs


1 + μR(x)( 11138573

1113872 1113873λ


1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ 1 minus μR(x)( 11138573

1113872 1113873λ

3

11139741113972

le

1 + μR(x)( 11138573

1113872 1113873λ

minus ]R(x)( 11138573

1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ ]R(x)( 11138573

1113872 1113873λ

3

11139741113972

23

radic]R(x)( 1113857

λ

2 minus ]R(x)( 11138573

1113872 1113873λ

+ ]R(x)3

1113872 1113873λ3

1113970

le23

radic]R(x)( 1113857

λ

1 + μR(x)( 11138573

1113872 1113873λ

+ ]R(x)( 11138573

1113872 1113873λ3

1113970

(5)

ampus

1 + μR(x)( 11138573

1113872 1113873λ


1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ 1 minus μR(x)( 11138573

1113872 1113873λ

3

11139741113972

⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠

3

+

23

radic]R(x)( 1113857

λ

2 minus ]R(x)( 11138573

1113872 1113873λ

+ ]R(x)( 11138573

1113872 1113873λ3

1113970⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

3

le 1

(6)

Furthermore

1 + μR(x)( 11138573

1113872 1113873λ


1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ 1 minus μR(x)( 11138573

1113872 1113873λ

3

11139741113972

⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠

3

+

23

radic]R(x)( 1113857

λ

2 minus ]R(x)( 11138573

1113872 1113873λ

+ ]R(x)31113872 1113873λ3

1113970⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

3

0

(7)


1 + μR(x)( 11138573

1113872 1113873λ


1113872 1113873λ

1 + μR(x)( 11138573

1113872 1113873λ

+ 1 minus μR(x)( 11138573

1113872 1113873λ

3

11139741113972

⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠

3

+

23

radic]R(x)( 1113857

λ

2 minus ]R(x)( 11138573

1113872 1113873λ

+ ]R(x)31113872 1113873λ3

1113970⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

3

1

(8)










1 otimes εRλ2



Proof(i)

R1oplusεR2


32

3

11139741113972

]1 middot ε]2


11139691113898 1113899


31

3

11139741113972

]2 middot ε]1


11139691113898 1113899

R2oplusεR1

(9)

(ii)

R1oplusεR2


32

3

11139741113972

]1 middot ε]2


11139691113898 1113899

(10)

is equivalent to

R1oplusεR2



3

11139741113972

1113898

23

radic]1 middot ε]2


31113969 1113899

(11)


R1oplusεR2

a minus b

a + b

31113971

2c3

radic

d + c

3radic1113898 1113899 (12)



a minus b

a + b

31113971

2c3

radic

d + c

3radic1113898 1113899

(1 +((a minus b)(a + b)))λ


(1 +((a minus b)(a + b)))λ


3

11139741113972

23

radicmiddot

2c

3radic

d + c

3radic

1113872 1113873λ


+(2c(d + c))λ3

11139691113898 1113899

aλ

minus bλ

aλ

+ bλ

3

1113971

2c

λ31113968

dλ

+ cλ3

11139681113898 1113899

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

minus 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

+ 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

3

11139741113972

23


λ2

2 minus ]311113872 1113873λ

middot ε 2 minus ]321113872 1113873λ

+ ]311113872 1113873λ

middot ε ]321113872 1113873λ3

11139691113898 1113899

(13)

On the other hand

λ middot εR1

1 + μ311113872 1113873λ

minus 1 minus μ311113872 1113873λ

1 + μ311113872 1113873λ

+ 1 minus μ311113872 1113873λ

3

11139741113972

23

radic]λ1

2 minus ]311113872 1113873λ

+ ]311113872 1113873λ3

11139691113898 1113899

a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899

λ middot εR2

1 + μ321113872 1113873λ

minus 1 minus μ321113872 1113873λ

1 + μ321113872 1113873λ

+ 1 minus μ321113872 1113873λ

3

11139741113972

23

radic]λ2

2 minus ]321113872 1113873λ

+ ]321113872 1113873λ3

11139691113898 1113899

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899

(14)



λ c1 (]31)λ

d1 (2 minus ]31)λ a2 (1 + μ32)

λ b2 (1 minus μ32)λ c2 (]32)


λ therefore


a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899oplusε

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899



31113971

223


1113969


311139691113898 1113899


31113971

2c1 middot εc2

31113968


311139681113898 1113899

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

minus 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

+ 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

3

11139741113972

23


λ2

2 minus ]311113872 1113873λ

middot ε 2 minus ]321113872 1113873λ

+ ]311113872 1113873λ

middot ε ]321113872 1113873λ3

11139691113898 1113899

(15)


λ1 middot εR

1 + μ31113872 1113873λ1

minus 1 minus μ31113872 1113873λ1

1 + μ31113872 1113873λ1

+ 1 minus μ31113872 1113873λ1

3

11139741113972

23

radic]λ1

2 minus ]31113872 1113873λ1

+ ]31113872 1113873λ13

11139691113898 1113899

a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899

λ2 middot εR

1 + μ31113872 1113873λ2

minus 1 minus μ31113872 1113873λ2

1 + μ31113872 1113873λ2

+ 1 minus μ31113872 1113873λ2

3

11139741113972

23

radic]λ2

2 minus ]31113872 1113873λ2

+ ]31113872 1113873λ23

11139691113898 1113899

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899

(16)



a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899oplusε

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899



31113971

223


1113969


311139691113898 1113899


31113971

2c1 middot εc2

31113968


311139681113898 1113899

1 + μ31113872 1113873λ1+λ2

minus 1 minus μ31113872 1113873λ1+λ2

1 + μ31113872 1113873λ1+λ2

+ 1 minus μ31113872 1113873λ1+λ2

3

11139741113972

23

radic]λ1+λ2

2 minus ]31113872 1113873λ1+λ2

+ ]311113872 1113873λ1+λ23

11139691113898 1113899

λ1 + λ2( 1113857 middot εR

(17)




(i) Rc1andεR

c2 (R1orεR2)

c

(ii) Rc1orεR

c2 (R1andεR2)

c

(iii) Rc1oplusεR

c2 (R1 otimes εR2)

c

(iv) Rc1 otimes εR

c2 (R1oplusεR2)

c



Proof It is obvious
















(18)


FFA R1R2 Rs( 1113857 1s




1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 + μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 + μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj311139691113898 1113899 (20)


When s 2



w1 middot εR1

1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

3

11139741113972

2]w11

31113969

2 minus ]311113872 1113873w1

+ ]311113872 1113873w13

11139691113898 1113899

w2 middot εR2

1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

3

11139741113972

23

radic]w22

2 minus ]321113872 1113873w2

+ ]321113872 1113873w23

11139691113898 1113899

(22)


ampen


1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 + 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

1 + 1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 middot ε 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

3

11139741113972

1113898

23

radic]w11

2 minus ]311113872 1113873w1

+ ]311113872 1113873w13

1113969

1113874 1113875 middot ε23

radic]w21

2 minus ]321113872 1113873w2

+ ]321113872 1113873w23

1113969

1113874 1113875

1 + 1 minus 2]3w11 2 minus ]311113872 1113873

w1+ ]311113872 1113873

w11113872 11138731113872 11138731113872 1113873 middot ε 1 minus 2]3w2

2 2 minus ]321113872 1113873w2

+ ]321113872 1113873w2

1113872 11138731113872 11138731113872 11138733

1113969 1113899

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

minus 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

+ 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

3

11139741113972

23


w22

2 minus ]311113872 1113873w1

middot ε 2 minus ]321113872 1113873w2

+ ]311113872 1113873w1

middot ε ]321113872 1113873w23

11139691113898 1113899

(23)



1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899 (24)

Now for s k + 1

FFEWA R1R2 Rk+1( 1113857

1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899

oplusε

1 + μ3k+11113872 1113873wk+1

minus 1 minus μ3k+11113872 1113873wk+1

1 + μ3k+11113872 1113873wk+1

+ 1 minus μ3k+11113872 1113873wk+1

3

11139741113972

23

radic]wk+1

k+1

2 minus ]3k+11113872 1113873wk+1

+ ]3k+11113872 1113873wk+13

11139691113898 1113899

1113937k+1j1 1 + μ3j1113872 1113873

wjminus 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

1113937k+1j1 1 + μ3j1113872 1113873

wj+ 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

k+1j1]

wj

j

1113937k+1j1 2 minus ]3j1113872 1113873

wj+ 1113937

k+1j1 ]3j1113872 1113873

wj311139691113898 1113899

(25)



then

1113945

s

j1R

wj

j le 1113944s

j1wjRj (26)





1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj 1

minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le 1 minus 1113945s

j11 minus μ3j1113872 1113873

wj le 1

(27)


sj1


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjge 0 (28)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le1113945s

j11 minus μ3j1113872 1113873

wj le 1

(29)

Also

1113945

s

j1]3j1113872 1113873

wj ge 0⟺21113937

sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge 0

(30)



sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le 1 minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

1

(31)




(32)



sj1 (1 + μ3j)wj + 1113937


sj1 (1 + μ3j)wj+


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le

1 minus 1113945s

j11 minus μ3j1113872 1113873

wj⟺ μβR le μR3

11139741113972

(33)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge

21113937sj1 ]3j1113872 1113873

wj

1113936sj1 wj 2 minus ]3j1113872 1113873 + 1113936

sj1 wj]

3j

ge1113945s

j1]3j1113872 1113873

wjrArr

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

3

11139741113972

ge1113945s

j1]wj

j rArr]βR le ]R

(34)



S Rβ

1113872 1113873 μβR1113872 11138733

minus ]βR1113872 11138733le μR( 1113857

3minus ]R( 1113857

3 S(R) (35)



(36)





(37)

Hence


(38)



FFEWA R1R2R3R4( 1113857

11139374j1 1 + μ3j1113872 1113873

wjminus 1113937

4j1 1 minus μ3j1113872 1113873

wj

11139374j1 1 + μ3j1113872 1113873

wj+ 1113937

4j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

j

11139374j1 2 minus ]3j1113872 1113873

wj+ 1113937

4j1 ]3j1113872 1113873

wj311139691113898 1113899

149 minus 048149 + 048

31113970

23

radictimes 055

180 + 0163

radic1113898 1113899 080 055lang rang

(39)

Now

FFWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3j1113872 1113873

wj 1113945

4

j1]j1113872 1113873


3

11139741113972

1113898(40)


sj1 wj 1


FFEWA R1R2 Rs( 1113857 Ro (41)



(42)



(43)



1113937sj1 1 + μ3o1113872 1113873

wjminus 1113937

sj1 1 minus μ3o1113872 1113873

wj

1113937sj1 1 + μ3o1113872 1113873

wj+ 1113937

sj1 1 minus μ3o1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

o

1113937sj1 2 minus ]3o1113872 1113873

wj+ 1113937

sj1 ]3o1113872 1113873

wj311139691113898 1113899

1 + μ3o1113872 11138731113936


1113936sj1 wj

1 + μ3o1113872 11138731113936

sj1 wj + 1 minus μ3o1113872 1113873

1113936sj1 wj

3

111397411139731113972

23

radic]1113936

sj1 wj

o

2 minus ]3o1113872 11138731113936

sj1 wj + ]3o1113872 1113873

1113936sj1 wj

311139701113898 1113899 μo ]o1113866 1113867

(44)





have

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

1113945

s

j1

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj

le1113945s

j1


1113888 1113889

wj

le1113945s

j1

1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

⟺1 minus μ3jmax

1 + μ3jmax1113888 1113889

1113944s

j1wj

le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

1113944s

j1wj

⟺1 minus μ3jmax

1 + μ3jmax

⎛⎝ ⎞⎠le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin

⎛⎝ ⎞⎠

⟺2

1 + μ3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1


1113888 1113889

wj

le2

1 + μ3jmin

⎛⎝ ⎞⎠

⟺1 + μ3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle

1 + μ3jmax

2⎛⎝ ⎞⎠

⟺ 1 + μ3jmin1113872 1113873le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle 1 + μ3jmax1113872 1113873

⟺ μ3jmin le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjminus 1le μ3jmax

⟺ μ3jmin le1113937

sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjle μ3jmax

(45)

ampus

μjmin le

1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le μjmax (46)



jmin le ]3

j le ]3

jmax

forallj then g(]3

jmax)leg(]3

j)le g(]3


]3

jmax)]3

jmax)le ((2minus ]3

j)]3

j)le ((2 minus ]3

jmin)]3

jmin) ampen


2 minus ]3jmax

]3jmax1113888 1113889

wj

le2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

wj

1113945

s

j1

2 minus ]3jmax

]3jmax1113888 1113889

wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le1113945s

j1

2 minus ]3jmin

]3jmin1113888 1113889

wj

rArr2 minus ]3jmax

]3jmax1113888 1113889

1113944

s

j1wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

1113944

s

j1wj

rArr2 minus ]3jmax

]3jmax

⎛⎝ ⎞⎠le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin

⎛⎝ ⎞⎠

rArr2

]3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2

]3jmin

⎛⎝ ⎞⎠

rArr]3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle

]3jmax

2⎛⎝ ⎞⎠

rArr ]3jmin1113872 1113873le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax1113872 1113873

rArr]3jmin le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax


sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjle ]3jmax

rArr]jmin le

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj31113969 le ]jmax

(47)








(49)









1 2 s



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

1113898

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj31113969 1113899

(51)





(52)


(03 03 02 02)T as

S R1( 1113857 (06)3

minus (07)3

minus 013

S R2( 1113857 (08)3

minus (07)3

017

S R3( 1113857 (06)3

minus (09)3

minus 051

S R4( 1113857 (09)3

minus (04)3

067

(53)


Rϱ(1) R4 (09 04)

Rϱ(2) R2 (08 07)

Rϱ(3) R1 (06 07)

Rϱ(4) R3 (06 09)

(54)



11139374j1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

11139374j1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

ϱ(j)

11139374j1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

4j1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

144 minus 049144 + 049

31113970

23

radictimes 066

162 + 0663

radic1113898 1113899 079 055lang rang

(55)

Now

FFOWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3ϱ(j)1113872 1113873

wj3

11139741113972

11139454

j1]ϱ(j)1113872 1113873

wj1113898 1113899 080 066lang rang


(56)


sj1 wj 1


FFEOWA R1R2 Rs( 1113857 Ro (57)



(58)



(59)



sj1 wj 1 then



1λ

(60)









23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

middot

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(61)

Proof Since

Rλj

23

radicμλj

2 minus μ3j1113872 1113873λ

+ μ3j1113872 1113873λ3

1113969

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

3

11139741113972

1113898 1113899


1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1113874 1113875 1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

1113874 11138753

1113970

1113888 1113889

wj

1113937sj1 2 minus 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 1113875

wj3

1113970 1113899

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ3

1113970

1113888 1113889

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj3

1113970 1113899

(62)

amperefore


1λ

2 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

1113971

2 minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

11139711113898

1 + 21113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

minus 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

1 + 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

3

1113974111397311139731113972

1113899

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

11139711113898

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(63)


When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)



sj1 wj 1



1λ

(65)

where λgt 0





23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)








23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod






























































1 otimes εRλ2



Proof(i)

R1oplusεR2


32

3

11139741113972

]1 middot ε]2


11139691113898 1113899


31

3

11139741113972

]2 middot ε]1


11139691113898 1113899

R2oplusεR1

(9)

(ii)

R1oplusεR2


32

3

11139741113972

]1 middot ε]2


11139691113898 1113899

(10)

is equivalent to

R1oplusεR2



3

11139741113972

1113898

23

radic]1 middot ε]2


31113969 1113899

(11)


R1oplusεR2

a minus b

a + b

31113971

2c3

radic

d + c

3radic1113898 1113899 (12)



a minus b

a + b

31113971

2c3

radic

d + c

3radic1113898 1113899

(1 +((a minus b)(a + b)))λ


(1 +((a minus b)(a + b)))λ


3

11139741113972

23

radicmiddot

2c

3radic

d + c

3radic

1113872 1113873λ


+(2c(d + c))λ3

11139691113898 1113899

aλ

minus bλ

aλ

+ bλ

3

1113971

2c

λ31113968

dλ

+ cλ3

11139681113898 1113899

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

minus 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

+ 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

3

11139741113972

23


λ2

2 minus ]311113872 1113873λ

middot ε 2 minus ]321113872 1113873λ

+ ]311113872 1113873λ

middot ε ]321113872 1113873λ3

11139691113898 1113899

(13)

On the other hand

λ middot εR1

1 + μ311113872 1113873λ

minus 1 minus μ311113872 1113873λ

1 + μ311113872 1113873λ

+ 1 minus μ311113872 1113873λ

3

11139741113972

23

radic]λ1

2 minus ]311113872 1113873λ

+ ]311113872 1113873λ3

11139691113898 1113899

a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899

λ middot εR2

1 + μ321113872 1113873λ

minus 1 minus μ321113872 1113873λ

1 + μ321113872 1113873λ

+ 1 minus μ321113872 1113873λ

3

11139741113972

23

radic]λ2

2 minus ]321113872 1113873λ

+ ]321113872 1113873λ3

11139691113898 1113899

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899

(14)



λ c1 (]31)λ

d1 (2 minus ]31)λ a2 (1 + μ32)

λ b2 (1 minus μ32)λ c2 (]32)


λ therefore


a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899oplusε

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899



31113971

223


1113969


311139691113898 1113899


31113971

2c1 middot εc2

31113968


311139681113898 1113899

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

minus 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

+ 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

3

11139741113972

23


λ2

2 minus ]311113872 1113873λ

middot ε 2 minus ]321113872 1113873λ

+ ]311113872 1113873λ

middot ε ]321113872 1113873λ3

11139691113898 1113899

(15)


λ1 middot εR

1 + μ31113872 1113873λ1

minus 1 minus μ31113872 1113873λ1

1 + μ31113872 1113873λ1

+ 1 minus μ31113872 1113873λ1

3

11139741113972

23

radic]λ1

2 minus ]31113872 1113873λ1

+ ]31113872 1113873λ13

11139691113898 1113899

a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899

λ2 middot εR

1 + μ31113872 1113873λ2

minus 1 minus μ31113872 1113873λ2

1 + μ31113872 1113873λ2

+ 1 minus μ31113872 1113873λ2

3

11139741113972

23

radic]λ2

2 minus ]31113872 1113873λ2

+ ]31113872 1113873λ23

11139691113898 1113899

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899

(16)



a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899oplusε

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899



31113971

223


1113969


311139691113898 1113899


31113971

2c1 middot εc2

31113968


311139681113898 1113899

1 + μ31113872 1113873λ1+λ2

minus 1 minus μ31113872 1113873λ1+λ2

1 + μ31113872 1113873λ1+λ2

+ 1 minus μ31113872 1113873λ1+λ2

3

11139741113972

23

radic]λ1+λ2

2 minus ]31113872 1113873λ1+λ2

+ ]311113872 1113873λ1+λ23

11139691113898 1113899

λ1 + λ2( 1113857 middot εR

(17)




(i) Rc1andεR

c2 (R1orεR2)

c

(ii) Rc1orεR

c2 (R1andεR2)

c

(iii) Rc1oplusεR

c2 (R1 otimes εR2)

c

(iv) Rc1 otimes εR

c2 (R1oplusεR2)

c



Proof It is obvious
















(18)


FFA R1R2 Rs( 1113857 1s




1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 + μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 + μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj311139691113898 1113899 (20)


When s 2



w1 middot εR1

1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

3

11139741113972

2]w11

31113969

2 minus ]311113872 1113873w1

+ ]311113872 1113873w13

11139691113898 1113899

w2 middot εR2

1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

3

11139741113972

23

radic]w22

2 minus ]321113872 1113873w2

+ ]321113872 1113873w23

11139691113898 1113899

(22)


ampen


1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 + 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

1 + 1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 middot ε 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

3

11139741113972

1113898

23

radic]w11

2 minus ]311113872 1113873w1

+ ]311113872 1113873w13

1113969

1113874 1113875 middot ε23

radic]w21

2 minus ]321113872 1113873w2

+ ]321113872 1113873w23

1113969

1113874 1113875

1 + 1 minus 2]3w11 2 minus ]311113872 1113873

w1+ ]311113872 1113873

w11113872 11138731113872 11138731113872 1113873 middot ε 1 minus 2]3w2

2 2 minus ]321113872 1113873w2

+ ]321113872 1113873w2

1113872 11138731113872 11138731113872 11138733

1113969 1113899

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

minus 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

+ 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

3

11139741113972

23


w22

2 minus ]311113872 1113873w1

middot ε 2 minus ]321113872 1113873w2

+ ]311113872 1113873w1

middot ε ]321113872 1113873w23

11139691113898 1113899

(23)



1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899 (24)

Now for s k + 1

FFEWA R1R2 Rk+1( 1113857

1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899

oplusε

1 + μ3k+11113872 1113873wk+1

minus 1 minus μ3k+11113872 1113873wk+1

1 + μ3k+11113872 1113873wk+1

+ 1 minus μ3k+11113872 1113873wk+1

3

11139741113972

23

radic]wk+1

k+1

2 minus ]3k+11113872 1113873wk+1

+ ]3k+11113872 1113873wk+13

11139691113898 1113899

1113937k+1j1 1 + μ3j1113872 1113873

wjminus 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

1113937k+1j1 1 + μ3j1113872 1113873

wj+ 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

k+1j1]

wj

j

1113937k+1j1 2 minus ]3j1113872 1113873

wj+ 1113937

k+1j1 ]3j1113872 1113873

wj311139691113898 1113899

(25)



then

1113945

s

j1R

wj

j le 1113944s

j1wjRj (26)





1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj 1

minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le 1 minus 1113945s

j11 minus μ3j1113872 1113873

wj le 1

(27)


sj1


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjge 0 (28)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le1113945s

j11 minus μ3j1113872 1113873

wj le 1

(29)

Also

1113945

s

j1]3j1113872 1113873

wj ge 0⟺21113937

sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge 0

(30)



sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le 1 minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

1

(31)




(32)



sj1 (1 + μ3j)wj + 1113937


sj1 (1 + μ3j)wj+


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le

1 minus 1113945s

j11 minus μ3j1113872 1113873

wj⟺ μβR le μR3

11139741113972

(33)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge

21113937sj1 ]3j1113872 1113873

wj

1113936sj1 wj 2 minus ]3j1113872 1113873 + 1113936

sj1 wj]

3j

ge1113945s

j1]3j1113872 1113873

wjrArr

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

3

11139741113972

ge1113945s

j1]wj

j rArr]βR le ]R

(34)



S Rβ

1113872 1113873 μβR1113872 11138733

minus ]βR1113872 11138733le μR( 1113857

3minus ]R( 1113857

3 S(R) (35)



(36)





(37)

Hence


(38)



FFEWA R1R2R3R4( 1113857

11139374j1 1 + μ3j1113872 1113873

wjminus 1113937

4j1 1 minus μ3j1113872 1113873

wj

11139374j1 1 + μ3j1113872 1113873

wj+ 1113937

4j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

j

11139374j1 2 minus ]3j1113872 1113873

wj+ 1113937

4j1 ]3j1113872 1113873

wj311139691113898 1113899

149 minus 048149 + 048

31113970

23

radictimes 055

180 + 0163

radic1113898 1113899 080 055lang rang

(39)

Now

FFWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3j1113872 1113873

wj 1113945

4

j1]j1113872 1113873


3

11139741113972

1113898(40)


sj1 wj 1


FFEWA R1R2 Rs( 1113857 Ro (41)



(42)



(43)



1113937sj1 1 + μ3o1113872 1113873

wjminus 1113937

sj1 1 minus μ3o1113872 1113873

wj

1113937sj1 1 + μ3o1113872 1113873

wj+ 1113937

sj1 1 minus μ3o1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

o

1113937sj1 2 minus ]3o1113872 1113873

wj+ 1113937

sj1 ]3o1113872 1113873

wj311139691113898 1113899

1 + μ3o1113872 11138731113936


1113936sj1 wj

1 + μ3o1113872 11138731113936

sj1 wj + 1 minus μ3o1113872 1113873

1113936sj1 wj

3

111397411139731113972

23

radic]1113936

sj1 wj

o

2 minus ]3o1113872 11138731113936

sj1 wj + ]3o1113872 1113873

1113936sj1 wj

311139701113898 1113899 μo ]o1113866 1113867

(44)





have

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

1113945

s

j1

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj

le1113945s

j1


1113888 1113889

wj

le1113945s

j1

1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

⟺1 minus μ3jmax

1 + μ3jmax1113888 1113889

1113944s

j1wj

le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

1113944s

j1wj

⟺1 minus μ3jmax

1 + μ3jmax

⎛⎝ ⎞⎠le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin

⎛⎝ ⎞⎠

⟺2

1 + μ3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1


1113888 1113889

wj

le2

1 + μ3jmin

⎛⎝ ⎞⎠

⟺1 + μ3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle

1 + μ3jmax

2⎛⎝ ⎞⎠

⟺ 1 + μ3jmin1113872 1113873le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle 1 + μ3jmax1113872 1113873

⟺ μ3jmin le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjminus 1le μ3jmax

⟺ μ3jmin le1113937

sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjle μ3jmax

(45)

ampus

μjmin le

1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le μjmax (46)



jmin le ]3

j le ]3

jmax

forallj then g(]3

jmax)leg(]3

j)le g(]3


]3

jmax)]3

jmax)le ((2minus ]3

j)]3

j)le ((2 minus ]3

jmin)]3

jmin) ampen


2 minus ]3jmax

]3jmax1113888 1113889

wj

le2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

wj

1113945

s

j1

2 minus ]3jmax

]3jmax1113888 1113889

wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le1113945s

j1

2 minus ]3jmin

]3jmin1113888 1113889

wj

rArr2 minus ]3jmax

]3jmax1113888 1113889

1113944

s

j1wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

1113944

s

j1wj

rArr2 minus ]3jmax

]3jmax

⎛⎝ ⎞⎠le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin

⎛⎝ ⎞⎠

rArr2

]3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2

]3jmin

⎛⎝ ⎞⎠

rArr]3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle

]3jmax

2⎛⎝ ⎞⎠

rArr ]3jmin1113872 1113873le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax1113872 1113873

rArr]3jmin le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax


sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjle ]3jmax

rArr]jmin le

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj31113969 le ]jmax

(47)








(49)









1 2 s



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

1113898

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj31113969 1113899

(51)





(52)


(03 03 02 02)T as

S R1( 1113857 (06)3

minus (07)3

minus 013

S R2( 1113857 (08)3

minus (07)3

017

S R3( 1113857 (06)3

minus (09)3

minus 051

S R4( 1113857 (09)3

minus (04)3

067

(53)


Rϱ(1) R4 (09 04)

Rϱ(2) R2 (08 07)

Rϱ(3) R1 (06 07)

Rϱ(4) R3 (06 09)

(54)



11139374j1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

11139374j1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

ϱ(j)

11139374j1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

4j1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

144 minus 049144 + 049

31113970

23

radictimes 066

162 + 0663

radic1113898 1113899 079 055lang rang

(55)

Now

FFOWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3ϱ(j)1113872 1113873

wj3

11139741113972

11139454

j1]ϱ(j)1113872 1113873

wj1113898 1113899 080 066lang rang


(56)


sj1 wj 1


FFEOWA R1R2 Rs( 1113857 Ro (57)



(58)



(59)



sj1 wj 1 then



1λ

(60)









23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

middot

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(61)

Proof Since

Rλj

23

radicμλj

2 minus μ3j1113872 1113873λ

+ μ3j1113872 1113873λ3

1113969

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

3

11139741113972

1113898 1113899


1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1113874 1113875 1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

1113874 11138753

1113970

1113888 1113889

wj

1113937sj1 2 minus 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 1113875

wj3

1113970 1113899

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ3

1113970

1113888 1113889

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj3

1113970 1113899

(62)

amperefore


1λ

2 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

1113971

2 minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

11139711113898

1 + 21113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

minus 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

1 + 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

3

1113974111397311139731113972

1113899

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

11139711113898

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(63)


When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)



sj1 wj 1



1λ

(65)

where λgt 0





23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)








23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod






























































λ c1 (]31)λ

d1 (2 minus ]31)λ a2 (1 + μ32)

λ b2 (1 minus μ32)λ c2 (]32)


λ therefore


a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899oplusε

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899



31113971

223


1113969


311139691113898 1113899


31113971

2c1 middot εc2

31113968


311139681113898 1113899

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

minus 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

1 + μ311113872 1113873λ

middot ε 1 + μ321113872 1113873λ

+ 1 minus μ311113872 1113873λ

middot ε 1 minus μ321113872 1113873λ

3

11139741113972

23


λ2

2 minus ]311113872 1113873λ

middot ε 2 minus ]321113872 1113873λ

+ ]311113872 1113873λ

middot ε ]321113872 1113873λ3

11139691113898 1113899

(15)


λ1 middot εR

1 + μ31113872 1113873λ1

minus 1 minus μ31113872 1113873λ1

1 + μ31113872 1113873λ1

+ 1 minus μ31113872 1113873λ1

3

11139741113972

23

radic]λ1

2 minus ]31113872 1113873λ1

+ ]31113872 1113873λ13

11139691113898 1113899

a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899

λ2 middot εR

1 + μ31113872 1113873λ2

minus 1 minus μ31113872 1113873λ2

1 + μ31113872 1113873λ2

+ 1 minus μ31113872 1113873λ2

3

11139741113972

23

radic]λ2

2 minus ]31113872 1113873λ2

+ ]31113872 1113873λ23

11139691113898 1113899

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899

(16)



a1 minus b1

a1 + b1

31113971

2c1

31113968

d1 + c1

311139681113898 1113899oplusε

a2 minus b2

a2 + b2

31113971

2c2

31113968

d2 + c2

311139681113898 1113899



31113971

223


1113969


311139691113898 1113899


31113971

2c1 middot εc2

31113968


311139681113898 1113899

1 + μ31113872 1113873λ1+λ2

minus 1 minus μ31113872 1113873λ1+λ2

1 + μ31113872 1113873λ1+λ2

+ 1 minus μ31113872 1113873λ1+λ2

3

11139741113972

23

radic]λ1+λ2

2 minus ]31113872 1113873λ1+λ2

+ ]311113872 1113873λ1+λ23

11139691113898 1113899

λ1 + λ2( 1113857 middot εR

(17)




(i) Rc1andεR

c2 (R1orεR2)

c

(ii) Rc1orεR

c2 (R1andεR2)

c

(iii) Rc1oplusεR

c2 (R1 otimes εR2)

c

(iv) Rc1 otimes εR

c2 (R1oplusεR2)

c



Proof It is obvious
















(18)


FFA R1R2 Rs( 1113857 1s




1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 + μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 + μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj311139691113898 1113899 (20)


When s 2



w1 middot εR1

1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

3

11139741113972

2]w11

31113969

2 minus ]311113872 1113873w1

+ ]311113872 1113873w13

11139691113898 1113899

w2 middot εR2

1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

3

11139741113972

23

radic]w22

2 minus ]321113872 1113873w2

+ ]321113872 1113873w23

11139691113898 1113899

(22)


ampen


1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 + 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

1 + 1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 middot ε 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

3

11139741113972

1113898

23

radic]w11

2 minus ]311113872 1113873w1

+ ]311113872 1113873w13

1113969

1113874 1113875 middot ε23

radic]w21

2 minus ]321113872 1113873w2

+ ]321113872 1113873w23

1113969

1113874 1113875

1 + 1 minus 2]3w11 2 minus ]311113872 1113873

w1+ ]311113872 1113873

w11113872 11138731113872 11138731113872 1113873 middot ε 1 minus 2]3w2

2 2 minus ]321113872 1113873w2

+ ]321113872 1113873w2

1113872 11138731113872 11138731113872 11138733

1113969 1113899

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

minus 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

+ 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

3

11139741113972

23


w22

2 minus ]311113872 1113873w1

middot ε 2 minus ]321113872 1113873w2

+ ]311113872 1113873w1

middot ε ]321113872 1113873w23

11139691113898 1113899

(23)



1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899 (24)

Now for s k + 1

FFEWA R1R2 Rk+1( 1113857

1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899

oplusε

1 + μ3k+11113872 1113873wk+1

minus 1 minus μ3k+11113872 1113873wk+1

1 + μ3k+11113872 1113873wk+1

+ 1 minus μ3k+11113872 1113873wk+1

3

11139741113972

23

radic]wk+1

k+1

2 minus ]3k+11113872 1113873wk+1

+ ]3k+11113872 1113873wk+13

11139691113898 1113899

1113937k+1j1 1 + μ3j1113872 1113873

wjminus 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

1113937k+1j1 1 + μ3j1113872 1113873

wj+ 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

k+1j1]

wj

j

1113937k+1j1 2 minus ]3j1113872 1113873

wj+ 1113937

k+1j1 ]3j1113872 1113873

wj311139691113898 1113899

(25)



then

1113945

s

j1R

wj

j le 1113944s

j1wjRj (26)





1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj 1

minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le 1 minus 1113945s

j11 minus μ3j1113872 1113873

wj le 1

(27)


sj1


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjge 0 (28)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le1113945s

j11 minus μ3j1113872 1113873

wj le 1

(29)

Also

1113945

s

j1]3j1113872 1113873

wj ge 0⟺21113937

sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge 0

(30)



sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le 1 minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

1

(31)




(32)



sj1 (1 + μ3j)wj + 1113937


sj1 (1 + μ3j)wj+


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le

1 minus 1113945s

j11 minus μ3j1113872 1113873

wj⟺ μβR le μR3

11139741113972

(33)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge

21113937sj1 ]3j1113872 1113873

wj

1113936sj1 wj 2 minus ]3j1113872 1113873 + 1113936

sj1 wj]

3j

ge1113945s

j1]3j1113872 1113873

wjrArr

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

3

11139741113972

ge1113945s

j1]wj

j rArr]βR le ]R

(34)



S Rβ

1113872 1113873 μβR1113872 11138733

minus ]βR1113872 11138733le μR( 1113857

3minus ]R( 1113857

3 S(R) (35)



(36)





(37)

Hence


(38)



FFEWA R1R2R3R4( 1113857

11139374j1 1 + μ3j1113872 1113873

wjminus 1113937

4j1 1 minus μ3j1113872 1113873

wj

11139374j1 1 + μ3j1113872 1113873

wj+ 1113937

4j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

j

11139374j1 2 minus ]3j1113872 1113873

wj+ 1113937

4j1 ]3j1113872 1113873

wj311139691113898 1113899

149 minus 048149 + 048

31113970

23

radictimes 055

180 + 0163

radic1113898 1113899 080 055lang rang

(39)

Now

FFWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3j1113872 1113873

wj 1113945

4

j1]j1113872 1113873


3

11139741113972

1113898(40)


sj1 wj 1


FFEWA R1R2 Rs( 1113857 Ro (41)



(42)



(43)



1113937sj1 1 + μ3o1113872 1113873

wjminus 1113937

sj1 1 minus μ3o1113872 1113873

wj

1113937sj1 1 + μ3o1113872 1113873

wj+ 1113937

sj1 1 minus μ3o1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

o

1113937sj1 2 minus ]3o1113872 1113873

wj+ 1113937

sj1 ]3o1113872 1113873

wj311139691113898 1113899

1 + μ3o1113872 11138731113936


1113936sj1 wj

1 + μ3o1113872 11138731113936

sj1 wj + 1 minus μ3o1113872 1113873

1113936sj1 wj

3

111397411139731113972

23

radic]1113936

sj1 wj

o

2 minus ]3o1113872 11138731113936

sj1 wj + ]3o1113872 1113873

1113936sj1 wj

311139701113898 1113899 μo ]o1113866 1113867

(44)





have

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

1113945

s

j1

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj

le1113945s

j1


1113888 1113889

wj

le1113945s

j1

1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

⟺1 minus μ3jmax

1 + μ3jmax1113888 1113889

1113944s

j1wj

le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

1113944s

j1wj

⟺1 minus μ3jmax

1 + μ3jmax

⎛⎝ ⎞⎠le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin

⎛⎝ ⎞⎠

⟺2

1 + μ3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1


1113888 1113889

wj

le2

1 + μ3jmin

⎛⎝ ⎞⎠

⟺1 + μ3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle

1 + μ3jmax

2⎛⎝ ⎞⎠

⟺ 1 + μ3jmin1113872 1113873le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle 1 + μ3jmax1113872 1113873

⟺ μ3jmin le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjminus 1le μ3jmax

⟺ μ3jmin le1113937

sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjle μ3jmax

(45)

ampus

μjmin le

1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le μjmax (46)



jmin le ]3

j le ]3

jmax

forallj then g(]3

jmax)leg(]3

j)le g(]3


]3

jmax)]3

jmax)le ((2minus ]3

j)]3

j)le ((2 minus ]3

jmin)]3

jmin) ampen


2 minus ]3jmax

]3jmax1113888 1113889

wj

le2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

wj

1113945

s

j1

2 minus ]3jmax

]3jmax1113888 1113889

wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le1113945s

j1

2 minus ]3jmin

]3jmin1113888 1113889

wj

rArr2 minus ]3jmax

]3jmax1113888 1113889

1113944

s

j1wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

1113944

s

j1wj

rArr2 minus ]3jmax

]3jmax

⎛⎝ ⎞⎠le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin

⎛⎝ ⎞⎠

rArr2

]3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2

]3jmin

⎛⎝ ⎞⎠

rArr]3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle

]3jmax

2⎛⎝ ⎞⎠

rArr ]3jmin1113872 1113873le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax1113872 1113873

rArr]3jmin le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax


sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjle ]3jmax

rArr]jmin le

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj31113969 le ]jmax

(47)








(49)









1 2 s



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

1113898

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj31113969 1113899

(51)





(52)


(03 03 02 02)T as

S R1( 1113857 (06)3

minus (07)3

minus 013

S R2( 1113857 (08)3

minus (07)3

017

S R3( 1113857 (06)3

minus (09)3

minus 051

S R4( 1113857 (09)3

minus (04)3

067

(53)


Rϱ(1) R4 (09 04)

Rϱ(2) R2 (08 07)

Rϱ(3) R1 (06 07)

Rϱ(4) R3 (06 09)

(54)



11139374j1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

11139374j1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

ϱ(j)

11139374j1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

4j1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

144 minus 049144 + 049

31113970

23

radictimes 066

162 + 0663

radic1113898 1113899 079 055lang rang

(55)

Now

FFOWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3ϱ(j)1113872 1113873

wj3

11139741113972

11139454

j1]ϱ(j)1113872 1113873

wj1113898 1113899 080 066lang rang


(56)


sj1 wj 1


FFEOWA R1R2 Rs( 1113857 Ro (57)



(58)



(59)



sj1 wj 1 then



1λ

(60)









23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

middot

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(61)

Proof Since

Rλj

23

radicμλj

2 minus μ3j1113872 1113873λ

+ μ3j1113872 1113873λ3

1113969

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

3

11139741113972

1113898 1113899


1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1113874 1113875 1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

1113874 11138753

1113970

1113888 1113889

wj

1113937sj1 2 minus 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 1113875

wj3

1113970 1113899

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ3

1113970

1113888 1113889

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj3

1113970 1113899

(62)

amperefore


1λ

2 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

1113971

2 minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

11139711113898

1 + 21113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

minus 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

1 + 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

3

1113974111397311139731113972

1113899

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

11139711113898

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(63)


When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)



sj1 wj 1



1λ

(65)

where λgt 0





23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)








23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod































































(i) Rc1andεR

c2 (R1orεR2)

c

(ii) Rc1orεR

c2 (R1andεR2)

c

(iii) Rc1oplusεR

c2 (R1 otimes εR2)

c

(iv) Rc1 otimes εR

c2 (R1oplusεR2)

c



Proof It is obvious
















(18)


FFA R1R2 Rs( 1113857 1s




1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 + μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 + μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj311139691113898 1113899 (20)


When s 2



w1 middot εR1

1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

3

11139741113972

2]w11

31113969

2 minus ]311113872 1113873w1

+ ]311113872 1113873w13

11139691113898 1113899

w2 middot εR2

1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

3

11139741113972

23

radic]w22

2 minus ]321113872 1113873w2

+ ]321113872 1113873w23

11139691113898 1113899

(22)


ampen


1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 + 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

1 + 1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 middot ε 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

3

11139741113972

1113898

23

radic]w11

2 minus ]311113872 1113873w1

+ ]311113872 1113873w13

1113969

1113874 1113875 middot ε23

radic]w21

2 minus ]321113872 1113873w2

+ ]321113872 1113873w23

1113969

1113874 1113875

1 + 1 minus 2]3w11 2 minus ]311113872 1113873

w1+ ]311113872 1113873

w11113872 11138731113872 11138731113872 1113873 middot ε 1 minus 2]3w2

2 2 minus ]321113872 1113873w2

+ ]321113872 1113873w2

1113872 11138731113872 11138731113872 11138733

1113969 1113899

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

minus 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

+ 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

3

11139741113972

23


w22

2 minus ]311113872 1113873w1

middot ε 2 minus ]321113872 1113873w2

+ ]311113872 1113873w1

middot ε ]321113872 1113873w23

11139691113898 1113899

(23)



1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899 (24)

Now for s k + 1

FFEWA R1R2 Rk+1( 1113857

1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899

oplusε

1 + μ3k+11113872 1113873wk+1

minus 1 minus μ3k+11113872 1113873wk+1

1 + μ3k+11113872 1113873wk+1

+ 1 minus μ3k+11113872 1113873wk+1

3

11139741113972

23

radic]wk+1

k+1

2 minus ]3k+11113872 1113873wk+1

+ ]3k+11113872 1113873wk+13

11139691113898 1113899

1113937k+1j1 1 + μ3j1113872 1113873

wjminus 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

1113937k+1j1 1 + μ3j1113872 1113873

wj+ 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

k+1j1]

wj

j

1113937k+1j1 2 minus ]3j1113872 1113873

wj+ 1113937

k+1j1 ]3j1113872 1113873

wj311139691113898 1113899

(25)



then

1113945

s

j1R

wj

j le 1113944s

j1wjRj (26)





1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj 1

minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le 1 minus 1113945s

j11 minus μ3j1113872 1113873

wj le 1

(27)


sj1


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjge 0 (28)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le1113945s

j11 minus μ3j1113872 1113873

wj le 1

(29)

Also

1113945

s

j1]3j1113872 1113873

wj ge 0⟺21113937

sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge 0

(30)



sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le 1 minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

1

(31)




(32)



sj1 (1 + μ3j)wj + 1113937


sj1 (1 + μ3j)wj+


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le

1 minus 1113945s

j11 minus μ3j1113872 1113873

wj⟺ μβR le μR3

11139741113972

(33)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge

21113937sj1 ]3j1113872 1113873

wj

1113936sj1 wj 2 minus ]3j1113872 1113873 + 1113936

sj1 wj]

3j

ge1113945s

j1]3j1113872 1113873

wjrArr

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

3

11139741113972

ge1113945s

j1]wj

j rArr]βR le ]R

(34)



S Rβ

1113872 1113873 μβR1113872 11138733

minus ]βR1113872 11138733le μR( 1113857

3minus ]R( 1113857

3 S(R) (35)



(36)





(37)

Hence


(38)



FFEWA R1R2R3R4( 1113857

11139374j1 1 + μ3j1113872 1113873

wjminus 1113937

4j1 1 minus μ3j1113872 1113873

wj

11139374j1 1 + μ3j1113872 1113873

wj+ 1113937

4j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

j

11139374j1 2 minus ]3j1113872 1113873

wj+ 1113937

4j1 ]3j1113872 1113873

wj311139691113898 1113899

149 minus 048149 + 048

31113970

23

radictimes 055

180 + 0163

radic1113898 1113899 080 055lang rang

(39)

Now

FFWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3j1113872 1113873

wj 1113945

4

j1]j1113872 1113873


3

11139741113972

1113898(40)


sj1 wj 1


FFEWA R1R2 Rs( 1113857 Ro (41)



(42)



(43)



1113937sj1 1 + μ3o1113872 1113873

wjminus 1113937

sj1 1 minus μ3o1113872 1113873

wj

1113937sj1 1 + μ3o1113872 1113873

wj+ 1113937

sj1 1 minus μ3o1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

o

1113937sj1 2 minus ]3o1113872 1113873

wj+ 1113937

sj1 ]3o1113872 1113873

wj311139691113898 1113899

1 + μ3o1113872 11138731113936


1113936sj1 wj

1 + μ3o1113872 11138731113936

sj1 wj + 1 minus μ3o1113872 1113873

1113936sj1 wj

3

111397411139731113972

23

radic]1113936

sj1 wj

o

2 minus ]3o1113872 11138731113936

sj1 wj + ]3o1113872 1113873

1113936sj1 wj

311139701113898 1113899 μo ]o1113866 1113867

(44)





have

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

1113945

s

j1

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj

le1113945s

j1


1113888 1113889

wj

le1113945s

j1

1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

⟺1 minus μ3jmax

1 + μ3jmax1113888 1113889

1113944s

j1wj

le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

1113944s

j1wj

⟺1 minus μ3jmax

1 + μ3jmax

⎛⎝ ⎞⎠le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin

⎛⎝ ⎞⎠

⟺2

1 + μ3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1


1113888 1113889

wj

le2

1 + μ3jmin

⎛⎝ ⎞⎠

⟺1 + μ3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle

1 + μ3jmax

2⎛⎝ ⎞⎠

⟺ 1 + μ3jmin1113872 1113873le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle 1 + μ3jmax1113872 1113873

⟺ μ3jmin le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjminus 1le μ3jmax

⟺ μ3jmin le1113937

sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjle μ3jmax

(45)

ampus

μjmin le

1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le μjmax (46)



jmin le ]3

j le ]3

jmax

forallj then g(]3

jmax)leg(]3

j)le g(]3


]3

jmax)]3

jmax)le ((2minus ]3

j)]3

j)le ((2 minus ]3

jmin)]3

jmin) ampen


2 minus ]3jmax

]3jmax1113888 1113889

wj

le2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

wj

1113945

s

j1

2 minus ]3jmax

]3jmax1113888 1113889

wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le1113945s

j1

2 minus ]3jmin

]3jmin1113888 1113889

wj

rArr2 minus ]3jmax

]3jmax1113888 1113889

1113944

s

j1wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

1113944

s

j1wj

rArr2 minus ]3jmax

]3jmax

⎛⎝ ⎞⎠le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin

⎛⎝ ⎞⎠

rArr2

]3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2

]3jmin

⎛⎝ ⎞⎠

rArr]3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle

]3jmax

2⎛⎝ ⎞⎠

rArr ]3jmin1113872 1113873le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax1113872 1113873

rArr]3jmin le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax


sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjle ]3jmax

rArr]jmin le

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj31113969 le ]jmax

(47)








(49)









1 2 s



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

1113898

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj31113969 1113899

(51)





(52)


(03 03 02 02)T as

S R1( 1113857 (06)3

minus (07)3

minus 013

S R2( 1113857 (08)3

minus (07)3

017

S R3( 1113857 (06)3

minus (09)3

minus 051

S R4( 1113857 (09)3

minus (04)3

067

(53)


Rϱ(1) R4 (09 04)

Rϱ(2) R2 (08 07)

Rϱ(3) R1 (06 07)

Rϱ(4) R3 (06 09)

(54)



11139374j1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

11139374j1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

ϱ(j)

11139374j1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

4j1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

144 minus 049144 + 049

31113970

23

radictimes 066

162 + 0663

radic1113898 1113899 079 055lang rang

(55)

Now

FFOWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3ϱ(j)1113872 1113873

wj3

11139741113972

11139454

j1]ϱ(j)1113872 1113873

wj1113898 1113899 080 066lang rang


(56)


sj1 wj 1


FFEOWA R1R2 Rs( 1113857 Ro (57)



(58)



(59)



sj1 wj 1 then



1λ

(60)









23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

middot

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(61)

Proof Since

Rλj

23

radicμλj

2 minus μ3j1113872 1113873λ

+ μ3j1113872 1113873λ3

1113969

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

3

11139741113972

1113898 1113899


1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1113874 1113875 1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

1113874 11138753

1113970

1113888 1113889

wj

1113937sj1 2 minus 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 1113875

wj3

1113970 1113899

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ3

1113970

1113888 1113889

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj3

1113970 1113899

(62)

amperefore


1λ

2 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

1113971

2 minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

11139711113898

1 + 21113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

minus 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

1 + 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

3

1113974111397311139731113972

1113899

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

11139711113898

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(63)


When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)



sj1 wj 1



1λ

(65)

where λgt 0





23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)








23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod





























































ampen


1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 + 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

1 + 1 + μ311113872 1113873w1

minus 1 minus μ311113872 1113873w1

1113872 1113873 1 + μ311113872 1113873w1

+ 1 minus μ311113872 1113873w1

1113872 11138731113872 1113873 middot ε 1 + μ321113872 1113873w2

minus 1 minus μ321113872 1113873w2

1113872 1113873 1 + μ321113872 1113873w2

+ 1 minus μ321113872 1113873w2

1113872 11138731113872 1113873

3

11139741113972

1113898

23

radic]w11

2 minus ]311113872 1113873w1

+ ]311113872 1113873w13

1113969

1113874 1113875 middot ε23

radic]w21

2 minus ]321113872 1113873w2

+ ]321113872 1113873w23

1113969

1113874 1113875

1 + 1 minus 2]3w11 2 minus ]311113872 1113873

w1+ ]311113872 1113873

w11113872 11138731113872 11138731113872 1113873 middot ε 1 minus 2]3w2

2 2 minus ]321113872 1113873w2

+ ]321113872 1113873w2

1113872 11138731113872 11138731113872 11138733

1113969 1113899

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

minus 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

1 + μ311113872 1113873w1

middotε 1 + μ321113872 1113873w2

+ 1 minus μ311113872 1113873w1

middotε 1 minus μ321113872 1113873w2

3

11139741113972

23


w22

2 minus ]311113872 1113873w1

middot ε 2 minus ]321113872 1113873w2

+ ]311113872 1113873w1

middot ε ]321113872 1113873w23

11139691113898 1113899

(23)



1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899 (24)

Now for s k + 1

FFEWA R1R2 Rk+1( 1113857

1113937kj1 1 + μ3j1113872 1113873

wjminus 1113937

kj1 1 minus μ3j1113872 1113873

wj

1113937kj1 1 + μ3j1113872 1113873

wj+ 1113937

kj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

kj1 ]

wj

j

1113937kj1 2 minus ]3j1113872 1113873

wj+ 1113937

kj1 ]3j1113872 1113873

wj311139691113898 1113899

oplusε

1 + μ3k+11113872 1113873wk+1

minus 1 minus μ3k+11113872 1113873wk+1

1 + μ3k+11113872 1113873wk+1

+ 1 minus μ3k+11113872 1113873wk+1

3

11139741113972

23

radic]wk+1

k+1

2 minus ]3k+11113872 1113873wk+1

+ ]3k+11113872 1113873wk+13

11139691113898 1113899

1113937k+1j1 1 + μ3j1113872 1113873

wjminus 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

1113937k+1j1 1 + μ3j1113872 1113873

wj+ 1113937

k+1j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

k+1j1]

wj

j

1113937k+1j1 2 minus ]3j1113872 1113873

wj+ 1113937

k+1j1 ]3j1113872 1113873

wj311139691113898 1113899

(25)



then

1113945

s

j1R

wj

j le 1113944s

j1wjRj (26)





1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj 1

minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le 1 minus 1113945s

j11 minus μ3j1113872 1113873

wj le 1

(27)


sj1


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjge 0 (28)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le1113945s

j11 minus μ3j1113872 1113873

wj le 1

(29)

Also

1113945

s

j1]3j1113872 1113873

wj ge 0⟺21113937

sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge 0

(30)



sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le 1 minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

1

(31)




(32)



sj1 (1 + μ3j)wj + 1113937


sj1 (1 + μ3j)wj+


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le

1 minus 1113945s

j11 minus μ3j1113872 1113873

wj⟺ μβR le μR3

11139741113972

(33)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge

21113937sj1 ]3j1113872 1113873

wj

1113936sj1 wj 2 minus ]3j1113872 1113873 + 1113936

sj1 wj]

3j

ge1113945s

j1]3j1113872 1113873

wjrArr

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

3

11139741113972

ge1113945s

j1]wj

j rArr]βR le ]R

(34)



S Rβ

1113872 1113873 μβR1113872 11138733

minus ]βR1113872 11138733le μR( 1113857

3minus ]R( 1113857

3 S(R) (35)



(36)





(37)

Hence


(38)



FFEWA R1R2R3R4( 1113857

11139374j1 1 + μ3j1113872 1113873

wjminus 1113937

4j1 1 minus μ3j1113872 1113873

wj

11139374j1 1 + μ3j1113872 1113873

wj+ 1113937

4j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

j

11139374j1 2 minus ]3j1113872 1113873

wj+ 1113937

4j1 ]3j1113872 1113873

wj311139691113898 1113899

149 minus 048149 + 048

31113970

23

radictimes 055

180 + 0163

radic1113898 1113899 080 055lang rang

(39)

Now

FFWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3j1113872 1113873

wj 1113945

4

j1]j1113872 1113873


3

11139741113972

1113898(40)


sj1 wj 1


FFEWA R1R2 Rs( 1113857 Ro (41)



(42)



(43)



1113937sj1 1 + μ3o1113872 1113873

wjminus 1113937

sj1 1 minus μ3o1113872 1113873

wj

1113937sj1 1 + μ3o1113872 1113873

wj+ 1113937

sj1 1 minus μ3o1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

o

1113937sj1 2 minus ]3o1113872 1113873

wj+ 1113937

sj1 ]3o1113872 1113873

wj311139691113898 1113899

1 + μ3o1113872 11138731113936


1113936sj1 wj

1 + μ3o1113872 11138731113936

sj1 wj + 1 minus μ3o1113872 1113873

1113936sj1 wj

3

111397411139731113972

23

radic]1113936

sj1 wj

o

2 minus ]3o1113872 11138731113936

sj1 wj + ]3o1113872 1113873

1113936sj1 wj

311139701113898 1113899 μo ]o1113866 1113867

(44)





have

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

1113945

s

j1

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj

le1113945s

j1


1113888 1113889

wj

le1113945s

j1

1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

⟺1 minus μ3jmax

1 + μ3jmax1113888 1113889

1113944s

j1wj

le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

1113944s

j1wj

⟺1 minus μ3jmax

1 + μ3jmax

⎛⎝ ⎞⎠le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin

⎛⎝ ⎞⎠

⟺2

1 + μ3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1


1113888 1113889

wj

le2

1 + μ3jmin

⎛⎝ ⎞⎠

⟺1 + μ3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle

1 + μ3jmax

2⎛⎝ ⎞⎠

⟺ 1 + μ3jmin1113872 1113873le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle 1 + μ3jmax1113872 1113873

⟺ μ3jmin le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjminus 1le μ3jmax

⟺ μ3jmin le1113937

sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjle μ3jmax

(45)

ampus

μjmin le

1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le μjmax (46)



jmin le ]3

j le ]3

jmax

forallj then g(]3

jmax)leg(]3

j)le g(]3


]3

jmax)]3

jmax)le ((2minus ]3

j)]3

j)le ((2 minus ]3

jmin)]3

jmin) ampen


2 minus ]3jmax

]3jmax1113888 1113889

wj

le2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

wj

1113945

s

j1

2 minus ]3jmax

]3jmax1113888 1113889

wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le1113945s

j1

2 minus ]3jmin

]3jmin1113888 1113889

wj

rArr2 minus ]3jmax

]3jmax1113888 1113889

1113944

s

j1wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

1113944

s

j1wj

rArr2 minus ]3jmax

]3jmax

⎛⎝ ⎞⎠le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin

⎛⎝ ⎞⎠

rArr2

]3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2

]3jmin

⎛⎝ ⎞⎠

rArr]3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle

]3jmax

2⎛⎝ ⎞⎠

rArr ]3jmin1113872 1113873le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax1113872 1113873

rArr]3jmin le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax


sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjle ]3jmax

rArr]jmin le

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj31113969 le ]jmax

(47)








(49)









1 2 s



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

1113898

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj31113969 1113899

(51)





(52)


(03 03 02 02)T as

S R1( 1113857 (06)3

minus (07)3

minus 013

S R2( 1113857 (08)3

minus (07)3

017

S R3( 1113857 (06)3

minus (09)3

minus 051

S R4( 1113857 (09)3

minus (04)3

067

(53)


Rϱ(1) R4 (09 04)

Rϱ(2) R2 (08 07)

Rϱ(3) R1 (06 07)

Rϱ(4) R3 (06 09)

(54)



11139374j1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

11139374j1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

ϱ(j)

11139374j1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

4j1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

144 minus 049144 + 049

31113970

23

radictimes 066

162 + 0663

radic1113898 1113899 079 055lang rang

(55)

Now

FFOWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3ϱ(j)1113872 1113873

wj3

11139741113972

11139454

j1]ϱ(j)1113872 1113873

wj1113898 1113899 080 066lang rang


(56)


sj1 wj 1


FFEOWA R1R2 Rs( 1113857 Ro (57)



(58)



(59)



sj1 wj 1 then



1λ

(60)









23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

middot

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(61)

Proof Since

Rλj

23

radicμλj

2 minus μ3j1113872 1113873λ

+ μ3j1113872 1113873λ3

1113969

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

3

11139741113972

1113898 1113899


1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1113874 1113875 1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

1113874 11138753

1113970

1113888 1113889

wj

1113937sj1 2 minus 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 1113875

wj3

1113970 1113899

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ3

1113970

1113888 1113889

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj3

1113970 1113899

(62)

amperefore


1λ

2 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

1113971

2 minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

11139711113898

1 + 21113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

minus 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

1 + 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

3

1113974111397311139731113972

1113899

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

11139711113898

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(63)


When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)



sj1 wj 1



1λ

(65)

where λgt 0





23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)








23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod





























































1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj 1

minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le 1 minus 1113945s

j11 minus μ3j1113872 1113873

wj le 1

(27)


sj1


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjge 0 (28)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

le1113945s

j11 minus μ3j1113872 1113873

wj le 1

(29)

Also

1113945

s

j1]3j1113872 1113873

wj ge 0⟺21113937

sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge 0

(30)



sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

le 1 minus21113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj+

21113937sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

1

(31)




(32)



sj1 (1 + μ3j)wj + 1113937


sj1 (1 + μ3j)wj+


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le

1 minus 1113945s

j11 minus μ3j1113872 1113873

wj⟺ μβR le μR3

11139741113972

(33)


21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjge

21113937sj1 ]3j1113872 1113873

wj

1113936sj1 wj 2 minus ]3j1113872 1113873 + 1113936

sj1 wj]

3j

ge1113945s

j1]3j1113872 1113873

wjrArr

21113937sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj

3

11139741113972

ge1113945s

j1]wj

j rArr]βR le ]R

(34)



S Rβ

1113872 1113873 μβR1113872 11138733

minus ]βR1113872 11138733le μR( 1113857

3minus ]R( 1113857

3 S(R) (35)



(36)





(37)

Hence


(38)



FFEWA R1R2R3R4( 1113857

11139374j1 1 + μ3j1113872 1113873

wjminus 1113937

4j1 1 minus μ3j1113872 1113873

wj

11139374j1 1 + μ3j1113872 1113873

wj+ 1113937

4j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

j

11139374j1 2 minus ]3j1113872 1113873

wj+ 1113937

4j1 ]3j1113872 1113873

wj311139691113898 1113899

149 minus 048149 + 048

31113970

23

radictimes 055

180 + 0163

radic1113898 1113899 080 055lang rang

(39)

Now

FFWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3j1113872 1113873

wj 1113945

4

j1]j1113872 1113873


3

11139741113972

1113898(40)


sj1 wj 1


FFEWA R1R2 Rs( 1113857 Ro (41)



(42)



(43)



1113937sj1 1 + μ3o1113872 1113873

wjminus 1113937

sj1 1 minus μ3o1113872 1113873

wj

1113937sj1 1 + μ3o1113872 1113873

wj+ 1113937

sj1 1 minus μ3o1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

o

1113937sj1 2 minus ]3o1113872 1113873

wj+ 1113937

sj1 ]3o1113872 1113873

wj311139691113898 1113899

1 + μ3o1113872 11138731113936


1113936sj1 wj

1 + μ3o1113872 11138731113936

sj1 wj + 1 minus μ3o1113872 1113873

1113936sj1 wj

3

111397411139731113972

23

radic]1113936

sj1 wj

o

2 minus ]3o1113872 11138731113936

sj1 wj + ]3o1113872 1113873

1113936sj1 wj

311139701113898 1113899 μo ]o1113866 1113867

(44)





have

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

1113945

s

j1

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj

le1113945s

j1


1113888 1113889

wj

le1113945s

j1

1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

⟺1 minus μ3jmax

1 + μ3jmax1113888 1113889

1113944s

j1wj

le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

1113944s

j1wj

⟺1 minus μ3jmax

1 + μ3jmax

⎛⎝ ⎞⎠le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin

⎛⎝ ⎞⎠

⟺2

1 + μ3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1


1113888 1113889

wj

le2

1 + μ3jmin

⎛⎝ ⎞⎠

⟺1 + μ3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle

1 + μ3jmax

2⎛⎝ ⎞⎠

⟺ 1 + μ3jmin1113872 1113873le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle 1 + μ3jmax1113872 1113873

⟺ μ3jmin le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjminus 1le μ3jmax

⟺ μ3jmin le1113937

sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjle μ3jmax

(45)

ampus

μjmin le

1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le μjmax (46)



jmin le ]3

j le ]3

jmax

forallj then g(]3

jmax)leg(]3

j)le g(]3


]3

jmax)]3

jmax)le ((2minus ]3

j)]3

j)le ((2 minus ]3

jmin)]3

jmin) ampen


2 minus ]3jmax

]3jmax1113888 1113889

wj

le2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

wj

1113945

s

j1

2 minus ]3jmax

]3jmax1113888 1113889

wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le1113945s

j1

2 minus ]3jmin

]3jmin1113888 1113889

wj

rArr2 minus ]3jmax

]3jmax1113888 1113889

1113944

s

j1wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

1113944

s

j1wj

rArr2 minus ]3jmax

]3jmax

⎛⎝ ⎞⎠le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin

⎛⎝ ⎞⎠

rArr2

]3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2

]3jmin

⎛⎝ ⎞⎠

rArr]3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle

]3jmax

2⎛⎝ ⎞⎠

rArr ]3jmin1113872 1113873le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax1113872 1113873

rArr]3jmin le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax


sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjle ]3jmax

rArr]jmin le

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj31113969 le ]jmax

(47)








(49)









1 2 s



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

1113898

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj31113969 1113899

(51)





(52)


(03 03 02 02)T as

S R1( 1113857 (06)3

minus (07)3

minus 013

S R2( 1113857 (08)3

minus (07)3

017

S R3( 1113857 (06)3

minus (09)3

minus 051

S R4( 1113857 (09)3

minus (04)3

067

(53)


Rϱ(1) R4 (09 04)

Rϱ(2) R2 (08 07)

Rϱ(3) R1 (06 07)

Rϱ(4) R3 (06 09)

(54)



11139374j1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

11139374j1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

ϱ(j)

11139374j1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

4j1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

144 minus 049144 + 049

31113970

23

radictimes 066

162 + 0663

radic1113898 1113899 079 055lang rang

(55)

Now

FFOWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3ϱ(j)1113872 1113873

wj3

11139741113972

11139454

j1]ϱ(j)1113872 1113873

wj1113898 1113899 080 066lang rang


(56)


sj1 wj 1


FFEOWA R1R2 Rs( 1113857 Ro (57)



(58)



(59)



sj1 wj 1 then



1λ

(60)









23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

middot

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(61)

Proof Since

Rλj

23

radicμλj

2 minus μ3j1113872 1113873λ

+ μ3j1113872 1113873λ3

1113969

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

3

11139741113972

1113898 1113899


1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1113874 1113875 1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

1113874 11138753

1113970

1113888 1113889

wj

1113937sj1 2 minus 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 1113875

wj3

1113970 1113899

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ3

1113970

1113888 1113889

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj3

1113970 1113899

(62)

amperefore


1λ

2 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

1113971

2 minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

11139711113898

1 + 21113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

minus 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

1 + 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

3

1113974111397311139731113972

1113899

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

11139711113898

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(63)


When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)



sj1 wj 1



1λ

(65)

where λgt 0





23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)








23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod






























































S Rβ

1113872 1113873 μβR1113872 11138733

minus ]βR1113872 11138733le μR( 1113857

3minus ]R( 1113857

3 S(R) (35)



(36)





(37)

Hence


(38)



FFEWA R1R2R3R4( 1113857

11139374j1 1 + μ3j1113872 1113873

wjminus 1113937

4j1 1 minus μ3j1113872 1113873

wj

11139374j1 1 + μ3j1113872 1113873

wj+ 1113937

4j1 1 minus μ3j1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

j

11139374j1 2 minus ]3j1113872 1113873

wj+ 1113937

4j1 ]3j1113872 1113873

wj311139691113898 1113899

149 minus 048149 + 048

31113970

23

radictimes 055

180 + 0163

radic1113898 1113899 080 055lang rang

(39)

Now

FFWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3j1113872 1113873

wj 1113945

4

j1]j1113872 1113873


3

11139741113972

1113898(40)


sj1 wj 1


FFEWA R1R2 Rs( 1113857 Ro (41)



(42)



(43)



1113937sj1 1 + μ3o1113872 1113873

wjminus 1113937

sj1 1 minus μ3o1113872 1113873

wj

1113937sj1 1 + μ3o1113872 1113873

wj+ 1113937

sj1 1 minus μ3o1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

o

1113937sj1 2 minus ]3o1113872 1113873

wj+ 1113937

sj1 ]3o1113872 1113873

wj311139691113898 1113899

1 + μ3o1113872 11138731113936


1113936sj1 wj

1 + μ3o1113872 11138731113936

sj1 wj + 1 minus μ3o1113872 1113873

1113936sj1 wj

3

111397411139731113972

23

radic]1113936

sj1 wj

o

2 minus ]3o1113872 11138731113936

sj1 wj + ]3o1113872 1113873

1113936sj1 wj

311139701113898 1113899 μo ]o1113866 1113867

(44)





have

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

1113945

s

j1

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj

le1113945s

j1


1113888 1113889

wj

le1113945s

j1

1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

⟺1 minus μ3jmax

1 + μ3jmax1113888 1113889

1113944s

j1wj

le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

1113944s

j1wj

⟺1 minus μ3jmax

1 + μ3jmax

⎛⎝ ⎞⎠le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin

⎛⎝ ⎞⎠

⟺2

1 + μ3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1


1113888 1113889

wj

le2

1 + μ3jmin

⎛⎝ ⎞⎠

⟺1 + μ3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle

1 + μ3jmax

2⎛⎝ ⎞⎠

⟺ 1 + μ3jmin1113872 1113873le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle 1 + μ3jmax1113872 1113873

⟺ μ3jmin le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjminus 1le μ3jmax

⟺ μ3jmin le1113937

sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjle μ3jmax

(45)

ampus

μjmin le

1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le μjmax (46)



jmin le ]3

j le ]3

jmax

forallj then g(]3

jmax)leg(]3

j)le g(]3


]3

jmax)]3

jmax)le ((2minus ]3

j)]3

j)le ((2 minus ]3

jmin)]3

jmin) ampen


2 minus ]3jmax

]3jmax1113888 1113889

wj

le2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

wj

1113945

s

j1

2 minus ]3jmax

]3jmax1113888 1113889

wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le1113945s

j1

2 minus ]3jmin

]3jmin1113888 1113889

wj

rArr2 minus ]3jmax

]3jmax1113888 1113889

1113944

s

j1wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

1113944

s

j1wj

rArr2 minus ]3jmax

]3jmax

⎛⎝ ⎞⎠le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin

⎛⎝ ⎞⎠

rArr2

]3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2

]3jmin

⎛⎝ ⎞⎠

rArr]3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle

]3jmax

2⎛⎝ ⎞⎠

rArr ]3jmin1113872 1113873le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax1113872 1113873

rArr]3jmin le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax


sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjle ]3jmax

rArr]jmin le

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj31113969 le ]jmax

(47)








(49)









1 2 s



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

1113898

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj31113969 1113899

(51)





(52)


(03 03 02 02)T as

S R1( 1113857 (06)3

minus (07)3

minus 013

S R2( 1113857 (08)3

minus (07)3

017

S R3( 1113857 (06)3

minus (09)3

minus 051

S R4( 1113857 (09)3

minus (04)3

067

(53)


Rϱ(1) R4 (09 04)

Rϱ(2) R2 (08 07)

Rϱ(3) R1 (06 07)

Rϱ(4) R3 (06 09)

(54)



11139374j1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

11139374j1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

ϱ(j)

11139374j1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

4j1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

144 minus 049144 + 049

31113970

23

radictimes 066

162 + 0663

radic1113898 1113899 079 055lang rang

(55)

Now

FFOWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3ϱ(j)1113872 1113873

wj3

11139741113972

11139454

j1]ϱ(j)1113872 1113873

wj1113898 1113899 080 066lang rang


(56)


sj1 wj 1


FFEOWA R1R2 Rs( 1113857 Ro (57)



(58)



(59)



sj1 wj 1 then



1λ

(60)









23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

middot

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(61)

Proof Since

Rλj

23

radicμλj

2 minus μ3j1113872 1113873λ

+ μ3j1113872 1113873λ3

1113969

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

3

11139741113972

1113898 1113899


1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1113874 1113875 1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

1113874 11138753

1113970

1113888 1113889

wj

1113937sj1 2 minus 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 1113875

wj3

1113970 1113899

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ3

1113970

1113888 1113889

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj3

1113970 1113899

(62)

amperefore


1λ

2 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

1113971

2 minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

11139711113898

1 + 21113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

minus 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

1 + 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

3

1113974111397311139731113972

1113899

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

11139711113898

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(63)


When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)



sj1 wj 1



1λ

(65)

where λgt 0





23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)








23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod
































































have

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

1113945

s

j1

1 minus μ3jmax

1 + μ3jmax1113888 1113889

wj

le1113945s

j1


1113888 1113889

wj

le1113945s

j1

1 minus μ3jmin

1 + μ3jmin1113888 1113889

wj

⟺1 minus μ3jmax

1 + μ3jmax1113888 1113889

1113944s

j1wj

le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin1113888 1113889

1113944s

j1wj

⟺1 minus μ3jmax

1 + μ3jmax

⎛⎝ ⎞⎠le1113945s

j1


1113888 1113889

wj

le1 minus μ3jmin

1 + μ3jmin

⎛⎝ ⎞⎠

⟺2

1 + μ3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1


1113888 1113889

wj

le2

1 + μ3jmin

⎛⎝ ⎞⎠

⟺1 + μ3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle

1 + μ3jmax

2⎛⎝ ⎞⎠

⟺ 1 + μ3jmin1113872 1113873le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjle 1 + μ3jmax1113872 1113873

⟺ μ3jmin le2

1 + 1113937sj1 1 minus μ3j1113872 1113873 1 + μ3j1113872 11138731113872 1113873

wjminus 1le μ3jmax

⟺ μ3jmin le1113937

sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wjle μ3jmax

(45)

ampus

μjmin le

1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

3

11139741113972

le μjmax (46)



jmin le ]3

j le ]3

jmax

forallj then g(]3

jmax)leg(]3

j)le g(]3


]3

jmax)]3

jmax)le ((2minus ]3

j)]3

j)le ((2 minus ]3

jmin)]3

jmin) ampen


2 minus ]3jmax

]3jmax1113888 1113889

wj

le2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

wj

1113945

s

j1

2 minus ]3jmax

]3jmax1113888 1113889

wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le1113945s

j1

2 minus ]3jmin

]3jmin1113888 1113889

wj

rArr2 minus ]3jmax

]3jmax1113888 1113889

1113944

s

j1wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

1113944

s

j1wj

rArr2 minus ]3jmax

]3jmax

⎛⎝ ⎞⎠le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin

⎛⎝ ⎞⎠

rArr2

]3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2

]3jmin

⎛⎝ ⎞⎠

rArr]3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle

]3jmax

2⎛⎝ ⎞⎠

rArr ]3jmin1113872 1113873le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax1113872 1113873

rArr]3jmin le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax


sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjle ]3jmax

rArr]jmin le

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj31113969 le ]jmax

(47)








(49)









1 2 s



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

1113898

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj31113969 1113899

(51)





(52)


(03 03 02 02)T as

S R1( 1113857 (06)3

minus (07)3

minus 013

S R2( 1113857 (08)3

minus (07)3

017

S R3( 1113857 (06)3

minus (09)3

minus 051

S R4( 1113857 (09)3

minus (04)3

067

(53)


Rϱ(1) R4 (09 04)

Rϱ(2) R2 (08 07)

Rϱ(3) R1 (06 07)

Rϱ(4) R3 (06 09)

(54)



11139374j1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

11139374j1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

ϱ(j)

11139374j1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

4j1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

144 minus 049144 + 049

31113970

23

radictimes 066

162 + 0663

radic1113898 1113899 079 055lang rang

(55)

Now

FFOWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3ϱ(j)1113872 1113873

wj3

11139741113972

11139454

j1]ϱ(j)1113872 1113873

wj1113898 1113899 080 066lang rang


(56)


sj1 wj 1


FFEOWA R1R2 Rs( 1113857 Ro (57)



(58)



(59)



sj1 wj 1 then



1λ

(60)









23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

middot

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(61)

Proof Since

Rλj

23

radicμλj

2 minus μ3j1113872 1113873λ

+ μ3j1113872 1113873λ3

1113969

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

3

11139741113972

1113898 1113899


1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1113874 1113875 1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

1113874 11138753

1113970

1113888 1113889

wj

1113937sj1 2 minus 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 1113875

wj3

1113970 1113899

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ3

1113970

1113888 1113889

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj3

1113970 1113899

(62)

amperefore


1λ

2 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

1113971

2 minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

11139711113898

1 + 21113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

minus 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

1 + 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

3

1113974111397311139731113972

1113899

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

11139711113898

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(63)


When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)



sj1 wj 1



1λ

(65)

where λgt 0





23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)








23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod





























































2 minus ]3jmax

]3jmax1113888 1113889

wj

le2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

wj

1113945

s

j1

2 minus ]3jmax

]3jmax1113888 1113889

wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le1113945s

j1

2 minus ]3jmin

]3jmin1113888 1113889

wj

rArr2 minus ]3jmax

]3jmax1113888 1113889

1113944

s

j1wj

le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin1113888 1113889

1113944

s

j1wj

rArr2 minus ]3jmax

]3jmax

⎛⎝ ⎞⎠le1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2 minus ]3jmin

]3jmin

⎛⎝ ⎞⎠

rArr2

]3jmax

⎛⎝ ⎞⎠le 1 + 1113945s

j1

2 minus ]3j]3j

1113888 1113889

wj

le2

]3jmin

⎛⎝ ⎞⎠

rArr]3jmin

2⎛⎝ ⎞⎠le

11 + 1113937

sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle

]3jmax

2⎛⎝ ⎞⎠

rArr ]3jmin1113872 1113873le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax1113872 1113873

rArr]3jmin le2

1 + 1113937sj1 2 minus ]3j1113872 1113873]3j1113872 1113873

wjle ]3jmax


sj1 ]3j1113872 1113873

wj

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wjle ]3jmax

rArr]jmin le

23

radic1113937

sj1 ]

wj

j

1113937sj1 2 minus ]3j1113872 1113873

wj+ 1113937

sj1 ]3j1113872 1113873

wj31113969 le ]jmax

(47)








(49)









1 2 s



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

1113898

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj31113969 1113899

(51)





(52)


(03 03 02 02)T as

S R1( 1113857 (06)3

minus (07)3

minus 013

S R2( 1113857 (08)3

minus (07)3

017

S R3( 1113857 (06)3

minus (09)3

minus 051

S R4( 1113857 (09)3

minus (04)3

067

(53)


Rϱ(1) R4 (09 04)

Rϱ(2) R2 (08 07)

Rϱ(3) R1 (06 07)

Rϱ(4) R3 (06 09)

(54)



11139374j1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

11139374j1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

ϱ(j)

11139374j1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

4j1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

144 minus 049144 + 049

31113970

23

radictimes 066

162 + 0663

radic1113898 1113899 079 055lang rang

(55)

Now

FFOWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3ϱ(j)1113872 1113873

wj3

11139741113972

11139454

j1]ϱ(j)1113872 1113873

wj1113898 1113899 080 066lang rang


(56)


sj1 wj 1


FFEOWA R1R2 Rs( 1113857 Ro (57)



(58)



(59)



sj1 wj 1 then



1λ

(60)









23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

middot

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(61)

Proof Since

Rλj

23

radicμλj

2 minus μ3j1113872 1113873λ

+ μ3j1113872 1113873λ3

1113969

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

3

11139741113972

1113898 1113899


1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1113874 1113875 1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

1113874 11138753

1113970

1113888 1113889

wj

1113937sj1 2 minus 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 1113875

wj3

1113970 1113899

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ3

1113970

1113888 1113889

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj3

1113970 1113899

(62)

amperefore


1λ

2 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

1113971

2 minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

11139711113898

1 + 21113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

minus 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

1 + 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

3

1113974111397311139731113972

1113899

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

11139711113898

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(63)


When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)



sj1 wj 1



1λ

(65)

where λgt 0





23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)








23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod
































































(52)


(03 03 02 02)T as

S R1( 1113857 (06)3

minus (07)3

minus 013

S R2( 1113857 (08)3

minus (07)3

017

S R3( 1113857 (06)3

minus (09)3

minus 051

S R4( 1113857 (09)3

minus (04)3

067

(53)


Rϱ(1) R4 (09 04)

Rϱ(2) R2 (08 07)

Rϱ(3) R1 (06 07)

Rϱ(4) R3 (06 09)

(54)



11139374j1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

11139374j1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

4j1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

4j1 ]

wj

ϱ(j)

11139374j1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

4j1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

144 minus 049144 + 049

31113970

23

radictimes 066

162 + 0663

radic1113898 1113899 079 055lang rang

(55)

Now

FFOWA R1R2R3R4( 1113857

1 minus 11139454

j11 minus μ3ϱ(j)1113872 1113873

wj3

11139741113972

11139454

j1]ϱ(j)1113872 1113873

wj1113898 1113899 080 066lang rang


(56)


sj1 wj 1


FFEOWA R1R2 Rs( 1113857 Ro (57)



(58)



(59)



sj1 wj 1 then



1λ

(60)









23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

middot

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(61)

Proof Since

Rλj

23

radicμλj

2 minus μ3j1113872 1113873λ

+ μ3j1113872 1113873λ3

1113969

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

3

11139741113972

1113898 1113899


1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1113874 1113875 1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

1113874 11138753

1113970

1113888 1113889

wj

1113937sj1 2 minus 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 1113875

wj3

1113970 1113899

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ3

1113970

1113888 1113889

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj3

1113970 1113899

(62)

amperefore


1λ

2 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

1113971

2 minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

11139711113898

1 + 21113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

minus 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

1 + 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

3

1113974111397311139731113972

1113899

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

11139711113898

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(63)


When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)



sj1 wj 1



1λ

(65)

where λgt 0





23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)








23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod






























































23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

middot

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(61)

Proof Since

Rλj

23

radicμλj

2 minus μ3j1113872 1113873λ

+ μ3j1113872 1113873λ3

1113969

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

3

11139741113972

1113898 1113899


1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

1113937sj1 1 + 2μ3λj 2 minus μ3j1113872 1113873

λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj


λ+ μ3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ

1113874 1113875 1 + ]3j1113872 1113873λ

+ 1 minus ]3j1113872 1113873λ

1113874 11138753

1113970

1113888 1113889

wj

1113937sj1 2 minus 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 11138751113874 1113875

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113874 1113875 1 + ]3j1113872 1113873

λ+ 1 minus ]3j1113872 1113873

λ1113874 11138751113874 1113875

wj3

1113970 1113899

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

3

111397411139731113972

1113898

23

radic1113937

sj1

1 + ]3j1113872 1113873λ

minus 1 minus ]3j1113872 1113873λ3

1113970

1113888 1113889

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj3

1113970 1113899

(62)

amperefore


1λ

2 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

1113971

2 minus 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 1113875 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751λ3

11139711113898

1 + 21113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

minus 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

1 + 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

+ 1 minus 21113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751113874 11138751113874 11138751λ

3

1113974111397311139731113972

1113899

23

radic1113937

sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3j1113872 1113873

λ+ 3 μ3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 2 minus μ3j1113872 1113873

λminus μ3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

11139711113898

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3j1113872 1113873

λ+ 3 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3j1113872 1113873

λminus 1 minus ]3j1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(63)


When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)



sj1 wj 1



1λ

(65)

where λgt 0





23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)








23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod





























































When λ 1 then


1113937sj1 1 + μ3j1113872 1113873

wjminus 1113937

sj1 1 minus μ3j1113872 1113873

wj1113937

sj1 1 minus μ3j1113872 1113873

wj

1113937sj1 1 + μ3j1113872 1113873

wj+ 1113937

sj1 1 minus μ3j1113872 1113873

wj

2

3radic1113945

s

j1]wj

j

1113945

s

j12 minus ]3j1113872 1113873

wj+ 1113945

s

j1]3j1113872 1113873

wj3

11139741113972 11138993

111397411139731113973111397311139731113972

1113898

(64)



sj1 wj 1



1λ

(65)

where λgt 0





23

radic1113937

sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113896 1113897

13λ

1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 2 minus μ3ϱ(j)1113874 1113875

λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 2 minus μ3ϱ(j)1113874 1113875

λ+ 3 μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj


λminus μ3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ3

11139741113972 1113898

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

+ 31113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

+ 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λ+ 3 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

minus 1113937sj1 1 + ]3ϱ(j)1113874 1113875

λminus 1 minus ]3ϱ(j)1113874 1113875

λ1113896 1113897

wj

1113888 1113889

1λ

3

111397411139731113973111397311139731113972 1113899

(66)



1113937sj1 1 + μ3ϱ(j)1113872 1113873

wjminus 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

1113937sj1 1 + μ3ϱ(j)1113872 1113873

wj+ 1113937

sj1 1 minus μ3ϱ(j)1113872 1113873

wj

3

11139741113972

23

radic1113937

sj1 ]

wj

ϱ(j)

1113937sj1 2 minus ]3ϱ(j)1113872 1113873

wj+ 1113937

sj1 ]3ϱ(j)1113872 1113873

wj311139691113898 1113899

(67)








23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod






























































23

radic1113937

sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113882 111388313λ

1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 2 minus μ3lj1113872 1113873

λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 2 minus μ3lj1113872 1113873

λ+ 3 μ3lj1113872 1113873

λ1113882 1113883

wj


λminus μ3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ3

1113971 1113898

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

minus 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

+ 31113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

+ 1113937sj1 1 + ]3lj1113872 1113873

λ+ 3 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

minus 1113937sj1 1 + ]3lj1113872 1113873

λminus 1 minus ]3lj1113872 1113873

λ1113882 1113883

wj

1113874 11138751λ

3

1113974111397311139731113972

1113899

(68)









(69)


w1 060

w2 025

w3 015

1113944

3

j1wj 1

(70)



B1 (064 048)

B2 (046 028)

B3 (072 031)

B4 (077 034)

(71)



ALGORITHM 1


S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod





























































S B1( 1113857 015

S B2( 1113857 008

S B3( 1113857 034

S B4( 1113857 042

(72)




9 Validity Test







S B1( 1113857 015

S 1113957B2) minus 003S B3( 1113857 034S B4( 1113857 0421113872(74)








S+

(08 03) (08 01) (09 06)

Sminus

(06 07) (05 03) (03 06) (75)



ξ K1( 1113857 minus 305

ξ K2( 1113857 minus 411

ξ K3( 1113857 minus 128

ξ K4( 1113857 010

(76)




Table 2 FFDM

1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)K2 (04 02) (06 04) (03 06)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


1113957E J1 J2 J3

K1 (06 07) (06 02) (08 04)1113957K2 (03 04) (05 04) (02 08)K3 (08 03) (05 03) (06 04)K4 (07 05) (08 01) (09 06)


D(KlS+) D(KlS

minus )

030 008046 029018 021009 032






11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod

































































11 Conclusions




Data Availability




References





ndash45ndash4

ndash35ndash3

ndash25ndash2

ndash15ndash1

ndash050

051

015 008034 042

01

ndash305

ndash411

ndash128

K_1 K_2 K_3 K_4



FF decision matrix

FF decision matrix


FFPIS and FFNIS






FF Einstein AOs

FF TOPSISmethod





























































Decision-MakingFrameworkforanEffectiveSanitizertoReduce...

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