Review ArticleAn Exhaustive Study of Possibility Measures ofInterval-Valued Intuitionistic Fuzzy Sets and Application toMulticriteria Decision Making

Fatma Dammak Leila Baccour and Adel M Alimi

REGIM-Lab Research Groups in Intelligent Machines University of Sfax ENIS BP 1173 3038 Sfax Tunisia

Correspondence should be addressed to Fatma Dammak fatmadammaktnieeeorg

Received 14 April 2016 Revised 7 June 2016 Accepted 9 June 2016

Academic Editor RustomM Mamlook

Copyright copy 2016 Fatma Dammak et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This work is interested in showing the importance of possibility theory in multicriteria decision making (MCDM) Thus we applysome possibility measures from literature to the MCDM method using interval-valued intuitionistic fuzzy sets (IVIFSs) Thesemeasures are applied to a decision matrix after being transformed with aggregation operators The results are compared betweeneach other and concluding remarks are drawn

1 Introduction

Multicriteria decision making methods have been developedwidely using fuzzy sets and their generalizations Park et al[1] extended TOPSIS method for decision making problemsunder interval-valued intuitionistic fuzzy environment Parket al [2] generalized the concepts of correlation coefficientof intuitionistic fuzzy sets into interval-valued intuitionisticfuzzy sets Ye [3] proposed weighted correlation coefficientsusing entropy weights under interval-valued intuitionisticfuzzy environment to rank alternatives Zhang and Yu [4]extended TOPSIS method using cross entropy and gen-eralized an MCMD approach with interval-valued intu-itionistic fuzzy sets The possibility theory has also beenapplied in many research topics To rank alternatives acomparison between the obtained matrix and aggregatedIVIFS is mandatory This is applied by an accuracy func-tion in [5ndash8] or a possibility measure To apply possibilitymeasures to a decision matrix of IVIFS an aggregation isneeded Some aggregation methods under interval-valuedintuitionistic fuzzy information are given in [7 9 10] Inthe same way Xu in [11 12] developed some aggregationoperators intuitionistic fuzzy weighted averaging operatorintuitionistic fuzzy ordered weighted averaging (IFOWA)intuitionistic fuzzy weighted averaging (IFWA) operator andintuitionistic fuzzy hybrid aggregation (IFHA) operator Wei

and Tang [13] extended the possibility method of interval-valued numbers defined by [14 15] to intuitionistic fuzzysets [16] and defined a possibility formula to compare twointuitionistic fuzzy numbers (IFNs) In addition Xu and Da[14] presented a possibility formula to compare two intervalfuzzy numbers and applied possibility measures of interval-valued intuitionistic fuzzy numbers to multicriteria decisionmaking Gao [17] presented four possibility measures andproved their equivalence Liu and Lv [18] used possibilitymeasures for the ranking of interval fuzzy numbers

Our aim is to present and compare several possibilitymeasures under intuitionistic fuzzy and interval-valued intu-itionistic fuzzy environment The remaining of this paper isorganized as follows in Section 2 the possibility theory andmeasures of IFS are detailed In Section 3 some preliminariesabout IVIFS and the possibility measures are introduced InSection 4 an IVIF MCDM method is adopted In Section 5aggregation operators and possibility measures are appliedand their results are compared In Section 6 the conclusionis drawn

2 Possibility Theory

The possibility theory proposed by Zadeh [19] defines apair of dual set functions possibility and necessity measuresTherefore a possibility degree prod(119906) quantifies the extent

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2016 Article ID 9185706 10 pageshttpdxdoiorg10115520169185706

2 Advances in Fuzzy Systems

an event 119906 is plausible and the necessity degree 119873(119906)quantifies the certainty of 119906 The model of imperfect data inthe possibility theory is the possibility distribution 120587

119861(119909)isin

[0 1] which characterizes the uncertain membership of anelement 119909 in a (well-defined) known class 119861

(i) prod(119860) = max119906isin119860

120587(119887) the possibility degree that 119906 isin119860

(ii) 119873(119860) = 1 minus prod(119860) = min119906notin119860

1 minus 120587(119887) the certaintydegree (necessity) that 119906 isin 119860 and 119860 complement ofevent 119860

Then the possibility distribution 120587 can be easily recoveredfrom the possibility measureprod

(i) Possibility distribution 120587(119906) = inf119860119906isin119860

prod(119860)(ii) Necessity measures 119873(119860) = 1 minus prod(119860) = inf

119906notin1198601 minus

120587(119906) If119873(119860) = 1119860 is certainly true and if119873(119860) = 0119860 is not certain (119860 is still possible)

21 Possibility Measures to Compare Intuitionistic Fuzzy Num-bers (IFNs) Existing in Literature The comparison betweenIFNs (see appendices) can be solved by using the possibilitydegree formula of the interval values Thus the possibilitymeasures have to satisfy the following properties [20]

Let 119886 = [119886minus 119886+] 119887 = [119887minus 119887+] and 119888 = [119888minus 119888+] then wehave the following

(i) 0 le 119901(119886 ge 119887) le 1(ii) complementary 119901(119886 ge 119887) + 119901(119887 ge 119886) = 1(iii) transitivity if 119901(119886 ge 119887) ge 05 and 119901(119887 ge 119888) ge 05 then

119901(119886 ge 119888) ge 05(iv) 119901(119886 ge 119887) ge 05 if and only if 119886minus + 119886+ ge 119887minus + 119887+(v) 119901(119886 ge 119887) = 05 if and only if 119886minus + 119886+ = 119887minus + 119887+

In what follows we present the different formulas of possi-bility measures 119901(119886 ge 119887) with 119886 = [119886minus 119886+] and 119887 = [119887minus 119887+]being two intuitionistic fuzzy numbers and120587(119886) = 1minus119886minusminus119886+120587(119887) = 1 minus 119887minus minus 119887+

(i) Yuan and Qu [20] presented some formulas to com-pare two interval numbers and denoted 119897

119886= 119886+ minus 119886minus 119897

119887=

119887+ minus 119887minus The possibility measures of 119901(119886 ge 119887) are presented asfollows

Definition 1 [21]

119901 (119886 ge 119887) = minmax(119886+ minus 119887minus

119897119886+ 119897119887

0) 1 (1)

Definition 2 [22]

119901 (119886 ge 119887) =max 0 119897

119886+ 119897119887minusmax (119887+ minus 119886minus)119897119886+ 119897119887

(2)

Definition 3 [23]

119901 (119886 ge 119887) =min 119897

119886+ 119897119887max (119886+ minus 119887minus 0)119897119886+ 119897119887

(3)

In [14 17] the authors have proved the equivalence ofthe above three formulas In [20] these formulas are used tocompare two countries Reference [18] used possibility degreeformula (3) to rank interval rough numbers

(ii) In [4 24 25] the authors defined the possibilitymeasures and called them likelihood measures as follows

119901 (119886 ge 119887) = max1 minusmax(119887+ minus 119886minus

119897119886+ 119897119887

0) 0 (4)

119901 (119887 ge 119886) = max1 minusmax(119886+ minus 119887minus

119897119886+ 119897119887

0) 0 (5)

where 119897119886= 119886+ minus 119886minus and 119897

119887= 119887+ minus 119887minus

(iii) In [15] the possibility measure is shown as follows

119901 (119886 ge 119887) =max (0 119886+ minus 119887minus) minusmax (0 119886minus minus 119887+)

(119886+ minus 119886minus) + (119887+ minus 119887minus) (6)

(iv) Wei and Tang [13] generalized possibility measure ofinterval-valued numbers to intuitionistic fuzzy sets

If 120587(119886) and 120587(119887) are different from zero the possibilitymeasure of 119886 ge 119887 is

119901 (119886 gt 119887) =max 0 (119886minus + 120587 (119886)) minus 119887minus

120587 (119886) + 120587 (119887)

minusmax 0 119886minus minus (119887minus + 120587 (119887))

120587 (119886) + 120587 (119887)

(7)

(v) Gao [17] presented some formulas of possibility (1)(2) (4) and (8) and proved their equivalence

1199013 (119886 ge 119887)

=

1 119887+ lt 119886minus

119886+ minus 119887minus

(119886+ minus 119886minus) + (119887+ minus 119887minus)119887minus le 119886+ 119886minus le 119887+

0 119887minus gt 119886+

(8)

(vi) Gao [17] determined the equivalence between formu-las (1) (2) (4) and (8) and integrated the following possibilitymeasures

(a) The first formula [26]

119901 (119886 ge 119887) =1

2(1 +

(119886+ + 119887+) + (119886minus minus 119887minus)

|119886+ minus 119887+| + |119886minus minus 119887minus| + 119897119886119887) (9)

where 119897119886119887= |119886+ minus 119886minus| + |119887+ minus 119887minus|

(b) The second formula [27]

119901 (119886 ge 119887)

=

1 119887+ le 119886minus

(119886+ minus 119887minus)2

(119886+ minus 119887minus)2 + (119887+ minus 119886minus)2119887minus lt 119886+ 119886minus lt 119887+

0 119887minus ge 119886+

(10)

Advances in Fuzzy Systems 3

(c) Results of integrated formulas (9) and (10)

119901 (119886 ge 119887)

=

1 119887+ le 119886minus

(119886+ minus 119887minus)2

2 (119886+ minus 119886minus) (119887+ minus 119887minus)119886minus lt 119887minus le 119886+ lt 119887+

1 minus(119887+ minus 119886minus)

2

2 (119886+ minus 119886minus) (119887+ minus 119887minus)119887minus lt 119886minus le 119887+ lt 119886+

119886minus + 119886+ minus 2119887minus

2 (119887+ minus 119887minus)119887minus le 119886minus le 119886+ le 119887+

2119886+ minus 119887+ minus 119887minus

2 (119886+ minus 119886minus)119886minus le 119887minus le 119887+ le 119886+

0 119887minus ge 119886+

(11)

(vii) According to Chen [28] the possibility measure ofthe event 119886 ge 119887 is presented as follows

119901 (119886 ge 119887) = max1 minusmax(1 minus 119887+) minus 119886minus

1198971+ 1198972

0 0 (12)

where 1198971= 1 minus 119886minus minus 119886+ and 119897

2= 1 minus 119887minus minus 119887+

3 Interval-Valued Intuitionistic Fuzzy Sets

There are some basic concepts related to the interval-valuedintuitionistic fuzzy sets (IVIFS) [29] Let119883 = 119909

1 1199092 119909

119899

be a nonempty set of the universe An IVIFS is defined as = ⟨119909

119894 [120583119871(119909119894) 120583119880(119909119894)] []119871(119909119894) ]119880(119909119894)]⟩ | 119909

119894isin 119883 where

[120583119871(119909119894) 120583119880(119909119894)] and []119871

(119909119894) ]119880(119909119894)] denote the intervals of

the membership degree and nonmembership degree of theelement 119909

119894isin satisfying the following

(i) 120583119880(119909119894) + ]119880(119909119894) le 1

(ii) 0 le ]119871(119909119894) le ]119880(119909119894) le 1 and 120583119871

(119909119894) le 120583119880(119909119894) for all

119909119894isin 119883

(iii) if 120583119871(119909119894) = 120583119880

(119909119894) and ]119871

(119909119894) = ]119880

(119909119894) then is

reduced to an IFS

31 Aggregation Operators Existing in Literature The aggre-gation operators are necessary to reduce the IVIFS valuesthus we can compare them using an accuracy function or apossibility measure In the following we present two existingaggregation operators

(i) Xu and Wei [6 30] defined the interval-valued intu-itionistic fuzzy weighted geometric (IVIFWG) operator asfollows

IVIFWG119908(1205721 1205722 120572

119899)

= ([

[

119899

prod119895=1

(119886119895)119908119895119899

prod119895=1

(119887119895)119908119895]

]

[

[

1 minus119899

prod119895=1

(1 minus 119888119908119895

119895) 1 minus

119899

prod119895=1

(1 minus 119889119908119895

119895)]

]

)

(13)

where 119908 = (1199081 1199082 119908

119899)119879 is the weight vector of 120572

119895(119895 =

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

119895=1119908119895= 1

(ii) Wang et al [9] defined the optimal aggregatedinterval-valued intuitionistic fuzzy sets using this formula

120572119894= ([

[

119899

sum119895=1

119886119894119895119908119895119899

sum119895=1

119887119894119895119908119895]

]

[

[

119899

sum119895=1

119888119894119895119908119895119899

sum119895=1

119889119894119895119908119895]

]

) (14)

32 PossibilityMeasures to Compare Interval-Valued Intuition-istic Fuzzy Number (IVIFN) Existing in Literature Let 120572

1=

([1198861 1198871] [1198881 1198891]) and 120572

2= ([1198862 1198872] [1198882 1198892]) be two interval-

valued intuitionistic fuzzy numbers in Ω which is the set ofall IVIFNs [2 5 31 32] 119901(120572

1ge 1205722) is the possibility measure

of two interval-valued intuitionistic fuzzy numbers Let uspresent the existing possibility measures

(i) Zhang et al [7] defined two possibilitymeasures of twointerval-valued intuitionistic fuzzy numbers as follows

(a) First measure

1199011(1205721ge 1205722)

= min (max (119860 (1205721) minus 119860 (120572

2) + 05 0) 1)

(15)

where119860(1205721) = 120582((119886+119887)2)+ (1minus120582)((119886minus 119888+119887minus119889)2)

and 120582 isin [0 1] which represents the performance onthe mean value of its membership degreeThis possibility degree satisfies the following proper-ties

(1) 0 le 1199011(1205721ge 1205722) le 1

(2) 1199011(1205721ge 1205722) = 1 hArr 119860(120572

1) minus 119860(120572

2) ge 05

(3) 1199011(1205721ge 1205722) = 0 hArr 119860(120572

1) minus 119860(120572

2) le minus05

(4) 1199011(1205721ge 1205722) + 1199011(1205722ge 1205721) = 1

(b) Second measure

1199012(1205721ge 1205722)

= 120574min(max( 1198871minus 1198862

1198871minus 1198862+ 1198872minus 1198862

0) 1)

+ (1 minus 120574)min(max( 1198892minus 1198881

1198891minus 1198881+ 1198892minus 1198882

0) 1)

(16)

where 120574 isin [0 1] gives the decision makersrsquo preferenceon membership degree or nonmembership degreeWhen 120574 ge 05 the decision maker is optimal whereaswhen 120574 lt 05 the decision maker is pessimisticThenthe below properties are checked

(1) 0 le 1199012(1205721ge 1205722) le 1

(2) 1199012(1205721ge 1205722) = 1 hArr 119887

2le 1198861and 119889

1le 1198882

(3) 1199012(1205721ge 1205722) = 0 hArr 119887

1le 1198862and 119889

2le 1198881

(4) 1199012(1205721ge 1205722) + 1199012(1205722ge 1205721) = 1

4 Advances in Fuzzy Systems

(ii)Wan andDong [33] defined possibilitymeasure by thefollowing formula

1199013(1205721ge 1205722)

=1

2119901 ([119886

1 1198871] ge [119886

2 1198872]) + 119901 ([119888

2 1198892] ge [1198881 1198891])

(17)

where 119901([1198861 1198871] ge [119886

2 1198872]) and 119901([119888

2 1198892] ge [119888

1 1198891]) can be

calculated using (4)(iii) Chen [28] defined a lower likelihood 119871minus and an upper

likelihood 119871+ on IVIFSs as

119871minus (1205721ge 1205722)

= max1 minusmax(1 minus 1198882) minus 1198861

1198971198861+ 1198971198862

0 0 (18)

where 1198971198861= 1 minus 119886

1minus 1198891and 1198971198862= 1 minus 119887

2minus 1198882 and

119871+ (1205721ge 1205722)

= max1 minusmax(1 minus 119889

2) minus 1198871

11989710158401198861+ 11989710158401198862

0 0 (19)

where 11989710158401198861= 1 minus 119887

1minus 1198881and 11989710158401198862= 1 minus 119886

2minus 1198892

Then for two IVIFNs the likelihood1199014(1205721ge 1205722) is defined

as follows

1199014(1205721ge 1205722) =

1

2(119871minus (120572

1ge 1205722) + 119871+ (120572

1ge 1205722)) (20)

These measures are the same as those of the possibilitymeasures

4 MCDM Based on PossibilityDegree of Interval-Valued IntuitionisticFuzzy Numbers

For a multicriteria decision making problem let 119860 =1198601 1198602 119860

119899 be the set of alternatives and 119883 =

1198831 1198832 119883

119899 the set of criteria 119882 = (119908

1 1199082 119908

119899)119879

is the weight vector of criteria 119883119895 where 119908

119895isin [0 1] and

sum119899

119895=1119908119895= 1

Suppose the characteristic information of alternative 119860119894

over criterion119883119895is represented by interval-valued intuition-

istic fuzzy number = ([119886 119887] [119888 119889]) where [119886 119887] representsthe fuzzy membership degree of the alternative 119860

119894over

criterion 119883119895and [119888 119889] represents the fuzzy nonmembership

degree of the alternative 119860119894over criterion 119883

119895 Then the

decision matrix is obtained as

(

([11988611 11988711] [11988811 11988911]) sdot sdot sdot ([119886

1119898 1198871119898] [1198881119898 1198891119898])

([1198861198991 1198871198991] [1198881198991 1198891198991]) ([119886

119899119898 119887119899119898] [119888119899119898 119889119899119898])

) (21)

The ranking of the alternatives in the multicriteria decisionmaking can be solved using the possibility measure ofinterval-valued intuitionistic fuzzy numbers We chose toadopt a modified version of the method described in [4]following the steps below

Step 1 Construct the interval-valued intuitionistic fuzzydecision matrix = (

119894119895)119898times119899

= ([119886119894119895 119887119894119895] [119888119894119895 119889119894119895])

Step 2 Calculate the intuitionistic fuzzy decision matrix119863119894119895

= [119863119871119894119895 119863119880119894119895] to derive 119863119871

119894119895and 119863119880

119894119895 and 119863

119894119895is the

transformed IFN decision matrix obtained from usingformulas (22)

119863119871119894119895=

119886119894119895ln 2 + 119889

119894119895ln (2 lowast 119889

119894119895119889119894119895+ 1) + ln (2119889

119894119895+ 1)

(119886119894119895+ 119889119894119895) ln 2 + 119886

119894119895ln (2119886

119894119895119886119894119895+ 1) + 119889

119894119895ln (2119889

119894119895119889119894119895+ 1) + ln (2119886

119894119895+ 1) + ln (2119889

119894119895+ 1)

119863119880119894119895=

119887119894119895ln 2 + 119888

119894119895ln (2 lowast 119888

119894119895119888119894119895+ 1) + ln (2119888

119894119895+ 1)

(119887119894119895+ 119888119894119895) ln 2 + 119887

119894119895ln (2119887

119894119895119887119894119895+ 1) + 119888

119894119895ln (2119888119894119895119888119894119895+ 1) + ln (2119887

119894119895+ 1) + ln (2119888

119894119895+ 1)

(22)

Step 3 Assign weights to criteria we use the followingstandard deviation (IF-SD) formula presented in [34] insteadof that used in [4]

119882119895=

120590119895

sum119899

119895=1120590119895

119895 = 1 119899 (23)

where

120590119895= radic119878 (120583

119894119895) + 119878 (]

119894119895)

119878 (120583119894119895) =

sum119898

119894=1(120583119894119895(119862119895) minus 120583119895(119862119895))2

119898

120583119895(119862119895) =

sum119898

119894=1120583119894119895(119862119895)

119898

119878 (]119894119895) =

sum119898

119894=1(]119894119895(119862119895) minus ]119895(119862119895))2

119898

(24)

where sum119899119895=1119908119895= 1

Step 4 Compute the performance of each alternative

119863119894= [

[

119899

sum119895=1

119882119895lowast 119863119871119894119895119899

sum119895=1

119882119895lowast 119863119880119894119895]

]

(25)

Advances in Fuzzy Systems 5

Step 5 Compute the likelihood matrix [25] To comparebetween tow interval fuzzy numbers we propose to usea possibility measure instead of the formula used in [4]to obtain a possibility matrix Therefore each possibilitymeasure presented in Section 21 is applied and all theachieved results are compared in Section 5

Step 6 Determine the alternatives ranking order accordingto the decreasing order of119882

119894[25] defined as

119882119894=sum119899

119895=1119901119894119895+ 1198982 minus 1

119898 (119898 minus 1) 119894 = 1 2 119898 (26)

5 Illustrative Example51 Application of Possibility Measure of IFS in Decision Mak-ing Problem This section described the data set presented in[4 9] to evaluate the four potential investment opportunities119860 = 1198601 1198602 1198603 1198604 The fund manager should evaluateeach investment considering four criteria risk (1198621) growth(1198622) sociopolitical issues (1198623) and environmental impacts(1198624) The fund manager is satisfied once he provides hisassessment of each alternative on each criterion

Step 1 The following interval-valued intuitionistic fuzzy sets(IVIFSs) decision making matrix (27) presents the relation-ship between criteria and alternatives of data set as follows

(

[042 048] [04 05] [06 07] [005 025] [04 05] [02 05] [055 075] [015 025]

[04 05] [04 05] [05 08] [01 02] [03 06] [03 04] [06 07] [01 03]

[03 05] [04 05] [01 03] [02 04] [07 08] [01 02] [05 07] [01 02]

[02 04] [04 05] [06 07] [02 03] [05 06] [02 03] [07 08] [01 02]

) (27)

Each element of this matrix is presented with IVIFSgiving the fund managerrsquos satisfaction or dissatisfactiondegree with an alternative The element represented for thefirst alternative [042 048] [04 05] where the interval 42ndash48 [4] reflects that the fund manager has an excellent

opportunity to respect the risk criterion (1198881) although theinterval 40ndash50 does not really represent an excellent choiceof 1198601 for risk (1198881)

Step 2 The intuitionistic fuzzy decision matrix (28) isobtained using (22)

(

[04452 05568] [07458 09429] [04304 07404] [07220 08773]

[04304 05696] [07404 09243] [04172 07040] [07040 09049]

[03494 05696] [02229 05966] [08245 09243] [07404 09049]

[02596 05000] [07040 08245] [06506 07889] [08245 09243]

) (28)

Step 3 Compute weights 119908 of the criteria based on (28) andusing (23)

119908 = 01884 02634 02439 03043 (29)

Step 4 We compute the performance of each alternativeusing (25) to obtain the interval fuzzy number

1198631= [06050 08008]

1198632= [05921 07978]

1198633= [05509 07653]

1198634= [06439 07851]

(30)

Step 5 In this step we apply each possibility measure anddetermine the achieved results These are then compared todefine the differences between them

Using the possibility measures (1) (2) (3) and (8) weachieved the results presented in Table 1 The best alternativeto be ranked first is 1198603

(a) For the possibility measures (4) and (5) the obtainedpossibility matrix is

(

05000 05198 06093 04656

04802 05000 05878 04437

04122 03907 05000 03413

06587 05563 05344 05000

) (31)

The results presented in Table 2 show 1198604 is the bestalternative and ranks first

(b) As for the possibility measure (6) the obtainedpossibility matrix is

(

05000 04802 04122 06587

05198 05000 03907 05563

06093 05878 05000 05344

04656 04437 03413 05000

) (32)

6 Advances in Fuzzy Systems

Table 1 Possibility degrees using (1) (2) (3) and (8)

Alternatives 1198601 1198602 1198603 1198604

Weights 02543 02472 02693 02292Ranking 2 3 1 4

Table 2 Possibility degrees using (4) and (5)

Alternatives 1198601 1198602 1198603 1198604

Weights 02579 02510 02204 02708Ranking 2 3 4 1

Table 3 Possibility degrees using (6)

Alternatives 1198601 1198602 1198603 1198604

Weights 02543 02472 02693 02292Ranking 2 3 1 4

Table 4 Possibility degrees using (7)

Alternatives 1198601 1198602 1198603 1198604

Weights 02535 02531 02569 02366Ranking 2 3 1 4

The results presented in Table 3 show 1198603 is the bestalternative and ranks first

(c) For the possibility measure (7) the obtained possibil-ity matrix is

(

05000 04941 04840 05634

05059 05000 04789 05522

05211 05160 05000 05458

04542 04478 04366 05000

) (33)

Table 4 shows that the best alternative is1198603 achievingthe first rank

(d) For the possibility measure (9) the obtained possibil-ity matrix is

(

05000 04810 04253 06204

05190 05000 04103 05474

05897 05747 05000 05296

04704 04526 03796 05000

) (34)

Table 5 shows that the best alternative is1198603 that ranksfirst

(e) For possibility measure (10) the obtained possibilitymatrix is

(

05000 04605 03297 07883

05395 05000 02914 06111

07086 06703 05000 05684

04316 03889 02117 05000

) (35)

Table 5 Possibility degrees using (9)

Alternatives 1198601 1198602 1198603 1198604

Weights 02522 02481 02662 02335Ranking 2 3 1 4

Table 6 Possibility degrees using (10)

Alternatives 1198601 1198602 1198603 1198604

Weights 02565 02452 02873 02110Ranking 2 3 1 4

Table 7 Possibility degrees using (12)

Alternatives 1198601 1198602 1198603 1198604

Weights 02511 02533 02557 02399Ranking 3 2 1 4

Table 8 Alternatives ranking order for different possibility mea-sures under IFN

Possibility measure Ranking Best alternative(1) (2) (3) and (8) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(4) and (5) 1198604 gt 1198601 gt 1198602 gt 1198603 1198604

(6) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(7) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(9) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(10) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(12) 1198603 gt 1198602 gt 1198601 gt 1198604 1198603

Table 6 shows that the best alternative that ranks firstis 1198602

(f) For the possibility measure (12) the obtained possi-bility matrix is

(

05000 04938 04872 05327

05062 05000 04939 05395

05061 05128 05000 05491

04509 04605 04673 05000

) (36)

Table 7 shows that the best alternative is 1198603 thatclearly ranks first

Table 8 presents a comparison of the obtained results applyingdifferent possibility measures under intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility measures (4) and (5) gave the same bestalternative 1198604 and the worst alternative 1198603 However usingformulas (1) (2) (3) (6) (7) (8) (9) (10) and (12) the bestalternative is1198603 and the worst alternative is1198604 These resultsshow that the measures (4) and (2) are different althoughthey are demonstrated to be equivalent (the operators lead tovalue 1) in [17] but they do not produce the same result

52 Application of Possibility Measures of IVIFS in DecisionMaking Problem We apply possibility measures of IVIFS

Advances in Fuzzy Systems 7

Table 9 Ranking IVIFSs alternatives using possibility measure (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02111 02161 02540 03200Ranking 4 3 2 1

Table 10 Ranking IVIFSs alternatives using possibility measures(15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02469 02401 02429 02640Ranking 2 4 3 1

presented in Section 32 to rank IVIFS data sets describedin Section 5 It is worth reminding that there are fouralternatives 1198601 1198602 1198603 and 1198604 and four criteria We use theIVIF matrix of alternatives (27) and the following criteriarsquosweight 119908

119895= [013 017 039 031] given in [9]

521 Case 1 Application of Interval-Valued Intuitionistic FuzzyWeighted Geometric (IVIFWG) Operator (13) The possibilitymeasures are applied in two cases In each case an aggregationoperator is also applied to the matrix (27)

Step 1 Compute the comprehensive evaluation of eachinvestment (alternative) using the geometric weighted aver-age operator (13) to aggregate the evaluation of each alterna-tive Thus we transform the IVIFS decision matrix to IVIFsfor each alternative presented as follows

1198631= [04760 05970] [01915 03926]

1198632= [04211 06454] [02260 03546]

1198633= [04057 06112] [01632 02833]

1198634= [05081 06389] [02007 03016]

(37)

Step 2 Each possibility measure presented in Section 32 isapplied to the obtained IVIFNs119863

1119863211986331198634

(a) For the possibility degree (16) the obtained possibilitymatrix is

(

05000 04671 03840 01823

05443 05000 03844 01647

06142 06210 05000 03131

06869 08353 08177 05000

) (38)

Table 9 presents the obtained results and shows thatthe best alternative is 1198604

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05264 05183 04185

04976 05000 04918 03921

05087 05063 05000 04003

05510 05597 05573 05000

) (39)

Table 10 presents the obtained results and shows thatthe best alternative is 1198604

Table 11 Ranking IVFISs alternatives using possibilitymeasure (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02627 02654 02105 02614Ranking 2 1 4 3

Table 12 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02254 02448 02613 02686Ranking 4 3 2 1

Table 13 Ranking order of alternatives for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198601 gt 1198603 gt 1198602 1198604

(16) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(17) 1198602 gt 1198601 gt 1198604 gt 1198603 1198602

(20) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(c) For the possibility measure (17) the obtained possi-bility matrix is

(

05000 05075 06501 04944

04925 05000 06635 05285

03365 03499 05000 03401

06599 04715 05056 05000

) (40)

The obtained results are presented in Table 11 showingthat the best alternative is 1198602

(d) For the possibility measure (20) the obtained possi-bility matrix is

(

05000 04446 03856 03740

05554 05000 04570 04256

05430 06144 05000 04778

05222 05744 06260 05000

) (41)

The obtained results are presented in Table 12 showingthat the best alternative is 1198604

Table 13 presents all the obtained results applying differentpossibility methods using the interval-valued intuitionisticfuzzy sets and shows the alternatives ranking results Weremark that the possibility formulas (15) (16) and (20)provide the same best alternative 1198604 However (17) providesthe best alternative 1198602

522 Case 2 Application of Optimal Aggregated Interval-Valued Intuitionistic Fuzzy Sets (14) Using the optimal aggre-gated operator (14) to IVIF decision matrix we obtain

8 Advances in Fuzzy Systems

Table 14 Ranking IVFSs using possibility degree (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02256 02352 02779 02599Ranking 4 3 1 2

Table 15 Ranking IVFSs using possibility measure (15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02369 02322 02557 02629Ranking 3 4 2 1

four interval-valued intuitionistic fuzzy numbers (IVIFNs)representing the alternatives as follows

1205721= [04831 06089] [01850 03800]

1205722= [04400 06520] [02170 03480]

1205723= [04840 06450] [01560 02730]

1205724= [05400 06530] [01950 02950]

(42)

(a) For the possibility measure (16) the obtained possi-bility degree matrix is

(

05000 04717 03595 03758

05366 05000 03682 04177

06329 06403 05000 05611

04358 05818 06008 05000

) (43)

The alternatives weight 119882119894is computed using (26)

and then ranked in a decreasing orderThe results aredisplayed in Table 14 showing that the best alternativethat ranks first is 119860

3

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05188 04308 03938

05000 05000 04120 03750

05525 05525 05000 04630

05168 05693 05693 05000

) (44)

We compute the weight 119882119894of the alternative using

(26) and we rank 119882119894in a decreasing order The

results are shown in Table 15 revealing that the bestalternative is 1198604 which ranks first

(c) For possibility measures (17) the obtained possibilitydegree matrix is

(

05000 05000 05767 04578

05000 05000 06123 05035

03877 04233 05000 03713

06287 04965 05422 05000

) (45)

Table 16 Ranking IVFSs using possibility degree (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02529 02596 02235 02639Ranking 3 2 4 1

Table 17 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02240 02392 02695 02673Ranking 4 3 1 2

Table 18 Alternatives ranking order for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198603 gt 1198601 gt 1198602 1198604

(16) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

(17) 1198604 gt 1198602 gt 1198601 gt 1198603 1198604

(20) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

We compute the weights119882119894of the alternatives using

(26) and we rank 119882119894in a decreasing order The

results are displayed in Table 16 showing that the bestalternative that rank first is 1198604

(d) For possibility measure (20) the obtained possibilitymatrix is

(

05000 04485 03685 03705

05515 05000 04082 04105

05918 06315 05000 05113

04887 05895 06295 05000

) (46)

The obtained results are presented in Table 17 showingthat the best alternative is 1198603

Table 18 presents the results of all applied possibilitymeasures using the interval-valued intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility formulas (15) and (17) provide the same bestalternative 1198604 However (16) and (20) provide the bestalternative 1198603 We note that the latter is the worst alternativeusing (17)

6 Conclusion

In this study we presented different formulas of possibilitymeasures The formulas exist in literature with IFN andIVIFN We also presented an MCDM method from theliterature We gave an illustrative examples for applicationsof different possibility measures and compared their resultsFirst we used an MCDM matrix with intuitionistic fuzzynumbers and then anMCDMmatrixwith IVIFNsThe valuesof the latter are aggregated with an aggregation operator intwo cases In each case a different aggregation operator wasused Thus the appropriate possibility measures are applied

Advances in Fuzzy Systems 9

The results show that the ranked alternatives can be differentfor each possibility measure even though some of thesemeasures have already been demonstrated to be equivalentin the literature

Appendix

Intuitionistic Fuzzy Sets

Intuitionistic fuzzy sets are introduced by Atanassov [16] whodefined a degree of membership 120583 a degree of nonmember-ship ] and a degree of hesitation 120587 of an element 119909 of an IFS

119860 = (120583119860 ]119860) denotes an intuitionistic fuzzy number if 120583

119860

and ]119860are fuzzy numbers with ]

119860le 120583119888119860 where 120583119888

119860denotes

the complement of 120583119860

If119883 is a discourse universe and 119860 a set in119883 then

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883 (A1)

with the conditions 0 le 120583119860(119909) le 1 0 le ]

119860(119909) le 1 0 le

120583119860(119909) + ]

119860(119909) le 1 and 120587

119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) Also for

each 119909 isin 119883 0 le Π119860(119909) le 1

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof this work by grants from General Direction of ScientificResearch (DGRST) Tunisia under the ARUB programThey would like to thank Mr Abdelmajid Dammak for hisproofreading and correction of the English of the paper

References

[1] J H Park I Y Park Y C L Kwun andX Tan ldquoExtension of theTOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environmentrdquo Applied MathematicalModelling vol 35 no 5 pp 2544ndash2556 2011

[2] D G Park Y C Kwun J H Park and I Y Park ldquoCorrelationcoefficient of interval-valued intuitionistic fuzzy sets and itsapplication to multiple attribute group decision making prob-lemsrdquoMathematical and Computer Modelling vol 50 no 9-10pp 1279ndash1293 2009

[3] J Ye ldquoMulticriteria fuzzy decision-making method usingentropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy setsrdquo Applied Mathematical Mod-elling vol 34 no 12 pp 3864ndash3870 2010

[4] H Zhang and L Yu ldquoMADM method based on cross-entropyand extended TOPSIS with interval-valued intuitionistic fuzzysetsrdquo Knowledge-Based Systems vol 30 pp 115ndash120 2012

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

[6] Z-S Xu and J Chen ldquoApproach to groupdecisionmaking basedon interval-valued intuitionistic judgment matricesrdquo SystemsEngineeringmdashTheory amp Practice vol 27 no 4 pp 126ndash133 2007

[7] X Zhang G Yue and Z Teng ldquoPossibility degree of interval-valued intuitionistic fuzzy numbers and its applicationrdquo inProceedings of the International Symposium on InformationProcessing (ISIP rsquo09) pp 33ndash36 Huangshan China 2009

[8] J Wu Q Cao and H Li ldquoAn approach for MADM problemswith interval-valued intuitionistic fuzzy sets based on nonlinearfunctionsrdquo Technological and Economic Development of Econ-omy vol 22 no 3 pp 336ndash356 2016

[9] Z Wang K W Li and W Wang ldquoAn approach to multi-attribute decision making with interval-valued intuitionisticfuzzy assessments and incomplete weightsrdquo Information Sci-ences vol 179 no 17 pp 3026ndash3040 2009

[10] S C Onar B Oztaysi I Otay and C Kahraman ldquoMulti-expert wind energy technology selection using interval-valuedintuitionistic fuzzy setsrdquo Energy vol 90 part 1 pp 274ndash2852015

[11] Z Xu and R R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006

[12] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[13] C-P Wei and X Tang ldquoPossibility degree method for rank-ing intuitionistic fuzzy numbersrdquo in Proceedings of the 3rdIEEEWICACM International Conference on Web Intelligenceand Intelligent Agent Technology (WI-IAT rsquo10) pp 142ndash145IEEE Toronto Canada August 2010

[14] Z S Xu and Q L Da ldquoPossibility degree method for rankinginterval numbers and its applicationrdquo Journal of Systems Engi-neering vol 18 pp 67ndash70 2003

[15] Y-M Wang J-B Yang and D-L Xu ldquoInterval weight gen-eration approaches based on consistency test and intervalcomparison matricesrdquo Applied Mathematics and Computationvol 167 no 1 pp 252ndash273 2005

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

[17] F Gao ldquoPossibility degree and comprehensive priority ofinterval numbersrdquo Systems EngineeringmdashTheoryamp Practice vol33 no 8 pp 2033ndash2040 2013

[18] Y-Y Liu and Y-J Lv ldquoA multiple attribute decision makingmethod with interval rough numbers based on the possibilitydegreerdquo in Proceedings of the 10th International Conference onNatural Computation (ICNC rsquo14) pp 407ndash411 IEEE XiamenChina August 2014

[19] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[20] H Yuan and Y Qu ldquoModel for conflict resolution withpreference represented as interval numbersrdquo Proceedings of theMATECWeb of Conferences 2015

[21] G Facchinetti R G Ricci and S Muzzioli ldquoNote on rankingfuzzy triangular numbersrdquo International Journal of IntelligentSystems vol 13 no 7 pp 613ndash622 1998

[22] Q L Da and X W Liu ldquoInterval number linear programmingand its satisfactory solutionrdquo Systems Engineering Theory ampPractice vol 19 pp 3ndash7 1999

[23] C-P Wei and X Tang ldquoPossibility degree method for rankingintuitionistic fuzzy numbersrdquo Journal of Systems Engineeringvol 18 pp 67ndash70 2003

[24] Z S Xu and Q L Da ldquoThe uncertain ow a operatorrdquo Interna-tional Journal of Intelligent Systems vol 17 no 6 pp 569ndash5752002

10 Advances in Fuzzy Systems

[25] Z S Xu and Q L Da ldquoA possibility based method for prioritiesof interval judjment matricesrdquo Chinese Journal of ManagementScience vol 11 pp 63ndash65 2003

[26] D Q Li and Y D Gu ldquomethods for ranking interval numberbased on possibility degreerdquo Journal of Systems Engineering vol23 pp 223ndash226 2008

[27] J B Lan L J Cao and J Lin ldquoMethod for rinking interval num-bers on two-dimensional priority degreerdquo Journal of ChongqingInstitute of Technology Natural Science Edition vol 21 pp 63ndash66 2007

[28] T-Y Chen ldquoInterval-valued intuitionistic fuzzy QUALIFLEXmethod with a likelihood-based comparison approach formultiple criteria decision analysisrdquo Information Sciences vol261 pp 149ndash169 2014

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

[30] G W Wei and X R Wang ldquoSome geometric aggregationoperators on interval-valued intuitionistic fuzzy sets and theirapplication to group decision makingrdquo in Proceedings of theInternational Conference on Computational Intelligence andSecurity (ICCIS rsquo07) pp 495ndash499 Harbin China December2007

[31] Z S Xu and J Chen ldquoOn geometric aggregation over interval-valued intuitionistic fuzzy informationrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) vol 2 pp 466ndash471 Haikou ChinaAugust 2007

[32] Y He H Chen L Zhou J Liu and Z Tao ldquoGeneralizedinterval-valuedAtanassovrsquos intuitionistic fuzzy power operatorsand their application to group decision makingrdquo InternationalJournal of Fuzzy Systems vol 15 no 4 pp 401ndash411 2013

[33] S Wan and J Dong ldquoA possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision mak-ingrdquo Journal of Computer and System Sciences vol 80 no 1 pp237ndash256 2014

[34] F Dammak L Baccour and A M Alimi ldquoThe impact ofcriterion weights techniques in topsis method of multi-criteriadecision making in crisp and intuitionistic fuzzy domainsrdquoin Proceedings of the IEEE International Conference on FuzzySystems (FUZZ-IEEE rsquo15) pp 1ndash8 Istanbul Turkey August 2015

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 Advances in Fuzzy Systems

an event 119906 is plausible and the necessity degree 119873(119906)quantifies the certainty of 119906 The model of imperfect data inthe possibility theory is the possibility distribution 120587

119861(119909)isin

[0 1] which characterizes the uncertain membership of anelement 119909 in a (well-defined) known class 119861

(i) prod(119860) = max119906isin119860

120587(119887) the possibility degree that 119906 isin119860

(ii) 119873(119860) = 1 minus prod(119860) = min119906notin119860

1 minus 120587(119887) the certaintydegree (necessity) that 119906 isin 119860 and 119860 complement ofevent 119860

Then the possibility distribution 120587 can be easily recoveredfrom the possibility measureprod

(i) Possibility distribution 120587(119906) = inf119860119906isin119860

prod(119860)(ii) Necessity measures 119873(119860) = 1 minus prod(119860) = inf

119906notin1198601 minus

120587(119906) If119873(119860) = 1119860 is certainly true and if119873(119860) = 0119860 is not certain (119860 is still possible)

21 Possibility Measures to Compare Intuitionistic Fuzzy Num-bers (IFNs) Existing in Literature The comparison betweenIFNs (see appendices) can be solved by using the possibilitydegree formula of the interval values Thus the possibilitymeasures have to satisfy the following properties [20]

Let 119886 = [119886minus 119886+] 119887 = [119887minus 119887+] and 119888 = [119888minus 119888+] then wehave the following

(i) 0 le 119901(119886 ge 119887) le 1(ii) complementary 119901(119886 ge 119887) + 119901(119887 ge 119886) = 1(iii) transitivity if 119901(119886 ge 119887) ge 05 and 119901(119887 ge 119888) ge 05 then

119901(119886 ge 119888) ge 05(iv) 119901(119886 ge 119887) ge 05 if and only if 119886minus + 119886+ ge 119887minus + 119887+(v) 119901(119886 ge 119887) = 05 if and only if 119886minus + 119886+ = 119887minus + 119887+

In what follows we present the different formulas of possi-bility measures 119901(119886 ge 119887) with 119886 = [119886minus 119886+] and 119887 = [119887minus 119887+]being two intuitionistic fuzzy numbers and120587(119886) = 1minus119886minusminus119886+120587(119887) = 1 minus 119887minus minus 119887+

(i) Yuan and Qu [20] presented some formulas to com-pare two interval numbers and denoted 119897

119886= 119886+ minus 119886minus 119897

119887=

119887+ minus 119887minus The possibility measures of 119901(119886 ge 119887) are presented asfollows

Definition 1 [21]

119901 (119886 ge 119887) = minmax(119886+ minus 119887minus

119897119886+ 119897119887

0) 1 (1)

Definition 2 [22]

119901 (119886 ge 119887) =max 0 119897

119886+ 119897119887minusmax (119887+ minus 119886minus)119897119886+ 119897119887

(2)

Definition 3 [23]

119901 (119886 ge 119887) =min 119897

119886+ 119897119887max (119886+ minus 119887minus 0)119897119886+ 119897119887

(3)

In [14 17] the authors have proved the equivalence ofthe above three formulas In [20] these formulas are used tocompare two countries Reference [18] used possibility degreeformula (3) to rank interval rough numbers

(ii) In [4 24 25] the authors defined the possibilitymeasures and called them likelihood measures as follows

119901 (119886 ge 119887) = max1 minusmax(119887+ minus 119886minus

119897119886+ 119897119887

0) 0 (4)

119901 (119887 ge 119886) = max1 minusmax(119886+ minus 119887minus

119897119886+ 119897119887

0) 0 (5)

where 119897119886= 119886+ minus 119886minus and 119897

119887= 119887+ minus 119887minus

(iii) In [15] the possibility measure is shown as follows

119901 (119886 ge 119887) =max (0 119886+ minus 119887minus) minusmax (0 119886minus minus 119887+)

(119886+ minus 119886minus) + (119887+ minus 119887minus) (6)

(iv) Wei and Tang [13] generalized possibility measure ofinterval-valued numbers to intuitionistic fuzzy sets

If 120587(119886) and 120587(119887) are different from zero the possibilitymeasure of 119886 ge 119887 is

119901 (119886 gt 119887) =max 0 (119886minus + 120587 (119886)) minus 119887minus

120587 (119886) + 120587 (119887)

minusmax 0 119886minus minus (119887minus + 120587 (119887))

120587 (119886) + 120587 (119887)

(7)

(v) Gao [17] presented some formulas of possibility (1)(2) (4) and (8) and proved their equivalence

1199013 (119886 ge 119887)

=

1 119887+ lt 119886minus

119886+ minus 119887minus

(119886+ minus 119886minus) + (119887+ minus 119887minus)119887minus le 119886+ 119886minus le 119887+

0 119887minus gt 119886+

(8)

(vi) Gao [17] determined the equivalence between formu-las (1) (2) (4) and (8) and integrated the following possibilitymeasures

(a) The first formula [26]

119901 (119886 ge 119887) =1

2(1 +

(119886+ + 119887+) + (119886minus minus 119887minus)

|119886+ minus 119887+| + |119886minus minus 119887minus| + 119897119886119887) (9)

where 119897119886119887= |119886+ minus 119886minus| + |119887+ minus 119887minus|

(b) The second formula [27]

119901 (119886 ge 119887)

=

1 119887+ le 119886minus

(119886+ minus 119887minus)2

(119886+ minus 119887minus)2 + (119887+ minus 119886minus)2119887minus lt 119886+ 119886minus lt 119887+

0 119887minus ge 119886+

(10)

Advances in Fuzzy Systems 3

(c) Results of integrated formulas (9) and (10)

119901 (119886 ge 119887)

=

1 119887+ le 119886minus

(119886+ minus 119887minus)2

2 (119886+ minus 119886minus) (119887+ minus 119887minus)119886minus lt 119887minus le 119886+ lt 119887+

1 minus(119887+ minus 119886minus)

2

2 (119886+ minus 119886minus) (119887+ minus 119887minus)119887minus lt 119886minus le 119887+ lt 119886+

119886minus + 119886+ minus 2119887minus

2 (119887+ minus 119887minus)119887minus le 119886minus le 119886+ le 119887+

2119886+ minus 119887+ minus 119887minus

2 (119886+ minus 119886minus)119886minus le 119887minus le 119887+ le 119886+

0 119887minus ge 119886+

(11)

(vii) According to Chen [28] the possibility measure ofthe event 119886 ge 119887 is presented as follows

119901 (119886 ge 119887) = max1 minusmax(1 minus 119887+) minus 119886minus

1198971+ 1198972

0 0 (12)

where 1198971= 1 minus 119886minus minus 119886+ and 119897

2= 1 minus 119887minus minus 119887+

3 Interval-Valued Intuitionistic Fuzzy Sets

There are some basic concepts related to the interval-valuedintuitionistic fuzzy sets (IVIFS) [29] Let119883 = 119909

1 1199092 119909

119899

be a nonempty set of the universe An IVIFS is defined as = ⟨119909

119894 [120583119871(119909119894) 120583119880(119909119894)] []119871(119909119894) ]119880(119909119894)]⟩ | 119909

119894isin 119883 where

[120583119871(119909119894) 120583119880(119909119894)] and []119871

(119909119894) ]119880(119909119894)] denote the intervals of

the membership degree and nonmembership degree of theelement 119909

119894isin satisfying the following

(i) 120583119880(119909119894) + ]119880(119909119894) le 1

(ii) 0 le ]119871(119909119894) le ]119880(119909119894) le 1 and 120583119871

(119909119894) le 120583119880(119909119894) for all

119909119894isin 119883

(iii) if 120583119871(119909119894) = 120583119880

(119909119894) and ]119871

(119909119894) = ]119880

(119909119894) then is

reduced to an IFS

31 Aggregation Operators Existing in Literature The aggre-gation operators are necessary to reduce the IVIFS valuesthus we can compare them using an accuracy function or apossibility measure In the following we present two existingaggregation operators

(i) Xu and Wei [6 30] defined the interval-valued intu-itionistic fuzzy weighted geometric (IVIFWG) operator asfollows

IVIFWG119908(1205721 1205722 120572

119899)

= ([

[

119899

prod119895=1

(119886119895)119908119895119899

prod119895=1

(119887119895)119908119895]

]

[

[

1 minus119899

prod119895=1

(1 minus 119888119908119895

119895) 1 minus

119899

prod119895=1

(1 minus 119889119908119895

119895)]

]

)

(13)

where 119908 = (1199081 1199082 119908

119899)119879 is the weight vector of 120572

119895(119895 =

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

119895=1119908119895= 1

(ii) Wang et al [9] defined the optimal aggregatedinterval-valued intuitionistic fuzzy sets using this formula

120572119894= ([

[

119899

sum119895=1

119886119894119895119908119895119899

sum119895=1

119887119894119895119908119895]

]

[

[

119899

sum119895=1

119888119894119895119908119895119899

sum119895=1

119889119894119895119908119895]

]

) (14)

32 PossibilityMeasures to Compare Interval-Valued Intuition-istic Fuzzy Number (IVIFN) Existing in Literature Let 120572

1=

([1198861 1198871] [1198881 1198891]) and 120572

2= ([1198862 1198872] [1198882 1198892]) be two interval-

valued intuitionistic fuzzy numbers in Ω which is the set ofall IVIFNs [2 5 31 32] 119901(120572

1ge 1205722) is the possibility measure

of two interval-valued intuitionistic fuzzy numbers Let uspresent the existing possibility measures

(i) Zhang et al [7] defined two possibilitymeasures of twointerval-valued intuitionistic fuzzy numbers as follows

(a) First measure

1199011(1205721ge 1205722)

= min (max (119860 (1205721) minus 119860 (120572

2) + 05 0) 1)

(15)

where119860(1205721) = 120582((119886+119887)2)+ (1minus120582)((119886minus 119888+119887minus119889)2)

and 120582 isin [0 1] which represents the performance onthe mean value of its membership degreeThis possibility degree satisfies the following proper-ties

(1) 0 le 1199011(1205721ge 1205722) le 1

(2) 1199011(1205721ge 1205722) = 1 hArr 119860(120572

1) minus 119860(120572

2) ge 05

(3) 1199011(1205721ge 1205722) = 0 hArr 119860(120572

1) minus 119860(120572

2) le minus05

(4) 1199011(1205721ge 1205722) + 1199011(1205722ge 1205721) = 1

(b) Second measure

1199012(1205721ge 1205722)

= 120574min(max( 1198871minus 1198862

1198871minus 1198862+ 1198872minus 1198862

0) 1)

+ (1 minus 120574)min(max( 1198892minus 1198881

1198891minus 1198881+ 1198892minus 1198882

0) 1)

(16)

where 120574 isin [0 1] gives the decision makersrsquo preferenceon membership degree or nonmembership degreeWhen 120574 ge 05 the decision maker is optimal whereaswhen 120574 lt 05 the decision maker is pessimisticThenthe below properties are checked

(1) 0 le 1199012(1205721ge 1205722) le 1

(2) 1199012(1205721ge 1205722) = 1 hArr 119887

2le 1198861and 119889

1le 1198882

(3) 1199012(1205721ge 1205722) = 0 hArr 119887

1le 1198862and 119889

2le 1198881

(4) 1199012(1205721ge 1205722) + 1199012(1205722ge 1205721) = 1

4 Advances in Fuzzy Systems

(ii)Wan andDong [33] defined possibilitymeasure by thefollowing formula

1199013(1205721ge 1205722)

=1

2119901 ([119886

1 1198871] ge [119886

2 1198872]) + 119901 ([119888

2 1198892] ge [1198881 1198891])

(17)

where 119901([1198861 1198871] ge [119886

2 1198872]) and 119901([119888

2 1198892] ge [119888

1 1198891]) can be

calculated using (4)(iii) Chen [28] defined a lower likelihood 119871minus and an upper

likelihood 119871+ on IVIFSs as

119871minus (1205721ge 1205722)

= max1 minusmax(1 minus 1198882) minus 1198861

1198971198861+ 1198971198862

0 0 (18)

where 1198971198861= 1 minus 119886

1minus 1198891and 1198971198862= 1 minus 119887

2minus 1198882 and

119871+ (1205721ge 1205722)

= max1 minusmax(1 minus 119889

2) minus 1198871

11989710158401198861+ 11989710158401198862

0 0 (19)

where 11989710158401198861= 1 minus 119887

1minus 1198881and 11989710158401198862= 1 minus 119886

2minus 1198892

Then for two IVIFNs the likelihood1199014(1205721ge 1205722) is defined

as follows

1199014(1205721ge 1205722) =

1

2(119871minus (120572

1ge 1205722) + 119871+ (120572

1ge 1205722)) (20)

These measures are the same as those of the possibilitymeasures

4 MCDM Based on PossibilityDegree of Interval-Valued IntuitionisticFuzzy Numbers

For a multicriteria decision making problem let 119860 =1198601 1198602 119860

119899 be the set of alternatives and 119883 =

1198831 1198832 119883

119899 the set of criteria 119882 = (119908

1 1199082 119908

119899)119879

is the weight vector of criteria 119883119895 where 119908

119895isin [0 1] and

sum119899

119895=1119908119895= 1

Suppose the characteristic information of alternative 119860119894

over criterion119883119895is represented by interval-valued intuition-

istic fuzzy number = ([119886 119887] [119888 119889]) where [119886 119887] representsthe fuzzy membership degree of the alternative 119860

119894over

criterion 119883119895and [119888 119889] represents the fuzzy nonmembership

degree of the alternative 119860119894over criterion 119883

119895 Then the

decision matrix is obtained as

(

([11988611 11988711] [11988811 11988911]) sdot sdot sdot ([119886

1119898 1198871119898] [1198881119898 1198891119898])

([1198861198991 1198871198991] [1198881198991 1198891198991]) ([119886

119899119898 119887119899119898] [119888119899119898 119889119899119898])

) (21)

The ranking of the alternatives in the multicriteria decisionmaking can be solved using the possibility measure ofinterval-valued intuitionistic fuzzy numbers We chose toadopt a modified version of the method described in [4]following the steps below

Step 1 Construct the interval-valued intuitionistic fuzzydecision matrix = (

119894119895)119898times119899

= ([119886119894119895 119887119894119895] [119888119894119895 119889119894119895])

Step 2 Calculate the intuitionistic fuzzy decision matrix119863119894119895

= [119863119871119894119895 119863119880119894119895] to derive 119863119871

119894119895and 119863119880

119894119895 and 119863

119894119895is the

transformed IFN decision matrix obtained from usingformulas (22)

119863119871119894119895=

119886119894119895ln 2 + 119889

119894119895ln (2 lowast 119889

119894119895119889119894119895+ 1) + ln (2119889

119894119895+ 1)

(119886119894119895+ 119889119894119895) ln 2 + 119886

119894119895ln (2119886

119894119895119886119894119895+ 1) + 119889

119894119895ln (2119889

119894119895119889119894119895+ 1) + ln (2119886

119894119895+ 1) + ln (2119889

119894119895+ 1)

119863119880119894119895=

119887119894119895ln 2 + 119888

119894119895ln (2 lowast 119888

119894119895119888119894119895+ 1) + ln (2119888

119894119895+ 1)

(119887119894119895+ 119888119894119895) ln 2 + 119887

119894119895ln (2119887

119894119895119887119894119895+ 1) + 119888

119894119895ln (2119888119894119895119888119894119895+ 1) + ln (2119887

119894119895+ 1) + ln (2119888

119894119895+ 1)

(22)

Step 3 Assign weights to criteria we use the followingstandard deviation (IF-SD) formula presented in [34] insteadof that used in [4]

119882119895=

120590119895

sum119899

119895=1120590119895

119895 = 1 119899 (23)

where

120590119895= radic119878 (120583

119894119895) + 119878 (]

119894119895)

119878 (120583119894119895) =

sum119898

119894=1(120583119894119895(119862119895) minus 120583119895(119862119895))2

119898

120583119895(119862119895) =

sum119898

119894=1120583119894119895(119862119895)

119898

119878 (]119894119895) =

sum119898

119894=1(]119894119895(119862119895) minus ]119895(119862119895))2

119898

(24)

where sum119899119895=1119908119895= 1

Step 4 Compute the performance of each alternative

119863119894= [

[

119899

sum119895=1

119882119895lowast 119863119871119894119895119899

sum119895=1

119882119895lowast 119863119880119894119895]

]

(25)

Advances in Fuzzy Systems 5

Step 5 Compute the likelihood matrix [25] To comparebetween tow interval fuzzy numbers we propose to usea possibility measure instead of the formula used in [4]to obtain a possibility matrix Therefore each possibilitymeasure presented in Section 21 is applied and all theachieved results are compared in Section 5

Step 6 Determine the alternatives ranking order accordingto the decreasing order of119882

119894[25] defined as

119882119894=sum119899

119895=1119901119894119895+ 1198982 minus 1

119898 (119898 minus 1) 119894 = 1 2 119898 (26)

5 Illustrative Example51 Application of Possibility Measure of IFS in Decision Mak-ing Problem This section described the data set presented in[4 9] to evaluate the four potential investment opportunities119860 = 1198601 1198602 1198603 1198604 The fund manager should evaluateeach investment considering four criteria risk (1198621) growth(1198622) sociopolitical issues (1198623) and environmental impacts(1198624) The fund manager is satisfied once he provides hisassessment of each alternative on each criterion

Step 1 The following interval-valued intuitionistic fuzzy sets(IVIFSs) decision making matrix (27) presents the relation-ship between criteria and alternatives of data set as follows

(

[042 048] [04 05] [06 07] [005 025] [04 05] [02 05] [055 075] [015 025]

[04 05] [04 05] [05 08] [01 02] [03 06] [03 04] [06 07] [01 03]

[03 05] [04 05] [01 03] [02 04] [07 08] [01 02] [05 07] [01 02]

[02 04] [04 05] [06 07] [02 03] [05 06] [02 03] [07 08] [01 02]

) (27)

Each element of this matrix is presented with IVIFSgiving the fund managerrsquos satisfaction or dissatisfactiondegree with an alternative The element represented for thefirst alternative [042 048] [04 05] where the interval 42ndash48 [4] reflects that the fund manager has an excellent

opportunity to respect the risk criterion (1198881) although theinterval 40ndash50 does not really represent an excellent choiceof 1198601 for risk (1198881)

Step 2 The intuitionistic fuzzy decision matrix (28) isobtained using (22)

(

[04452 05568] [07458 09429] [04304 07404] [07220 08773]

[04304 05696] [07404 09243] [04172 07040] [07040 09049]

[03494 05696] [02229 05966] [08245 09243] [07404 09049]

[02596 05000] [07040 08245] [06506 07889] [08245 09243]

) (28)

Step 3 Compute weights 119908 of the criteria based on (28) andusing (23)

119908 = 01884 02634 02439 03043 (29)

Step 4 We compute the performance of each alternativeusing (25) to obtain the interval fuzzy number

1198631= [06050 08008]

1198632= [05921 07978]

1198633= [05509 07653]

1198634= [06439 07851]

(30)

Step 5 In this step we apply each possibility measure anddetermine the achieved results These are then compared todefine the differences between them

Using the possibility measures (1) (2) (3) and (8) weachieved the results presented in Table 1 The best alternativeto be ranked first is 1198603

(a) For the possibility measures (4) and (5) the obtainedpossibility matrix is

(

05000 05198 06093 04656

04802 05000 05878 04437

04122 03907 05000 03413

06587 05563 05344 05000

) (31)

The results presented in Table 2 show 1198604 is the bestalternative and ranks first

(b) As for the possibility measure (6) the obtainedpossibility matrix is

(

05000 04802 04122 06587

05198 05000 03907 05563

06093 05878 05000 05344

04656 04437 03413 05000

) (32)

6 Advances in Fuzzy Systems

Table 1 Possibility degrees using (1) (2) (3) and (8)

Alternatives 1198601 1198602 1198603 1198604

Weights 02543 02472 02693 02292Ranking 2 3 1 4

Table 2 Possibility degrees using (4) and (5)

Alternatives 1198601 1198602 1198603 1198604

Weights 02579 02510 02204 02708Ranking 2 3 4 1

Table 3 Possibility degrees using (6)

Alternatives 1198601 1198602 1198603 1198604

Weights 02543 02472 02693 02292Ranking 2 3 1 4

Table 4 Possibility degrees using (7)

Alternatives 1198601 1198602 1198603 1198604

Weights 02535 02531 02569 02366Ranking 2 3 1 4

The results presented in Table 3 show 1198603 is the bestalternative and ranks first

(c) For the possibility measure (7) the obtained possibil-ity matrix is

(

05000 04941 04840 05634

05059 05000 04789 05522

05211 05160 05000 05458

04542 04478 04366 05000

) (33)

Table 4 shows that the best alternative is1198603 achievingthe first rank

(d) For the possibility measure (9) the obtained possibil-ity matrix is

(

05000 04810 04253 06204

05190 05000 04103 05474

05897 05747 05000 05296

04704 04526 03796 05000

) (34)

Table 5 shows that the best alternative is1198603 that ranksfirst

(e) For possibility measure (10) the obtained possibilitymatrix is

(

05000 04605 03297 07883

05395 05000 02914 06111

07086 06703 05000 05684

04316 03889 02117 05000

) (35)

Table 5 Possibility degrees using (9)

Alternatives 1198601 1198602 1198603 1198604

Weights 02522 02481 02662 02335Ranking 2 3 1 4

Table 6 Possibility degrees using (10)

Alternatives 1198601 1198602 1198603 1198604

Weights 02565 02452 02873 02110Ranking 2 3 1 4

Table 7 Possibility degrees using (12)

Alternatives 1198601 1198602 1198603 1198604

Weights 02511 02533 02557 02399Ranking 3 2 1 4

Table 8 Alternatives ranking order for different possibility mea-sures under IFN

Possibility measure Ranking Best alternative(1) (2) (3) and (8) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(4) and (5) 1198604 gt 1198601 gt 1198602 gt 1198603 1198604

(6) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(7) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(9) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(10) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(12) 1198603 gt 1198602 gt 1198601 gt 1198604 1198603

Table 6 shows that the best alternative that ranks firstis 1198602

(f) For the possibility measure (12) the obtained possi-bility matrix is

(

05000 04938 04872 05327

05062 05000 04939 05395

05061 05128 05000 05491

04509 04605 04673 05000

) (36)

Table 7 shows that the best alternative is 1198603 thatclearly ranks first

Table 8 presents a comparison of the obtained results applyingdifferent possibility measures under intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility measures (4) and (5) gave the same bestalternative 1198604 and the worst alternative 1198603 However usingformulas (1) (2) (3) (6) (7) (8) (9) (10) and (12) the bestalternative is1198603 and the worst alternative is1198604 These resultsshow that the measures (4) and (2) are different althoughthey are demonstrated to be equivalent (the operators lead tovalue 1) in [17] but they do not produce the same result

52 Application of Possibility Measures of IVIFS in DecisionMaking Problem We apply possibility measures of IVIFS

Advances in Fuzzy Systems 7

Table 9 Ranking IVIFSs alternatives using possibility measure (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02111 02161 02540 03200Ranking 4 3 2 1

Table 10 Ranking IVIFSs alternatives using possibility measures(15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02469 02401 02429 02640Ranking 2 4 3 1

presented in Section 32 to rank IVIFS data sets describedin Section 5 It is worth reminding that there are fouralternatives 1198601 1198602 1198603 and 1198604 and four criteria We use theIVIF matrix of alternatives (27) and the following criteriarsquosweight 119908

119895= [013 017 039 031] given in [9]

521 Case 1 Application of Interval-Valued Intuitionistic FuzzyWeighted Geometric (IVIFWG) Operator (13) The possibilitymeasures are applied in two cases In each case an aggregationoperator is also applied to the matrix (27)

Step 1 Compute the comprehensive evaluation of eachinvestment (alternative) using the geometric weighted aver-age operator (13) to aggregate the evaluation of each alterna-tive Thus we transform the IVIFS decision matrix to IVIFsfor each alternative presented as follows

1198631= [04760 05970] [01915 03926]

1198632= [04211 06454] [02260 03546]

1198633= [04057 06112] [01632 02833]

1198634= [05081 06389] [02007 03016]

(37)

Step 2 Each possibility measure presented in Section 32 isapplied to the obtained IVIFNs119863

1119863211986331198634

(a) For the possibility degree (16) the obtained possibilitymatrix is

(

05000 04671 03840 01823

05443 05000 03844 01647

06142 06210 05000 03131

06869 08353 08177 05000

) (38)

Table 9 presents the obtained results and shows thatthe best alternative is 1198604

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05264 05183 04185

04976 05000 04918 03921

05087 05063 05000 04003

05510 05597 05573 05000

) (39)

Table 10 presents the obtained results and shows thatthe best alternative is 1198604

Table 11 Ranking IVFISs alternatives using possibilitymeasure (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02627 02654 02105 02614Ranking 2 1 4 3

Table 12 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02254 02448 02613 02686Ranking 4 3 2 1

Table 13 Ranking order of alternatives for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198601 gt 1198603 gt 1198602 1198604

(16) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(17) 1198602 gt 1198601 gt 1198604 gt 1198603 1198602

(20) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(c) For the possibility measure (17) the obtained possi-bility matrix is

(

05000 05075 06501 04944

04925 05000 06635 05285

03365 03499 05000 03401

06599 04715 05056 05000

) (40)

The obtained results are presented in Table 11 showingthat the best alternative is 1198602

(d) For the possibility measure (20) the obtained possi-bility matrix is

(

05000 04446 03856 03740

05554 05000 04570 04256

05430 06144 05000 04778

05222 05744 06260 05000

) (41)

The obtained results are presented in Table 12 showingthat the best alternative is 1198604

Table 13 presents all the obtained results applying differentpossibility methods using the interval-valued intuitionisticfuzzy sets and shows the alternatives ranking results Weremark that the possibility formulas (15) (16) and (20)provide the same best alternative 1198604 However (17) providesthe best alternative 1198602

522 Case 2 Application of Optimal Aggregated Interval-Valued Intuitionistic Fuzzy Sets (14) Using the optimal aggre-gated operator (14) to IVIF decision matrix we obtain

8 Advances in Fuzzy Systems

Table 14 Ranking IVFSs using possibility degree (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02256 02352 02779 02599Ranking 4 3 1 2

Table 15 Ranking IVFSs using possibility measure (15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02369 02322 02557 02629Ranking 3 4 2 1

four interval-valued intuitionistic fuzzy numbers (IVIFNs)representing the alternatives as follows

1205721= [04831 06089] [01850 03800]

1205722= [04400 06520] [02170 03480]

1205723= [04840 06450] [01560 02730]

1205724= [05400 06530] [01950 02950]

(42)

(a) For the possibility measure (16) the obtained possi-bility degree matrix is

(

05000 04717 03595 03758

05366 05000 03682 04177

06329 06403 05000 05611

04358 05818 06008 05000

) (43)

The alternatives weight 119882119894is computed using (26)

and then ranked in a decreasing orderThe results aredisplayed in Table 14 showing that the best alternativethat ranks first is 119860

3

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05188 04308 03938

05000 05000 04120 03750

05525 05525 05000 04630

05168 05693 05693 05000

) (44)

We compute the weight 119882119894of the alternative using

(26) and we rank 119882119894in a decreasing order The

results are shown in Table 15 revealing that the bestalternative is 1198604 which ranks first

(c) For possibility measures (17) the obtained possibilitydegree matrix is

(

05000 05000 05767 04578

05000 05000 06123 05035

03877 04233 05000 03713

06287 04965 05422 05000

) (45)

Table 16 Ranking IVFSs using possibility degree (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02529 02596 02235 02639Ranking 3 2 4 1

Table 17 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02240 02392 02695 02673Ranking 4 3 1 2

Table 18 Alternatives ranking order for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198603 gt 1198601 gt 1198602 1198604

(16) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

(17) 1198604 gt 1198602 gt 1198601 gt 1198603 1198604

(20) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

We compute the weights119882119894of the alternatives using

(26) and we rank 119882119894in a decreasing order The

results are displayed in Table 16 showing that the bestalternative that rank first is 1198604

(d) For possibility measure (20) the obtained possibilitymatrix is

(

05000 04485 03685 03705

05515 05000 04082 04105

05918 06315 05000 05113

04887 05895 06295 05000

) (46)

The obtained results are presented in Table 17 showingthat the best alternative is 1198603

Table 18 presents the results of all applied possibilitymeasures using the interval-valued intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility formulas (15) and (17) provide the same bestalternative 1198604 However (16) and (20) provide the bestalternative 1198603 We note that the latter is the worst alternativeusing (17)

6 Conclusion

In this study we presented different formulas of possibilitymeasures The formulas exist in literature with IFN andIVIFN We also presented an MCDM method from theliterature We gave an illustrative examples for applicationsof different possibility measures and compared their resultsFirst we used an MCDM matrix with intuitionistic fuzzynumbers and then anMCDMmatrixwith IVIFNsThe valuesof the latter are aggregated with an aggregation operator intwo cases In each case a different aggregation operator wasused Thus the appropriate possibility measures are applied

Advances in Fuzzy Systems 9

The results show that the ranked alternatives can be differentfor each possibility measure even though some of thesemeasures have already been demonstrated to be equivalentin the literature

Appendix

Intuitionistic Fuzzy Sets

Intuitionistic fuzzy sets are introduced by Atanassov [16] whodefined a degree of membership 120583 a degree of nonmember-ship ] and a degree of hesitation 120587 of an element 119909 of an IFS

119860 = (120583119860 ]119860) denotes an intuitionistic fuzzy number if 120583

119860

and ]119860are fuzzy numbers with ]

119860le 120583119888119860 where 120583119888

119860denotes

the complement of 120583119860

If119883 is a discourse universe and 119860 a set in119883 then

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883 (A1)

with the conditions 0 le 120583119860(119909) le 1 0 le ]

119860(119909) le 1 0 le

120583119860(119909) + ]

119860(119909) le 1 and 120587

119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) Also for

each 119909 isin 119883 0 le Π119860(119909) le 1

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof this work by grants from General Direction of ScientificResearch (DGRST) Tunisia under the ARUB programThey would like to thank Mr Abdelmajid Dammak for hisproofreading and correction of the English of the paper

References

[1] J H Park I Y Park Y C L Kwun andX Tan ldquoExtension of theTOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environmentrdquo Applied MathematicalModelling vol 35 no 5 pp 2544ndash2556 2011

[2] D G Park Y C Kwun J H Park and I Y Park ldquoCorrelationcoefficient of interval-valued intuitionistic fuzzy sets and itsapplication to multiple attribute group decision making prob-lemsrdquoMathematical and Computer Modelling vol 50 no 9-10pp 1279ndash1293 2009

[3] J Ye ldquoMulticriteria fuzzy decision-making method usingentropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy setsrdquo Applied Mathematical Mod-elling vol 34 no 12 pp 3864ndash3870 2010

[4] H Zhang and L Yu ldquoMADM method based on cross-entropyand extended TOPSIS with interval-valued intuitionistic fuzzysetsrdquo Knowledge-Based Systems vol 30 pp 115ndash120 2012

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

[6] Z-S Xu and J Chen ldquoApproach to groupdecisionmaking basedon interval-valued intuitionistic judgment matricesrdquo SystemsEngineeringmdashTheory amp Practice vol 27 no 4 pp 126ndash133 2007

[7] X Zhang G Yue and Z Teng ldquoPossibility degree of interval-valued intuitionistic fuzzy numbers and its applicationrdquo inProceedings of the International Symposium on InformationProcessing (ISIP rsquo09) pp 33ndash36 Huangshan China 2009

[8] J Wu Q Cao and H Li ldquoAn approach for MADM problemswith interval-valued intuitionistic fuzzy sets based on nonlinearfunctionsrdquo Technological and Economic Development of Econ-omy vol 22 no 3 pp 336ndash356 2016

[9] Z Wang K W Li and W Wang ldquoAn approach to multi-attribute decision making with interval-valued intuitionisticfuzzy assessments and incomplete weightsrdquo Information Sci-ences vol 179 no 17 pp 3026ndash3040 2009

[10] S C Onar B Oztaysi I Otay and C Kahraman ldquoMulti-expert wind energy technology selection using interval-valuedintuitionistic fuzzy setsrdquo Energy vol 90 part 1 pp 274ndash2852015

[11] Z Xu and R R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006

[12] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[13] C-P Wei and X Tang ldquoPossibility degree method for rank-ing intuitionistic fuzzy numbersrdquo in Proceedings of the 3rdIEEEWICACM International Conference on Web Intelligenceand Intelligent Agent Technology (WI-IAT rsquo10) pp 142ndash145IEEE Toronto Canada August 2010

[14] Z S Xu and Q L Da ldquoPossibility degree method for rankinginterval numbers and its applicationrdquo Journal of Systems Engi-neering vol 18 pp 67ndash70 2003

[15] Y-M Wang J-B Yang and D-L Xu ldquoInterval weight gen-eration approaches based on consistency test and intervalcomparison matricesrdquo Applied Mathematics and Computationvol 167 no 1 pp 252ndash273 2005

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

[17] F Gao ldquoPossibility degree and comprehensive priority ofinterval numbersrdquo Systems EngineeringmdashTheoryamp Practice vol33 no 8 pp 2033ndash2040 2013

[18] Y-Y Liu and Y-J Lv ldquoA multiple attribute decision makingmethod with interval rough numbers based on the possibilitydegreerdquo in Proceedings of the 10th International Conference onNatural Computation (ICNC rsquo14) pp 407ndash411 IEEE XiamenChina August 2014

[19] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[20] H Yuan and Y Qu ldquoModel for conflict resolution withpreference represented as interval numbersrdquo Proceedings of theMATECWeb of Conferences 2015

[21] G Facchinetti R G Ricci and S Muzzioli ldquoNote on rankingfuzzy triangular numbersrdquo International Journal of IntelligentSystems vol 13 no 7 pp 613ndash622 1998

[22] Q L Da and X W Liu ldquoInterval number linear programmingand its satisfactory solutionrdquo Systems Engineering Theory ampPractice vol 19 pp 3ndash7 1999

[23] C-P Wei and X Tang ldquoPossibility degree method for rankingintuitionistic fuzzy numbersrdquo Journal of Systems Engineeringvol 18 pp 67ndash70 2003

[24] Z S Xu and Q L Da ldquoThe uncertain ow a operatorrdquo Interna-tional Journal of Intelligent Systems vol 17 no 6 pp 569ndash5752002

10 Advances in Fuzzy Systems

[25] Z S Xu and Q L Da ldquoA possibility based method for prioritiesof interval judjment matricesrdquo Chinese Journal of ManagementScience vol 11 pp 63ndash65 2003

[26] D Q Li and Y D Gu ldquomethods for ranking interval numberbased on possibility degreerdquo Journal of Systems Engineering vol23 pp 223ndash226 2008

[27] J B Lan L J Cao and J Lin ldquoMethod for rinking interval num-bers on two-dimensional priority degreerdquo Journal of ChongqingInstitute of Technology Natural Science Edition vol 21 pp 63ndash66 2007

[28] T-Y Chen ldquoInterval-valued intuitionistic fuzzy QUALIFLEXmethod with a likelihood-based comparison approach formultiple criteria decision analysisrdquo Information Sciences vol261 pp 149ndash169 2014

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

[30] G W Wei and X R Wang ldquoSome geometric aggregationoperators on interval-valued intuitionistic fuzzy sets and theirapplication to group decision makingrdquo in Proceedings of theInternational Conference on Computational Intelligence andSecurity (ICCIS rsquo07) pp 495ndash499 Harbin China December2007

[31] Z S Xu and J Chen ldquoOn geometric aggregation over interval-valued intuitionistic fuzzy informationrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) vol 2 pp 466ndash471 Haikou ChinaAugust 2007

[32] Y He H Chen L Zhou J Liu and Z Tao ldquoGeneralizedinterval-valuedAtanassovrsquos intuitionistic fuzzy power operatorsand their application to group decision makingrdquo InternationalJournal of Fuzzy Systems vol 15 no 4 pp 401ndash411 2013

[33] S Wan and J Dong ldquoA possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision mak-ingrdquo Journal of Computer and System Sciences vol 80 no 1 pp237ndash256 2014

[34] F Dammak L Baccour and A M Alimi ldquoThe impact ofcriterion weights techniques in topsis method of multi-criteriadecision making in crisp and intuitionistic fuzzy domainsrdquoin Proceedings of the IEEE International Conference on FuzzySystems (FUZZ-IEEE rsquo15) pp 1ndash8 Istanbul Turkey August 2015

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 3

(c) Results of integrated formulas (9) and (10)

119901 (119886 ge 119887)

=

1 119887+ le 119886minus

(119886+ minus 119887minus)2

2 (119886+ minus 119886minus) (119887+ minus 119887minus)119886minus lt 119887minus le 119886+ lt 119887+

1 minus(119887+ minus 119886minus)

2

2 (119886+ minus 119886minus) (119887+ minus 119887minus)119887minus lt 119886minus le 119887+ lt 119886+

119886minus + 119886+ minus 2119887minus

2 (119887+ minus 119887minus)119887minus le 119886minus le 119886+ le 119887+

2119886+ minus 119887+ minus 119887minus

2 (119886+ minus 119886minus)119886minus le 119887minus le 119887+ le 119886+

0 119887minus ge 119886+

(11)

(vii) According to Chen [28] the possibility measure ofthe event 119886 ge 119887 is presented as follows

119901 (119886 ge 119887) = max1 minusmax(1 minus 119887+) minus 119886minus

1198971+ 1198972

0 0 (12)

where 1198971= 1 minus 119886minus minus 119886+ and 119897

2= 1 minus 119887minus minus 119887+

3 Interval-Valued Intuitionistic Fuzzy Sets

There are some basic concepts related to the interval-valuedintuitionistic fuzzy sets (IVIFS) [29] Let119883 = 119909

1 1199092 119909

119899

be a nonempty set of the universe An IVIFS is defined as = ⟨119909

119894 [120583119871(119909119894) 120583119880(119909119894)] []119871(119909119894) ]119880(119909119894)]⟩ | 119909

119894isin 119883 where

[120583119871(119909119894) 120583119880(119909119894)] and []119871

(119909119894) ]119880(119909119894)] denote the intervals of

the membership degree and nonmembership degree of theelement 119909

119894isin satisfying the following

(i) 120583119880(119909119894) + ]119880(119909119894) le 1

(ii) 0 le ]119871(119909119894) le ]119880(119909119894) le 1 and 120583119871

(119909119894) le 120583119880(119909119894) for all

119909119894isin 119883

(iii) if 120583119871(119909119894) = 120583119880

(119909119894) and ]119871

(119909119894) = ]119880

(119909119894) then is

reduced to an IFS

31 Aggregation Operators Existing in Literature The aggre-gation operators are necessary to reduce the IVIFS valuesthus we can compare them using an accuracy function or apossibility measure In the following we present two existingaggregation operators

(i) Xu and Wei [6 30] defined the interval-valued intu-itionistic fuzzy weighted geometric (IVIFWG) operator asfollows

IVIFWG119908(1205721 1205722 120572

119899)

= ([

[

119899

prod119895=1

(119886119895)119908119895119899

prod119895=1

(119887119895)119908119895]

]

[

[

1 minus119899

prod119895=1

(1 minus 119888119908119895

119895) 1 minus

119899

prod119895=1

(1 minus 119889119908119895

119895)]

]

)

(13)

where 119908 = (1199081 1199082 119908

119899)119879 is the weight vector of 120572

119895(119895 =

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

119895=1119908119895= 1

(ii) Wang et al [9] defined the optimal aggregatedinterval-valued intuitionistic fuzzy sets using this formula

120572119894= ([

[

119899

sum119895=1

119886119894119895119908119895119899

sum119895=1

119887119894119895119908119895]

]

[

[

119899

sum119895=1

119888119894119895119908119895119899

sum119895=1

119889119894119895119908119895]

]

) (14)

32 PossibilityMeasures to Compare Interval-Valued Intuition-istic Fuzzy Number (IVIFN) Existing in Literature Let 120572

1=

([1198861 1198871] [1198881 1198891]) and 120572

2= ([1198862 1198872] [1198882 1198892]) be two interval-

valued intuitionistic fuzzy numbers in Ω which is the set ofall IVIFNs [2 5 31 32] 119901(120572

1ge 1205722) is the possibility measure

of two interval-valued intuitionistic fuzzy numbers Let uspresent the existing possibility measures

(i) Zhang et al [7] defined two possibilitymeasures of twointerval-valued intuitionistic fuzzy numbers as follows

(a) First measure

1199011(1205721ge 1205722)

= min (max (119860 (1205721) minus 119860 (120572

2) + 05 0) 1)

(15)

where119860(1205721) = 120582((119886+119887)2)+ (1minus120582)((119886minus 119888+119887minus119889)2)

and 120582 isin [0 1] which represents the performance onthe mean value of its membership degreeThis possibility degree satisfies the following proper-ties

(1) 0 le 1199011(1205721ge 1205722) le 1

(2) 1199011(1205721ge 1205722) = 1 hArr 119860(120572

1) minus 119860(120572

2) ge 05

(3) 1199011(1205721ge 1205722) = 0 hArr 119860(120572

1) minus 119860(120572

2) le minus05

(4) 1199011(1205721ge 1205722) + 1199011(1205722ge 1205721) = 1

(b) Second measure

1199012(1205721ge 1205722)

= 120574min(max( 1198871minus 1198862

1198871minus 1198862+ 1198872minus 1198862

0) 1)

+ (1 minus 120574)min(max( 1198892minus 1198881

1198891minus 1198881+ 1198892minus 1198882

0) 1)

(16)

where 120574 isin [0 1] gives the decision makersrsquo preferenceon membership degree or nonmembership degreeWhen 120574 ge 05 the decision maker is optimal whereaswhen 120574 lt 05 the decision maker is pessimisticThenthe below properties are checked

(1) 0 le 1199012(1205721ge 1205722) le 1

(2) 1199012(1205721ge 1205722) = 1 hArr 119887

2le 1198861and 119889

1le 1198882

(3) 1199012(1205721ge 1205722) = 0 hArr 119887

1le 1198862and 119889

2le 1198881

(4) 1199012(1205721ge 1205722) + 1199012(1205722ge 1205721) = 1

4 Advances in Fuzzy Systems

(ii)Wan andDong [33] defined possibilitymeasure by thefollowing formula

1199013(1205721ge 1205722)

=1

2119901 ([119886

1 1198871] ge [119886

2 1198872]) + 119901 ([119888

2 1198892] ge [1198881 1198891])

(17)

where 119901([1198861 1198871] ge [119886

2 1198872]) and 119901([119888

2 1198892] ge [119888

1 1198891]) can be

calculated using (4)(iii) Chen [28] defined a lower likelihood 119871minus and an upper

likelihood 119871+ on IVIFSs as

119871minus (1205721ge 1205722)

= max1 minusmax(1 minus 1198882) minus 1198861

1198971198861+ 1198971198862

0 0 (18)

where 1198971198861= 1 minus 119886

1minus 1198891and 1198971198862= 1 minus 119887

2minus 1198882 and

119871+ (1205721ge 1205722)

= max1 minusmax(1 minus 119889

2) minus 1198871

11989710158401198861+ 11989710158401198862

0 0 (19)

where 11989710158401198861= 1 minus 119887

1minus 1198881and 11989710158401198862= 1 minus 119886

2minus 1198892

Then for two IVIFNs the likelihood1199014(1205721ge 1205722) is defined

as follows

1199014(1205721ge 1205722) =

1

2(119871minus (120572

1ge 1205722) + 119871+ (120572

1ge 1205722)) (20)

These measures are the same as those of the possibilitymeasures

4 MCDM Based on PossibilityDegree of Interval-Valued IntuitionisticFuzzy Numbers

For a multicriteria decision making problem let 119860 =1198601 1198602 119860

119899 be the set of alternatives and 119883 =

1198831 1198832 119883

119899 the set of criteria 119882 = (119908

1 1199082 119908

119899)119879

is the weight vector of criteria 119883119895 where 119908

119895isin [0 1] and

sum119899

119895=1119908119895= 1

Suppose the characteristic information of alternative 119860119894

over criterion119883119895is represented by interval-valued intuition-

istic fuzzy number = ([119886 119887] [119888 119889]) where [119886 119887] representsthe fuzzy membership degree of the alternative 119860

119894over

criterion 119883119895and [119888 119889] represents the fuzzy nonmembership

degree of the alternative 119860119894over criterion 119883

119895 Then the

decision matrix is obtained as

(

([11988611 11988711] [11988811 11988911]) sdot sdot sdot ([119886

1119898 1198871119898] [1198881119898 1198891119898])

([1198861198991 1198871198991] [1198881198991 1198891198991]) ([119886

119899119898 119887119899119898] [119888119899119898 119889119899119898])

) (21)

The ranking of the alternatives in the multicriteria decisionmaking can be solved using the possibility measure ofinterval-valued intuitionistic fuzzy numbers We chose toadopt a modified version of the method described in [4]following the steps below

Step 1 Construct the interval-valued intuitionistic fuzzydecision matrix = (

119894119895)119898times119899

= ([119886119894119895 119887119894119895] [119888119894119895 119889119894119895])

Step 2 Calculate the intuitionistic fuzzy decision matrix119863119894119895

= [119863119871119894119895 119863119880119894119895] to derive 119863119871

119894119895and 119863119880

119894119895 and 119863

119894119895is the

transformed IFN decision matrix obtained from usingformulas (22)

119863119871119894119895=

119886119894119895ln 2 + 119889

119894119895ln (2 lowast 119889

119894119895119889119894119895+ 1) + ln (2119889

119894119895+ 1)

(119886119894119895+ 119889119894119895) ln 2 + 119886

119894119895ln (2119886

119894119895119886119894119895+ 1) + 119889

119894119895ln (2119889

119894119895119889119894119895+ 1) + ln (2119886

119894119895+ 1) + ln (2119889

119894119895+ 1)

119863119880119894119895=

119887119894119895ln 2 + 119888

119894119895ln (2 lowast 119888

119894119895119888119894119895+ 1) + ln (2119888

119894119895+ 1)

(119887119894119895+ 119888119894119895) ln 2 + 119887

119894119895ln (2119887

119894119895119887119894119895+ 1) + 119888

119894119895ln (2119888119894119895119888119894119895+ 1) + ln (2119887

119894119895+ 1) + ln (2119888

119894119895+ 1)

(22)

Step 3 Assign weights to criteria we use the followingstandard deviation (IF-SD) formula presented in [34] insteadof that used in [4]

119882119895=

120590119895

sum119899

119895=1120590119895

119895 = 1 119899 (23)

where

120590119895= radic119878 (120583

119894119895) + 119878 (]

119894119895)

119878 (120583119894119895) =

sum119898

119894=1(120583119894119895(119862119895) minus 120583119895(119862119895))2

119898

120583119895(119862119895) =

sum119898

119894=1120583119894119895(119862119895)

119898

119878 (]119894119895) =

sum119898

119894=1(]119894119895(119862119895) minus ]119895(119862119895))2

119898

(24)

where sum119899119895=1119908119895= 1

Step 4 Compute the performance of each alternative

119863119894= [

[

119899

sum119895=1

119882119895lowast 119863119871119894119895119899

sum119895=1

119882119895lowast 119863119880119894119895]

]

(25)

Advances in Fuzzy Systems 5

Step 5 Compute the likelihood matrix [25] To comparebetween tow interval fuzzy numbers we propose to usea possibility measure instead of the formula used in [4]to obtain a possibility matrix Therefore each possibilitymeasure presented in Section 21 is applied and all theachieved results are compared in Section 5

Step 6 Determine the alternatives ranking order accordingto the decreasing order of119882

119894[25] defined as

119882119894=sum119899

119895=1119901119894119895+ 1198982 minus 1

119898 (119898 minus 1) 119894 = 1 2 119898 (26)

5 Illustrative Example51 Application of Possibility Measure of IFS in Decision Mak-ing Problem This section described the data set presented in[4 9] to evaluate the four potential investment opportunities119860 = 1198601 1198602 1198603 1198604 The fund manager should evaluateeach investment considering four criteria risk (1198621) growth(1198622) sociopolitical issues (1198623) and environmental impacts(1198624) The fund manager is satisfied once he provides hisassessment of each alternative on each criterion

Step 1 The following interval-valued intuitionistic fuzzy sets(IVIFSs) decision making matrix (27) presents the relation-ship between criteria and alternatives of data set as follows

(

[042 048] [04 05] [06 07] [005 025] [04 05] [02 05] [055 075] [015 025]

[04 05] [04 05] [05 08] [01 02] [03 06] [03 04] [06 07] [01 03]

[03 05] [04 05] [01 03] [02 04] [07 08] [01 02] [05 07] [01 02]

[02 04] [04 05] [06 07] [02 03] [05 06] [02 03] [07 08] [01 02]

) (27)

Each element of this matrix is presented with IVIFSgiving the fund managerrsquos satisfaction or dissatisfactiondegree with an alternative The element represented for thefirst alternative [042 048] [04 05] where the interval 42ndash48 [4] reflects that the fund manager has an excellent

opportunity to respect the risk criterion (1198881) although theinterval 40ndash50 does not really represent an excellent choiceof 1198601 for risk (1198881)

Step 2 The intuitionistic fuzzy decision matrix (28) isobtained using (22)

(

[04452 05568] [07458 09429] [04304 07404] [07220 08773]

[04304 05696] [07404 09243] [04172 07040] [07040 09049]

[03494 05696] [02229 05966] [08245 09243] [07404 09049]

[02596 05000] [07040 08245] [06506 07889] [08245 09243]

) (28)

Step 3 Compute weights 119908 of the criteria based on (28) andusing (23)

119908 = 01884 02634 02439 03043 (29)

Step 4 We compute the performance of each alternativeusing (25) to obtain the interval fuzzy number

1198631= [06050 08008]

1198632= [05921 07978]

1198633= [05509 07653]

1198634= [06439 07851]

(30)

Step 5 In this step we apply each possibility measure anddetermine the achieved results These are then compared todefine the differences between them

Using the possibility measures (1) (2) (3) and (8) weachieved the results presented in Table 1 The best alternativeto be ranked first is 1198603

(a) For the possibility measures (4) and (5) the obtainedpossibility matrix is

(

05000 05198 06093 04656

04802 05000 05878 04437

04122 03907 05000 03413

06587 05563 05344 05000

) (31)

The results presented in Table 2 show 1198604 is the bestalternative and ranks first

(b) As for the possibility measure (6) the obtainedpossibility matrix is

(

05000 04802 04122 06587

05198 05000 03907 05563

06093 05878 05000 05344

04656 04437 03413 05000

) (32)

6 Advances in Fuzzy Systems

Table 1 Possibility degrees using (1) (2) (3) and (8)

Alternatives 1198601 1198602 1198603 1198604

Weights 02543 02472 02693 02292Ranking 2 3 1 4

Table 2 Possibility degrees using (4) and (5)

Alternatives 1198601 1198602 1198603 1198604

Weights 02579 02510 02204 02708Ranking 2 3 4 1

Table 3 Possibility degrees using (6)

Alternatives 1198601 1198602 1198603 1198604

Weights 02543 02472 02693 02292Ranking 2 3 1 4

Table 4 Possibility degrees using (7)

Alternatives 1198601 1198602 1198603 1198604

Weights 02535 02531 02569 02366Ranking 2 3 1 4

The results presented in Table 3 show 1198603 is the bestalternative and ranks first

(c) For the possibility measure (7) the obtained possibil-ity matrix is

(

05000 04941 04840 05634

05059 05000 04789 05522

05211 05160 05000 05458

04542 04478 04366 05000

) (33)

Table 4 shows that the best alternative is1198603 achievingthe first rank

(d) For the possibility measure (9) the obtained possibil-ity matrix is

(

05000 04810 04253 06204

05190 05000 04103 05474

05897 05747 05000 05296

04704 04526 03796 05000

) (34)

Table 5 shows that the best alternative is1198603 that ranksfirst

(e) For possibility measure (10) the obtained possibilitymatrix is

(

05000 04605 03297 07883

05395 05000 02914 06111

07086 06703 05000 05684

04316 03889 02117 05000

) (35)

Table 5 Possibility degrees using (9)

Alternatives 1198601 1198602 1198603 1198604

Weights 02522 02481 02662 02335Ranking 2 3 1 4

Table 6 Possibility degrees using (10)

Alternatives 1198601 1198602 1198603 1198604

Weights 02565 02452 02873 02110Ranking 2 3 1 4

Table 7 Possibility degrees using (12)

Alternatives 1198601 1198602 1198603 1198604

Weights 02511 02533 02557 02399Ranking 3 2 1 4

Table 8 Alternatives ranking order for different possibility mea-sures under IFN

Possibility measure Ranking Best alternative(1) (2) (3) and (8) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(4) and (5) 1198604 gt 1198601 gt 1198602 gt 1198603 1198604

(6) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(7) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(9) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(10) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(12) 1198603 gt 1198602 gt 1198601 gt 1198604 1198603

Table 6 shows that the best alternative that ranks firstis 1198602

(f) For the possibility measure (12) the obtained possi-bility matrix is

(

05000 04938 04872 05327

05062 05000 04939 05395

05061 05128 05000 05491

04509 04605 04673 05000

) (36)

Table 7 shows that the best alternative is 1198603 thatclearly ranks first

Table 8 presents a comparison of the obtained results applyingdifferent possibility measures under intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility measures (4) and (5) gave the same bestalternative 1198604 and the worst alternative 1198603 However usingformulas (1) (2) (3) (6) (7) (8) (9) (10) and (12) the bestalternative is1198603 and the worst alternative is1198604 These resultsshow that the measures (4) and (2) are different althoughthey are demonstrated to be equivalent (the operators lead tovalue 1) in [17] but they do not produce the same result

52 Application of Possibility Measures of IVIFS in DecisionMaking Problem We apply possibility measures of IVIFS

Advances in Fuzzy Systems 7

Table 9 Ranking IVIFSs alternatives using possibility measure (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02111 02161 02540 03200Ranking 4 3 2 1

Table 10 Ranking IVIFSs alternatives using possibility measures(15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02469 02401 02429 02640Ranking 2 4 3 1

presented in Section 32 to rank IVIFS data sets describedin Section 5 It is worth reminding that there are fouralternatives 1198601 1198602 1198603 and 1198604 and four criteria We use theIVIF matrix of alternatives (27) and the following criteriarsquosweight 119908

119895= [013 017 039 031] given in [9]

521 Case 1 Application of Interval-Valued Intuitionistic FuzzyWeighted Geometric (IVIFWG) Operator (13) The possibilitymeasures are applied in two cases In each case an aggregationoperator is also applied to the matrix (27)

Step 1 Compute the comprehensive evaluation of eachinvestment (alternative) using the geometric weighted aver-age operator (13) to aggregate the evaluation of each alterna-tive Thus we transform the IVIFS decision matrix to IVIFsfor each alternative presented as follows

1198631= [04760 05970] [01915 03926]

1198632= [04211 06454] [02260 03546]

1198633= [04057 06112] [01632 02833]

1198634= [05081 06389] [02007 03016]

(37)

Step 2 Each possibility measure presented in Section 32 isapplied to the obtained IVIFNs119863

1119863211986331198634

(a) For the possibility degree (16) the obtained possibilitymatrix is

(

05000 04671 03840 01823

05443 05000 03844 01647

06142 06210 05000 03131

06869 08353 08177 05000

) (38)

Table 9 presents the obtained results and shows thatthe best alternative is 1198604

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05264 05183 04185

04976 05000 04918 03921

05087 05063 05000 04003

05510 05597 05573 05000

) (39)

Table 10 presents the obtained results and shows thatthe best alternative is 1198604

Table 11 Ranking IVFISs alternatives using possibilitymeasure (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02627 02654 02105 02614Ranking 2 1 4 3

Table 12 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02254 02448 02613 02686Ranking 4 3 2 1

Table 13 Ranking order of alternatives for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198601 gt 1198603 gt 1198602 1198604

(16) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(17) 1198602 gt 1198601 gt 1198604 gt 1198603 1198602

(20) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(c) For the possibility measure (17) the obtained possi-bility matrix is

(

05000 05075 06501 04944

04925 05000 06635 05285

03365 03499 05000 03401

06599 04715 05056 05000

) (40)

The obtained results are presented in Table 11 showingthat the best alternative is 1198602

(d) For the possibility measure (20) the obtained possi-bility matrix is

(

05000 04446 03856 03740

05554 05000 04570 04256

05430 06144 05000 04778

05222 05744 06260 05000

) (41)

The obtained results are presented in Table 12 showingthat the best alternative is 1198604

Table 13 presents all the obtained results applying differentpossibility methods using the interval-valued intuitionisticfuzzy sets and shows the alternatives ranking results Weremark that the possibility formulas (15) (16) and (20)provide the same best alternative 1198604 However (17) providesthe best alternative 1198602

522 Case 2 Application of Optimal Aggregated Interval-Valued Intuitionistic Fuzzy Sets (14) Using the optimal aggre-gated operator (14) to IVIF decision matrix we obtain

8 Advances in Fuzzy Systems

Table 14 Ranking IVFSs using possibility degree (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02256 02352 02779 02599Ranking 4 3 1 2

Table 15 Ranking IVFSs using possibility measure (15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02369 02322 02557 02629Ranking 3 4 2 1

four interval-valued intuitionistic fuzzy numbers (IVIFNs)representing the alternatives as follows

1205721= [04831 06089] [01850 03800]

1205722= [04400 06520] [02170 03480]

1205723= [04840 06450] [01560 02730]

1205724= [05400 06530] [01950 02950]

(42)

(a) For the possibility measure (16) the obtained possi-bility degree matrix is

(

05000 04717 03595 03758

05366 05000 03682 04177

06329 06403 05000 05611

04358 05818 06008 05000

) (43)

The alternatives weight 119882119894is computed using (26)

and then ranked in a decreasing orderThe results aredisplayed in Table 14 showing that the best alternativethat ranks first is 119860

3

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05188 04308 03938

05000 05000 04120 03750

05525 05525 05000 04630

05168 05693 05693 05000

) (44)

We compute the weight 119882119894of the alternative using

(26) and we rank 119882119894in a decreasing order The

results are shown in Table 15 revealing that the bestalternative is 1198604 which ranks first

(c) For possibility measures (17) the obtained possibilitydegree matrix is

(

05000 05000 05767 04578

05000 05000 06123 05035

03877 04233 05000 03713

06287 04965 05422 05000

) (45)

Table 16 Ranking IVFSs using possibility degree (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02529 02596 02235 02639Ranking 3 2 4 1

Table 17 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02240 02392 02695 02673Ranking 4 3 1 2

Table 18 Alternatives ranking order for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198603 gt 1198601 gt 1198602 1198604

(16) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

(17) 1198604 gt 1198602 gt 1198601 gt 1198603 1198604

(20) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

We compute the weights119882119894of the alternatives using

(26) and we rank 119882119894in a decreasing order The

results are displayed in Table 16 showing that the bestalternative that rank first is 1198604

(d) For possibility measure (20) the obtained possibilitymatrix is

(

05000 04485 03685 03705

05515 05000 04082 04105

05918 06315 05000 05113

04887 05895 06295 05000

) (46)

The obtained results are presented in Table 17 showingthat the best alternative is 1198603

Table 18 presents the results of all applied possibilitymeasures using the interval-valued intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility formulas (15) and (17) provide the same bestalternative 1198604 However (16) and (20) provide the bestalternative 1198603 We note that the latter is the worst alternativeusing (17)

6 Conclusion

In this study we presented different formulas of possibilitymeasures The formulas exist in literature with IFN andIVIFN We also presented an MCDM method from theliterature We gave an illustrative examples for applicationsof different possibility measures and compared their resultsFirst we used an MCDM matrix with intuitionistic fuzzynumbers and then anMCDMmatrixwith IVIFNsThe valuesof the latter are aggregated with an aggregation operator intwo cases In each case a different aggregation operator wasused Thus the appropriate possibility measures are applied

Advances in Fuzzy Systems 9

The results show that the ranked alternatives can be differentfor each possibility measure even though some of thesemeasures have already been demonstrated to be equivalentin the literature

Appendix

Intuitionistic Fuzzy Sets

Intuitionistic fuzzy sets are introduced by Atanassov [16] whodefined a degree of membership 120583 a degree of nonmember-ship ] and a degree of hesitation 120587 of an element 119909 of an IFS

119860 = (120583119860 ]119860) denotes an intuitionistic fuzzy number if 120583

119860

and ]119860are fuzzy numbers with ]

119860le 120583119888119860 where 120583119888

119860denotes

the complement of 120583119860

If119883 is a discourse universe and 119860 a set in119883 then

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883 (A1)

with the conditions 0 le 120583119860(119909) le 1 0 le ]

119860(119909) le 1 0 le

120583119860(119909) + ]

119860(119909) le 1 and 120587

119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) Also for

each 119909 isin 119883 0 le Π119860(119909) le 1

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof this work by grants from General Direction of ScientificResearch (DGRST) Tunisia under the ARUB programThey would like to thank Mr Abdelmajid Dammak for hisproofreading and correction of the English of the paper

References

[1] J H Park I Y Park Y C L Kwun andX Tan ldquoExtension of theTOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environmentrdquo Applied MathematicalModelling vol 35 no 5 pp 2544ndash2556 2011

[2] D G Park Y C Kwun J H Park and I Y Park ldquoCorrelationcoefficient of interval-valued intuitionistic fuzzy sets and itsapplication to multiple attribute group decision making prob-lemsrdquoMathematical and Computer Modelling vol 50 no 9-10pp 1279ndash1293 2009

[3] J Ye ldquoMulticriteria fuzzy decision-making method usingentropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy setsrdquo Applied Mathematical Mod-elling vol 34 no 12 pp 3864ndash3870 2010

[4] H Zhang and L Yu ldquoMADM method based on cross-entropyand extended TOPSIS with interval-valued intuitionistic fuzzysetsrdquo Knowledge-Based Systems vol 30 pp 115ndash120 2012

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

[6] Z-S Xu and J Chen ldquoApproach to groupdecisionmaking basedon interval-valued intuitionistic judgment matricesrdquo SystemsEngineeringmdashTheory amp Practice vol 27 no 4 pp 126ndash133 2007

[7] X Zhang G Yue and Z Teng ldquoPossibility degree of interval-valued intuitionistic fuzzy numbers and its applicationrdquo inProceedings of the International Symposium on InformationProcessing (ISIP rsquo09) pp 33ndash36 Huangshan China 2009

[8] J Wu Q Cao and H Li ldquoAn approach for MADM problemswith interval-valued intuitionistic fuzzy sets based on nonlinearfunctionsrdquo Technological and Economic Development of Econ-omy vol 22 no 3 pp 336ndash356 2016

[9] Z Wang K W Li and W Wang ldquoAn approach to multi-attribute decision making with interval-valued intuitionisticfuzzy assessments and incomplete weightsrdquo Information Sci-ences vol 179 no 17 pp 3026ndash3040 2009

[10] S C Onar B Oztaysi I Otay and C Kahraman ldquoMulti-expert wind energy technology selection using interval-valuedintuitionistic fuzzy setsrdquo Energy vol 90 part 1 pp 274ndash2852015

[11] Z Xu and R R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006

[12] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[13] C-P Wei and X Tang ldquoPossibility degree method for rank-ing intuitionistic fuzzy numbersrdquo in Proceedings of the 3rdIEEEWICACM International Conference on Web Intelligenceand Intelligent Agent Technology (WI-IAT rsquo10) pp 142ndash145IEEE Toronto Canada August 2010

[14] Z S Xu and Q L Da ldquoPossibility degree method for rankinginterval numbers and its applicationrdquo Journal of Systems Engi-neering vol 18 pp 67ndash70 2003

[15] Y-M Wang J-B Yang and D-L Xu ldquoInterval weight gen-eration approaches based on consistency test and intervalcomparison matricesrdquo Applied Mathematics and Computationvol 167 no 1 pp 252ndash273 2005

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

[17] F Gao ldquoPossibility degree and comprehensive priority ofinterval numbersrdquo Systems EngineeringmdashTheoryamp Practice vol33 no 8 pp 2033ndash2040 2013

[18] Y-Y Liu and Y-J Lv ldquoA multiple attribute decision makingmethod with interval rough numbers based on the possibilitydegreerdquo in Proceedings of the 10th International Conference onNatural Computation (ICNC rsquo14) pp 407ndash411 IEEE XiamenChina August 2014

[19] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[20] H Yuan and Y Qu ldquoModel for conflict resolution withpreference represented as interval numbersrdquo Proceedings of theMATECWeb of Conferences 2015

[21] G Facchinetti R G Ricci and S Muzzioli ldquoNote on rankingfuzzy triangular numbersrdquo International Journal of IntelligentSystems vol 13 no 7 pp 613ndash622 1998

[22] Q L Da and X W Liu ldquoInterval number linear programmingand its satisfactory solutionrdquo Systems Engineering Theory ampPractice vol 19 pp 3ndash7 1999

[23] C-P Wei and X Tang ldquoPossibility degree method for rankingintuitionistic fuzzy numbersrdquo Journal of Systems Engineeringvol 18 pp 67ndash70 2003

[24] Z S Xu and Q L Da ldquoThe uncertain ow a operatorrdquo Interna-tional Journal of Intelligent Systems vol 17 no 6 pp 569ndash5752002

10 Advances in Fuzzy Systems

[25] Z S Xu and Q L Da ldquoA possibility based method for prioritiesof interval judjment matricesrdquo Chinese Journal of ManagementScience vol 11 pp 63ndash65 2003

[26] D Q Li and Y D Gu ldquomethods for ranking interval numberbased on possibility degreerdquo Journal of Systems Engineering vol23 pp 223ndash226 2008

[27] J B Lan L J Cao and J Lin ldquoMethod for rinking interval num-bers on two-dimensional priority degreerdquo Journal of ChongqingInstitute of Technology Natural Science Edition vol 21 pp 63ndash66 2007

[28] T-Y Chen ldquoInterval-valued intuitionistic fuzzy QUALIFLEXmethod with a likelihood-based comparison approach formultiple criteria decision analysisrdquo Information Sciences vol261 pp 149ndash169 2014

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

[30] G W Wei and X R Wang ldquoSome geometric aggregationoperators on interval-valued intuitionistic fuzzy sets and theirapplication to group decision makingrdquo in Proceedings of theInternational Conference on Computational Intelligence andSecurity (ICCIS rsquo07) pp 495ndash499 Harbin China December2007

[31] Z S Xu and J Chen ldquoOn geometric aggregation over interval-valued intuitionistic fuzzy informationrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) vol 2 pp 466ndash471 Haikou ChinaAugust 2007

[32] Y He H Chen L Zhou J Liu and Z Tao ldquoGeneralizedinterval-valuedAtanassovrsquos intuitionistic fuzzy power operatorsand their application to group decision makingrdquo InternationalJournal of Fuzzy Systems vol 15 no 4 pp 401ndash411 2013

[33] S Wan and J Dong ldquoA possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision mak-ingrdquo Journal of Computer and System Sciences vol 80 no 1 pp237ndash256 2014

[34] F Dammak L Baccour and A M Alimi ldquoThe impact ofcriterion weights techniques in topsis method of multi-criteriadecision making in crisp and intuitionistic fuzzy domainsrdquoin Proceedings of the IEEE International Conference on FuzzySystems (FUZZ-IEEE rsquo15) pp 1ndash8 Istanbul Turkey August 2015

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 Advances in Fuzzy Systems

(ii)Wan andDong [33] defined possibilitymeasure by thefollowing formula

1199013(1205721ge 1205722)

=1

2119901 ([119886

1 1198871] ge [119886

2 1198872]) + 119901 ([119888

2 1198892] ge [1198881 1198891])

(17)

where 119901([1198861 1198871] ge [119886

2 1198872]) and 119901([119888

2 1198892] ge [119888

1 1198891]) can be

calculated using (4)(iii) Chen [28] defined a lower likelihood 119871minus and an upper

likelihood 119871+ on IVIFSs as

119871minus (1205721ge 1205722)

= max1 minusmax(1 minus 1198882) minus 1198861

1198971198861+ 1198971198862

0 0 (18)

where 1198971198861= 1 minus 119886

1minus 1198891and 1198971198862= 1 minus 119887

2minus 1198882 and

119871+ (1205721ge 1205722)

= max1 minusmax(1 minus 119889

2) minus 1198871

11989710158401198861+ 11989710158401198862

0 0 (19)

where 11989710158401198861= 1 minus 119887

1minus 1198881and 11989710158401198862= 1 minus 119886

2minus 1198892

Then for two IVIFNs the likelihood1199014(1205721ge 1205722) is defined

as follows

1199014(1205721ge 1205722) =

1

2(119871minus (120572

1ge 1205722) + 119871+ (120572

1ge 1205722)) (20)

These measures are the same as those of the possibilitymeasures

4 MCDM Based on PossibilityDegree of Interval-Valued IntuitionisticFuzzy Numbers

For a multicriteria decision making problem let 119860 =1198601 1198602 119860

119899 be the set of alternatives and 119883 =

1198831 1198832 119883

119899 the set of criteria 119882 = (119908

1 1199082 119908

119899)119879

is the weight vector of criteria 119883119895 where 119908

119895isin [0 1] and

sum119899

119895=1119908119895= 1

Suppose the characteristic information of alternative 119860119894

over criterion119883119895is represented by interval-valued intuition-

istic fuzzy number = ([119886 119887] [119888 119889]) where [119886 119887] representsthe fuzzy membership degree of the alternative 119860

119894over

criterion 119883119895and [119888 119889] represents the fuzzy nonmembership

degree of the alternative 119860119894over criterion 119883

119895 Then the

decision matrix is obtained as

(

([11988611 11988711] [11988811 11988911]) sdot sdot sdot ([119886

1119898 1198871119898] [1198881119898 1198891119898])

([1198861198991 1198871198991] [1198881198991 1198891198991]) ([119886

119899119898 119887119899119898] [119888119899119898 119889119899119898])

) (21)

The ranking of the alternatives in the multicriteria decisionmaking can be solved using the possibility measure ofinterval-valued intuitionistic fuzzy numbers We chose toadopt a modified version of the method described in [4]following the steps below

Step 1 Construct the interval-valued intuitionistic fuzzydecision matrix = (

119894119895)119898times119899

= ([119886119894119895 119887119894119895] [119888119894119895 119889119894119895])

Step 2 Calculate the intuitionistic fuzzy decision matrix119863119894119895

= [119863119871119894119895 119863119880119894119895] to derive 119863119871

119894119895and 119863119880

119894119895 and 119863

119894119895is the

transformed IFN decision matrix obtained from usingformulas (22)

119863119871119894119895=

119886119894119895ln 2 + 119889

119894119895ln (2 lowast 119889

119894119895119889119894119895+ 1) + ln (2119889

119894119895+ 1)

(119886119894119895+ 119889119894119895) ln 2 + 119886

119894119895ln (2119886

119894119895119886119894119895+ 1) + 119889

119894119895ln (2119889

119894119895119889119894119895+ 1) + ln (2119886

119894119895+ 1) + ln (2119889

119894119895+ 1)

119863119880119894119895=

119887119894119895ln 2 + 119888

119894119895ln (2 lowast 119888

119894119895119888119894119895+ 1) + ln (2119888

119894119895+ 1)

(119887119894119895+ 119888119894119895) ln 2 + 119887

119894119895ln (2119887

119894119895119887119894119895+ 1) + 119888

119894119895ln (2119888119894119895119888119894119895+ 1) + ln (2119887

119894119895+ 1) + ln (2119888

119894119895+ 1)

(22)

Step 3 Assign weights to criteria we use the followingstandard deviation (IF-SD) formula presented in [34] insteadof that used in [4]

119882119895=

120590119895

sum119899

119895=1120590119895

119895 = 1 119899 (23)

where

120590119895= radic119878 (120583

119894119895) + 119878 (]

119894119895)

119878 (120583119894119895) =

sum119898

119894=1(120583119894119895(119862119895) minus 120583119895(119862119895))2

119898

120583119895(119862119895) =

sum119898

119894=1120583119894119895(119862119895)

119898

119878 (]119894119895) =

sum119898

119894=1(]119894119895(119862119895) minus ]119895(119862119895))2

119898

(24)

where sum119899119895=1119908119895= 1

Step 4 Compute the performance of each alternative

119863119894= [

[

119899

sum119895=1

119882119895lowast 119863119871119894119895119899

sum119895=1

119882119895lowast 119863119880119894119895]

]

(25)

Advances in Fuzzy Systems 5

Step 5 Compute the likelihood matrix [25] To comparebetween tow interval fuzzy numbers we propose to usea possibility measure instead of the formula used in [4]to obtain a possibility matrix Therefore each possibilitymeasure presented in Section 21 is applied and all theachieved results are compared in Section 5

Step 6 Determine the alternatives ranking order accordingto the decreasing order of119882

119894[25] defined as

119882119894=sum119899

119895=1119901119894119895+ 1198982 minus 1

119898 (119898 minus 1) 119894 = 1 2 119898 (26)

5 Illustrative Example51 Application of Possibility Measure of IFS in Decision Mak-ing Problem This section described the data set presented in[4 9] to evaluate the four potential investment opportunities119860 = 1198601 1198602 1198603 1198604 The fund manager should evaluateeach investment considering four criteria risk (1198621) growth(1198622) sociopolitical issues (1198623) and environmental impacts(1198624) The fund manager is satisfied once he provides hisassessment of each alternative on each criterion

Step 1 The following interval-valued intuitionistic fuzzy sets(IVIFSs) decision making matrix (27) presents the relation-ship between criteria and alternatives of data set as follows

(

[042 048] [04 05] [06 07] [005 025] [04 05] [02 05] [055 075] [015 025]

[04 05] [04 05] [05 08] [01 02] [03 06] [03 04] [06 07] [01 03]

[03 05] [04 05] [01 03] [02 04] [07 08] [01 02] [05 07] [01 02]

[02 04] [04 05] [06 07] [02 03] [05 06] [02 03] [07 08] [01 02]

) (27)

Each element of this matrix is presented with IVIFSgiving the fund managerrsquos satisfaction or dissatisfactiondegree with an alternative The element represented for thefirst alternative [042 048] [04 05] where the interval 42ndash48 [4] reflects that the fund manager has an excellent

opportunity to respect the risk criterion (1198881) although theinterval 40ndash50 does not really represent an excellent choiceof 1198601 for risk (1198881)

Step 2 The intuitionistic fuzzy decision matrix (28) isobtained using (22)

(

[04452 05568] [07458 09429] [04304 07404] [07220 08773]

[04304 05696] [07404 09243] [04172 07040] [07040 09049]

[03494 05696] [02229 05966] [08245 09243] [07404 09049]

[02596 05000] [07040 08245] [06506 07889] [08245 09243]

) (28)

Step 3 Compute weights 119908 of the criteria based on (28) andusing (23)

119908 = 01884 02634 02439 03043 (29)

Step 4 We compute the performance of each alternativeusing (25) to obtain the interval fuzzy number

1198631= [06050 08008]

1198632= [05921 07978]

1198633= [05509 07653]

1198634= [06439 07851]

(30)

Step 5 In this step we apply each possibility measure anddetermine the achieved results These are then compared todefine the differences between them

Using the possibility measures (1) (2) (3) and (8) weachieved the results presented in Table 1 The best alternativeto be ranked first is 1198603

(a) For the possibility measures (4) and (5) the obtainedpossibility matrix is

(

05000 05198 06093 04656

04802 05000 05878 04437

04122 03907 05000 03413

06587 05563 05344 05000

) (31)

The results presented in Table 2 show 1198604 is the bestalternative and ranks first

(b) As for the possibility measure (6) the obtainedpossibility matrix is

(

05000 04802 04122 06587

05198 05000 03907 05563

06093 05878 05000 05344

04656 04437 03413 05000

) (32)

6 Advances in Fuzzy Systems

Table 1 Possibility degrees using (1) (2) (3) and (8)

Alternatives 1198601 1198602 1198603 1198604

Weights 02543 02472 02693 02292Ranking 2 3 1 4

Table 2 Possibility degrees using (4) and (5)

Alternatives 1198601 1198602 1198603 1198604

Weights 02579 02510 02204 02708Ranking 2 3 4 1

Table 3 Possibility degrees using (6)

Alternatives 1198601 1198602 1198603 1198604

Weights 02543 02472 02693 02292Ranking 2 3 1 4

Table 4 Possibility degrees using (7)

Alternatives 1198601 1198602 1198603 1198604

Weights 02535 02531 02569 02366Ranking 2 3 1 4

The results presented in Table 3 show 1198603 is the bestalternative and ranks first

(c) For the possibility measure (7) the obtained possibil-ity matrix is

(

05000 04941 04840 05634

05059 05000 04789 05522

05211 05160 05000 05458

04542 04478 04366 05000

) (33)

Table 4 shows that the best alternative is1198603 achievingthe first rank

(d) For the possibility measure (9) the obtained possibil-ity matrix is

(

05000 04810 04253 06204

05190 05000 04103 05474

05897 05747 05000 05296

04704 04526 03796 05000

) (34)

Table 5 shows that the best alternative is1198603 that ranksfirst

(e) For possibility measure (10) the obtained possibilitymatrix is

(

05000 04605 03297 07883

05395 05000 02914 06111

07086 06703 05000 05684

04316 03889 02117 05000

) (35)

Table 5 Possibility degrees using (9)

Alternatives 1198601 1198602 1198603 1198604

Weights 02522 02481 02662 02335Ranking 2 3 1 4

Table 6 Possibility degrees using (10)

Alternatives 1198601 1198602 1198603 1198604

Weights 02565 02452 02873 02110Ranking 2 3 1 4

Table 7 Possibility degrees using (12)

Alternatives 1198601 1198602 1198603 1198604

Weights 02511 02533 02557 02399Ranking 3 2 1 4

Table 8 Alternatives ranking order for different possibility mea-sures under IFN

Possibility measure Ranking Best alternative(1) (2) (3) and (8) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(4) and (5) 1198604 gt 1198601 gt 1198602 gt 1198603 1198604

(6) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(7) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(9) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(10) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(12) 1198603 gt 1198602 gt 1198601 gt 1198604 1198603

Table 6 shows that the best alternative that ranks firstis 1198602

(f) For the possibility measure (12) the obtained possi-bility matrix is

(

05000 04938 04872 05327

05062 05000 04939 05395

05061 05128 05000 05491

04509 04605 04673 05000

) (36)

Table 7 shows that the best alternative is 1198603 thatclearly ranks first

Table 8 presents a comparison of the obtained results applyingdifferent possibility measures under intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility measures (4) and (5) gave the same bestalternative 1198604 and the worst alternative 1198603 However usingformulas (1) (2) (3) (6) (7) (8) (9) (10) and (12) the bestalternative is1198603 and the worst alternative is1198604 These resultsshow that the measures (4) and (2) are different althoughthey are demonstrated to be equivalent (the operators lead tovalue 1) in [17] but they do not produce the same result

52 Application of Possibility Measures of IVIFS in DecisionMaking Problem We apply possibility measures of IVIFS

Advances in Fuzzy Systems 7

Table 9 Ranking IVIFSs alternatives using possibility measure (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02111 02161 02540 03200Ranking 4 3 2 1

Table 10 Ranking IVIFSs alternatives using possibility measures(15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02469 02401 02429 02640Ranking 2 4 3 1

presented in Section 32 to rank IVIFS data sets describedin Section 5 It is worth reminding that there are fouralternatives 1198601 1198602 1198603 and 1198604 and four criteria We use theIVIF matrix of alternatives (27) and the following criteriarsquosweight 119908

119895= [013 017 039 031] given in [9]

521 Case 1 Application of Interval-Valued Intuitionistic FuzzyWeighted Geometric (IVIFWG) Operator (13) The possibilitymeasures are applied in two cases In each case an aggregationoperator is also applied to the matrix (27)

Step 1 Compute the comprehensive evaluation of eachinvestment (alternative) using the geometric weighted aver-age operator (13) to aggregate the evaluation of each alterna-tive Thus we transform the IVIFS decision matrix to IVIFsfor each alternative presented as follows

1198631= [04760 05970] [01915 03926]

1198632= [04211 06454] [02260 03546]

1198633= [04057 06112] [01632 02833]

1198634= [05081 06389] [02007 03016]

(37)

Step 2 Each possibility measure presented in Section 32 isapplied to the obtained IVIFNs119863

1119863211986331198634

(a) For the possibility degree (16) the obtained possibilitymatrix is

(

05000 04671 03840 01823

05443 05000 03844 01647

06142 06210 05000 03131

06869 08353 08177 05000

) (38)

Table 9 presents the obtained results and shows thatthe best alternative is 1198604

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05264 05183 04185

04976 05000 04918 03921

05087 05063 05000 04003

05510 05597 05573 05000

) (39)

Table 10 presents the obtained results and shows thatthe best alternative is 1198604

Table 11 Ranking IVFISs alternatives using possibilitymeasure (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02627 02654 02105 02614Ranking 2 1 4 3

Table 12 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02254 02448 02613 02686Ranking 4 3 2 1

Table 13 Ranking order of alternatives for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198601 gt 1198603 gt 1198602 1198604

(16) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(17) 1198602 gt 1198601 gt 1198604 gt 1198603 1198602

(20) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(c) For the possibility measure (17) the obtained possi-bility matrix is

(

05000 05075 06501 04944

04925 05000 06635 05285

03365 03499 05000 03401

06599 04715 05056 05000

) (40)

The obtained results are presented in Table 11 showingthat the best alternative is 1198602

(d) For the possibility measure (20) the obtained possi-bility matrix is

(

05000 04446 03856 03740

05554 05000 04570 04256

05430 06144 05000 04778

05222 05744 06260 05000

) (41)

The obtained results are presented in Table 12 showingthat the best alternative is 1198604

Table 13 presents all the obtained results applying differentpossibility methods using the interval-valued intuitionisticfuzzy sets and shows the alternatives ranking results Weremark that the possibility formulas (15) (16) and (20)provide the same best alternative 1198604 However (17) providesthe best alternative 1198602

522 Case 2 Application of Optimal Aggregated Interval-Valued Intuitionistic Fuzzy Sets (14) Using the optimal aggre-gated operator (14) to IVIF decision matrix we obtain

8 Advances in Fuzzy Systems

Table 14 Ranking IVFSs using possibility degree (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02256 02352 02779 02599Ranking 4 3 1 2

Table 15 Ranking IVFSs using possibility measure (15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02369 02322 02557 02629Ranking 3 4 2 1

four interval-valued intuitionistic fuzzy numbers (IVIFNs)representing the alternatives as follows

1205721= [04831 06089] [01850 03800]

1205722= [04400 06520] [02170 03480]

1205723= [04840 06450] [01560 02730]

1205724= [05400 06530] [01950 02950]

(42)

(a) For the possibility measure (16) the obtained possi-bility degree matrix is

(

05000 04717 03595 03758

05366 05000 03682 04177

06329 06403 05000 05611

04358 05818 06008 05000

) (43)

The alternatives weight 119882119894is computed using (26)

and then ranked in a decreasing orderThe results aredisplayed in Table 14 showing that the best alternativethat ranks first is 119860

3

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05188 04308 03938

05000 05000 04120 03750

05525 05525 05000 04630

05168 05693 05693 05000

) (44)

We compute the weight 119882119894of the alternative using

(26) and we rank 119882119894in a decreasing order The

results are shown in Table 15 revealing that the bestalternative is 1198604 which ranks first

(c) For possibility measures (17) the obtained possibilitydegree matrix is

(

05000 05000 05767 04578

05000 05000 06123 05035

03877 04233 05000 03713

06287 04965 05422 05000

) (45)

Table 16 Ranking IVFSs using possibility degree (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02529 02596 02235 02639Ranking 3 2 4 1

Table 17 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02240 02392 02695 02673Ranking 4 3 1 2

Table 18 Alternatives ranking order for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198603 gt 1198601 gt 1198602 1198604

(16) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

(17) 1198604 gt 1198602 gt 1198601 gt 1198603 1198604

(20) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

We compute the weights119882119894of the alternatives using

(26) and we rank 119882119894in a decreasing order The

results are displayed in Table 16 showing that the bestalternative that rank first is 1198604

(d) For possibility measure (20) the obtained possibilitymatrix is

(

05000 04485 03685 03705

05515 05000 04082 04105

05918 06315 05000 05113

04887 05895 06295 05000

) (46)

The obtained results are presented in Table 17 showingthat the best alternative is 1198603

Table 18 presents the results of all applied possibilitymeasures using the interval-valued intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility formulas (15) and (17) provide the same bestalternative 1198604 However (16) and (20) provide the bestalternative 1198603 We note that the latter is the worst alternativeusing (17)

6 Conclusion

In this study we presented different formulas of possibilitymeasures The formulas exist in literature with IFN andIVIFN We also presented an MCDM method from theliterature We gave an illustrative examples for applicationsof different possibility measures and compared their resultsFirst we used an MCDM matrix with intuitionistic fuzzynumbers and then anMCDMmatrixwith IVIFNsThe valuesof the latter are aggregated with an aggregation operator intwo cases In each case a different aggregation operator wasused Thus the appropriate possibility measures are applied

Advances in Fuzzy Systems 9

The results show that the ranked alternatives can be differentfor each possibility measure even though some of thesemeasures have already been demonstrated to be equivalentin the literature

Appendix

Intuitionistic Fuzzy Sets

Intuitionistic fuzzy sets are introduced by Atanassov [16] whodefined a degree of membership 120583 a degree of nonmember-ship ] and a degree of hesitation 120587 of an element 119909 of an IFS

119860 = (120583119860 ]119860) denotes an intuitionistic fuzzy number if 120583

119860

and ]119860are fuzzy numbers with ]

119860le 120583119888119860 where 120583119888

119860denotes

the complement of 120583119860

If119883 is a discourse universe and 119860 a set in119883 then

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883 (A1)

with the conditions 0 le 120583119860(119909) le 1 0 le ]

119860(119909) le 1 0 le

120583119860(119909) + ]

119860(119909) le 1 and 120587

119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) Also for

each 119909 isin 119883 0 le Π119860(119909) le 1

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof this work by grants from General Direction of ScientificResearch (DGRST) Tunisia under the ARUB programThey would like to thank Mr Abdelmajid Dammak for hisproofreading and correction of the English of the paper

References

[1] J H Park I Y Park Y C L Kwun andX Tan ldquoExtension of theTOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environmentrdquo Applied MathematicalModelling vol 35 no 5 pp 2544ndash2556 2011

[2] D G Park Y C Kwun J H Park and I Y Park ldquoCorrelationcoefficient of interval-valued intuitionistic fuzzy sets and itsapplication to multiple attribute group decision making prob-lemsrdquoMathematical and Computer Modelling vol 50 no 9-10pp 1279ndash1293 2009

[3] J Ye ldquoMulticriteria fuzzy decision-making method usingentropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy setsrdquo Applied Mathematical Mod-elling vol 34 no 12 pp 3864ndash3870 2010

[4] H Zhang and L Yu ldquoMADM method based on cross-entropyand extended TOPSIS with interval-valued intuitionistic fuzzysetsrdquo Knowledge-Based Systems vol 30 pp 115ndash120 2012

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

[6] Z-S Xu and J Chen ldquoApproach to groupdecisionmaking basedon interval-valued intuitionistic judgment matricesrdquo SystemsEngineeringmdashTheory amp Practice vol 27 no 4 pp 126ndash133 2007

[7] X Zhang G Yue and Z Teng ldquoPossibility degree of interval-valued intuitionistic fuzzy numbers and its applicationrdquo inProceedings of the International Symposium on InformationProcessing (ISIP rsquo09) pp 33ndash36 Huangshan China 2009

[8] J Wu Q Cao and H Li ldquoAn approach for MADM problemswith interval-valued intuitionistic fuzzy sets based on nonlinearfunctionsrdquo Technological and Economic Development of Econ-omy vol 22 no 3 pp 336ndash356 2016

[9] Z Wang K W Li and W Wang ldquoAn approach to multi-attribute decision making with interval-valued intuitionisticfuzzy assessments and incomplete weightsrdquo Information Sci-ences vol 179 no 17 pp 3026ndash3040 2009

[10] S C Onar B Oztaysi I Otay and C Kahraman ldquoMulti-expert wind energy technology selection using interval-valuedintuitionistic fuzzy setsrdquo Energy vol 90 part 1 pp 274ndash2852015

[11] Z Xu and R R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006

[12] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[13] C-P Wei and X Tang ldquoPossibility degree method for rank-ing intuitionistic fuzzy numbersrdquo in Proceedings of the 3rdIEEEWICACM International Conference on Web Intelligenceand Intelligent Agent Technology (WI-IAT rsquo10) pp 142ndash145IEEE Toronto Canada August 2010

[14] Z S Xu and Q L Da ldquoPossibility degree method for rankinginterval numbers and its applicationrdquo Journal of Systems Engi-neering vol 18 pp 67ndash70 2003

[15] Y-M Wang J-B Yang and D-L Xu ldquoInterval weight gen-eration approaches based on consistency test and intervalcomparison matricesrdquo Applied Mathematics and Computationvol 167 no 1 pp 252ndash273 2005

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

[17] F Gao ldquoPossibility degree and comprehensive priority ofinterval numbersrdquo Systems EngineeringmdashTheoryamp Practice vol33 no 8 pp 2033ndash2040 2013

[18] Y-Y Liu and Y-J Lv ldquoA multiple attribute decision makingmethod with interval rough numbers based on the possibilitydegreerdquo in Proceedings of the 10th International Conference onNatural Computation (ICNC rsquo14) pp 407ndash411 IEEE XiamenChina August 2014

[19] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[20] H Yuan and Y Qu ldquoModel for conflict resolution withpreference represented as interval numbersrdquo Proceedings of theMATECWeb of Conferences 2015

[21] G Facchinetti R G Ricci and S Muzzioli ldquoNote on rankingfuzzy triangular numbersrdquo International Journal of IntelligentSystems vol 13 no 7 pp 613ndash622 1998

[22] Q L Da and X W Liu ldquoInterval number linear programmingand its satisfactory solutionrdquo Systems Engineering Theory ampPractice vol 19 pp 3ndash7 1999

[23] C-P Wei and X Tang ldquoPossibility degree method for rankingintuitionistic fuzzy numbersrdquo Journal of Systems Engineeringvol 18 pp 67ndash70 2003

[24] Z S Xu and Q L Da ldquoThe uncertain ow a operatorrdquo Interna-tional Journal of Intelligent Systems vol 17 no 6 pp 569ndash5752002

10 Advances in Fuzzy Systems

[25] Z S Xu and Q L Da ldquoA possibility based method for prioritiesof interval judjment matricesrdquo Chinese Journal of ManagementScience vol 11 pp 63ndash65 2003

[26] D Q Li and Y D Gu ldquomethods for ranking interval numberbased on possibility degreerdquo Journal of Systems Engineering vol23 pp 223ndash226 2008

[27] J B Lan L J Cao and J Lin ldquoMethod for rinking interval num-bers on two-dimensional priority degreerdquo Journal of ChongqingInstitute of Technology Natural Science Edition vol 21 pp 63ndash66 2007

[28] T-Y Chen ldquoInterval-valued intuitionistic fuzzy QUALIFLEXmethod with a likelihood-based comparison approach formultiple criteria decision analysisrdquo Information Sciences vol261 pp 149ndash169 2014

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

[30] G W Wei and X R Wang ldquoSome geometric aggregationoperators on interval-valued intuitionistic fuzzy sets and theirapplication to group decision makingrdquo in Proceedings of theInternational Conference on Computational Intelligence andSecurity (ICCIS rsquo07) pp 495ndash499 Harbin China December2007

[31] Z S Xu and J Chen ldquoOn geometric aggregation over interval-valued intuitionistic fuzzy informationrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) vol 2 pp 466ndash471 Haikou ChinaAugust 2007

[32] Y He H Chen L Zhou J Liu and Z Tao ldquoGeneralizedinterval-valuedAtanassovrsquos intuitionistic fuzzy power operatorsand their application to group decision makingrdquo InternationalJournal of Fuzzy Systems vol 15 no 4 pp 401ndash411 2013

[33] S Wan and J Dong ldquoA possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision mak-ingrdquo Journal of Computer and System Sciences vol 80 no 1 pp237ndash256 2014

[34] F Dammak L Baccour and A M Alimi ldquoThe impact ofcriterion weights techniques in topsis method of multi-criteriadecision making in crisp and intuitionistic fuzzy domainsrdquoin Proceedings of the IEEE International Conference on FuzzySystems (FUZZ-IEEE rsquo15) pp 1ndash8 Istanbul Turkey August 2015

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 5

Step 5 Compute the likelihood matrix [25] To comparebetween tow interval fuzzy numbers we propose to usea possibility measure instead of the formula used in [4]to obtain a possibility matrix Therefore each possibilitymeasure presented in Section 21 is applied and all theachieved results are compared in Section 5

Step 6 Determine the alternatives ranking order accordingto the decreasing order of119882

119894[25] defined as

119882119894=sum119899

119895=1119901119894119895+ 1198982 minus 1

119898 (119898 minus 1) 119894 = 1 2 119898 (26)

5 Illustrative Example51 Application of Possibility Measure of IFS in Decision Mak-ing Problem This section described the data set presented in[4 9] to evaluate the four potential investment opportunities119860 = 1198601 1198602 1198603 1198604 The fund manager should evaluateeach investment considering four criteria risk (1198621) growth(1198622) sociopolitical issues (1198623) and environmental impacts(1198624) The fund manager is satisfied once he provides hisassessment of each alternative on each criterion

Step 1 The following interval-valued intuitionistic fuzzy sets(IVIFSs) decision making matrix (27) presents the relation-ship between criteria and alternatives of data set as follows

(

[042 048] [04 05] [06 07] [005 025] [04 05] [02 05] [055 075] [015 025]

[04 05] [04 05] [05 08] [01 02] [03 06] [03 04] [06 07] [01 03]

[03 05] [04 05] [01 03] [02 04] [07 08] [01 02] [05 07] [01 02]

[02 04] [04 05] [06 07] [02 03] [05 06] [02 03] [07 08] [01 02]

) (27)

Each element of this matrix is presented with IVIFSgiving the fund managerrsquos satisfaction or dissatisfactiondegree with an alternative The element represented for thefirst alternative [042 048] [04 05] where the interval 42ndash48 [4] reflects that the fund manager has an excellent

opportunity to respect the risk criterion (1198881) although theinterval 40ndash50 does not really represent an excellent choiceof 1198601 for risk (1198881)

Step 2 The intuitionistic fuzzy decision matrix (28) isobtained using (22)

(

[04452 05568] [07458 09429] [04304 07404] [07220 08773]

[04304 05696] [07404 09243] [04172 07040] [07040 09049]

[03494 05696] [02229 05966] [08245 09243] [07404 09049]

[02596 05000] [07040 08245] [06506 07889] [08245 09243]

) (28)

Step 3 Compute weights 119908 of the criteria based on (28) andusing (23)

119908 = 01884 02634 02439 03043 (29)

Step 4 We compute the performance of each alternativeusing (25) to obtain the interval fuzzy number

1198631= [06050 08008]

1198632= [05921 07978]

1198633= [05509 07653]

1198634= [06439 07851]

(30)

Step 5 In this step we apply each possibility measure anddetermine the achieved results These are then compared todefine the differences between them

Using the possibility measures (1) (2) (3) and (8) weachieved the results presented in Table 1 The best alternativeto be ranked first is 1198603

(a) For the possibility measures (4) and (5) the obtainedpossibility matrix is

(

05000 05198 06093 04656

04802 05000 05878 04437

04122 03907 05000 03413

06587 05563 05344 05000

) (31)

The results presented in Table 2 show 1198604 is the bestalternative and ranks first

(b) As for the possibility measure (6) the obtainedpossibility matrix is

(

05000 04802 04122 06587

05198 05000 03907 05563

06093 05878 05000 05344

04656 04437 03413 05000

) (32)

6 Advances in Fuzzy Systems

Table 1 Possibility degrees using (1) (2) (3) and (8)

Alternatives 1198601 1198602 1198603 1198604

Weights 02543 02472 02693 02292Ranking 2 3 1 4

Table 2 Possibility degrees using (4) and (5)

Alternatives 1198601 1198602 1198603 1198604

Weights 02579 02510 02204 02708Ranking 2 3 4 1

Table 3 Possibility degrees using (6)

Alternatives 1198601 1198602 1198603 1198604

Weights 02543 02472 02693 02292Ranking 2 3 1 4

Table 4 Possibility degrees using (7)

Alternatives 1198601 1198602 1198603 1198604

Weights 02535 02531 02569 02366Ranking 2 3 1 4

The results presented in Table 3 show 1198603 is the bestalternative and ranks first

(c) For the possibility measure (7) the obtained possibil-ity matrix is

(

05000 04941 04840 05634

05059 05000 04789 05522

05211 05160 05000 05458

04542 04478 04366 05000

) (33)

Table 4 shows that the best alternative is1198603 achievingthe first rank

(d) For the possibility measure (9) the obtained possibil-ity matrix is

(

05000 04810 04253 06204

05190 05000 04103 05474

05897 05747 05000 05296

04704 04526 03796 05000

) (34)

Table 5 shows that the best alternative is1198603 that ranksfirst

(e) For possibility measure (10) the obtained possibilitymatrix is

(

05000 04605 03297 07883

05395 05000 02914 06111

07086 06703 05000 05684

04316 03889 02117 05000

) (35)

Table 5 Possibility degrees using (9)

Alternatives 1198601 1198602 1198603 1198604

Weights 02522 02481 02662 02335Ranking 2 3 1 4

Table 6 Possibility degrees using (10)

Alternatives 1198601 1198602 1198603 1198604

Weights 02565 02452 02873 02110Ranking 2 3 1 4

Table 7 Possibility degrees using (12)

Alternatives 1198601 1198602 1198603 1198604

Weights 02511 02533 02557 02399Ranking 3 2 1 4

Table 8 Alternatives ranking order for different possibility mea-sures under IFN

Possibility measure Ranking Best alternative(1) (2) (3) and (8) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(4) and (5) 1198604 gt 1198601 gt 1198602 gt 1198603 1198604

(6) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(7) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(9) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(10) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(12) 1198603 gt 1198602 gt 1198601 gt 1198604 1198603

Table 6 shows that the best alternative that ranks firstis 1198602

(f) For the possibility measure (12) the obtained possi-bility matrix is

(

05000 04938 04872 05327

05062 05000 04939 05395

05061 05128 05000 05491

04509 04605 04673 05000

) (36)

Table 7 shows that the best alternative is 1198603 thatclearly ranks first

Table 8 presents a comparison of the obtained results applyingdifferent possibility measures under intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility measures (4) and (5) gave the same bestalternative 1198604 and the worst alternative 1198603 However usingformulas (1) (2) (3) (6) (7) (8) (9) (10) and (12) the bestalternative is1198603 and the worst alternative is1198604 These resultsshow that the measures (4) and (2) are different althoughthey are demonstrated to be equivalent (the operators lead tovalue 1) in [17] but they do not produce the same result

52 Application of Possibility Measures of IVIFS in DecisionMaking Problem We apply possibility measures of IVIFS

Advances in Fuzzy Systems 7

Table 9 Ranking IVIFSs alternatives using possibility measure (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02111 02161 02540 03200Ranking 4 3 2 1

Table 10 Ranking IVIFSs alternatives using possibility measures(15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02469 02401 02429 02640Ranking 2 4 3 1

presented in Section 32 to rank IVIFS data sets describedin Section 5 It is worth reminding that there are fouralternatives 1198601 1198602 1198603 and 1198604 and four criteria We use theIVIF matrix of alternatives (27) and the following criteriarsquosweight 119908

119895= [013 017 039 031] given in [9]

521 Case 1 Application of Interval-Valued Intuitionistic FuzzyWeighted Geometric (IVIFWG) Operator (13) The possibilitymeasures are applied in two cases In each case an aggregationoperator is also applied to the matrix (27)

Step 1 Compute the comprehensive evaluation of eachinvestment (alternative) using the geometric weighted aver-age operator (13) to aggregate the evaluation of each alterna-tive Thus we transform the IVIFS decision matrix to IVIFsfor each alternative presented as follows

1198631= [04760 05970] [01915 03926]

1198632= [04211 06454] [02260 03546]

1198633= [04057 06112] [01632 02833]

1198634= [05081 06389] [02007 03016]

(37)

Step 2 Each possibility measure presented in Section 32 isapplied to the obtained IVIFNs119863

1119863211986331198634

(a) For the possibility degree (16) the obtained possibilitymatrix is

(

05000 04671 03840 01823

05443 05000 03844 01647

06142 06210 05000 03131

06869 08353 08177 05000

) (38)

Table 9 presents the obtained results and shows thatthe best alternative is 1198604

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05264 05183 04185

04976 05000 04918 03921

05087 05063 05000 04003

05510 05597 05573 05000

) (39)

Table 10 presents the obtained results and shows thatthe best alternative is 1198604

Table 11 Ranking IVFISs alternatives using possibilitymeasure (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02627 02654 02105 02614Ranking 2 1 4 3

Table 12 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02254 02448 02613 02686Ranking 4 3 2 1

Table 13 Ranking order of alternatives for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198601 gt 1198603 gt 1198602 1198604

(16) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(17) 1198602 gt 1198601 gt 1198604 gt 1198603 1198602

(20) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(c) For the possibility measure (17) the obtained possi-bility matrix is

(

05000 05075 06501 04944

04925 05000 06635 05285

03365 03499 05000 03401

06599 04715 05056 05000

) (40)

The obtained results are presented in Table 11 showingthat the best alternative is 1198602

(d) For the possibility measure (20) the obtained possi-bility matrix is

(

05000 04446 03856 03740

05554 05000 04570 04256

05430 06144 05000 04778

05222 05744 06260 05000

) (41)

The obtained results are presented in Table 12 showingthat the best alternative is 1198604

Table 13 presents all the obtained results applying differentpossibility methods using the interval-valued intuitionisticfuzzy sets and shows the alternatives ranking results Weremark that the possibility formulas (15) (16) and (20)provide the same best alternative 1198604 However (17) providesthe best alternative 1198602

522 Case 2 Application of Optimal Aggregated Interval-Valued Intuitionistic Fuzzy Sets (14) Using the optimal aggre-gated operator (14) to IVIF decision matrix we obtain

8 Advances in Fuzzy Systems

Table 14 Ranking IVFSs using possibility degree (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02256 02352 02779 02599Ranking 4 3 1 2

Table 15 Ranking IVFSs using possibility measure (15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02369 02322 02557 02629Ranking 3 4 2 1

four interval-valued intuitionistic fuzzy numbers (IVIFNs)representing the alternatives as follows

1205721= [04831 06089] [01850 03800]

1205722= [04400 06520] [02170 03480]

1205723= [04840 06450] [01560 02730]

1205724= [05400 06530] [01950 02950]

(42)

(a) For the possibility measure (16) the obtained possi-bility degree matrix is

(

05000 04717 03595 03758

05366 05000 03682 04177

06329 06403 05000 05611

04358 05818 06008 05000

) (43)

The alternatives weight 119882119894is computed using (26)

and then ranked in a decreasing orderThe results aredisplayed in Table 14 showing that the best alternativethat ranks first is 119860

3

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05188 04308 03938

05000 05000 04120 03750

05525 05525 05000 04630

05168 05693 05693 05000

) (44)

We compute the weight 119882119894of the alternative using

(26) and we rank 119882119894in a decreasing order The

results are shown in Table 15 revealing that the bestalternative is 1198604 which ranks first

(c) For possibility measures (17) the obtained possibilitydegree matrix is

(

05000 05000 05767 04578

05000 05000 06123 05035

03877 04233 05000 03713

06287 04965 05422 05000

) (45)

Table 16 Ranking IVFSs using possibility degree (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02529 02596 02235 02639Ranking 3 2 4 1

Table 17 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02240 02392 02695 02673Ranking 4 3 1 2

Table 18 Alternatives ranking order for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198603 gt 1198601 gt 1198602 1198604

(16) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

(17) 1198604 gt 1198602 gt 1198601 gt 1198603 1198604

(20) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

We compute the weights119882119894of the alternatives using

(26) and we rank 119882119894in a decreasing order The

results are displayed in Table 16 showing that the bestalternative that rank first is 1198604

(d) For possibility measure (20) the obtained possibilitymatrix is

(

05000 04485 03685 03705

05515 05000 04082 04105

05918 06315 05000 05113

04887 05895 06295 05000

) (46)

The obtained results are presented in Table 17 showingthat the best alternative is 1198603

Table 18 presents the results of all applied possibilitymeasures using the interval-valued intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility formulas (15) and (17) provide the same bestalternative 1198604 However (16) and (20) provide the bestalternative 1198603 We note that the latter is the worst alternativeusing (17)

6 Conclusion

In this study we presented different formulas of possibilitymeasures The formulas exist in literature with IFN andIVIFN We also presented an MCDM method from theliterature We gave an illustrative examples for applicationsof different possibility measures and compared their resultsFirst we used an MCDM matrix with intuitionistic fuzzynumbers and then anMCDMmatrixwith IVIFNsThe valuesof the latter are aggregated with an aggregation operator intwo cases In each case a different aggregation operator wasused Thus the appropriate possibility measures are applied

Advances in Fuzzy Systems 9

The results show that the ranked alternatives can be differentfor each possibility measure even though some of thesemeasures have already been demonstrated to be equivalentin the literature

Appendix

Intuitionistic Fuzzy Sets

Intuitionistic fuzzy sets are introduced by Atanassov [16] whodefined a degree of membership 120583 a degree of nonmember-ship ] and a degree of hesitation 120587 of an element 119909 of an IFS

119860 = (120583119860 ]119860) denotes an intuitionistic fuzzy number if 120583

119860

and ]119860are fuzzy numbers with ]

119860le 120583119888119860 where 120583119888

119860denotes

the complement of 120583119860

If119883 is a discourse universe and 119860 a set in119883 then

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883 (A1)

with the conditions 0 le 120583119860(119909) le 1 0 le ]

119860(119909) le 1 0 le

120583119860(119909) + ]

119860(119909) le 1 and 120587

119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) Also for

each 119909 isin 119883 0 le Π119860(119909) le 1

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof this work by grants from General Direction of ScientificResearch (DGRST) Tunisia under the ARUB programThey would like to thank Mr Abdelmajid Dammak for hisproofreading and correction of the English of the paper

References

[1] J H Park I Y Park Y C L Kwun andX Tan ldquoExtension of theTOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environmentrdquo Applied MathematicalModelling vol 35 no 5 pp 2544ndash2556 2011

[2] D G Park Y C Kwun J H Park and I Y Park ldquoCorrelationcoefficient of interval-valued intuitionistic fuzzy sets and itsapplication to multiple attribute group decision making prob-lemsrdquoMathematical and Computer Modelling vol 50 no 9-10pp 1279ndash1293 2009

[3] J Ye ldquoMulticriteria fuzzy decision-making method usingentropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy setsrdquo Applied Mathematical Mod-elling vol 34 no 12 pp 3864ndash3870 2010

[4] H Zhang and L Yu ldquoMADM method based on cross-entropyand extended TOPSIS with interval-valued intuitionistic fuzzysetsrdquo Knowledge-Based Systems vol 30 pp 115ndash120 2012

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

[6] Z-S Xu and J Chen ldquoApproach to groupdecisionmaking basedon interval-valued intuitionistic judgment matricesrdquo SystemsEngineeringmdashTheory amp Practice vol 27 no 4 pp 126ndash133 2007

[7] X Zhang G Yue and Z Teng ldquoPossibility degree of interval-valued intuitionistic fuzzy numbers and its applicationrdquo inProceedings of the International Symposium on InformationProcessing (ISIP rsquo09) pp 33ndash36 Huangshan China 2009

[8] J Wu Q Cao and H Li ldquoAn approach for MADM problemswith interval-valued intuitionistic fuzzy sets based on nonlinearfunctionsrdquo Technological and Economic Development of Econ-omy vol 22 no 3 pp 336ndash356 2016

[9] Z Wang K W Li and W Wang ldquoAn approach to multi-attribute decision making with interval-valued intuitionisticfuzzy assessments and incomplete weightsrdquo Information Sci-ences vol 179 no 17 pp 3026ndash3040 2009

[10] S C Onar B Oztaysi I Otay and C Kahraman ldquoMulti-expert wind energy technology selection using interval-valuedintuitionistic fuzzy setsrdquo Energy vol 90 part 1 pp 274ndash2852015

[11] Z Xu and R R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006

[12] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[13] C-P Wei and X Tang ldquoPossibility degree method for rank-ing intuitionistic fuzzy numbersrdquo in Proceedings of the 3rdIEEEWICACM International Conference on Web Intelligenceand Intelligent Agent Technology (WI-IAT rsquo10) pp 142ndash145IEEE Toronto Canada August 2010

[14] Z S Xu and Q L Da ldquoPossibility degree method for rankinginterval numbers and its applicationrdquo Journal of Systems Engi-neering vol 18 pp 67ndash70 2003

[15] Y-M Wang J-B Yang and D-L Xu ldquoInterval weight gen-eration approaches based on consistency test and intervalcomparison matricesrdquo Applied Mathematics and Computationvol 167 no 1 pp 252ndash273 2005

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

[17] F Gao ldquoPossibility degree and comprehensive priority ofinterval numbersrdquo Systems EngineeringmdashTheoryamp Practice vol33 no 8 pp 2033ndash2040 2013

[18] Y-Y Liu and Y-J Lv ldquoA multiple attribute decision makingmethod with interval rough numbers based on the possibilitydegreerdquo in Proceedings of the 10th International Conference onNatural Computation (ICNC rsquo14) pp 407ndash411 IEEE XiamenChina August 2014

[19] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[20] H Yuan and Y Qu ldquoModel for conflict resolution withpreference represented as interval numbersrdquo Proceedings of theMATECWeb of Conferences 2015

[21] G Facchinetti R G Ricci and S Muzzioli ldquoNote on rankingfuzzy triangular numbersrdquo International Journal of IntelligentSystems vol 13 no 7 pp 613ndash622 1998

[22] Q L Da and X W Liu ldquoInterval number linear programmingand its satisfactory solutionrdquo Systems Engineering Theory ampPractice vol 19 pp 3ndash7 1999

[23] C-P Wei and X Tang ldquoPossibility degree method for rankingintuitionistic fuzzy numbersrdquo Journal of Systems Engineeringvol 18 pp 67ndash70 2003

[24] Z S Xu and Q L Da ldquoThe uncertain ow a operatorrdquo Interna-tional Journal of Intelligent Systems vol 17 no 6 pp 569ndash5752002

10 Advances in Fuzzy Systems

[25] Z S Xu and Q L Da ldquoA possibility based method for prioritiesof interval judjment matricesrdquo Chinese Journal of ManagementScience vol 11 pp 63ndash65 2003

[26] D Q Li and Y D Gu ldquomethods for ranking interval numberbased on possibility degreerdquo Journal of Systems Engineering vol23 pp 223ndash226 2008

[27] J B Lan L J Cao and J Lin ldquoMethod for rinking interval num-bers on two-dimensional priority degreerdquo Journal of ChongqingInstitute of Technology Natural Science Edition vol 21 pp 63ndash66 2007

[28] T-Y Chen ldquoInterval-valued intuitionistic fuzzy QUALIFLEXmethod with a likelihood-based comparison approach formultiple criteria decision analysisrdquo Information Sciences vol261 pp 149ndash169 2014

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

[30] G W Wei and X R Wang ldquoSome geometric aggregationoperators on interval-valued intuitionistic fuzzy sets and theirapplication to group decision makingrdquo in Proceedings of theInternational Conference on Computational Intelligence andSecurity (ICCIS rsquo07) pp 495ndash499 Harbin China December2007

[31] Z S Xu and J Chen ldquoOn geometric aggregation over interval-valued intuitionistic fuzzy informationrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) vol 2 pp 466ndash471 Haikou ChinaAugust 2007

[32] Y He H Chen L Zhou J Liu and Z Tao ldquoGeneralizedinterval-valuedAtanassovrsquos intuitionistic fuzzy power operatorsand their application to group decision makingrdquo InternationalJournal of Fuzzy Systems vol 15 no 4 pp 401ndash411 2013

[33] S Wan and J Dong ldquoA possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision mak-ingrdquo Journal of Computer and System Sciences vol 80 no 1 pp237ndash256 2014

[34] F Dammak L Baccour and A M Alimi ldquoThe impact ofcriterion weights techniques in topsis method of multi-criteriadecision making in crisp and intuitionistic fuzzy domainsrdquoin Proceedings of the IEEE International Conference on FuzzySystems (FUZZ-IEEE rsquo15) pp 1ndash8 Istanbul Turkey August 2015

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Advances in Fuzzy Systems

Table 1 Possibility degrees using (1) (2) (3) and (8)

Alternatives 1198601 1198602 1198603 1198604

Weights 02543 02472 02693 02292Ranking 2 3 1 4

Table 2 Possibility degrees using (4) and (5)

Alternatives 1198601 1198602 1198603 1198604

Weights 02579 02510 02204 02708Ranking 2 3 4 1

Table 3 Possibility degrees using (6)

Alternatives 1198601 1198602 1198603 1198604

Weights 02543 02472 02693 02292Ranking 2 3 1 4

Table 4 Possibility degrees using (7)

Alternatives 1198601 1198602 1198603 1198604

Weights 02535 02531 02569 02366Ranking 2 3 1 4

The results presented in Table 3 show 1198603 is the bestalternative and ranks first

(c) For the possibility measure (7) the obtained possibil-ity matrix is

(

05000 04941 04840 05634

05059 05000 04789 05522

05211 05160 05000 05458

04542 04478 04366 05000

) (33)

Table 4 shows that the best alternative is1198603 achievingthe first rank

(d) For the possibility measure (9) the obtained possibil-ity matrix is

(

05000 04810 04253 06204

05190 05000 04103 05474

05897 05747 05000 05296

04704 04526 03796 05000

) (34)

Table 5 shows that the best alternative is1198603 that ranksfirst

(e) For possibility measure (10) the obtained possibilitymatrix is

(

05000 04605 03297 07883

05395 05000 02914 06111

07086 06703 05000 05684

04316 03889 02117 05000

) (35)

Table 5 Possibility degrees using (9)

Alternatives 1198601 1198602 1198603 1198604

Weights 02522 02481 02662 02335Ranking 2 3 1 4

Table 6 Possibility degrees using (10)

Alternatives 1198601 1198602 1198603 1198604

Weights 02565 02452 02873 02110Ranking 2 3 1 4

Table 7 Possibility degrees using (12)

Alternatives 1198601 1198602 1198603 1198604

Weights 02511 02533 02557 02399Ranking 3 2 1 4

Table 8 Alternatives ranking order for different possibility mea-sures under IFN

Possibility measure Ranking Best alternative(1) (2) (3) and (8) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(4) and (5) 1198604 gt 1198601 gt 1198602 gt 1198603 1198604

(6) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(7) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(9) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(10) 1198603 gt 1198601 gt 1198602 gt 1198604 1198603

(12) 1198603 gt 1198602 gt 1198601 gt 1198604 1198603

Table 6 shows that the best alternative that ranks firstis 1198602

(f) For the possibility measure (12) the obtained possi-bility matrix is

(

05000 04938 04872 05327

05062 05000 04939 05395

05061 05128 05000 05491

04509 04605 04673 05000

) (36)

Table 7 shows that the best alternative is 1198603 thatclearly ranks first

Table 8 presents a comparison of the obtained results applyingdifferent possibility measures under intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility measures (4) and (5) gave the same bestalternative 1198604 and the worst alternative 1198603 However usingformulas (1) (2) (3) (6) (7) (8) (9) (10) and (12) the bestalternative is1198603 and the worst alternative is1198604 These resultsshow that the measures (4) and (2) are different althoughthey are demonstrated to be equivalent (the operators lead tovalue 1) in [17] but they do not produce the same result

52 Application of Possibility Measures of IVIFS in DecisionMaking Problem We apply possibility measures of IVIFS

Advances in Fuzzy Systems 7

Table 9 Ranking IVIFSs alternatives using possibility measure (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02111 02161 02540 03200Ranking 4 3 2 1

Table 10 Ranking IVIFSs alternatives using possibility measures(15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02469 02401 02429 02640Ranking 2 4 3 1

presented in Section 32 to rank IVIFS data sets describedin Section 5 It is worth reminding that there are fouralternatives 1198601 1198602 1198603 and 1198604 and four criteria We use theIVIF matrix of alternatives (27) and the following criteriarsquosweight 119908

119895= [013 017 039 031] given in [9]

521 Case 1 Application of Interval-Valued Intuitionistic FuzzyWeighted Geometric (IVIFWG) Operator (13) The possibilitymeasures are applied in two cases In each case an aggregationoperator is also applied to the matrix (27)

Step 1 Compute the comprehensive evaluation of eachinvestment (alternative) using the geometric weighted aver-age operator (13) to aggregate the evaluation of each alterna-tive Thus we transform the IVIFS decision matrix to IVIFsfor each alternative presented as follows

1198631= [04760 05970] [01915 03926]

1198632= [04211 06454] [02260 03546]

1198633= [04057 06112] [01632 02833]

1198634= [05081 06389] [02007 03016]

(37)

Step 2 Each possibility measure presented in Section 32 isapplied to the obtained IVIFNs119863

1119863211986331198634

(a) For the possibility degree (16) the obtained possibilitymatrix is

(

05000 04671 03840 01823

05443 05000 03844 01647

06142 06210 05000 03131

06869 08353 08177 05000

) (38)

Table 9 presents the obtained results and shows thatthe best alternative is 1198604

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05264 05183 04185

04976 05000 04918 03921

05087 05063 05000 04003

05510 05597 05573 05000

) (39)

Table 10 presents the obtained results and shows thatthe best alternative is 1198604

Table 11 Ranking IVFISs alternatives using possibilitymeasure (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02627 02654 02105 02614Ranking 2 1 4 3

Table 12 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02254 02448 02613 02686Ranking 4 3 2 1

Table 13 Ranking order of alternatives for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198601 gt 1198603 gt 1198602 1198604

(16) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(17) 1198602 gt 1198601 gt 1198604 gt 1198603 1198602

(20) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(c) For the possibility measure (17) the obtained possi-bility matrix is

(

05000 05075 06501 04944

04925 05000 06635 05285

03365 03499 05000 03401

06599 04715 05056 05000

) (40)

The obtained results are presented in Table 11 showingthat the best alternative is 1198602

(d) For the possibility measure (20) the obtained possi-bility matrix is

(

05000 04446 03856 03740

05554 05000 04570 04256

05430 06144 05000 04778

05222 05744 06260 05000

) (41)

The obtained results are presented in Table 12 showingthat the best alternative is 1198604

Table 13 presents all the obtained results applying differentpossibility methods using the interval-valued intuitionisticfuzzy sets and shows the alternatives ranking results Weremark that the possibility formulas (15) (16) and (20)provide the same best alternative 1198604 However (17) providesthe best alternative 1198602

522 Case 2 Application of Optimal Aggregated Interval-Valued Intuitionistic Fuzzy Sets (14) Using the optimal aggre-gated operator (14) to IVIF decision matrix we obtain

8 Advances in Fuzzy Systems

Table 14 Ranking IVFSs using possibility degree (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02256 02352 02779 02599Ranking 4 3 1 2

Table 15 Ranking IVFSs using possibility measure (15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02369 02322 02557 02629Ranking 3 4 2 1

four interval-valued intuitionistic fuzzy numbers (IVIFNs)representing the alternatives as follows

1205721= [04831 06089] [01850 03800]

1205722= [04400 06520] [02170 03480]

1205723= [04840 06450] [01560 02730]

1205724= [05400 06530] [01950 02950]

(42)

(a) For the possibility measure (16) the obtained possi-bility degree matrix is

(

05000 04717 03595 03758

05366 05000 03682 04177

06329 06403 05000 05611

04358 05818 06008 05000

) (43)

The alternatives weight 119882119894is computed using (26)

and then ranked in a decreasing orderThe results aredisplayed in Table 14 showing that the best alternativethat ranks first is 119860

3

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05188 04308 03938

05000 05000 04120 03750

05525 05525 05000 04630

05168 05693 05693 05000

) (44)

We compute the weight 119882119894of the alternative using

(26) and we rank 119882119894in a decreasing order The

results are shown in Table 15 revealing that the bestalternative is 1198604 which ranks first

(c) For possibility measures (17) the obtained possibilitydegree matrix is

(

05000 05000 05767 04578

05000 05000 06123 05035

03877 04233 05000 03713

06287 04965 05422 05000

) (45)

Table 16 Ranking IVFSs using possibility degree (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02529 02596 02235 02639Ranking 3 2 4 1

Table 17 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02240 02392 02695 02673Ranking 4 3 1 2

Table 18 Alternatives ranking order for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198603 gt 1198601 gt 1198602 1198604

(16) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

(17) 1198604 gt 1198602 gt 1198601 gt 1198603 1198604

(20) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

We compute the weights119882119894of the alternatives using

(26) and we rank 119882119894in a decreasing order The

results are displayed in Table 16 showing that the bestalternative that rank first is 1198604

(d) For possibility measure (20) the obtained possibilitymatrix is

(

05000 04485 03685 03705

05515 05000 04082 04105

05918 06315 05000 05113

04887 05895 06295 05000

) (46)

The obtained results are presented in Table 17 showingthat the best alternative is 1198603

Table 18 presents the results of all applied possibilitymeasures using the interval-valued intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility formulas (15) and (17) provide the same bestalternative 1198604 However (16) and (20) provide the bestalternative 1198603 We note that the latter is the worst alternativeusing (17)

6 Conclusion

In this study we presented different formulas of possibilitymeasures The formulas exist in literature with IFN andIVIFN We also presented an MCDM method from theliterature We gave an illustrative examples for applicationsof different possibility measures and compared their resultsFirst we used an MCDM matrix with intuitionistic fuzzynumbers and then anMCDMmatrixwith IVIFNsThe valuesof the latter are aggregated with an aggregation operator intwo cases In each case a different aggregation operator wasused Thus the appropriate possibility measures are applied

Advances in Fuzzy Systems 9

The results show that the ranked alternatives can be differentfor each possibility measure even though some of thesemeasures have already been demonstrated to be equivalentin the literature

Appendix

Intuitionistic Fuzzy Sets

Intuitionistic fuzzy sets are introduced by Atanassov [16] whodefined a degree of membership 120583 a degree of nonmember-ship ] and a degree of hesitation 120587 of an element 119909 of an IFS

119860 = (120583119860 ]119860) denotes an intuitionistic fuzzy number if 120583

119860

and ]119860are fuzzy numbers with ]

119860le 120583119888119860 where 120583119888

119860denotes

the complement of 120583119860

If119883 is a discourse universe and 119860 a set in119883 then

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883 (A1)

with the conditions 0 le 120583119860(119909) le 1 0 le ]

119860(119909) le 1 0 le

120583119860(119909) + ]

119860(119909) le 1 and 120587

119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) Also for

each 119909 isin 119883 0 le Π119860(119909) le 1

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof this work by grants from General Direction of ScientificResearch (DGRST) Tunisia under the ARUB programThey would like to thank Mr Abdelmajid Dammak for hisproofreading and correction of the English of the paper

References

[1] J H Park I Y Park Y C L Kwun andX Tan ldquoExtension of theTOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environmentrdquo Applied MathematicalModelling vol 35 no 5 pp 2544ndash2556 2011

[2] D G Park Y C Kwun J H Park and I Y Park ldquoCorrelationcoefficient of interval-valued intuitionistic fuzzy sets and itsapplication to multiple attribute group decision making prob-lemsrdquoMathematical and Computer Modelling vol 50 no 9-10pp 1279ndash1293 2009

[3] J Ye ldquoMulticriteria fuzzy decision-making method usingentropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy setsrdquo Applied Mathematical Mod-elling vol 34 no 12 pp 3864ndash3870 2010

[4] H Zhang and L Yu ldquoMADM method based on cross-entropyand extended TOPSIS with interval-valued intuitionistic fuzzysetsrdquo Knowledge-Based Systems vol 30 pp 115ndash120 2012

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

[6] Z-S Xu and J Chen ldquoApproach to groupdecisionmaking basedon interval-valued intuitionistic judgment matricesrdquo SystemsEngineeringmdashTheory amp Practice vol 27 no 4 pp 126ndash133 2007

[7] X Zhang G Yue and Z Teng ldquoPossibility degree of interval-valued intuitionistic fuzzy numbers and its applicationrdquo inProceedings of the International Symposium on InformationProcessing (ISIP rsquo09) pp 33ndash36 Huangshan China 2009

[8] J Wu Q Cao and H Li ldquoAn approach for MADM problemswith interval-valued intuitionistic fuzzy sets based on nonlinearfunctionsrdquo Technological and Economic Development of Econ-omy vol 22 no 3 pp 336ndash356 2016

[9] Z Wang K W Li and W Wang ldquoAn approach to multi-attribute decision making with interval-valued intuitionisticfuzzy assessments and incomplete weightsrdquo Information Sci-ences vol 179 no 17 pp 3026ndash3040 2009

[10] S C Onar B Oztaysi I Otay and C Kahraman ldquoMulti-expert wind energy technology selection using interval-valuedintuitionistic fuzzy setsrdquo Energy vol 90 part 1 pp 274ndash2852015

[11] Z Xu and R R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006

[12] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[13] C-P Wei and X Tang ldquoPossibility degree method for rank-ing intuitionistic fuzzy numbersrdquo in Proceedings of the 3rdIEEEWICACM International Conference on Web Intelligenceand Intelligent Agent Technology (WI-IAT rsquo10) pp 142ndash145IEEE Toronto Canada August 2010

[14] Z S Xu and Q L Da ldquoPossibility degree method for rankinginterval numbers and its applicationrdquo Journal of Systems Engi-neering vol 18 pp 67ndash70 2003

[15] Y-M Wang J-B Yang and D-L Xu ldquoInterval weight gen-eration approaches based on consistency test and intervalcomparison matricesrdquo Applied Mathematics and Computationvol 167 no 1 pp 252ndash273 2005

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

[17] F Gao ldquoPossibility degree and comprehensive priority ofinterval numbersrdquo Systems EngineeringmdashTheoryamp Practice vol33 no 8 pp 2033ndash2040 2013

[18] Y-Y Liu and Y-J Lv ldquoA multiple attribute decision makingmethod with interval rough numbers based on the possibilitydegreerdquo in Proceedings of the 10th International Conference onNatural Computation (ICNC rsquo14) pp 407ndash411 IEEE XiamenChina August 2014

[19] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[20] H Yuan and Y Qu ldquoModel for conflict resolution withpreference represented as interval numbersrdquo Proceedings of theMATECWeb of Conferences 2015

[21] G Facchinetti R G Ricci and S Muzzioli ldquoNote on rankingfuzzy triangular numbersrdquo International Journal of IntelligentSystems vol 13 no 7 pp 613ndash622 1998

[22] Q L Da and X W Liu ldquoInterval number linear programmingand its satisfactory solutionrdquo Systems Engineering Theory ampPractice vol 19 pp 3ndash7 1999

[23] C-P Wei and X Tang ldquoPossibility degree method for rankingintuitionistic fuzzy numbersrdquo Journal of Systems Engineeringvol 18 pp 67ndash70 2003

[24] Z S Xu and Q L Da ldquoThe uncertain ow a operatorrdquo Interna-tional Journal of Intelligent Systems vol 17 no 6 pp 569ndash5752002

10 Advances in Fuzzy Systems

[25] Z S Xu and Q L Da ldquoA possibility based method for prioritiesof interval judjment matricesrdquo Chinese Journal of ManagementScience vol 11 pp 63ndash65 2003

[26] D Q Li and Y D Gu ldquomethods for ranking interval numberbased on possibility degreerdquo Journal of Systems Engineering vol23 pp 223ndash226 2008

[27] J B Lan L J Cao and J Lin ldquoMethod for rinking interval num-bers on two-dimensional priority degreerdquo Journal of ChongqingInstitute of Technology Natural Science Edition vol 21 pp 63ndash66 2007

[28] T-Y Chen ldquoInterval-valued intuitionistic fuzzy QUALIFLEXmethod with a likelihood-based comparison approach formultiple criteria decision analysisrdquo Information Sciences vol261 pp 149ndash169 2014

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

[30] G W Wei and X R Wang ldquoSome geometric aggregationoperators on interval-valued intuitionistic fuzzy sets and theirapplication to group decision makingrdquo in Proceedings of theInternational Conference on Computational Intelligence andSecurity (ICCIS rsquo07) pp 495ndash499 Harbin China December2007

[31] Z S Xu and J Chen ldquoOn geometric aggregation over interval-valued intuitionistic fuzzy informationrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) vol 2 pp 466ndash471 Haikou ChinaAugust 2007

[32] Y He H Chen L Zhou J Liu and Z Tao ldquoGeneralizedinterval-valuedAtanassovrsquos intuitionistic fuzzy power operatorsand their application to group decision makingrdquo InternationalJournal of Fuzzy Systems vol 15 no 4 pp 401ndash411 2013

[33] S Wan and J Dong ldquoA possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision mak-ingrdquo Journal of Computer and System Sciences vol 80 no 1 pp237ndash256 2014

[34] F Dammak L Baccour and A M Alimi ldquoThe impact ofcriterion weights techniques in topsis method of multi-criteriadecision making in crisp and intuitionistic fuzzy domainsrdquoin Proceedings of the IEEE International Conference on FuzzySystems (FUZZ-IEEE rsquo15) pp 1ndash8 Istanbul Turkey August 2015

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 7

Table 9 Ranking IVIFSs alternatives using possibility measure (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02111 02161 02540 03200Ranking 4 3 2 1

Table 10 Ranking IVIFSs alternatives using possibility measures(15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02469 02401 02429 02640Ranking 2 4 3 1

presented in Section 32 to rank IVIFS data sets describedin Section 5 It is worth reminding that there are fouralternatives 1198601 1198602 1198603 and 1198604 and four criteria We use theIVIF matrix of alternatives (27) and the following criteriarsquosweight 119908

119895= [013 017 039 031] given in [9]

521 Case 1 Application of Interval-Valued Intuitionistic FuzzyWeighted Geometric (IVIFWG) Operator (13) The possibilitymeasures are applied in two cases In each case an aggregationoperator is also applied to the matrix (27)

Step 1 Compute the comprehensive evaluation of eachinvestment (alternative) using the geometric weighted aver-age operator (13) to aggregate the evaluation of each alterna-tive Thus we transform the IVIFS decision matrix to IVIFsfor each alternative presented as follows

1198631= [04760 05970] [01915 03926]

1198632= [04211 06454] [02260 03546]

1198633= [04057 06112] [01632 02833]

1198634= [05081 06389] [02007 03016]

(37)

Step 2 Each possibility measure presented in Section 32 isapplied to the obtained IVIFNs119863

1119863211986331198634

(a) For the possibility degree (16) the obtained possibilitymatrix is

(

05000 04671 03840 01823

05443 05000 03844 01647

06142 06210 05000 03131

06869 08353 08177 05000

) (38)

Table 9 presents the obtained results and shows thatthe best alternative is 1198604

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05264 05183 04185

04976 05000 04918 03921

05087 05063 05000 04003

05510 05597 05573 05000

) (39)

Table 10 presents the obtained results and shows thatthe best alternative is 1198604

Table 11 Ranking IVFISs alternatives using possibilitymeasure (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02627 02654 02105 02614Ranking 2 1 4 3

Table 12 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02254 02448 02613 02686Ranking 4 3 2 1

Table 13 Ranking order of alternatives for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198601 gt 1198603 gt 1198602 1198604

(16) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(17) 1198602 gt 1198601 gt 1198604 gt 1198603 1198602

(20) 1198604 gt 1198603 gt 1198602 gt 1198601 1198604

(c) For the possibility measure (17) the obtained possi-bility matrix is

(

05000 05075 06501 04944

04925 05000 06635 05285

03365 03499 05000 03401

06599 04715 05056 05000

) (40)

The obtained results are presented in Table 11 showingthat the best alternative is 1198602

(d) For the possibility measure (20) the obtained possi-bility matrix is

(

05000 04446 03856 03740

05554 05000 04570 04256

05430 06144 05000 04778

05222 05744 06260 05000

) (41)

The obtained results are presented in Table 12 showingthat the best alternative is 1198604

Table 13 presents all the obtained results applying differentpossibility methods using the interval-valued intuitionisticfuzzy sets and shows the alternatives ranking results Weremark that the possibility formulas (15) (16) and (20)provide the same best alternative 1198604 However (17) providesthe best alternative 1198602

522 Case 2 Application of Optimal Aggregated Interval-Valued Intuitionistic Fuzzy Sets (14) Using the optimal aggre-gated operator (14) to IVIF decision matrix we obtain

8 Advances in Fuzzy Systems

Table 14 Ranking IVFSs using possibility degree (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02256 02352 02779 02599Ranking 4 3 1 2

Table 15 Ranking IVFSs using possibility measure (15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02369 02322 02557 02629Ranking 3 4 2 1

four interval-valued intuitionistic fuzzy numbers (IVIFNs)representing the alternatives as follows

1205721= [04831 06089] [01850 03800]

1205722= [04400 06520] [02170 03480]

1205723= [04840 06450] [01560 02730]

1205724= [05400 06530] [01950 02950]

(42)

(a) For the possibility measure (16) the obtained possi-bility degree matrix is

(

05000 04717 03595 03758

05366 05000 03682 04177

06329 06403 05000 05611

04358 05818 06008 05000

) (43)

The alternatives weight 119882119894is computed using (26)

and then ranked in a decreasing orderThe results aredisplayed in Table 14 showing that the best alternativethat ranks first is 119860

3

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05188 04308 03938

05000 05000 04120 03750

05525 05525 05000 04630

05168 05693 05693 05000

) (44)

We compute the weight 119882119894of the alternative using

(26) and we rank 119882119894in a decreasing order The

results are shown in Table 15 revealing that the bestalternative is 1198604 which ranks first

(c) For possibility measures (17) the obtained possibilitydegree matrix is

(

05000 05000 05767 04578

05000 05000 06123 05035

03877 04233 05000 03713

06287 04965 05422 05000

) (45)

Table 16 Ranking IVFSs using possibility degree (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02529 02596 02235 02639Ranking 3 2 4 1

Table 17 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02240 02392 02695 02673Ranking 4 3 1 2

Table 18 Alternatives ranking order for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198603 gt 1198601 gt 1198602 1198604

(16) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

(17) 1198604 gt 1198602 gt 1198601 gt 1198603 1198604

(20) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

We compute the weights119882119894of the alternatives using

(26) and we rank 119882119894in a decreasing order The

results are displayed in Table 16 showing that the bestalternative that rank first is 1198604

(d) For possibility measure (20) the obtained possibilitymatrix is

(

05000 04485 03685 03705

05515 05000 04082 04105

05918 06315 05000 05113

04887 05895 06295 05000

) (46)

The obtained results are presented in Table 17 showingthat the best alternative is 1198603

Table 18 presents the results of all applied possibilitymeasures using the interval-valued intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility formulas (15) and (17) provide the same bestalternative 1198604 However (16) and (20) provide the bestalternative 1198603 We note that the latter is the worst alternativeusing (17)

6 Conclusion

In this study we presented different formulas of possibilitymeasures The formulas exist in literature with IFN andIVIFN We also presented an MCDM method from theliterature We gave an illustrative examples for applicationsof different possibility measures and compared their resultsFirst we used an MCDM matrix with intuitionistic fuzzynumbers and then anMCDMmatrixwith IVIFNsThe valuesof the latter are aggregated with an aggregation operator intwo cases In each case a different aggregation operator wasused Thus the appropriate possibility measures are applied

Advances in Fuzzy Systems 9

The results show that the ranked alternatives can be differentfor each possibility measure even though some of thesemeasures have already been demonstrated to be equivalentin the literature

Appendix

Intuitionistic Fuzzy Sets

Intuitionistic fuzzy sets are introduced by Atanassov [16] whodefined a degree of membership 120583 a degree of nonmember-ship ] and a degree of hesitation 120587 of an element 119909 of an IFS

119860 = (120583119860 ]119860) denotes an intuitionistic fuzzy number if 120583

119860

and ]119860are fuzzy numbers with ]

119860le 120583119888119860 where 120583119888

119860denotes

the complement of 120583119860

If119883 is a discourse universe and 119860 a set in119883 then

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883 (A1)

with the conditions 0 le 120583119860(119909) le 1 0 le ]

119860(119909) le 1 0 le

120583119860(119909) + ]

119860(119909) le 1 and 120587

119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) Also for

each 119909 isin 119883 0 le Π119860(119909) le 1

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof this work by grants from General Direction of ScientificResearch (DGRST) Tunisia under the ARUB programThey would like to thank Mr Abdelmajid Dammak for hisproofreading and correction of the English of the paper

References

[1] J H Park I Y Park Y C L Kwun andX Tan ldquoExtension of theTOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environmentrdquo Applied MathematicalModelling vol 35 no 5 pp 2544ndash2556 2011

[2] D G Park Y C Kwun J H Park and I Y Park ldquoCorrelationcoefficient of interval-valued intuitionistic fuzzy sets and itsapplication to multiple attribute group decision making prob-lemsrdquoMathematical and Computer Modelling vol 50 no 9-10pp 1279ndash1293 2009

[3] J Ye ldquoMulticriteria fuzzy decision-making method usingentropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy setsrdquo Applied Mathematical Mod-elling vol 34 no 12 pp 3864ndash3870 2010

[4] H Zhang and L Yu ldquoMADM method based on cross-entropyand extended TOPSIS with interval-valued intuitionistic fuzzysetsrdquo Knowledge-Based Systems vol 30 pp 115ndash120 2012

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

[6] Z-S Xu and J Chen ldquoApproach to groupdecisionmaking basedon interval-valued intuitionistic judgment matricesrdquo SystemsEngineeringmdashTheory amp Practice vol 27 no 4 pp 126ndash133 2007

[7] X Zhang G Yue and Z Teng ldquoPossibility degree of interval-valued intuitionistic fuzzy numbers and its applicationrdquo inProceedings of the International Symposium on InformationProcessing (ISIP rsquo09) pp 33ndash36 Huangshan China 2009

[8] J Wu Q Cao and H Li ldquoAn approach for MADM problemswith interval-valued intuitionistic fuzzy sets based on nonlinearfunctionsrdquo Technological and Economic Development of Econ-omy vol 22 no 3 pp 336ndash356 2016

[9] Z Wang K W Li and W Wang ldquoAn approach to multi-attribute decision making with interval-valued intuitionisticfuzzy assessments and incomplete weightsrdquo Information Sci-ences vol 179 no 17 pp 3026ndash3040 2009

[10] S C Onar B Oztaysi I Otay and C Kahraman ldquoMulti-expert wind energy technology selection using interval-valuedintuitionistic fuzzy setsrdquo Energy vol 90 part 1 pp 274ndash2852015

[11] Z Xu and R R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006

[12] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[13] C-P Wei and X Tang ldquoPossibility degree method for rank-ing intuitionistic fuzzy numbersrdquo in Proceedings of the 3rdIEEEWICACM International Conference on Web Intelligenceand Intelligent Agent Technology (WI-IAT rsquo10) pp 142ndash145IEEE Toronto Canada August 2010

[14] Z S Xu and Q L Da ldquoPossibility degree method for rankinginterval numbers and its applicationrdquo Journal of Systems Engi-neering vol 18 pp 67ndash70 2003

[15] Y-M Wang J-B Yang and D-L Xu ldquoInterval weight gen-eration approaches based on consistency test and intervalcomparison matricesrdquo Applied Mathematics and Computationvol 167 no 1 pp 252ndash273 2005

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

[17] F Gao ldquoPossibility degree and comprehensive priority ofinterval numbersrdquo Systems EngineeringmdashTheoryamp Practice vol33 no 8 pp 2033ndash2040 2013

[18] Y-Y Liu and Y-J Lv ldquoA multiple attribute decision makingmethod with interval rough numbers based on the possibilitydegreerdquo in Proceedings of the 10th International Conference onNatural Computation (ICNC rsquo14) pp 407ndash411 IEEE XiamenChina August 2014

[19] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[20] H Yuan and Y Qu ldquoModel for conflict resolution withpreference represented as interval numbersrdquo Proceedings of theMATECWeb of Conferences 2015

[21] G Facchinetti R G Ricci and S Muzzioli ldquoNote on rankingfuzzy triangular numbersrdquo International Journal of IntelligentSystems vol 13 no 7 pp 613ndash622 1998

[22] Q L Da and X W Liu ldquoInterval number linear programmingand its satisfactory solutionrdquo Systems Engineering Theory ampPractice vol 19 pp 3ndash7 1999

[23] C-P Wei and X Tang ldquoPossibility degree method for rankingintuitionistic fuzzy numbersrdquo Journal of Systems Engineeringvol 18 pp 67ndash70 2003

[24] Z S Xu and Q L Da ldquoThe uncertain ow a operatorrdquo Interna-tional Journal of Intelligent Systems vol 17 no 6 pp 569ndash5752002

10 Advances in Fuzzy Systems

[25] Z S Xu and Q L Da ldquoA possibility based method for prioritiesof interval judjment matricesrdquo Chinese Journal of ManagementScience vol 11 pp 63ndash65 2003

[26] D Q Li and Y D Gu ldquomethods for ranking interval numberbased on possibility degreerdquo Journal of Systems Engineering vol23 pp 223ndash226 2008

[27] J B Lan L J Cao and J Lin ldquoMethod for rinking interval num-bers on two-dimensional priority degreerdquo Journal of ChongqingInstitute of Technology Natural Science Edition vol 21 pp 63ndash66 2007

[28] T-Y Chen ldquoInterval-valued intuitionistic fuzzy QUALIFLEXmethod with a likelihood-based comparison approach formultiple criteria decision analysisrdquo Information Sciences vol261 pp 149ndash169 2014

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

[30] G W Wei and X R Wang ldquoSome geometric aggregationoperators on interval-valued intuitionistic fuzzy sets and theirapplication to group decision makingrdquo in Proceedings of theInternational Conference on Computational Intelligence andSecurity (ICCIS rsquo07) pp 495ndash499 Harbin China December2007

[31] Z S Xu and J Chen ldquoOn geometric aggregation over interval-valued intuitionistic fuzzy informationrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) vol 2 pp 466ndash471 Haikou ChinaAugust 2007

[32] Y He H Chen L Zhou J Liu and Z Tao ldquoGeneralizedinterval-valuedAtanassovrsquos intuitionistic fuzzy power operatorsand their application to group decision makingrdquo InternationalJournal of Fuzzy Systems vol 15 no 4 pp 401ndash411 2013

[33] S Wan and J Dong ldquoA possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision mak-ingrdquo Journal of Computer and System Sciences vol 80 no 1 pp237ndash256 2014

[34] F Dammak L Baccour and A M Alimi ldquoThe impact ofcriterion weights techniques in topsis method of multi-criteriadecision making in crisp and intuitionistic fuzzy domainsrdquoin Proceedings of the IEEE International Conference on FuzzySystems (FUZZ-IEEE rsquo15) pp 1ndash8 Istanbul Turkey August 2015

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 Advances in Fuzzy Systems

Table 14 Ranking IVFSs using possibility degree (16)

Alternatives 1198601 1198602 1198603 1198604

Weights 02256 02352 02779 02599Ranking 4 3 1 2

Table 15 Ranking IVFSs using possibility measure (15)

Alternatives 1198601 1198602 1198603 1198604

Weights 02369 02322 02557 02629Ranking 3 4 2 1

four interval-valued intuitionistic fuzzy numbers (IVIFNs)representing the alternatives as follows

1205721= [04831 06089] [01850 03800]

1205722= [04400 06520] [02170 03480]

1205723= [04840 06450] [01560 02730]

1205724= [05400 06530] [01950 02950]

(42)

(a) For the possibility measure (16) the obtained possi-bility degree matrix is

(

05000 04717 03595 03758

05366 05000 03682 04177

06329 06403 05000 05611

04358 05818 06008 05000

) (43)

The alternatives weight 119882119894is computed using (26)

and then ranked in a decreasing orderThe results aredisplayed in Table 14 showing that the best alternativethat ranks first is 119860

3

(b) For possibility measure (15) the obtained possibilitydegree matrix is

(

05000 05188 04308 03938

05000 05000 04120 03750

05525 05525 05000 04630

05168 05693 05693 05000

) (44)

We compute the weight 119882119894of the alternative using

(26) and we rank 119882119894in a decreasing order The

results are shown in Table 15 revealing that the bestalternative is 1198604 which ranks first

(c) For possibility measures (17) the obtained possibilitydegree matrix is

(

05000 05000 05767 04578

05000 05000 06123 05035

03877 04233 05000 03713

06287 04965 05422 05000

) (45)

Table 16 Ranking IVFSs using possibility degree (17)

Alternatives 1198601 1198602 1198603 1198604

Weights 02529 02596 02235 02639Ranking 3 2 4 1

Table 17 Ranking IVFISs alternatives using possibility measure(20)

Alternatives 1198601 1198602 1198603 1198604

Weights 02240 02392 02695 02673Ranking 4 3 1 2

Table 18 Alternatives ranking order for each possibility measureusing IVIFS

Possibility measures Ranking Best alternative(15) 1198604 gt 1198603 gt 1198601 gt 1198602 1198604

(16) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

(17) 1198604 gt 1198602 gt 1198601 gt 1198603 1198604

(20) 1198603 gt 1198604 gt 1198602 gt 1198601 1198603

We compute the weights119882119894of the alternatives using

(26) and we rank 119882119894in a decreasing order The

results are displayed in Table 16 showing that the bestalternative that rank first is 1198604

(d) For possibility measure (20) the obtained possibilitymatrix is

(

05000 04485 03685 03705

05515 05000 04082 04105

05918 06315 05000 05113

04887 05895 06295 05000

) (46)

The obtained results are presented in Table 17 showingthat the best alternative is 1198603

Table 18 presents the results of all applied possibilitymeasures using the interval-valued intuitionistic fuzzy setsand shows the alternatives ranking results We remark thatthe possibility formulas (15) and (17) provide the same bestalternative 1198604 However (16) and (20) provide the bestalternative 1198603 We note that the latter is the worst alternativeusing (17)

6 Conclusion

In this study we presented different formulas of possibilitymeasures The formulas exist in literature with IFN andIVIFN We also presented an MCDM method from theliterature We gave an illustrative examples for applicationsof different possibility measures and compared their resultsFirst we used an MCDM matrix with intuitionistic fuzzynumbers and then anMCDMmatrixwith IVIFNsThe valuesof the latter are aggregated with an aggregation operator intwo cases In each case a different aggregation operator wasused Thus the appropriate possibility measures are applied

Advances in Fuzzy Systems 9

The results show that the ranked alternatives can be differentfor each possibility measure even though some of thesemeasures have already been demonstrated to be equivalentin the literature

Appendix

Intuitionistic Fuzzy Sets

Intuitionistic fuzzy sets are introduced by Atanassov [16] whodefined a degree of membership 120583 a degree of nonmember-ship ] and a degree of hesitation 120587 of an element 119909 of an IFS

119860 = (120583119860 ]119860) denotes an intuitionistic fuzzy number if 120583

119860

and ]119860are fuzzy numbers with ]

119860le 120583119888119860 where 120583119888

119860denotes

the complement of 120583119860

If119883 is a discourse universe and 119860 a set in119883 then

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883 (A1)

with the conditions 0 le 120583119860(119909) le 1 0 le ]

119860(119909) le 1 0 le

120583119860(119909) + ]

119860(119909) le 1 and 120587

119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) Also for

each 119909 isin 119883 0 le Π119860(119909) le 1

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof this work by grants from General Direction of ScientificResearch (DGRST) Tunisia under the ARUB programThey would like to thank Mr Abdelmajid Dammak for hisproofreading and correction of the English of the paper

References

[1] J H Park I Y Park Y C L Kwun andX Tan ldquoExtension of theTOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environmentrdquo Applied MathematicalModelling vol 35 no 5 pp 2544ndash2556 2011

[2] D G Park Y C Kwun J H Park and I Y Park ldquoCorrelationcoefficient of interval-valued intuitionistic fuzzy sets and itsapplication to multiple attribute group decision making prob-lemsrdquoMathematical and Computer Modelling vol 50 no 9-10pp 1279ndash1293 2009

[3] J Ye ldquoMulticriteria fuzzy decision-making method usingentropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy setsrdquo Applied Mathematical Mod-elling vol 34 no 12 pp 3864ndash3870 2010

[4] H Zhang and L Yu ldquoMADM method based on cross-entropyand extended TOPSIS with interval-valued intuitionistic fuzzysetsrdquo Knowledge-Based Systems vol 30 pp 115ndash120 2012

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

[6] Z-S Xu and J Chen ldquoApproach to groupdecisionmaking basedon interval-valued intuitionistic judgment matricesrdquo SystemsEngineeringmdashTheory amp Practice vol 27 no 4 pp 126ndash133 2007

[7] X Zhang G Yue and Z Teng ldquoPossibility degree of interval-valued intuitionistic fuzzy numbers and its applicationrdquo inProceedings of the International Symposium on InformationProcessing (ISIP rsquo09) pp 33ndash36 Huangshan China 2009

[8] J Wu Q Cao and H Li ldquoAn approach for MADM problemswith interval-valued intuitionistic fuzzy sets based on nonlinearfunctionsrdquo Technological and Economic Development of Econ-omy vol 22 no 3 pp 336ndash356 2016

[9] Z Wang K W Li and W Wang ldquoAn approach to multi-attribute decision making with interval-valued intuitionisticfuzzy assessments and incomplete weightsrdquo Information Sci-ences vol 179 no 17 pp 3026ndash3040 2009

[10] S C Onar B Oztaysi I Otay and C Kahraman ldquoMulti-expert wind energy technology selection using interval-valuedintuitionistic fuzzy setsrdquo Energy vol 90 part 1 pp 274ndash2852015

[11] Z Xu and R R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006

[12] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[13] C-P Wei and X Tang ldquoPossibility degree method for rank-ing intuitionistic fuzzy numbersrdquo in Proceedings of the 3rdIEEEWICACM International Conference on Web Intelligenceand Intelligent Agent Technology (WI-IAT rsquo10) pp 142ndash145IEEE Toronto Canada August 2010

[14] Z S Xu and Q L Da ldquoPossibility degree method for rankinginterval numbers and its applicationrdquo Journal of Systems Engi-neering vol 18 pp 67ndash70 2003

[15] Y-M Wang J-B Yang and D-L Xu ldquoInterval weight gen-eration approaches based on consistency test and intervalcomparison matricesrdquo Applied Mathematics and Computationvol 167 no 1 pp 252ndash273 2005

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

[17] F Gao ldquoPossibility degree and comprehensive priority ofinterval numbersrdquo Systems EngineeringmdashTheoryamp Practice vol33 no 8 pp 2033ndash2040 2013

[18] Y-Y Liu and Y-J Lv ldquoA multiple attribute decision makingmethod with interval rough numbers based on the possibilitydegreerdquo in Proceedings of the 10th International Conference onNatural Computation (ICNC rsquo14) pp 407ndash411 IEEE XiamenChina August 2014

[19] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[20] H Yuan and Y Qu ldquoModel for conflict resolution withpreference represented as interval numbersrdquo Proceedings of theMATECWeb of Conferences 2015

[21] G Facchinetti R G Ricci and S Muzzioli ldquoNote on rankingfuzzy triangular numbersrdquo International Journal of IntelligentSystems vol 13 no 7 pp 613ndash622 1998

[22] Q L Da and X W Liu ldquoInterval number linear programmingand its satisfactory solutionrdquo Systems Engineering Theory ampPractice vol 19 pp 3ndash7 1999

[23] C-P Wei and X Tang ldquoPossibility degree method for rankingintuitionistic fuzzy numbersrdquo Journal of Systems Engineeringvol 18 pp 67ndash70 2003

[24] Z S Xu and Q L Da ldquoThe uncertain ow a operatorrdquo Interna-tional Journal of Intelligent Systems vol 17 no 6 pp 569ndash5752002

10 Advances in Fuzzy Systems

[25] Z S Xu and Q L Da ldquoA possibility based method for prioritiesof interval judjment matricesrdquo Chinese Journal of ManagementScience vol 11 pp 63ndash65 2003

[26] D Q Li and Y D Gu ldquomethods for ranking interval numberbased on possibility degreerdquo Journal of Systems Engineering vol23 pp 223ndash226 2008

[27] J B Lan L J Cao and J Lin ldquoMethod for rinking interval num-bers on two-dimensional priority degreerdquo Journal of ChongqingInstitute of Technology Natural Science Edition vol 21 pp 63ndash66 2007

[28] T-Y Chen ldquoInterval-valued intuitionistic fuzzy QUALIFLEXmethod with a likelihood-based comparison approach formultiple criteria decision analysisrdquo Information Sciences vol261 pp 149ndash169 2014

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

[30] G W Wei and X R Wang ldquoSome geometric aggregationoperators on interval-valued intuitionistic fuzzy sets and theirapplication to group decision makingrdquo in Proceedings of theInternational Conference on Computational Intelligence andSecurity (ICCIS rsquo07) pp 495ndash499 Harbin China December2007

[31] Z S Xu and J Chen ldquoOn geometric aggregation over interval-valued intuitionistic fuzzy informationrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) vol 2 pp 466ndash471 Haikou ChinaAugust 2007

[32] Y He H Chen L Zhou J Liu and Z Tao ldquoGeneralizedinterval-valuedAtanassovrsquos intuitionistic fuzzy power operatorsand their application to group decision makingrdquo InternationalJournal of Fuzzy Systems vol 15 no 4 pp 401ndash411 2013

[33] S Wan and J Dong ldquoA possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision mak-ingrdquo Journal of Computer and System Sciences vol 80 no 1 pp237ndash256 2014

[34] F Dammak L Baccour and A M Alimi ldquoThe impact ofcriterion weights techniques in topsis method of multi-criteriadecision making in crisp and intuitionistic fuzzy domainsrdquoin Proceedings of the IEEE International Conference on FuzzySystems (FUZZ-IEEE rsquo15) pp 1ndash8 Istanbul Turkey August 2015

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 9

The results show that the ranked alternatives can be differentfor each possibility measure even though some of thesemeasures have already been demonstrated to be equivalentin the literature

Appendix

Intuitionistic Fuzzy Sets

Intuitionistic fuzzy sets are introduced by Atanassov [16] whodefined a degree of membership 120583 a degree of nonmember-ship ] and a degree of hesitation 120587 of an element 119909 of an IFS

119860 = (120583119860 ]119860) denotes an intuitionistic fuzzy number if 120583

119860

and ]119860are fuzzy numbers with ]

119860le 120583119888119860 where 120583119888

119860denotes

the complement of 120583119860

If119883 is a discourse universe and 119860 a set in119883 then

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883 (A1)

with the conditions 0 le 120583119860(119909) le 1 0 le ]

119860(119909) le 1 0 le

120583119860(119909) + ]

119860(119909) le 1 and 120587

119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) Also for

each 119909 isin 119883 0 le Π119860(119909) le 1

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof this work by grants from General Direction of ScientificResearch (DGRST) Tunisia under the ARUB programThey would like to thank Mr Abdelmajid Dammak for hisproofreading and correction of the English of the paper

References

[1] J H Park I Y Park Y C L Kwun andX Tan ldquoExtension of theTOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environmentrdquo Applied MathematicalModelling vol 35 no 5 pp 2544ndash2556 2011

[2] D G Park Y C Kwun J H Park and I Y Park ldquoCorrelationcoefficient of interval-valued intuitionistic fuzzy sets and itsapplication to multiple attribute group decision making prob-lemsrdquoMathematical and Computer Modelling vol 50 no 9-10pp 1279ndash1293 2009

[3] J Ye ldquoMulticriteria fuzzy decision-making method usingentropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy setsrdquo Applied Mathematical Mod-elling vol 34 no 12 pp 3864ndash3870 2010

[4] H Zhang and L Yu ldquoMADM method based on cross-entropyand extended TOPSIS with interval-valued intuitionistic fuzzysetsrdquo Knowledge-Based Systems vol 30 pp 115ndash120 2012

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

[6] Z-S Xu and J Chen ldquoApproach to groupdecisionmaking basedon interval-valued intuitionistic judgment matricesrdquo SystemsEngineeringmdashTheory amp Practice vol 27 no 4 pp 126ndash133 2007

[7] X Zhang G Yue and Z Teng ldquoPossibility degree of interval-valued intuitionistic fuzzy numbers and its applicationrdquo inProceedings of the International Symposium on InformationProcessing (ISIP rsquo09) pp 33ndash36 Huangshan China 2009

[8] J Wu Q Cao and H Li ldquoAn approach for MADM problemswith interval-valued intuitionistic fuzzy sets based on nonlinearfunctionsrdquo Technological and Economic Development of Econ-omy vol 22 no 3 pp 336ndash356 2016

[9] Z Wang K W Li and W Wang ldquoAn approach to multi-attribute decision making with interval-valued intuitionisticfuzzy assessments and incomplete weightsrdquo Information Sci-ences vol 179 no 17 pp 3026ndash3040 2009

[10] S C Onar B Oztaysi I Otay and C Kahraman ldquoMulti-expert wind energy technology selection using interval-valuedintuitionistic fuzzy setsrdquo Energy vol 90 part 1 pp 274ndash2852015

[11] Z Xu and R R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006

[12] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[13] C-P Wei and X Tang ldquoPossibility degree method for rank-ing intuitionistic fuzzy numbersrdquo in Proceedings of the 3rdIEEEWICACM International Conference on Web Intelligenceand Intelligent Agent Technology (WI-IAT rsquo10) pp 142ndash145IEEE Toronto Canada August 2010

[14] Z S Xu and Q L Da ldquoPossibility degree method for rankinginterval numbers and its applicationrdquo Journal of Systems Engi-neering vol 18 pp 67ndash70 2003

[15] Y-M Wang J-B Yang and D-L Xu ldquoInterval weight gen-eration approaches based on consistency test and intervalcomparison matricesrdquo Applied Mathematics and Computationvol 167 no 1 pp 252ndash273 2005

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

[17] F Gao ldquoPossibility degree and comprehensive priority ofinterval numbersrdquo Systems EngineeringmdashTheoryamp Practice vol33 no 8 pp 2033ndash2040 2013

[18] Y-Y Liu and Y-J Lv ldquoA multiple attribute decision makingmethod with interval rough numbers based on the possibilitydegreerdquo in Proceedings of the 10th International Conference onNatural Computation (ICNC rsquo14) pp 407ndash411 IEEE XiamenChina August 2014

[19] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978

[20] H Yuan and Y Qu ldquoModel for conflict resolution withpreference represented as interval numbersrdquo Proceedings of theMATECWeb of Conferences 2015

[21] G Facchinetti R G Ricci and S Muzzioli ldquoNote on rankingfuzzy triangular numbersrdquo International Journal of IntelligentSystems vol 13 no 7 pp 613ndash622 1998

[22] Q L Da and X W Liu ldquoInterval number linear programmingand its satisfactory solutionrdquo Systems Engineering Theory ampPractice vol 19 pp 3ndash7 1999

[23] C-P Wei and X Tang ldquoPossibility degree method for rankingintuitionistic fuzzy numbersrdquo Journal of Systems Engineeringvol 18 pp 67ndash70 2003

[24] Z S Xu and Q L Da ldquoThe uncertain ow a operatorrdquo Interna-tional Journal of Intelligent Systems vol 17 no 6 pp 569ndash5752002

10 Advances in Fuzzy Systems

[25] Z S Xu and Q L Da ldquoA possibility based method for prioritiesof interval judjment matricesrdquo Chinese Journal of ManagementScience vol 11 pp 63ndash65 2003

[26] D Q Li and Y D Gu ldquomethods for ranking interval numberbased on possibility degreerdquo Journal of Systems Engineering vol23 pp 223ndash226 2008

[27] J B Lan L J Cao and J Lin ldquoMethod for rinking interval num-bers on two-dimensional priority degreerdquo Journal of ChongqingInstitute of Technology Natural Science Edition vol 21 pp 63ndash66 2007

[28] T-Y Chen ldquoInterval-valued intuitionistic fuzzy QUALIFLEXmethod with a likelihood-based comparison approach formultiple criteria decision analysisrdquo Information Sciences vol261 pp 149ndash169 2014

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

[30] G W Wei and X R Wang ldquoSome geometric aggregationoperators on interval-valued intuitionistic fuzzy sets and theirapplication to group decision makingrdquo in Proceedings of theInternational Conference on Computational Intelligence andSecurity (ICCIS rsquo07) pp 495ndash499 Harbin China December2007

[31] Z S Xu and J Chen ldquoOn geometric aggregation over interval-valued intuitionistic fuzzy informationrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) vol 2 pp 466ndash471 Haikou ChinaAugust 2007

[32] Y He H Chen L Zhou J Liu and Z Tao ldquoGeneralizedinterval-valuedAtanassovrsquos intuitionistic fuzzy power operatorsand their application to group decision makingrdquo InternationalJournal of Fuzzy Systems vol 15 no 4 pp 401ndash411 2013

[33] S Wan and J Dong ldquoA possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision mak-ingrdquo Journal of Computer and System Sciences vol 80 no 1 pp237ndash256 2014

[34] F Dammak L Baccour and A M Alimi ldquoThe impact ofcriterion weights techniques in topsis method of multi-criteriadecision making in crisp and intuitionistic fuzzy domainsrdquoin Proceedings of the IEEE International Conference on FuzzySystems (FUZZ-IEEE rsquo15) pp 1ndash8 Istanbul Turkey August 2015

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 Advances in Fuzzy Systems

[25] Z S Xu and Q L Da ldquoA possibility based method for prioritiesof interval judjment matricesrdquo Chinese Journal of ManagementScience vol 11 pp 63ndash65 2003

[26] D Q Li and Y D Gu ldquomethods for ranking interval numberbased on possibility degreerdquo Journal of Systems Engineering vol23 pp 223ndash226 2008

[27] J B Lan L J Cao and J Lin ldquoMethod for rinking interval num-bers on two-dimensional priority degreerdquo Journal of ChongqingInstitute of Technology Natural Science Edition vol 21 pp 63ndash66 2007

[28] T-Y Chen ldquoInterval-valued intuitionistic fuzzy QUALIFLEXmethod with a likelihood-based comparison approach formultiple criteria decision analysisrdquo Information Sciences vol261 pp 149ndash169 2014

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

[30] G W Wei and X R Wang ldquoSome geometric aggregationoperators on interval-valued intuitionistic fuzzy sets and theirapplication to group decision makingrdquo in Proceedings of theInternational Conference on Computational Intelligence andSecurity (ICCIS rsquo07) pp 495ndash499 Harbin China December2007

[31] Z S Xu and J Chen ldquoOn geometric aggregation over interval-valued intuitionistic fuzzy informationrdquo in Proceedings of the4th International Conference on Fuzzy Systems and KnowledgeDiscovery (FSKD rsquo07) vol 2 pp 466ndash471 Haikou ChinaAugust 2007

[32] Y He H Chen L Zhou J Liu and Z Tao ldquoGeneralizedinterval-valuedAtanassovrsquos intuitionistic fuzzy power operatorsand their application to group decision makingrdquo InternationalJournal of Fuzzy Systems vol 15 no 4 pp 401ndash411 2013

[33] S Wan and J Dong ldquoA possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision mak-ingrdquo Journal of Computer and System Sciences vol 80 no 1 pp237ndash256 2014

[34] F Dammak L Baccour and A M Alimi ldquoThe impact ofcriterion weights techniques in topsis method of multi-criteriadecision making in crisp and intuitionistic fuzzy domainsrdquoin Proceedings of the IEEE International Conference on FuzzySystems (FUZZ-IEEE rsquo15) pp 1ndash8 Istanbul Turkey August 2015

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014