Research ArticleA New Approach to Solve Intuitionistic Fuzzy OptimizationProblem Using Possibility Necessity and Credibility Measures

Dipankar Chakraborty1 Dipak Kumar Jana2 and Tapan Kumar Roy3

1 Department of Mathematics Heritage Institute of Technology Anandapur Kolkata West Bengal 700107 India2Department of Applied Science Haldia Institute of Technology Haldia Purba Midnapur West Bengal 721657 India3 Department of Mathematics Indian Institute of Engineering Science and Technology Shibpur Howrah West Bengal 711103 India

Correspondence should be addressed to Dipak Kumar Jana dipakjanagmailcom

Received 27 February 2014 Revised 23 July 2014 Accepted 9 September 2014 Published 25 September 2014

Academic Editor Wansheng Tang

Copyright copy 2014 Dipankar Chakraborty et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Corresponding to chance constraints real-life possibility necessity and credibility measures on intuitionistic fuzzy set are definedFor the first time the mathematical and graphical representations of different types of measures in trapezoidal intuitionistic fuzzyenvironment are defined in this paperWe have developed intuitionistic fuzzy chance constraintsmodel (CCM) based on possibilityand necessity measures We have also proposed a new method for solving an intuitionistic fuzzy CCM using chance operators Tovalidate the proposed method we have discussed three different approaches to solve the intuitionistic fuzzy linear programming(IFLPP) using possibility necessity and credibility measures Numerical and graphical representations of optimal solutions of thegiven example at different possibility and necessity levels have been discussed

1 Introduction

In the real world some data often provide imprecisionand vagueness at certain level Such vagueness has beenrepresented through fuzzy sets Zadeh [1] first introducedthe fuzzy sets The perception of intuitionistic fuzzy set (IFS)can be analysed as an unconventional approach to define afuzzy set where available information is not adequate for thedefinition of an imprecise concept by means of a usual fuzzyset This IFS was first introduced by Atanassov [2] Manyresearchers have shown their interest in the study of intuition-istic fuzzy setsnumbers [3ndash7] Fuzzy sets are defined by themembership function in all its entirety (cf Pramanik et al[8 9]) but IFS is characterized by amembership function anda nonmembership function so that the sum of both values liesbetween zero and one [10] Esmailzadeh and Esmailzadeh [11]provided newdistance between triangular intuitionistic fuzzynumbers

Recently the IFN has also found its application in fuzzyoptimization Angelov [12] proposed the optimization inan intuitionistic fuzzy environment Dubey and Mehra [13]solved linear programming with triangular intuitionistic

fuzzy number Parvathi andMalathi [14] developed intuition-istic fuzzy simplex method Hussain and Kumar [15] andNagoor Gani and Abbas [16] proposed a method for solvingintuitionistic fuzzy transportation problem Ye [17] discussedexpected value method for intuitionistic trapezoidal fuzzymulticriteria decision-making problems Wan and Dong [18]used possibility degree method for interval-valued intuition-istic fuzzy for decision making

Possibility necessity and credibility measures have asignificant role in fuzzy and intuitionistic fuzzy optimiza-tion Buckley [19] introduced possibility and necessity inoptimization and Jamison and Lodwick [20] developed theconstruction of consistent possibility and necessity measuresDuality in fuzzy linear programming with possibility andnecessity relations has been developed by Ramık [21] Iskan-der [22] suggested an approach for possibility and necessitydominance indices in stochastic fuzzy linear programmingSakawa et al [23] used possibility and necessity to solvefuzzy random bilevel linear programming Pathak et al [24]discussed a possibility and necessity approach to solve fuzzyproduction inventory model for deteriorating items withshortages under the effect of time dependent learning and

Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2014 Article ID 593185 12 pageshttpdxdoiorg1011552014593185

2 International Journal of Engineering Mathematics

forgetting Maity [25] established possibility and necessityrepresentations of fuzzy inequality and its application totwo warehouse production-inventory problem Wu [26] pre-sented possibility and necessity measures fuzzy optimizationproblems based on the embedding theorem Xu and Zhou[27] discussed possibility necessity and credibility measuresfor fuzzy optimization Maity and Maiti [28] developed thepossibility and necessity constraints and their defuzzificationfor multiitem production-inventory scenario via optimalcontrol theory Das et al [29] presented a two-warehousesupply-chain model under possibility necessity and credibil-itymeasures Panda et al [30] proposed a single period inven-torymodelwith imperfect production and stochastic demandunder chance and imprecise constraints Intuitionistic fuzzy-valued possibility and necessitymeasures have been devolvedby Ban [31] using measure theory With our best knowledgehowever none of them introduced chance constraints modelbased on possibility necessity and credibility measures onintuitionistic fuzzy set for membership and nonmembershipfunctions

The rest of this paper is organized into different sectionas follows demonstrating the deduction of our theory and itsapplication In Section 2 we recall some preliminary knowl-edge about intuitionistic fuzzy and its arithmetic operationSection 3 has provided possibility necessity and credibilitymeasures in trapezoidal intuitionistic fuzzy number and itsgraphical representation In Section 4 we have proposedintuitionistic fuzzy chance constraint models based on pos-sibility necessity and credibility measures The solutionmethodology of the proposed models using chance operatorhas been discussed in Section 5 In Section 6 a numericalexample is presented to validate the proposed methodThe numerical and graphical results at different possibilityand necessity levels of the given problems have also beendiscussed here Section 7 summarizes the paper and alsodiscusses about the scope of future work

2 Preliminaries

Definition 1 (intuitionistic fuzzy set [2 10]) Let 119864 be a givenset and let 119860 sub 119864 be a set An IFS 119860

lowast in 119864 is given by 119860lowast

=

⟨119909 120583119860(119909) ]119860(119909)⟩ 119909 isin 119864 where 120583

119860 119864 rarr [0 1] and ]

119860

119864 rarr [0 1] define the degree of membership and the degreeof nonmembership of the element 119909 isin 119864 to 119860 sub 119864 satisfyingthe condition 0 le 120583

119860(119909) + ]

119860(119909) le 1

Definition 2 (intuitionistic fuzzy number [7]) An IFN 119860119868 is

(i) an intuitionistic fuzzy subset on real line

(ii) there exist 119898 isin R such that 120583119860119868(119898) = 1 and

]119860119868(119898) = 0

(iii) convex for the membership function 120583119860119868 that is

120583119860119868(1205821199091+ (1 minus 120582)119909

2) ge min(120583

119860119868(1199091) 120583119860119868(1199092)) 1199091

1199092isin 119877 120582 isin [0 1]

(iv) concave for the nonmembership function ]119860119868 that is

]119860119868(1205821199091+(1minus120582)119909

2) le max(]

119860119868(1199091) ]119860119868(1199092)) 1199091 1199092isin

119877 120582 isin [0 1]

0

1

a9984001 a9984004a1 a2 a3 a4

Figure 1 Membership and nonmembership functions of TIFN

Definition 3 (trapezoidal intuitionistic fuzzy number(TIFN)) Let 1198861015840

1le 1198861le 1198862le 1198863le 1198864le 1198861015840

4 A TIFN 119860

119868 inR written as (119886

1 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) has membership

function (cf Figure 1)

120583119860119868 (119909) =

119909 minus 1198861

1198862minus 1198861

1198861le 119909 le 119886

2

1 1198862le 119909 le 119886

3

1198864minus 119909

1198864minus 1198863

1198863le 119909 le 119886

4

0 otherwise

(1)

and nonmembership function

]119860119868 (119909) =

1198862minus 119909

1198862minus 1198861015840

1

1198861015840

1le 119909 le 119886

2

0 1198862le 119909 le 119886

3

119909 minus 1198863

1198861015840

4minus 1198863

1198863le 119909 le 119886

1015840

4

1 otherwise

(2)

Definition 4 (triangular intuitionistic fuzzy number (TrIFN))Let 11988610158401le 1198861le 1198862le 1198863le 1198861015840

3 A TrIFN 119860

119868 in R written as(1198861 1198862 1198863)(1198861015840

1 1198862 1198861015840

3) has membership function (cf Figure 2)

120583119860119868 (119909) =

119909 minus 1198861

1198862minus 1198861

1198861le 119909 le 119886

2

1 119909 = 1198862

1198863minus 119909

1198863minus 1198862

1198862le 119909 le 119886

3

0 otherwise

(3)

and nonmembership function

]119860119868 (119909) =

1198862minus 119909

1198862minus 1198861015840

1

1198861015840

1le 119909 le 119886

2

0 119909 = 1198862

119909 minus 1198862

1198861015840

3minus 1198862

1198862le 119909 le 119886

1015840

3

1 otherwise

(4)

Definition 5 A positive TIFN 119860119868 is denoted by 119860

119868

=

(1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) where all 119886

119894gt 0 for all 119894 = 1 2

3 4 and 1198861015840

119894gt 0 for 119894 = 1 4

International Journal of Engineering Mathematics 3

0

1

a9984001 a9984003a1 a2 a3

Figure 2 Membership and nonmembership functions of TrIFN

Definition 6 Two TIFN119860119868

= (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and

119861119868

= (1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) are said to be equal if and only

if 1198861= 1198871 1198862= 1198872 1198863= 1198873 1198864= 1198874 11988610158401= 1198871015840

1and 1198861015840

4= 1198871015840

4

Definition 7 Let 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) be two TIFN then

(i) 119860119868 oplus119861119868

= (1198861+ 1198871 1198862+ 1198872 1198863+ 1198873 1198864+ 1198874)(1198861015840

1+ 1198871015840

1 1198862+

1198872 1198863+ 1198873 1198861015840

4+ 1198871015840

4)

(ii) 119896119860119868 = 119896(1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) = (119896119886

1 1198961198862 1198961198863

1198961198864)(1198961198861015840

1 1198961198862 1198961198863 1198961198861015840

4) if 119896 ge 0

(iii) 119896119860119868 = 119896(1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) = (119896119886

4 1198961198863 1198961198862

1198961198861)(1198961198861015840

4 1198961198863 1198961198862 1198961198861015840

1) if 119896 lt 0

(iv) 119860119868 ⊖119861119868

= (1198861minus 1198874 1198862minus 1198873 1198863minus 1198872 1198864minus 1198871)(1198861015840

1minus 1198871015840

4 1198862minus

1198873 1198863minus 1198872 1198861015840

4minus 1198871015840

1)

(v) 119860119868 otimes 119861119868

= (11988611198871 11988621198872 11988631198873 11988641198874)(1198861015840

11198871015840

1 11988621198872 11988631198873 1198861015840

41198871015840

4)

3 Possibility Necessity and CredibilityMeasures of Intuitionistic Fuzzy Number

Definition 8 Let 119860119868 and 119861119868 be two IFN with membership

function 120583119860119868 120583119861119868 and nonmembership function ]

119860119868 ]119861119868

respectively and 119877 is the set of real numbers Then

Pos120583(119860119868

lowast 119861119868

) = sup min (120583119860119868 120583119861119868) 119909 119910 isin 119877 119909 lowast 119910

Pos] (119860119868

lowast 119861119868

) = sup min (]119860119868 ]119861119868) 119909 119910 isin 119877 119909 lowast 119910

Nes120583(119860119868

lowast 119861119868

) = inf max (120583119860119868 120583119861119868) 119909 119910 isin 119877 119909 lowast 119910

Nes] (119860119868

lowast 119861119868

) = inf max (]119860119868 ]119861119868) 119909 119910 isin 119877 119909 lowast 119910

(5)

where the abbreviations Pos120583and Pos] represent possibility

of membership and nonmembership function and Nes120583and

Nes] represent necessity ofmembership and nonmembershipfunction lowast is any of the relations lt gt le ge =

The dual relationship of possibility and necessity gives

Nes120583(119860119868

lowast 119861119868

) = 1 minus Pos120583(119860119868 lowast 119861119868)

Nes] (119860119868

lowast 119861119868

) = 1 minus Pos] (119860119868 lowast 119861119868)

(6)

where 119860119868 lowast 119861119868 represents complement of the event 119860119868 lowast 119861119868

Definition 9 Let 119860119868 be a IFN Then the intuitionistic fuzzymeasures of 119860119868 for membership and nonmembership func-tion are

Me120583119860119868

= 120582Pos120583119860119868

+ (1 minus 120582)Nec120583119860119868

Me] 119860119868

= 120582Pos] 119860119868

+ (1 minus 120582)Nec] 119860119868

(7)

where the abbreviation Me120583and Me] represent measures of

membership and nonmembership functions and 120582 (0 le 120582 le

1) is the optimistic-pessimistic parameter to determine thecombined attitude of a decision maker

If 120582 = 1 then Me120583

= Pos120583 Me] = Pos] it means

the decision maker is optimistic and maximum chance of 119860119868holds

If 120582 = 0 then Me120583

= Nes120583 Me] = Nes] it means

the decision maker is pessimistic and minimal chance of 119860119868holds

If 120582 = 05 then Me120583

= Cr120583 Me] = Cr] where Cr is

the credibility measure it means the decision maker takescompromise attitude

31 Measures of Trapezoidal Intuitionistic Fuzzy Number Let119860119868

= (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

= (1198871 1198872 1198873 1198874)

(1198871015840

1 1198872 1198873 1198871015840

4) be two TIFN FromDefinition 8 the possibilities

of 119860119868 le 119861119868 for membership and nonmembership functions

(cf Figures 3 and 4) are as follows

Pos120583(119860119868

le 119861119868

) =

1 1198862le 1198873

1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

1198861lt 1198874 1198862gt 1198873

0 1198874le 1198861

Pos] (119860119868

le 119861119868

) =

0 1198862le 1198873

1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

1198862gt 1198873 1198871015840

4gt 1198861015840

1

1 1198871015840

4le 1198861015840

1

(8)

4 International Journal of Engineering Mathematics

0

1

b1 b2 b3 b4a2a1 a3 a4

Figure 3 Membership function of TIFN 119860119868 and 119861

119868 and Pos120583(119860119868

le

119861119868

)

0

1

a9984001 a9984004b2 b3 a2 a3b9984001 b9984004

Figure 4 Nonmembership function of TIFN 119860119868 and 119861

119868 andPos](119860

119868

le 119861119868

)

From the Definition 8 the possibilities of 119860119868 ge 119861119868 for

membership andnonmembership function (cf Figures 5 and6) are as follows

Pos120583(119860119868

ge 119861119868

) =

1 1198863ge 1198872

1198864minus 1198871

1198864minus 1198863+ 1198872minus 1198871

1198863lt 1198872 1198864gt 1198871

0 1198864le 1198871

Pos] (119860119868

ge 119861119868

) =

0 1198872le 1198863

1198872minus 1198863

1198872minus 1198871015840

1+ 1198861015840

4minus 1198863

1198872gt 1198863 1198861015840

4gt 1198871015840

1

1 1198871015840

1ge 1198861015840

4

(9)

1

0b1 b2 b3 b4a2 a4a1 a3

Figure 5 Membership function of TIFN 119860119868 and 119861

119868 and Pos120583(119860119868

ge

119861119868

)

0

1

a9984001 a9984004b2 b3a2 a3b9984001 b9984004

Figure 6 Nonmembership function of TIFN 119860119868 and 119861

119868 andPos](119860

119868

ge 119861119868

)

Now by Definition 8 necessity of the event 119860119868 le 119861119868 are

as follows

Nes120583(119860119868

le 119861119868

) = 1 minus Pos120583(119860119868

gt 119861119868

)

=

0 1198873ge 1198861

1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Nes] (119860119868

le 119861119868

) = 1 minus Pos] (119860119868

gt 119861119868

)

=

0 1198872ge 1198861015840

4

1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

1198872lt 1198861015840

4 1198863lt 1198871

1 1198863ge 1198871015840

1

(10)

International Journal of Engineering Mathematics 5

By Definition 8 necessity of the event 119860119868 ge 119861119868 are as follows

Nes120583(119860119868

ge 119861119868

) = 1 minus Pos120583(119860119868

lt 119861119868

)

=

0 1198862le 1198874

1198862minus 1198874

1198862minus 1198861minus 1198874+ 1198873

1198862gt 1198874 1198861lt 1198873

1 1198861ge 1198873

Nes] (119860119868

ge 119861119868

) = 1 minus Pos] (119860119868

lt 119861119868

)

=

0 1198873le 1198861015840

1

1198873minus 1198861015840

1

1198862minus 1198861015840

1minus 1198871015840

4+ 1198873

1198873gt 1198861015840

1 1198862gt 1198871015840

4

1 1198862le 1198871015840

4

(11)

By Definition 9 measures of the event 119860119868 le 119861119868 are as follows

Me120583(119860119868

le 119861119868

)

= 120582Pos120583(119860119868

le 119861119868

) + (1 minus 120582)Nes120583(119860119868

le 119861119868

)

=

0 1198874le 1198861

1205821198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

1198873gt 1198862 1198871lt 1198862

120582 1198873gt 1198862 1198871lt 1198863

120582 + (1 minus 120582)1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Me] (119860119868

le 119861119868

)

= 120582Pos] (119860119868

le 119861119868

) + (1 minus 120582)Nes] (119860119868

le 119861119868

)

=

1 1198871015840

4le 1198861015840

1

1205821198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

+ (1 minus 120582) 1198871015840

4gt 1198861015840

1 1198862gt 1198873

(1 minus 120582) 1198862lt 1198873 1198871015840

1lt 1198863

(1 minus 120582)1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

1198863lt 1198871015840

1 1198861015840

4gt 1198872

0 1198872ge 1198861015840

4

(12)

By Definition 9 measures of the event 119860119868 ge 119861119868 are as follows

Me120583(119860119868

ge 119861119868

)

= 120582Pos120583(119860119868

ge 119861119868

) + (1 minus 120582)Nes120583(119860119868

ge 119861119868

)

=

1 1198873le 1198862

(1 minus 120582)1198862minus 1198874

1198862minus 1198861minus 1198874+ 1198873

+ 120582 1198873gt 1198861 1198874lt 1198862

120582 1198874gt 1198862 1198872lt 1198863

1205821198864minus 1198871

1198864minus 1198863+ 1198872minus 1198871

1198863lt 1198862 1198864gt 1198871

0 1198871ge 1198864

Me] (119860119868

ge 119861119868

)

= 120582Pos] (119860119868

ge 119861119868

) + (1 minus 120582)Nes] (119860119868

ge 119861119868

)

=

0 1198873le 1198861015840

1

(1 minus 120582)1198873minus 1198861015840

1

1198862minus 1198861015840

1minus 1198871015840

4+ 1198873

1198873gt 1198861015840

1 1198862gt 1198871015840

4

(1 minus 120582) 1198862lt 1198871015840

4 1198872lt 1198863

(1 minus 120582) + 1205821198872minus 1198863

1198872minus 1198871015840

1+ 1198861015840

4minus 1198863

1198863lt 1198872 1198861015840

4gt 1198871015840

1

1 1198871015840

1ge 1198861015840

4

(13)

For 120582 = 05

Cr120583(119860119868

le119861119868

)=

0 1198871le 1198861

1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)

1198874gt 1198861 1198862gt 1198873

1

2

1198873gt 1198862 1198871lt 1198863

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Cr] (119860119868

le119861119868

)=

1 1198871015840

4le 1198861015840

1

21198862minus 21198873+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)

1198871015840

4gt 1198861015840

1 1198862gt 1198873

1

2

1198862lt 1198873 1198871015840

1gt 1198863

1198861015840

4minus 1198872

2 (1198861015840

4minus 1198863minus 1198872+ 1198871015840

1)

1198863lt 1198871015840

1 1198872lt 1198861015840

4

0 1198872ge 1198861015840

4

Cr120583(119860119868

ge119861119868

)=

1 1198873le 1198862

21198862minus 21198874+ 1198873minus 1198861

2 (1198862minus 1198861minus 1198874+ 1198873)

1198873gt 1198861 1198862gt 1198874

1

2

1198874gt 1198862 1198872lt 1198863

1198864minus 1198871

2 (1198864minus 1198863+ 1198872minus 1198871)

1198872gt 1198863 1198864gt 1198871

0 1198871ge 1198864

Cr] (119860119868

ge119861119868

)=

0 1198871015840

3le 1198861015840

1

1198873minus 1198861015840

1

2 (1198862minus 1198861015840

1minus 1198871015840

4+ 1198873)

1198873gt 1198861015840

1 1198862gt 1198871015840

4

1

2

1198862lt 1198871015840

4 1198872lt 1198863

21198872minus 21198863+ 1198861015840

4minus 1198871015840

1

2 (1198872minus 1198871015840

1+ 1198861015840

4minus 1198863)

1198872gt 1198863 1198871015840

1lt 1198861015840

4

1 1198871015840

1ge 1198861015840

4

(14)

Lemma 10 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Pos120583(119860119868

le 119861119868

) ge 120572 Pos] (119860119868

le 119861119868

) le 120573

lArrrArr1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

ge 120572

1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

le 120573

(15)

6 International Journal of Engineering Mathematics

Proof Let us consider

Pos120583(119860119868

le 119861119868

) ge 120572 Pos] (119860119868

ge 119861119868

) le 120573 (16)

Now from (8)

Pos120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

ge 120572

Pos] (119860119868

le 119861119868

) le 120573 lArrrArr1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

le 120573

(17)

Note Pos120583(119860119868

le 119909) ge 120572 and Pos](119860119868

le 119909) le 120573 hArr (119909 minus

1198861)(1198862minus 1198861) ge 120572 and (119886

2minus 119909)(119886

2minus 1198861015840

1) le 120573

Lemma 11 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Nes120583(119860119868

le 119861119868

) ge 120572 Nes] (119860119868

le 119861119868

) le 120573

lArrrArr1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

ge 120572

1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

le 120573

(18)

Proof Let us consider

Nes120583(119860119868

le 119861119868

) ge 120572 Nes] (119860119868

le 119861119868

) le 120573 (19)

Now from (10)

Nes120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

ge 120572

Nes] (119860119868

le 119861119868

) le 120573 lArrrArr1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

le 120573

(20)

Note Nes120583(119860119868

le 119909) ge 120572 and Nes](119860119868

le 119909) le 120573 hArr (119909 minus

1198863)(1198864minus 1198863) ge 120572 and (119886

1015840

4minus 119909)(119886

1015840

4minus 1198863) le 120573

Lemma 12 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Cr120583(119860119868

le 119861119868

) ge 120572 Cr] (119860119868

le 119861119868

) le 120573

lArrrArr1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)ge 120572

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)ge 120572

21198862minus 21198871015840

3+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)le 120573

1198861015840

4minus 1198872

2 (1198861015840

4minus 1198863minus 1198872+ 1198871015840

1)le 120573

(21)

Proof Let us consider

Cr120583(119860119868

le 119861119868

) ge 120572 Cr] (119860119868

le 119861119868

) le 120573 (22)

Now from (14)

Cr120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)ge 120572

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)ge 120572

Cr] (119860119868

le 119861119868

) le 120573 lArrrArr21198862minus 21198871015840

3+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)le 120573

1198872minus 1198861015840

4

2 (1198872minus 1198871015840

1minus 1198861015840

4+ 1198863)le 120573

(23)

Note Cr120583(119860119868

le 119909) ge 120572 Cr](119860119868

le 119909) ge 120573 hArr (119909 minus 1198861)2(1198862minus

1198861) ge 120572 (119886

4minus21198863+119909)2(119886

4minus1198863) ge 120572 and (2119886

2minus119909minus119886

1015840

1)2(1198862minus

1198861015840

1) le 120573 (1198861015840

4minus 119909)2(119886

1015840

4minus 1198863) le 120573

4 Intuitionistic Fuzzy CCM

The chance operator is actually taken as possibility or neces-sity or credibility measures We can use chance operatorto transform the intuitionistic fuzzy problem into crispproblem which is called as CCM [27] A general single-objective mathematical programming problem with intu-itionistic fuzzy parameter should have the following form

Max 119891 (119909 120585119868

)

subject to 119892119894(119909 120585119868

) le 119868

119894 119894 = 1 2 119899

119909 ge 0

(24)

where 119909 is the decision vector 120585119868 and 119868

119894are intuitionistic

fuzzy parameters 119891(119909 120585119868) is an imprecise objective functionand 119892

119894(119909 120585119868

) are constraints function for 119894 = 1 2 119899The general chance-constraints model for problem (24) is

as followsMax 119891

1+ 1198912

subject to Ch120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Ch] 119891 (119909 120585119868

) ge 1198912 le 120573

Ch120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Ch] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(25)

The abbreviations Ch120583and Ch] represent chance operator

(ie Pos or Nec measure) for membership and nonmem-bership functions 120572 120573 120582

119894 and 120595

119894are the predetermined

confidence levels such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

International Journal of Engineering Mathematics 7

41 Intuitionistic Fuzzy CCM Based on Possibility MeasureThe CCM based on possibility measure is as follows

Max 1198911+ 1198912

Subject to Pos120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Pos] 119891 (119909 120585119868

) ge 1198912 le 120573

Pos120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Pos] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(26)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 13 A solution 119909lowast of the problem (26) satisfies

Pos120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Pos](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution at (120582119894 120595119894) possibility

levels 119894 = 1 2 119899

Definition 14 A feasible solution at (120582119894 120595119894) possibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (26)

if and only if there exists no other feasible solution at(120582119894 120595119894) possibility levels such that Pos

120583119891(119909 120585

119868

) ge 120572 andPos]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

42 Intuitionistic Fuzzy CCM Based on Necessity MeasureThe CCM based on necessity measure is as follows

Max 1198911+ 1198912

Subject to Nes120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Nes] 119891 (119909 120585119868

) ge 1198912 le 120573

Nes120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Nes] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(27)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 15 A solution 119909lowast of the problem (27) satisfies

Nes120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Nes](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) necessity

levels 119894 = 1 2 119899

Definition 16 A feasible solution at (120582119894 120595119894) necessity levels

119909lowast is said to be (120572 120573) efficient solution for problem (27)

if and only if there exists no other feasible solution at(120582119894 120595119894) necessity levels such that Nes

120583119891(119909 120585

119868

) ge 120572 andNes]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

43 Intuitionistic Fuzzy CCM Based on Credibility MeasureThe CCM based on credibility measure is as follows

Max 1198911+ 1198912

Subject to Cr120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Cr] 119891 (119909 120585119868

) ge 1198912 le 120573

Cr120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Cr] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(28)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 17 A solution 119909lowast of the problem (28) satisfies

Cr120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Cr](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) credibility

levels 119894 = 1 2 119899

Definition 18 A feasible solution at (120582119894 120595119894) credibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (28)

if and only if there exists no other feasible solution at(120582119894 120595119894) credibility levels such that Cr

120583119891(119909 120585

119868

) ge 120572 andCr]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

5 Proposed Method to Solve IFLPPUsing Chance Operator

To solve intuitionistic fuzzy CCM based on possibility ornecessity or credibility measures we propose the followingmethod

Step 1 Apply chance operator possibilitynecessitycredi-bility in intuitionistic fuzzy programming (24) Problem (24)can be converted into following problem

Max 1198911+ 1198912

Subject to Pos120583119891 (119909I

119868

) ge 1198911 ge 120572

or Nes120583119891 (119909I

119868

) ge 1198911 ge 120572

or Cr120583119891 (119909I

119868

) ge 1198911 ge 120572

(29)

Pos] 119891 (119909I119868

) ge 1198912 le 120573

or Nes] 119891 (119909I119868

) ge 1198912 le 120573

or Cr] 119891 (119909I119868

) ge 1198912 le 120573

(30)

8 International Journal of Engineering Mathematics

Pos120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Nes120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Cr120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

(31)

Pos] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Nes] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Cr] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

(32)

119909 ge 0 for 119894 = 1 2 119899 (33)

0 le 120582119894+ 120595119894le 1 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

(34)

where (120582119894 120595119894) and (120572 120573) are the predefined confidence levels

Step 2 Using Lemmas 10 11 andor Lemma 12 the aboveproblem in Step 1 can also be written as

Max 1198911+ 1198912

Subject to 1198911+ 1198912ge 119885

(31) ndash (34)

(35)

where 119885 is obtained by applying Lemmas 10 11 andorLemma 12 in (30) and (29)

Step 3 The above problem is equivalent to

Max 119885

Subject to (31) ndash (34)

(36)

Step 4 Crisp programming problem obtained in Step 2 canbe solved using any well-known method to get the optimalsolution

6 Numerical Example

Let us consider the following intuitionistic fuzzy mathemati-cal programming problem as

Maximize 120585119868

11199091oplus 120585119868

21199092

subject to 120585119868

31199091oplus 120585119868

41199092⪯ 120585119868

5

120585119868

61199091oplus 120585119868

71199092⪯ 120585119868

8

1199091 1199092ge 0

(37)

where 120585119868

1= (5 6 7 8)(4 6 7 9) 120585119868

2= (4 5 6 7)(3 5 6 8)

120585119868

3= (1 2 3 4)(05 2 3 6) 120585119868

4= (2 3 4 5)(1 3 4 6) 120585119868

5=

(6 7 8 9)(5 7 8 10) 1205851198686= (3 4 5 6)(2 4 5 7) 120585119868

7= (1 2 3

4)(0 2 3 4) and 120585119868

8= (10 11 12 14)(9 11 12 16)

61 Intuitionistic Fuzzy CCM Based on Possibility MeasureNowby using Step 2 of themethod explained in Section 4 and

Start Applychance operator

Convertinto crisp

Writeequivalent Solve

Figure 7 Flow chart of the proposed algorithm

Lemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (37) problem (37) is con-verted into the following crisp programming problem

Maximize (15 minus 120572 + 2120573) 1199091+ (13 minus 120572 + 2120573) 119909

2

subject to (1 + 1205821) 1199091+ (2 + 120582

1) 1199092le 9 minus 120582

1

(2 minus 151205951) 1199091+ (3 minus 2120595

1) 1199092le 8 + 2120595

1

(3 + 1205822) 1199091+ (1 + 120582

2) 1199092le 14 minus 2120582

2

(4 minus 21205952) 1199091+ (2 minus 2120595

2) 1199092le 12 + 4120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(38)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and different possibility levels we get differentoptimal solutions Optimal solution of (38) at differentpossibility levels (in Figure 7) are presented in Table 1 FromTable 1 we can observe that maximum value (= 7309) can beobtained at (120582

1= 040 120595

1= 035) and (120582

2= 030 120595

2= 040)

possibility levels

62 Intuitionistic Fuzzy CCM Based on Necessity MeasureNowby using Step 2 of themethod explained in Section 4 andLemma 11 if we apply the necessity measure in (37) problem(37) is converted into following crisp programming problem

Maximize (10 minus 120572 + 2120573) 1199091+ (8 minus 120572 + 2120573) 119909

2

subject to (3 + 21205821) 1199091+ (4 + 2120582

1) 1199092le 6 + 120582

1

(5 minus 21205951) 1199091+ (6 minus 2120595

1) 1199092le 7 minus 2120595

1

(5 + 1205822) 1199091+ (3 + 120582

2) 1199092le 6 + 120582

2

(7 minus 21205952) 1199091+ (4 minus 120595

2) 1199092le 11 minus 2120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(39)

Solving the above crisp linear programming problem forefficient levels (120572 = 06 120573 = 04) and different necessitylevels we get different optimal solutions Optimal solutionsof (39) at different necessity levels (in Figures 8 and 9) arepresented in Table 2 From Table 2 we can observed that at(1205821= 035 120595

1= 045) and (120582

2= 035 120595

2= 045) the decision

maker will get the maximum value = 1298

International Journal of Engineering Mathematics 9

Table 1 Optimal solution of (38) at different possibility levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)030 030 035 035 119909

1= 341 119909

2= 137 7009

030 035 035 040 1199091= 329 119909

2= 166 7214

040 035 030 040 1199091= 343 119909

2= 157 7309

035 040 040 030 1199091= 310 119909

2= 19 722

040 045 040 045 1199091= 316 119909

2= 173 7105

050 045 045 05 1199091= 316 119909

2= 150 6793

045 050 050 040 1199091= 297 119909

2= 173 6802

060 045 045 06 1199091= 329 119909

2= 120 6593

050 050 050 050 1199091= 303 119909

2= 157 6700

055 040 055 040 1199091= 297 119909

2= 150 6510

Table 2 Optimal solution of (39) at different necessity levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)035 030 030 035 119909

1= 091 119909

2= 043 1293

045 030 040 035 1199091= 090 119909

2= 045 1289

050 035 050 035 1199091= 087 119909

2= 047 1286

060 040 060 040 1199091= 085 119909

2= 050 1284

070 020 070 020 1199091= 087 119909

2= 045 1271

060 030 060 020 1199091= 087 119909

2= 047 1279

065 035 075 020 1199091= 084 119909

2= 050 1275

070 010 070 010 1199091= 089 119909

2= 043 1267

050 050 050 050 1199091= 085 119909

2= 051 1294

035 045 035 045 1199091= 088 119909

2= 048 1298

Table 3 Input data for IFTP

1198631

119868

1198632

119868

1198633

119868 Availability (119886119894

119868

)

1198781

119868 (2 4 6 7) (1 4 6 9) (4 6 7 8) (3 6 7 9) (3 7 9 12) (2 7 9 13) (4 6 8 9) (2 6 8 10)1198782

119868 (1 3 4 6) (05 3 4 7) (3 5 6 7) (2 5 6 9) (2 6 7 11) (1 6 7 12) (0 05 1 2) (0 05 1 5)1198783

119868 (3 4 5 8) (2 4 5 10) (1 2 3 4) (05 2 3 5) (2 4 5 10) (1 4 5 11) (8 95 10 11) (65 95 10 11)Demand (119887

119895

119868

) (6 7 8 10) (5 7 8 12) (4 5 6 9) (3 5 6 11) (2 4 5 7) (05 4 5 8)

646668707274

(030 030)

(035

035

)

(035

040

)

(030

040

) (055 040)

(050 050)

(060 045)

(045 050)

(050 045)(0

40 0

30)

(040

045

)

(045

050

)

(050

040

)

(045

060

)

(045

045

)

(030

030

)(040 045)

(035 040)

(040 035)

(030 035)(1205821 1205951

)(1205822 120595

2 )

flowast

Figure 8 Optimal solution at different possibility levels

10 International Journal of Engineering Mathematics

1265127

1275128

1285129

129513

(035 030)

(050

035

)

(060

040

)

(070

020

)

(060

020

)

(075

020

)

(070

010

)

(050

050

)

(035

045

)

(035 045)

(050 050)

(070 010)

(065 035)

(060 030)

(070 020)

(060 040)

(050 035)

(045 030)

(040

035

)

(030

035

)

(1205821 1205951)

(1205822 120595

2 )

flowast

Figure 9 Optimal solution at different necessity levels

7 Intuitionistic Fuzzy TransportationProblem Based on Possibility Measure

Let us consider the following intuitionistic fuzzy transporta-tion problem (IFTP) (in Table 3)

Above transportation problem is a balanced transporta-tion problem as ⨁3

119894=1119886119868

119894= ⨁3

119895=1119868

119895 The above IFTP can be

written as

Minimize (2 4 6 7) (1 4 6 9) 11990911

oplus (4 6 7 8) (3 6 7 9) 11990912

oplus (3 7 9 12) (2 7 9 13) 11990913

oplus (1 3 4 6) (05 3 4 7) 11990921

oplus (3 5 6 7) (2 5 6 9) 11990922

oplus (2 6 7 11) (1 6 7 12) 11990923

oplus (3 4 5 8) (2 4 5 10) 11990931

oplus (1 2 3 4) (05 2 3 5) 11990932

oplus (2 4 5 10) (1 4 5 11) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (4 6 8 9) (2 6 8 10)

11990921

+ 11990922

+ 11990923

le (0 05 1 2) (0 05 1 5)

11990931

+ 11990932

+ 11990933

le (8 95 10 11) (65 95 10 11)

11990911

+ 11990921

+ 11990931

ge (6 7 8 10) (5 7 8 12)

11990912

+ 11990922

+ 11990932

ge (4 5 6 9) (3 5 6 11)

11990913

+ 11990923

+ 11990933

ge (2 4 5 7) (05 4 5 8)

119909119894119895ge 0 forall119894 119895

(40)

Nowby using Step 2 of themethod explained in Section 4 andLemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (40) problem (40) isconverted into following crisp programming problem

Minimize (6 + 2120572 minus 3120573) 11990911

+ (10 + 2120572 minus 2120573) 11990912

+ (10 + 4120572 minus 5120573) 11990913

+ (4 + 2120572 minus 25120573) 11990921

+ (8 + 2120572 minus 3120573) 11990922

+ (8 + 4120572 minus 5120573) 11990923

+ (7 + 120572 minus 5120573) 11990931

+ (3 + 120572 minus 15120573) 11990932

+ (6 + 2120572 minus 3120573) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (17 minus 1205821+ 21205951)

11990921

+ 11990922

+ 11990923

le (3 minus 1205822+ 41205952)

11990931

+ 11990932

+ 11990933

le (21 minus 1205823+ 1205953)

11990911

+ 11990921

+ 11990931

ge (13 + 1205824minus 21205954)

11990912

+ 11990922

+ 11990932

ge (9 + 1205825minus 21205955)

11990913

+ 11990923

+ 11990933

ge (6 + 21205826minus 35120595

6)

119909119894119895ge 0 for 119894 119895 = 1 2 3 0 le 120572 + 120573 le 1

0 le 120582119896+ 120595119896le 1 for 119896 = 1 2 6

(41)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and possibility levels (120582119894= 05 120595

119894= 05) for

119894 = 1 2 6 using Lingo-110 we get 11990911

= 075 11990912

= 011990913

= 0 11990921

= 45 11990922

= 0 11990923

= 0 11990931

= 725 11990932

= 85 and

International Journal of Engineering Mathematics 11

11990933

= 525 Now the minimum intuitionstic fuzzy optimalcost is

119888119868

= (7225 109 136 2035) (4562 109 136 24625) (42)

8 Discussion

Intuitionistic fuzzy sets being a generalization of fuzzy setsgive us an additional possibility to represent imperfect knowl-edge making it possible to describe many real problems ina more adequate way So in this paper we have developedthe possibility and necessity measures on intuitionistic fuzzyset Here we have presented first time the mathematicalrepresentation of different types of measures in intuitionisticfuzzy environments and some graphical representations ofthemare depictedWehave also developed the theoretical cal-culation on possibility necessity and credibility measures fordefuzzify intuitionistic fuzzy linear programming problemusing chance operators To validate the proposed methodwe have discussed three different approaches to defuzzifythe intuitionistic fuzzy relations using possibility necessityand credibility measures Using chance operator we canconvert a problem under imprecise models to correspondingcrisp models At different levels of possibility necessity andcredibility we have achieved different optimal solution Anumerical example is presented and solved using LINGO-110to illustrate the proposed approaches The proposed methodcan be applied for multiobjective multiitem transportationproblem This method can be also extended to be appliedinto different types of optimization problem namely optimalcontrol and solid transportation problems

Conflict of Interests

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

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov Intuitionistic Fuzzy Sets VII ITKRrsquos SessionSofia Bulgarian 1983

[3] S K De and S S Sana ldquoA multi-periods production-inventorymodel with capacity constraints for multi-manufacturersmdashaglobal optimality in intuitionistic fuzzy environmentrdquo AppliedMathematics andComputation vol 242 no 1 pp 825ndash841 2014

[4] HGargM Rani S P Sharma andYVishwakarma ldquoIntuition-istic fuzzy optimization technique for solving multi-objectivereliability optimization problems in interval environmentrdquoExpert Systems with Applications vol 41 no 7 pp 3157ndash31672014

[5] J Wu and Y Liu ldquoAn approach for multiple attribute groupdecision making problems with interval-valued intuitionistictrapezoidal fuzzy numbersrdquoComputers and Industrial Engineer-ing vol 66 no 2 pp 311ndash324 2013

[6] A Nagoorgani and K Ponnalagu ldquoA new approach on solv-ing intuitionistic fuzzy linear programming problemrdquo AppliedMathematical Sciences vol 6 no 70 pp 3467ndash3474 2012

[7] G S Mahapatra and T K Roy ldquoIntuitionistic fuzzy numberand its arithmetic operation with application on system failurerdquoJournal of Uncertain Systems vol 7 no 2 pp 92ndash107 2013

[8] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[9] S Pramanik D K Jana and M Maiti ldquoA multi objectivesolid transportation problem in fuzzy Bi-fuzzy environmentvia genetic algorithmrdquo International Journal of Advanced Oper-ations Management vol 6 no 1 pp 4ndash26 2014

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

[11] M Esmailzadeh and M Esmailzadeh ldquoNew distance betweentriangular intuitionistic fuzzy numbersrdquo Advances in Computa-tional Mathematics and Its Applications vol 2 no 3 pp 3ndash102013

[12] P P Angelov ldquoOptimization in an intuitionistic fuzzy environ-mentrdquo Fuzzy Sets and Systems vol 86 no 3 pp 299ndash306 1997

[13] D Dubey and A Mehra ldquoLinear programming with triangularintuitionistic fuzzy numberrdquo Advances in Intelligent SystemsResearch vol 1 no 1 pp 563ndash569 2011

[14] R Parvathi and C Malathi ldquoIntuitionistic fuzzy simplexmethodrdquo International Journal of Computer Applications vol48 no 6 pp 39ndash48 2012

[15] R J Hussain and S P Kumar ldquoAlgorithmic approach forsolving intuitionistic fuzzy transportation problemrdquo AppliedMathematical Sciences vol 6 no 77-80 pp 3981ndash3989 2012

[16] ANagoorGani and S Abbas ldquoSolving intuitionstic fuzzy trans-portation problem using zero suffix algorithmrdquo InternationalJournal of Mathematics Sciences amp Enggineering Applicationsvol 6 pp 73ndash82 2012

[17] J Ye ldquoExpected value method for intuitionistic trapezoidalfuzzy multicriteria decision-making problemsrdquo Expert Systemswith Applications vol 38 no 9 pp 11730ndash11734 2011

[18] 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 2013

[19] J J Buckley ldquoPossibility and necessity in optimizationrdquo FuzzySets and Systems vol 25 no 1 pp 1ndash13 1988

[20] K D Jamison and W A Lodwick ldquoThe construction ofconsistent possibility and necessity measuresrdquo Fuzzy Sets andSystems vol 132 no 1 pp 1ndash10 2002

[21] J Ramık ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006

[22] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[23] M Sakawa H Katagiri and T Matsui ldquoFuzzy random bilevellinear programming through expectation optimization usingpossibility and necessityrdquo International Journal of MachineLearning and Cybernetics vol 3 no 3 pp 183ndash192 2012

[24] S Pathak S Kar and S Sarkar ldquoFuzzy production inventorymodel for deteriorating items with shortages under the effect oftime dependent learning and forgetting a possibilitynecessityapproachrdquo OPSEARCH vol 50 no 2 pp 149ndash181 2013

[25] K Maity ldquoPossibility and necessity representations of fuzzyinequality and its application to two warehouse production-inventory problemrdquo Applied Mathematical Modelling vol 35no 3 pp 1252ndash1263 2011

12 International Journal of Engineering Mathematics

[26] H-C Wu ldquoFuzzy optimization problems based on the embed-ding theorem and possibility and necessity measuresrdquo Mathe-matical and Computer Modelling vol 40 no 3-4 pp 329ndash3362004

[27] J Xu and X Zhou Fuzzy-Like Multiple Objective DecesionMaking Springer 2010

[28] K Maity and M Maiti ldquoPossibility and necessity constraintsand their defuzzificationmdasha multi-item production-inventoryscenario via optimal control theoryrdquo European Journal of Oper-ational Research vol 177 no 2 pp 882ndash896 2007

[29] B Das K Maity and M Maiti ldquoA two warehouse supply-chain model under possibilitynecessitycredibility measuresrdquoMathematical andComputerModelling vol 46 no 3-4 pp 398ndash409 2007

[30] D Panda S Kar K Maity and M Maiti ldquoA single periodinventory model with imperfect production and stochasticdemand under chance and imprecise constraintsrdquo EuropeanJournal of Operational Research vol 188 no 1 pp 121ndash139 2008

[31] A I Ban ldquoIntuitionistic fuzzy-valued possibility and necessitymeasuresrdquo in Proceedings of the 8th International Conference onIntuitionistic Fuzzy Sets vol 10 pp 1ndash7 Varna Bulgaria 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Engineering Mathematics

forgetting Maity [25] established possibility and necessityrepresentations of fuzzy inequality and its application totwo warehouse production-inventory problem Wu [26] pre-sented possibility and necessity measures fuzzy optimizationproblems based on the embedding theorem Xu and Zhou[27] discussed possibility necessity and credibility measuresfor fuzzy optimization Maity and Maiti [28] developed thepossibility and necessity constraints and their defuzzificationfor multiitem production-inventory scenario via optimalcontrol theory Das et al [29] presented a two-warehousesupply-chain model under possibility necessity and credibil-itymeasures Panda et al [30] proposed a single period inven-torymodelwith imperfect production and stochastic demandunder chance and imprecise constraints Intuitionistic fuzzy-valued possibility and necessitymeasures have been devolvedby Ban [31] using measure theory With our best knowledgehowever none of them introduced chance constraints modelbased on possibility necessity and credibility measures onintuitionistic fuzzy set for membership and nonmembershipfunctions

The rest of this paper is organized into different sectionas follows demonstrating the deduction of our theory and itsapplication In Section 2 we recall some preliminary knowl-edge about intuitionistic fuzzy and its arithmetic operationSection 3 has provided possibility necessity and credibilitymeasures in trapezoidal intuitionistic fuzzy number and itsgraphical representation In Section 4 we have proposedintuitionistic fuzzy chance constraint models based on pos-sibility necessity and credibility measures The solutionmethodology of the proposed models using chance operatorhas been discussed in Section 5 In Section 6 a numericalexample is presented to validate the proposed methodThe numerical and graphical results at different possibilityand necessity levels of the given problems have also beendiscussed here Section 7 summarizes the paper and alsodiscusses about the scope of future work

2 Preliminaries

Definition 1 (intuitionistic fuzzy set [2 10]) Let 119864 be a givenset and let 119860 sub 119864 be a set An IFS 119860

lowast in 119864 is given by 119860lowast

=

⟨119909 120583119860(119909) ]119860(119909)⟩ 119909 isin 119864 where 120583

119860 119864 rarr [0 1] and ]

119860

119864 rarr [0 1] define the degree of membership and the degreeof nonmembership of the element 119909 isin 119864 to 119860 sub 119864 satisfyingthe condition 0 le 120583

119860(119909) + ]

119860(119909) le 1

Definition 2 (intuitionistic fuzzy number [7]) An IFN 119860119868 is

(i) an intuitionistic fuzzy subset on real line

(ii) there exist 119898 isin R such that 120583119860119868(119898) = 1 and

]119860119868(119898) = 0

(iii) convex for the membership function 120583119860119868 that is

120583119860119868(1205821199091+ (1 minus 120582)119909

2) ge min(120583

119860119868(1199091) 120583119860119868(1199092)) 1199091

1199092isin 119877 120582 isin [0 1]

(iv) concave for the nonmembership function ]119860119868 that is

]119860119868(1205821199091+(1minus120582)119909

2) le max(]

119860119868(1199091) ]119860119868(1199092)) 1199091 1199092isin

119877 120582 isin [0 1]

0

1

a9984001 a9984004a1 a2 a3 a4

Figure 1 Membership and nonmembership functions of TIFN

Definition 3 (trapezoidal intuitionistic fuzzy number(TIFN)) Let 1198861015840

1le 1198861le 1198862le 1198863le 1198864le 1198861015840

4 A TIFN 119860

119868 inR written as (119886

1 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) has membership

function (cf Figure 1)

120583119860119868 (119909) =

119909 minus 1198861

1198862minus 1198861

1198861le 119909 le 119886

2

1 1198862le 119909 le 119886

3

1198864minus 119909

1198864minus 1198863

1198863le 119909 le 119886

4

0 otherwise

(1)

and nonmembership function

]119860119868 (119909) =

1198862minus 119909

1198862minus 1198861015840

1

1198861015840

1le 119909 le 119886

2

0 1198862le 119909 le 119886

3

119909 minus 1198863

1198861015840

4minus 1198863

1198863le 119909 le 119886

1015840

4

1 otherwise

(2)

Definition 4 (triangular intuitionistic fuzzy number (TrIFN))Let 11988610158401le 1198861le 1198862le 1198863le 1198861015840

3 A TrIFN 119860

119868 in R written as(1198861 1198862 1198863)(1198861015840

1 1198862 1198861015840

3) has membership function (cf Figure 2)

120583119860119868 (119909) =

119909 minus 1198861

1198862minus 1198861

1198861le 119909 le 119886

2

1 119909 = 1198862

1198863minus 119909

1198863minus 1198862

1198862le 119909 le 119886

3

0 otherwise

(3)

and nonmembership function

]119860119868 (119909) =

1198862minus 119909

1198862minus 1198861015840

1

1198861015840

1le 119909 le 119886

2

0 119909 = 1198862

119909 minus 1198862

1198861015840

3minus 1198862

1198862le 119909 le 119886

1015840

3

1 otherwise

(4)

Definition 5 A positive TIFN 119860119868 is denoted by 119860

119868

=

(1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) where all 119886

119894gt 0 for all 119894 = 1 2

3 4 and 1198861015840

119894gt 0 for 119894 = 1 4

International Journal of Engineering Mathematics 3

0

1

a9984001 a9984003a1 a2 a3

Figure 2 Membership and nonmembership functions of TrIFN

Definition 6 Two TIFN119860119868

= (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and

119861119868

= (1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) are said to be equal if and only

if 1198861= 1198871 1198862= 1198872 1198863= 1198873 1198864= 1198874 11988610158401= 1198871015840

1and 1198861015840

4= 1198871015840

4

Definition 7 Let 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) be two TIFN then

(i) 119860119868 oplus119861119868

= (1198861+ 1198871 1198862+ 1198872 1198863+ 1198873 1198864+ 1198874)(1198861015840

1+ 1198871015840

1 1198862+

1198872 1198863+ 1198873 1198861015840

4+ 1198871015840

4)

(ii) 119896119860119868 = 119896(1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) = (119896119886

1 1198961198862 1198961198863

1198961198864)(1198961198861015840

1 1198961198862 1198961198863 1198961198861015840

4) if 119896 ge 0

(iii) 119896119860119868 = 119896(1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) = (119896119886

4 1198961198863 1198961198862

1198961198861)(1198961198861015840

4 1198961198863 1198961198862 1198961198861015840

1) if 119896 lt 0

(iv) 119860119868 ⊖119861119868

= (1198861minus 1198874 1198862minus 1198873 1198863minus 1198872 1198864minus 1198871)(1198861015840

1minus 1198871015840

4 1198862minus

1198873 1198863minus 1198872 1198861015840

4minus 1198871015840

1)

(v) 119860119868 otimes 119861119868

= (11988611198871 11988621198872 11988631198873 11988641198874)(1198861015840

11198871015840

1 11988621198872 11988631198873 1198861015840

41198871015840

4)

3 Possibility Necessity and CredibilityMeasures of Intuitionistic Fuzzy Number

Definition 8 Let 119860119868 and 119861119868 be two IFN with membership

function 120583119860119868 120583119861119868 and nonmembership function ]

119860119868 ]119861119868

respectively and 119877 is the set of real numbers Then

Pos120583(119860119868

lowast 119861119868

) = sup min (120583119860119868 120583119861119868) 119909 119910 isin 119877 119909 lowast 119910

Pos] (119860119868

lowast 119861119868

) = sup min (]119860119868 ]119861119868) 119909 119910 isin 119877 119909 lowast 119910

Nes120583(119860119868

lowast 119861119868

) = inf max (120583119860119868 120583119861119868) 119909 119910 isin 119877 119909 lowast 119910

Nes] (119860119868

lowast 119861119868

) = inf max (]119860119868 ]119861119868) 119909 119910 isin 119877 119909 lowast 119910

(5)

where the abbreviations Pos120583and Pos] represent possibility

of membership and nonmembership function and Nes120583and

Nes] represent necessity ofmembership and nonmembershipfunction lowast is any of the relations lt gt le ge =

The dual relationship of possibility and necessity gives

Nes120583(119860119868

lowast 119861119868

) = 1 minus Pos120583(119860119868 lowast 119861119868)

Nes] (119860119868

lowast 119861119868

) = 1 minus Pos] (119860119868 lowast 119861119868)

(6)

where 119860119868 lowast 119861119868 represents complement of the event 119860119868 lowast 119861119868

Definition 9 Let 119860119868 be a IFN Then the intuitionistic fuzzymeasures of 119860119868 for membership and nonmembership func-tion are

Me120583119860119868

= 120582Pos120583119860119868

+ (1 minus 120582)Nec120583119860119868

Me] 119860119868

= 120582Pos] 119860119868

+ (1 minus 120582)Nec] 119860119868

(7)

where the abbreviation Me120583and Me] represent measures of

membership and nonmembership functions and 120582 (0 le 120582 le

1) is the optimistic-pessimistic parameter to determine thecombined attitude of a decision maker

If 120582 = 1 then Me120583

= Pos120583 Me] = Pos] it means

the decision maker is optimistic and maximum chance of 119860119868holds

If 120582 = 0 then Me120583

= Nes120583 Me] = Nes] it means

the decision maker is pessimistic and minimal chance of 119860119868holds

If 120582 = 05 then Me120583

= Cr120583 Me] = Cr] where Cr is

the credibility measure it means the decision maker takescompromise attitude

31 Measures of Trapezoidal Intuitionistic Fuzzy Number Let119860119868

= (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

= (1198871 1198872 1198873 1198874)

(1198871015840

1 1198872 1198873 1198871015840

4) be two TIFN FromDefinition 8 the possibilities

of 119860119868 le 119861119868 for membership and nonmembership functions

(cf Figures 3 and 4) are as follows

Pos120583(119860119868

le 119861119868

) =

1 1198862le 1198873

1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

1198861lt 1198874 1198862gt 1198873

0 1198874le 1198861

Pos] (119860119868

le 119861119868

) =

0 1198862le 1198873

1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

1198862gt 1198873 1198871015840

4gt 1198861015840

1

1 1198871015840

4le 1198861015840

1

(8)

4 International Journal of Engineering Mathematics

0

1

b1 b2 b3 b4a2a1 a3 a4

Figure 3 Membership function of TIFN 119860119868 and 119861

119868 and Pos120583(119860119868

le

119861119868

)

0

1

a9984001 a9984004b2 b3 a2 a3b9984001 b9984004

Figure 4 Nonmembership function of TIFN 119860119868 and 119861

119868 andPos](119860

119868

le 119861119868

)

From the Definition 8 the possibilities of 119860119868 ge 119861119868 for

membership andnonmembership function (cf Figures 5 and6) are as follows

Pos120583(119860119868

ge 119861119868

) =

1 1198863ge 1198872

1198864minus 1198871

1198864minus 1198863+ 1198872minus 1198871

1198863lt 1198872 1198864gt 1198871

0 1198864le 1198871

Pos] (119860119868

ge 119861119868

) =

0 1198872le 1198863

1198872minus 1198863

1198872minus 1198871015840

1+ 1198861015840

4minus 1198863

1198872gt 1198863 1198861015840

4gt 1198871015840

1

1 1198871015840

1ge 1198861015840

4

(9)

1

0b1 b2 b3 b4a2 a4a1 a3

Figure 5 Membership function of TIFN 119860119868 and 119861

119868 and Pos120583(119860119868

ge

119861119868

)

0

1

a9984001 a9984004b2 b3a2 a3b9984001 b9984004

Figure 6 Nonmembership function of TIFN 119860119868 and 119861

119868 andPos](119860

119868

ge 119861119868

)

Now by Definition 8 necessity of the event 119860119868 le 119861119868 are

as follows

Nes120583(119860119868

le 119861119868

) = 1 minus Pos120583(119860119868

gt 119861119868

)

=

0 1198873ge 1198861

1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Nes] (119860119868

le 119861119868

) = 1 minus Pos] (119860119868

gt 119861119868

)

=

0 1198872ge 1198861015840

4

1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

1198872lt 1198861015840

4 1198863lt 1198871

1 1198863ge 1198871015840

1

(10)

International Journal of Engineering Mathematics 5

By Definition 8 necessity of the event 119860119868 ge 119861119868 are as follows

Nes120583(119860119868

ge 119861119868

) = 1 minus Pos120583(119860119868

lt 119861119868

)

=

0 1198862le 1198874

1198862minus 1198874

1198862minus 1198861minus 1198874+ 1198873

1198862gt 1198874 1198861lt 1198873

1 1198861ge 1198873

Nes] (119860119868

ge 119861119868

) = 1 minus Pos] (119860119868

lt 119861119868

)

=

0 1198873le 1198861015840

1

1198873minus 1198861015840

1

1198862minus 1198861015840

1minus 1198871015840

4+ 1198873

1198873gt 1198861015840

1 1198862gt 1198871015840

4

1 1198862le 1198871015840

4

(11)

By Definition 9 measures of the event 119860119868 le 119861119868 are as follows

Me120583(119860119868

le 119861119868

)

= 120582Pos120583(119860119868

le 119861119868

) + (1 minus 120582)Nes120583(119860119868

le 119861119868

)

=

0 1198874le 1198861

1205821198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

1198873gt 1198862 1198871lt 1198862

120582 1198873gt 1198862 1198871lt 1198863

120582 + (1 minus 120582)1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Me] (119860119868

le 119861119868

)

= 120582Pos] (119860119868

le 119861119868

) + (1 minus 120582)Nes] (119860119868

le 119861119868

)

=

1 1198871015840

4le 1198861015840

1

1205821198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

+ (1 minus 120582) 1198871015840

4gt 1198861015840

1 1198862gt 1198873

(1 minus 120582) 1198862lt 1198873 1198871015840

1lt 1198863

(1 minus 120582)1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

1198863lt 1198871015840

1 1198861015840

4gt 1198872

0 1198872ge 1198861015840

4

(12)

By Definition 9 measures of the event 119860119868 ge 119861119868 are as follows

Me120583(119860119868

ge 119861119868

)

= 120582Pos120583(119860119868

ge 119861119868

) + (1 minus 120582)Nes120583(119860119868

ge 119861119868

)

=

1 1198873le 1198862

(1 minus 120582)1198862minus 1198874

1198862minus 1198861minus 1198874+ 1198873

+ 120582 1198873gt 1198861 1198874lt 1198862

120582 1198874gt 1198862 1198872lt 1198863

1205821198864minus 1198871

1198864minus 1198863+ 1198872minus 1198871

1198863lt 1198862 1198864gt 1198871

0 1198871ge 1198864

Me] (119860119868

ge 119861119868

)

= 120582Pos] (119860119868

ge 119861119868

) + (1 minus 120582)Nes] (119860119868

ge 119861119868

)

=

0 1198873le 1198861015840

1

(1 minus 120582)1198873minus 1198861015840

1

1198862minus 1198861015840

1minus 1198871015840

4+ 1198873

1198873gt 1198861015840

1 1198862gt 1198871015840

4

(1 minus 120582) 1198862lt 1198871015840

4 1198872lt 1198863

(1 minus 120582) + 1205821198872minus 1198863

1198872minus 1198871015840

1+ 1198861015840

4minus 1198863

1198863lt 1198872 1198861015840

4gt 1198871015840

1

1 1198871015840

1ge 1198861015840

4

(13)

For 120582 = 05

Cr120583(119860119868

le119861119868

)=

0 1198871le 1198861

1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)

1198874gt 1198861 1198862gt 1198873

1

2

1198873gt 1198862 1198871lt 1198863

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Cr] (119860119868

le119861119868

)=

1 1198871015840

4le 1198861015840

1

21198862minus 21198873+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)

1198871015840

4gt 1198861015840

1 1198862gt 1198873

1

2

1198862lt 1198873 1198871015840

1gt 1198863

1198861015840

4minus 1198872

2 (1198861015840

4minus 1198863minus 1198872+ 1198871015840

1)

1198863lt 1198871015840

1 1198872lt 1198861015840

4

0 1198872ge 1198861015840

4

Cr120583(119860119868

ge119861119868

)=

1 1198873le 1198862

21198862minus 21198874+ 1198873minus 1198861

2 (1198862minus 1198861minus 1198874+ 1198873)

1198873gt 1198861 1198862gt 1198874

1

2

1198874gt 1198862 1198872lt 1198863

1198864minus 1198871

2 (1198864minus 1198863+ 1198872minus 1198871)

1198872gt 1198863 1198864gt 1198871

0 1198871ge 1198864

Cr] (119860119868

ge119861119868

)=

0 1198871015840

3le 1198861015840

1

1198873minus 1198861015840

1

2 (1198862minus 1198861015840

1minus 1198871015840

4+ 1198873)

1198873gt 1198861015840

1 1198862gt 1198871015840

4

1

2

1198862lt 1198871015840

4 1198872lt 1198863

21198872minus 21198863+ 1198861015840

4minus 1198871015840

1

2 (1198872minus 1198871015840

1+ 1198861015840

4minus 1198863)

1198872gt 1198863 1198871015840

1lt 1198861015840

4

1 1198871015840

1ge 1198861015840

4

(14)

Lemma 10 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Pos120583(119860119868

le 119861119868

) ge 120572 Pos] (119860119868

le 119861119868

) le 120573

lArrrArr1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

ge 120572

1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

le 120573

(15)

6 International Journal of Engineering Mathematics

Proof Let us consider

Pos120583(119860119868

le 119861119868

) ge 120572 Pos] (119860119868

ge 119861119868

) le 120573 (16)

Now from (8)

Pos120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

ge 120572

Pos] (119860119868

le 119861119868

) le 120573 lArrrArr1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

le 120573

(17)

Note Pos120583(119860119868

le 119909) ge 120572 and Pos](119860119868

le 119909) le 120573 hArr (119909 minus

1198861)(1198862minus 1198861) ge 120572 and (119886

2minus 119909)(119886

2minus 1198861015840

1) le 120573

Lemma 11 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Nes120583(119860119868

le 119861119868

) ge 120572 Nes] (119860119868

le 119861119868

) le 120573

lArrrArr1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

ge 120572

1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

le 120573

(18)

Proof Let us consider

Nes120583(119860119868

le 119861119868

) ge 120572 Nes] (119860119868

le 119861119868

) le 120573 (19)

Now from (10)

Nes120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

ge 120572

Nes] (119860119868

le 119861119868

) le 120573 lArrrArr1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

le 120573

(20)

Note Nes120583(119860119868

le 119909) ge 120572 and Nes](119860119868

le 119909) le 120573 hArr (119909 minus

1198863)(1198864minus 1198863) ge 120572 and (119886

1015840

4minus 119909)(119886

1015840

4minus 1198863) le 120573

Lemma 12 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Cr120583(119860119868

le 119861119868

) ge 120572 Cr] (119860119868

le 119861119868

) le 120573

lArrrArr1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)ge 120572

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)ge 120572

21198862minus 21198871015840

3+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)le 120573

1198861015840

4minus 1198872

2 (1198861015840

4minus 1198863minus 1198872+ 1198871015840

1)le 120573

(21)

Proof Let us consider

Cr120583(119860119868

le 119861119868

) ge 120572 Cr] (119860119868

le 119861119868

) le 120573 (22)

Now from (14)

Cr120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)ge 120572

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)ge 120572

Cr] (119860119868

le 119861119868

) le 120573 lArrrArr21198862minus 21198871015840

3+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)le 120573

1198872minus 1198861015840

4

2 (1198872minus 1198871015840

1minus 1198861015840

4+ 1198863)le 120573

(23)

Note Cr120583(119860119868

le 119909) ge 120572 Cr](119860119868

le 119909) ge 120573 hArr (119909 minus 1198861)2(1198862minus

1198861) ge 120572 (119886

4minus21198863+119909)2(119886

4minus1198863) ge 120572 and (2119886

2minus119909minus119886

1015840

1)2(1198862minus

1198861015840

1) le 120573 (1198861015840

4minus 119909)2(119886

1015840

4minus 1198863) le 120573

4 Intuitionistic Fuzzy CCM

The chance operator is actually taken as possibility or neces-sity or credibility measures We can use chance operatorto transform the intuitionistic fuzzy problem into crispproblem which is called as CCM [27] A general single-objective mathematical programming problem with intu-itionistic fuzzy parameter should have the following form

Max 119891 (119909 120585119868

)

subject to 119892119894(119909 120585119868

) le 119868

119894 119894 = 1 2 119899

119909 ge 0

(24)

where 119909 is the decision vector 120585119868 and 119868

119894are intuitionistic

fuzzy parameters 119891(119909 120585119868) is an imprecise objective functionand 119892

119894(119909 120585119868

) are constraints function for 119894 = 1 2 119899The general chance-constraints model for problem (24) is

as followsMax 119891

1+ 1198912

subject to Ch120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Ch] 119891 (119909 120585119868

) ge 1198912 le 120573

Ch120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Ch] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(25)

The abbreviations Ch120583and Ch] represent chance operator

(ie Pos or Nec measure) for membership and nonmem-bership functions 120572 120573 120582

119894 and 120595

119894are the predetermined

confidence levels such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

International Journal of Engineering Mathematics 7

41 Intuitionistic Fuzzy CCM Based on Possibility MeasureThe CCM based on possibility measure is as follows

Max 1198911+ 1198912

Subject to Pos120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Pos] 119891 (119909 120585119868

) ge 1198912 le 120573

Pos120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Pos] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(26)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 13 A solution 119909lowast of the problem (26) satisfies

Pos120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Pos](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution at (120582119894 120595119894) possibility

levels 119894 = 1 2 119899

Definition 14 A feasible solution at (120582119894 120595119894) possibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (26)

if and only if there exists no other feasible solution at(120582119894 120595119894) possibility levels such that Pos

120583119891(119909 120585

119868

) ge 120572 andPos]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

42 Intuitionistic Fuzzy CCM Based on Necessity MeasureThe CCM based on necessity measure is as follows

Max 1198911+ 1198912

Subject to Nes120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Nes] 119891 (119909 120585119868

) ge 1198912 le 120573

Nes120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Nes] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(27)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 15 A solution 119909lowast of the problem (27) satisfies

Nes120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Nes](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) necessity

levels 119894 = 1 2 119899

Definition 16 A feasible solution at (120582119894 120595119894) necessity levels

119909lowast is said to be (120572 120573) efficient solution for problem (27)

if and only if there exists no other feasible solution at(120582119894 120595119894) necessity levels such that Nes

120583119891(119909 120585

119868

) ge 120572 andNes]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

43 Intuitionistic Fuzzy CCM Based on Credibility MeasureThe CCM based on credibility measure is as follows

Max 1198911+ 1198912

Subject to Cr120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Cr] 119891 (119909 120585119868

) ge 1198912 le 120573

Cr120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Cr] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(28)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 17 A solution 119909lowast of the problem (28) satisfies

Cr120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Cr](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) credibility

levels 119894 = 1 2 119899

Definition 18 A feasible solution at (120582119894 120595119894) credibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (28)

if and only if there exists no other feasible solution at(120582119894 120595119894) credibility levels such that Cr

120583119891(119909 120585

119868

) ge 120572 andCr]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

5 Proposed Method to Solve IFLPPUsing Chance Operator

To solve intuitionistic fuzzy CCM based on possibility ornecessity or credibility measures we propose the followingmethod

Step 1 Apply chance operator possibilitynecessitycredi-bility in intuitionistic fuzzy programming (24) Problem (24)can be converted into following problem

Max 1198911+ 1198912

Subject to Pos120583119891 (119909I

119868

) ge 1198911 ge 120572

or Nes120583119891 (119909I

119868

) ge 1198911 ge 120572

or Cr120583119891 (119909I

119868

) ge 1198911 ge 120572

(29)

Pos] 119891 (119909I119868

) ge 1198912 le 120573

or Nes] 119891 (119909I119868

) ge 1198912 le 120573

or Cr] 119891 (119909I119868

) ge 1198912 le 120573

(30)

8 International Journal of Engineering Mathematics

Pos120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Nes120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Cr120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

(31)

Pos] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Nes] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Cr] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

(32)

119909 ge 0 for 119894 = 1 2 119899 (33)

0 le 120582119894+ 120595119894le 1 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

(34)

where (120582119894 120595119894) and (120572 120573) are the predefined confidence levels

Step 2 Using Lemmas 10 11 andor Lemma 12 the aboveproblem in Step 1 can also be written as

Max 1198911+ 1198912

Subject to 1198911+ 1198912ge 119885

(31) ndash (34)

(35)

where 119885 is obtained by applying Lemmas 10 11 andorLemma 12 in (30) and (29)

Step 3 The above problem is equivalent to

Max 119885

Subject to (31) ndash (34)

(36)

Step 4 Crisp programming problem obtained in Step 2 canbe solved using any well-known method to get the optimalsolution

6 Numerical Example

Let us consider the following intuitionistic fuzzy mathemati-cal programming problem as

Maximize 120585119868

11199091oplus 120585119868

21199092

subject to 120585119868

31199091oplus 120585119868

41199092⪯ 120585119868

5

120585119868

61199091oplus 120585119868

71199092⪯ 120585119868

8

1199091 1199092ge 0

(37)

where 120585119868

1= (5 6 7 8)(4 6 7 9) 120585119868

2= (4 5 6 7)(3 5 6 8)

120585119868

3= (1 2 3 4)(05 2 3 6) 120585119868

4= (2 3 4 5)(1 3 4 6) 120585119868

5=

(6 7 8 9)(5 7 8 10) 1205851198686= (3 4 5 6)(2 4 5 7) 120585119868

7= (1 2 3

4)(0 2 3 4) and 120585119868

8= (10 11 12 14)(9 11 12 16)

61 Intuitionistic Fuzzy CCM Based on Possibility MeasureNowby using Step 2 of themethod explained in Section 4 and

Start Applychance operator

Convertinto crisp

Writeequivalent Solve

Figure 7 Flow chart of the proposed algorithm

Lemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (37) problem (37) is con-verted into the following crisp programming problem

Maximize (15 minus 120572 + 2120573) 1199091+ (13 minus 120572 + 2120573) 119909

2

subject to (1 + 1205821) 1199091+ (2 + 120582

1) 1199092le 9 minus 120582

1

(2 minus 151205951) 1199091+ (3 minus 2120595

1) 1199092le 8 + 2120595

1

(3 + 1205822) 1199091+ (1 + 120582

2) 1199092le 14 minus 2120582

2

(4 minus 21205952) 1199091+ (2 minus 2120595

2) 1199092le 12 + 4120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(38)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and different possibility levels we get differentoptimal solutions Optimal solution of (38) at differentpossibility levels (in Figure 7) are presented in Table 1 FromTable 1 we can observe that maximum value (= 7309) can beobtained at (120582

1= 040 120595

1= 035) and (120582

2= 030 120595

2= 040)

possibility levels

62 Intuitionistic Fuzzy CCM Based on Necessity MeasureNowby using Step 2 of themethod explained in Section 4 andLemma 11 if we apply the necessity measure in (37) problem(37) is converted into following crisp programming problem

Maximize (10 minus 120572 + 2120573) 1199091+ (8 minus 120572 + 2120573) 119909

2

subject to (3 + 21205821) 1199091+ (4 + 2120582

1) 1199092le 6 + 120582

1

(5 minus 21205951) 1199091+ (6 minus 2120595

1) 1199092le 7 minus 2120595

1

(5 + 1205822) 1199091+ (3 + 120582

2) 1199092le 6 + 120582

2

(7 minus 21205952) 1199091+ (4 minus 120595

2) 1199092le 11 minus 2120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(39)

Solving the above crisp linear programming problem forefficient levels (120572 = 06 120573 = 04) and different necessitylevels we get different optimal solutions Optimal solutionsof (39) at different necessity levels (in Figures 8 and 9) arepresented in Table 2 From Table 2 we can observed that at(1205821= 035 120595

1= 045) and (120582

2= 035 120595

2= 045) the decision

maker will get the maximum value = 1298

International Journal of Engineering Mathematics 9

Table 1 Optimal solution of (38) at different possibility levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)030 030 035 035 119909

1= 341 119909

2= 137 7009

030 035 035 040 1199091= 329 119909

2= 166 7214

040 035 030 040 1199091= 343 119909

2= 157 7309

035 040 040 030 1199091= 310 119909

2= 19 722

040 045 040 045 1199091= 316 119909

2= 173 7105

050 045 045 05 1199091= 316 119909

2= 150 6793

045 050 050 040 1199091= 297 119909

2= 173 6802

060 045 045 06 1199091= 329 119909

2= 120 6593

050 050 050 050 1199091= 303 119909

2= 157 6700

055 040 055 040 1199091= 297 119909

2= 150 6510

Table 2 Optimal solution of (39) at different necessity levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)035 030 030 035 119909

1= 091 119909

2= 043 1293

045 030 040 035 1199091= 090 119909

2= 045 1289

050 035 050 035 1199091= 087 119909

2= 047 1286

060 040 060 040 1199091= 085 119909

2= 050 1284

070 020 070 020 1199091= 087 119909

2= 045 1271

060 030 060 020 1199091= 087 119909

2= 047 1279

065 035 075 020 1199091= 084 119909

2= 050 1275

070 010 070 010 1199091= 089 119909

2= 043 1267

050 050 050 050 1199091= 085 119909

2= 051 1294

035 045 035 045 1199091= 088 119909

2= 048 1298

Table 3 Input data for IFTP

1198631

119868

1198632

119868

1198633

119868 Availability (119886119894

119868

)

1198781

119868 (2 4 6 7) (1 4 6 9) (4 6 7 8) (3 6 7 9) (3 7 9 12) (2 7 9 13) (4 6 8 9) (2 6 8 10)1198782

119868 (1 3 4 6) (05 3 4 7) (3 5 6 7) (2 5 6 9) (2 6 7 11) (1 6 7 12) (0 05 1 2) (0 05 1 5)1198783

119868 (3 4 5 8) (2 4 5 10) (1 2 3 4) (05 2 3 5) (2 4 5 10) (1 4 5 11) (8 95 10 11) (65 95 10 11)Demand (119887

119895

119868

) (6 7 8 10) (5 7 8 12) (4 5 6 9) (3 5 6 11) (2 4 5 7) (05 4 5 8)

646668707274

(030 030)

(035

035

)

(035

040

)

(030

040

) (055 040)

(050 050)

(060 045)

(045 050)

(050 045)(0

40 0

30)

(040

045

)

(045

050

)

(050

040

)

(045

060

)

(045

045

)

(030

030

)(040 045)

(035 040)

(040 035)

(030 035)(1205821 1205951

)(1205822 120595

2 )

flowast

Figure 8 Optimal solution at different possibility levels

10 International Journal of Engineering Mathematics

1265127

1275128

1285129

129513

(035 030)

(050

035

)

(060

040

)

(070

020

)

(060

020

)

(075

020

)

(070

010

)

(050

050

)

(035

045

)

(035 045)

(050 050)

(070 010)

(065 035)

(060 030)

(070 020)

(060 040)

(050 035)

(045 030)

(040

035

)

(030

035

)

(1205821 1205951)

(1205822 120595

2 )

flowast

Figure 9 Optimal solution at different necessity levels

7 Intuitionistic Fuzzy TransportationProblem Based on Possibility Measure

Let us consider the following intuitionistic fuzzy transporta-tion problem (IFTP) (in Table 3)

Above transportation problem is a balanced transporta-tion problem as ⨁3

119894=1119886119868

119894= ⨁3

119895=1119868

119895 The above IFTP can be

written as

Minimize (2 4 6 7) (1 4 6 9) 11990911

oplus (4 6 7 8) (3 6 7 9) 11990912

oplus (3 7 9 12) (2 7 9 13) 11990913

oplus (1 3 4 6) (05 3 4 7) 11990921

oplus (3 5 6 7) (2 5 6 9) 11990922

oplus (2 6 7 11) (1 6 7 12) 11990923

oplus (3 4 5 8) (2 4 5 10) 11990931

oplus (1 2 3 4) (05 2 3 5) 11990932

oplus (2 4 5 10) (1 4 5 11) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (4 6 8 9) (2 6 8 10)

11990921

+ 11990922

+ 11990923

le (0 05 1 2) (0 05 1 5)

11990931

+ 11990932

+ 11990933

le (8 95 10 11) (65 95 10 11)

11990911

+ 11990921

+ 11990931

ge (6 7 8 10) (5 7 8 12)

11990912

+ 11990922

+ 11990932

ge (4 5 6 9) (3 5 6 11)

11990913

+ 11990923

+ 11990933

ge (2 4 5 7) (05 4 5 8)

119909119894119895ge 0 forall119894 119895

(40)

Nowby using Step 2 of themethod explained in Section 4 andLemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (40) problem (40) isconverted into following crisp programming problem

Minimize (6 + 2120572 minus 3120573) 11990911

+ (10 + 2120572 minus 2120573) 11990912

+ (10 + 4120572 minus 5120573) 11990913

+ (4 + 2120572 minus 25120573) 11990921

+ (8 + 2120572 minus 3120573) 11990922

+ (8 + 4120572 minus 5120573) 11990923

+ (7 + 120572 minus 5120573) 11990931

+ (3 + 120572 minus 15120573) 11990932

+ (6 + 2120572 minus 3120573) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (17 minus 1205821+ 21205951)

11990921

+ 11990922

+ 11990923

le (3 minus 1205822+ 41205952)

11990931

+ 11990932

+ 11990933

le (21 minus 1205823+ 1205953)

11990911

+ 11990921

+ 11990931

ge (13 + 1205824minus 21205954)

11990912

+ 11990922

+ 11990932

ge (9 + 1205825minus 21205955)

11990913

+ 11990923

+ 11990933

ge (6 + 21205826minus 35120595

6)

119909119894119895ge 0 for 119894 119895 = 1 2 3 0 le 120572 + 120573 le 1

0 le 120582119896+ 120595119896le 1 for 119896 = 1 2 6

(41)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and possibility levels (120582119894= 05 120595

119894= 05) for

119894 = 1 2 6 using Lingo-110 we get 11990911

= 075 11990912

= 011990913

= 0 11990921

= 45 11990922

= 0 11990923

= 0 11990931

= 725 11990932

= 85 and

International Journal of Engineering Mathematics 11

11990933

= 525 Now the minimum intuitionstic fuzzy optimalcost is

119888119868

= (7225 109 136 2035) (4562 109 136 24625) (42)

8 Discussion

Intuitionistic fuzzy sets being a generalization of fuzzy setsgive us an additional possibility to represent imperfect knowl-edge making it possible to describe many real problems ina more adequate way So in this paper we have developedthe possibility and necessity measures on intuitionistic fuzzyset Here we have presented first time the mathematicalrepresentation of different types of measures in intuitionisticfuzzy environments and some graphical representations ofthemare depictedWehave also developed the theoretical cal-culation on possibility necessity and credibility measures fordefuzzify intuitionistic fuzzy linear programming problemusing chance operators To validate the proposed methodwe have discussed three different approaches to defuzzifythe intuitionistic fuzzy relations using possibility necessityand credibility measures Using chance operator we canconvert a problem under imprecise models to correspondingcrisp models At different levels of possibility necessity andcredibility we have achieved different optimal solution Anumerical example is presented and solved using LINGO-110to illustrate the proposed approaches The proposed methodcan be applied for multiobjective multiitem transportationproblem This method can be also extended to be appliedinto different types of optimization problem namely optimalcontrol and solid transportation problems

Conflict of Interests

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

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov Intuitionistic Fuzzy Sets VII ITKRrsquos SessionSofia Bulgarian 1983

[3] S K De and S S Sana ldquoA multi-periods production-inventorymodel with capacity constraints for multi-manufacturersmdashaglobal optimality in intuitionistic fuzzy environmentrdquo AppliedMathematics andComputation vol 242 no 1 pp 825ndash841 2014

[4] HGargM Rani S P Sharma andYVishwakarma ldquoIntuition-istic fuzzy optimization technique for solving multi-objectivereliability optimization problems in interval environmentrdquoExpert Systems with Applications vol 41 no 7 pp 3157ndash31672014

[5] J Wu and Y Liu ldquoAn approach for multiple attribute groupdecision making problems with interval-valued intuitionistictrapezoidal fuzzy numbersrdquoComputers and Industrial Engineer-ing vol 66 no 2 pp 311ndash324 2013

[6] A Nagoorgani and K Ponnalagu ldquoA new approach on solv-ing intuitionistic fuzzy linear programming problemrdquo AppliedMathematical Sciences vol 6 no 70 pp 3467ndash3474 2012

[7] G S Mahapatra and T K Roy ldquoIntuitionistic fuzzy numberand its arithmetic operation with application on system failurerdquoJournal of Uncertain Systems vol 7 no 2 pp 92ndash107 2013

[8] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[9] S Pramanik D K Jana and M Maiti ldquoA multi objectivesolid transportation problem in fuzzy Bi-fuzzy environmentvia genetic algorithmrdquo International Journal of Advanced Oper-ations Management vol 6 no 1 pp 4ndash26 2014

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

[11] M Esmailzadeh and M Esmailzadeh ldquoNew distance betweentriangular intuitionistic fuzzy numbersrdquo Advances in Computa-tional Mathematics and Its Applications vol 2 no 3 pp 3ndash102013

[12] P P Angelov ldquoOptimization in an intuitionistic fuzzy environ-mentrdquo Fuzzy Sets and Systems vol 86 no 3 pp 299ndash306 1997

[13] D Dubey and A Mehra ldquoLinear programming with triangularintuitionistic fuzzy numberrdquo Advances in Intelligent SystemsResearch vol 1 no 1 pp 563ndash569 2011

[14] R Parvathi and C Malathi ldquoIntuitionistic fuzzy simplexmethodrdquo International Journal of Computer Applications vol48 no 6 pp 39ndash48 2012

[15] R J Hussain and S P Kumar ldquoAlgorithmic approach forsolving intuitionistic fuzzy transportation problemrdquo AppliedMathematical Sciences vol 6 no 77-80 pp 3981ndash3989 2012

[16] ANagoorGani and S Abbas ldquoSolving intuitionstic fuzzy trans-portation problem using zero suffix algorithmrdquo InternationalJournal of Mathematics Sciences amp Enggineering Applicationsvol 6 pp 73ndash82 2012

[17] J Ye ldquoExpected value method for intuitionistic trapezoidalfuzzy multicriteria decision-making problemsrdquo Expert Systemswith Applications vol 38 no 9 pp 11730ndash11734 2011

[18] 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 2013

[19] J J Buckley ldquoPossibility and necessity in optimizationrdquo FuzzySets and Systems vol 25 no 1 pp 1ndash13 1988

[20] K D Jamison and W A Lodwick ldquoThe construction ofconsistent possibility and necessity measuresrdquo Fuzzy Sets andSystems vol 132 no 1 pp 1ndash10 2002

[21] J Ramık ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006

[22] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[23] M Sakawa H Katagiri and T Matsui ldquoFuzzy random bilevellinear programming through expectation optimization usingpossibility and necessityrdquo International Journal of MachineLearning and Cybernetics vol 3 no 3 pp 183ndash192 2012

[24] S Pathak S Kar and S Sarkar ldquoFuzzy production inventorymodel for deteriorating items with shortages under the effect oftime dependent learning and forgetting a possibilitynecessityapproachrdquo OPSEARCH vol 50 no 2 pp 149ndash181 2013

[25] K Maity ldquoPossibility and necessity representations of fuzzyinequality and its application to two warehouse production-inventory problemrdquo Applied Mathematical Modelling vol 35no 3 pp 1252ndash1263 2011

12 International Journal of Engineering Mathematics

[26] H-C Wu ldquoFuzzy optimization problems based on the embed-ding theorem and possibility and necessity measuresrdquo Mathe-matical and Computer Modelling vol 40 no 3-4 pp 329ndash3362004

[27] J Xu and X Zhou Fuzzy-Like Multiple Objective DecesionMaking Springer 2010

[28] K Maity and M Maiti ldquoPossibility and necessity constraintsand their defuzzificationmdasha multi-item production-inventoryscenario via optimal control theoryrdquo European Journal of Oper-ational Research vol 177 no 2 pp 882ndash896 2007

[29] B Das K Maity and M Maiti ldquoA two warehouse supply-chain model under possibilitynecessitycredibility measuresrdquoMathematical andComputerModelling vol 46 no 3-4 pp 398ndash409 2007

[30] D Panda S Kar K Maity and M Maiti ldquoA single periodinventory model with imperfect production and stochasticdemand under chance and imprecise constraintsrdquo EuropeanJournal of Operational Research vol 188 no 1 pp 121ndash139 2008

[31] A I Ban ldquoIntuitionistic fuzzy-valued possibility and necessitymeasuresrdquo in Proceedings of the 8th International Conference onIntuitionistic Fuzzy Sets vol 10 pp 1ndash7 Varna Bulgaria 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Engineering Mathematics 3

0

1

a9984001 a9984003a1 a2 a3

Figure 2 Membership and nonmembership functions of TrIFN

Definition 6 Two TIFN119860119868

= (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and

119861119868

= (1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) are said to be equal if and only

if 1198861= 1198871 1198862= 1198872 1198863= 1198873 1198864= 1198874 11988610158401= 1198871015840

1and 1198861015840

4= 1198871015840

4

Definition 7 Let 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) be two TIFN then

(i) 119860119868 oplus119861119868

= (1198861+ 1198871 1198862+ 1198872 1198863+ 1198873 1198864+ 1198874)(1198861015840

1+ 1198871015840

1 1198862+

1198872 1198863+ 1198873 1198861015840

4+ 1198871015840

4)

(ii) 119896119860119868 = 119896(1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) = (119896119886

1 1198961198862 1198961198863

1198961198864)(1198961198861015840

1 1198961198862 1198961198863 1198961198861015840

4) if 119896 ge 0

(iii) 119896119860119868 = 119896(1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) = (119896119886

4 1198961198863 1198961198862

1198961198861)(1198961198861015840

4 1198961198863 1198961198862 1198961198861015840

1) if 119896 lt 0

(iv) 119860119868 ⊖119861119868

= (1198861minus 1198874 1198862minus 1198873 1198863minus 1198872 1198864minus 1198871)(1198861015840

1minus 1198871015840

4 1198862minus

1198873 1198863minus 1198872 1198861015840

4minus 1198871015840

1)

(v) 119860119868 otimes 119861119868

= (11988611198871 11988621198872 11988631198873 11988641198874)(1198861015840

11198871015840

1 11988621198872 11988631198873 1198861015840

41198871015840

4)

3 Possibility Necessity and CredibilityMeasures of Intuitionistic Fuzzy Number

Definition 8 Let 119860119868 and 119861119868 be two IFN with membership

function 120583119860119868 120583119861119868 and nonmembership function ]

119860119868 ]119861119868

respectively and 119877 is the set of real numbers Then

Pos120583(119860119868

lowast 119861119868

) = sup min (120583119860119868 120583119861119868) 119909 119910 isin 119877 119909 lowast 119910

Pos] (119860119868

lowast 119861119868

) = sup min (]119860119868 ]119861119868) 119909 119910 isin 119877 119909 lowast 119910

Nes120583(119860119868

lowast 119861119868

) = inf max (120583119860119868 120583119861119868) 119909 119910 isin 119877 119909 lowast 119910

Nes] (119860119868

lowast 119861119868

) = inf max (]119860119868 ]119861119868) 119909 119910 isin 119877 119909 lowast 119910

(5)

where the abbreviations Pos120583and Pos] represent possibility

of membership and nonmembership function and Nes120583and

Nes] represent necessity ofmembership and nonmembershipfunction lowast is any of the relations lt gt le ge =

The dual relationship of possibility and necessity gives

Nes120583(119860119868

lowast 119861119868

) = 1 minus Pos120583(119860119868 lowast 119861119868)

Nes] (119860119868

lowast 119861119868

) = 1 minus Pos] (119860119868 lowast 119861119868)

(6)

where 119860119868 lowast 119861119868 represents complement of the event 119860119868 lowast 119861119868

Definition 9 Let 119860119868 be a IFN Then the intuitionistic fuzzymeasures of 119860119868 for membership and nonmembership func-tion are

Me120583119860119868

= 120582Pos120583119860119868

+ (1 minus 120582)Nec120583119860119868

Me] 119860119868

= 120582Pos] 119860119868

+ (1 minus 120582)Nec] 119860119868

(7)

where the abbreviation Me120583and Me] represent measures of

membership and nonmembership functions and 120582 (0 le 120582 le

1) is the optimistic-pessimistic parameter to determine thecombined attitude of a decision maker

If 120582 = 1 then Me120583

= Pos120583 Me] = Pos] it means

the decision maker is optimistic and maximum chance of 119860119868holds

If 120582 = 0 then Me120583

= Nes120583 Me] = Nes] it means

the decision maker is pessimistic and minimal chance of 119860119868holds

If 120582 = 05 then Me120583

= Cr120583 Me] = Cr] where Cr is

the credibility measure it means the decision maker takescompromise attitude

31 Measures of Trapezoidal Intuitionistic Fuzzy Number Let119860119868

= (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

= (1198871 1198872 1198873 1198874)

(1198871015840

1 1198872 1198873 1198871015840

4) be two TIFN FromDefinition 8 the possibilities

of 119860119868 le 119861119868 for membership and nonmembership functions

(cf Figures 3 and 4) are as follows

Pos120583(119860119868

le 119861119868

) =

1 1198862le 1198873

1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

1198861lt 1198874 1198862gt 1198873

0 1198874le 1198861

Pos] (119860119868

le 119861119868

) =

0 1198862le 1198873

1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

1198862gt 1198873 1198871015840

4gt 1198861015840

1

1 1198871015840

4le 1198861015840

1

(8)

4 International Journal of Engineering Mathematics

0

1

b1 b2 b3 b4a2a1 a3 a4

Figure 3 Membership function of TIFN 119860119868 and 119861

119868 and Pos120583(119860119868

le

119861119868

)

0

1

a9984001 a9984004b2 b3 a2 a3b9984001 b9984004

Figure 4 Nonmembership function of TIFN 119860119868 and 119861

119868 andPos](119860

119868

le 119861119868

)

From the Definition 8 the possibilities of 119860119868 ge 119861119868 for

membership andnonmembership function (cf Figures 5 and6) are as follows

Pos120583(119860119868

ge 119861119868

) =

1 1198863ge 1198872

1198864minus 1198871

1198864minus 1198863+ 1198872minus 1198871

1198863lt 1198872 1198864gt 1198871

0 1198864le 1198871

Pos] (119860119868

ge 119861119868

) =

0 1198872le 1198863

1198872minus 1198863

1198872minus 1198871015840

1+ 1198861015840

4minus 1198863

1198872gt 1198863 1198861015840

4gt 1198871015840

1

1 1198871015840

1ge 1198861015840

4

(9)

1

0b1 b2 b3 b4a2 a4a1 a3

Figure 5 Membership function of TIFN 119860119868 and 119861

119868 and Pos120583(119860119868

ge

119861119868

)

0

1

a9984001 a9984004b2 b3a2 a3b9984001 b9984004

Figure 6 Nonmembership function of TIFN 119860119868 and 119861

119868 andPos](119860

119868

ge 119861119868

)

Now by Definition 8 necessity of the event 119860119868 le 119861119868 are

as follows

Nes120583(119860119868

le 119861119868

) = 1 minus Pos120583(119860119868

gt 119861119868

)

=

0 1198873ge 1198861

1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Nes] (119860119868

le 119861119868

) = 1 minus Pos] (119860119868

gt 119861119868

)

=

0 1198872ge 1198861015840

4

1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

1198872lt 1198861015840

4 1198863lt 1198871

1 1198863ge 1198871015840

1

(10)

International Journal of Engineering Mathematics 5

By Definition 8 necessity of the event 119860119868 ge 119861119868 are as follows

Nes120583(119860119868

ge 119861119868

) = 1 minus Pos120583(119860119868

lt 119861119868

)

=

0 1198862le 1198874

1198862minus 1198874

1198862minus 1198861minus 1198874+ 1198873

1198862gt 1198874 1198861lt 1198873

1 1198861ge 1198873

Nes] (119860119868

ge 119861119868

) = 1 minus Pos] (119860119868

lt 119861119868

)

=

0 1198873le 1198861015840

1

1198873minus 1198861015840

1

1198862minus 1198861015840

1minus 1198871015840

4+ 1198873

1198873gt 1198861015840

1 1198862gt 1198871015840

4

1 1198862le 1198871015840

4

(11)

By Definition 9 measures of the event 119860119868 le 119861119868 are as follows

Me120583(119860119868

le 119861119868

)

= 120582Pos120583(119860119868

le 119861119868

) + (1 minus 120582)Nes120583(119860119868

le 119861119868

)

=

0 1198874le 1198861

1205821198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

1198873gt 1198862 1198871lt 1198862

120582 1198873gt 1198862 1198871lt 1198863

120582 + (1 minus 120582)1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Me] (119860119868

le 119861119868

)

= 120582Pos] (119860119868

le 119861119868

) + (1 minus 120582)Nes] (119860119868

le 119861119868

)

=

1 1198871015840

4le 1198861015840

1

1205821198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

+ (1 minus 120582) 1198871015840

4gt 1198861015840

1 1198862gt 1198873

(1 minus 120582) 1198862lt 1198873 1198871015840

1lt 1198863

(1 minus 120582)1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

1198863lt 1198871015840

1 1198861015840

4gt 1198872

0 1198872ge 1198861015840

4

(12)

By Definition 9 measures of the event 119860119868 ge 119861119868 are as follows

Me120583(119860119868

ge 119861119868

)

= 120582Pos120583(119860119868

ge 119861119868

) + (1 minus 120582)Nes120583(119860119868

ge 119861119868

)

=

1 1198873le 1198862

(1 minus 120582)1198862minus 1198874

1198862minus 1198861minus 1198874+ 1198873

+ 120582 1198873gt 1198861 1198874lt 1198862

120582 1198874gt 1198862 1198872lt 1198863

1205821198864minus 1198871

1198864minus 1198863+ 1198872minus 1198871

1198863lt 1198862 1198864gt 1198871

0 1198871ge 1198864

Me] (119860119868

ge 119861119868

)

= 120582Pos] (119860119868

ge 119861119868

) + (1 minus 120582)Nes] (119860119868

ge 119861119868

)

=

0 1198873le 1198861015840

1

(1 minus 120582)1198873minus 1198861015840

1

1198862minus 1198861015840

1minus 1198871015840

4+ 1198873

1198873gt 1198861015840

1 1198862gt 1198871015840

4

(1 minus 120582) 1198862lt 1198871015840

4 1198872lt 1198863

(1 minus 120582) + 1205821198872minus 1198863

1198872minus 1198871015840

1+ 1198861015840

4minus 1198863

1198863lt 1198872 1198861015840

4gt 1198871015840

1

1 1198871015840

1ge 1198861015840

4

(13)

For 120582 = 05

Cr120583(119860119868

le119861119868

)=

0 1198871le 1198861

1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)

1198874gt 1198861 1198862gt 1198873

1

2

1198873gt 1198862 1198871lt 1198863

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Cr] (119860119868

le119861119868

)=

1 1198871015840

4le 1198861015840

1

21198862minus 21198873+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)

1198871015840

4gt 1198861015840

1 1198862gt 1198873

1

2

1198862lt 1198873 1198871015840

1gt 1198863

1198861015840

4minus 1198872

2 (1198861015840

4minus 1198863minus 1198872+ 1198871015840

1)

1198863lt 1198871015840

1 1198872lt 1198861015840

4

0 1198872ge 1198861015840

4

Cr120583(119860119868

ge119861119868

)=

1 1198873le 1198862

21198862minus 21198874+ 1198873minus 1198861

2 (1198862minus 1198861minus 1198874+ 1198873)

1198873gt 1198861 1198862gt 1198874

1

2

1198874gt 1198862 1198872lt 1198863

1198864minus 1198871

2 (1198864minus 1198863+ 1198872minus 1198871)

1198872gt 1198863 1198864gt 1198871

0 1198871ge 1198864

Cr] (119860119868

ge119861119868

)=

0 1198871015840

3le 1198861015840

1

1198873minus 1198861015840

1

2 (1198862minus 1198861015840

1minus 1198871015840

4+ 1198873)

1198873gt 1198861015840

1 1198862gt 1198871015840

4

1

2

1198862lt 1198871015840

4 1198872lt 1198863

21198872minus 21198863+ 1198861015840

4minus 1198871015840

1

2 (1198872minus 1198871015840

1+ 1198861015840

4minus 1198863)

1198872gt 1198863 1198871015840

1lt 1198861015840

4

1 1198871015840

1ge 1198861015840

4

(14)

Lemma 10 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Pos120583(119860119868

le 119861119868

) ge 120572 Pos] (119860119868

le 119861119868

) le 120573

lArrrArr1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

ge 120572

1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

le 120573

(15)

6 International Journal of Engineering Mathematics

Proof Let us consider

Pos120583(119860119868

le 119861119868

) ge 120572 Pos] (119860119868

ge 119861119868

) le 120573 (16)

Now from (8)

Pos120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

ge 120572

Pos] (119860119868

le 119861119868

) le 120573 lArrrArr1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

le 120573

(17)

Note Pos120583(119860119868

le 119909) ge 120572 and Pos](119860119868

le 119909) le 120573 hArr (119909 minus

1198861)(1198862minus 1198861) ge 120572 and (119886

2minus 119909)(119886

2minus 1198861015840

1) le 120573

Lemma 11 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Nes120583(119860119868

le 119861119868

) ge 120572 Nes] (119860119868

le 119861119868

) le 120573

lArrrArr1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

ge 120572

1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

le 120573

(18)

Proof Let us consider

Nes120583(119860119868

le 119861119868

) ge 120572 Nes] (119860119868

le 119861119868

) le 120573 (19)

Now from (10)

Nes120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

ge 120572

Nes] (119860119868

le 119861119868

) le 120573 lArrrArr1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

le 120573

(20)

Note Nes120583(119860119868

le 119909) ge 120572 and Nes](119860119868

le 119909) le 120573 hArr (119909 minus

1198863)(1198864minus 1198863) ge 120572 and (119886

1015840

4minus 119909)(119886

1015840

4minus 1198863) le 120573

Lemma 12 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Cr120583(119860119868

le 119861119868

) ge 120572 Cr] (119860119868

le 119861119868

) le 120573

lArrrArr1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)ge 120572

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)ge 120572

21198862minus 21198871015840

3+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)le 120573

1198861015840

4minus 1198872

2 (1198861015840

4minus 1198863minus 1198872+ 1198871015840

1)le 120573

(21)

Proof Let us consider

Cr120583(119860119868

le 119861119868

) ge 120572 Cr] (119860119868

le 119861119868

) le 120573 (22)

Now from (14)

Cr120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)ge 120572

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)ge 120572

Cr] (119860119868

le 119861119868

) le 120573 lArrrArr21198862minus 21198871015840

3+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)le 120573

1198872minus 1198861015840

4

2 (1198872minus 1198871015840

1minus 1198861015840

4+ 1198863)le 120573

(23)

Note Cr120583(119860119868

le 119909) ge 120572 Cr](119860119868

le 119909) ge 120573 hArr (119909 minus 1198861)2(1198862minus

1198861) ge 120572 (119886

4minus21198863+119909)2(119886

4minus1198863) ge 120572 and (2119886

2minus119909minus119886

1015840

1)2(1198862minus

1198861015840

1) le 120573 (1198861015840

4minus 119909)2(119886

1015840

4minus 1198863) le 120573

4 Intuitionistic Fuzzy CCM

The chance operator is actually taken as possibility or neces-sity or credibility measures We can use chance operatorto transform the intuitionistic fuzzy problem into crispproblem which is called as CCM [27] A general single-objective mathematical programming problem with intu-itionistic fuzzy parameter should have the following form

Max 119891 (119909 120585119868

)

subject to 119892119894(119909 120585119868

) le 119868

119894 119894 = 1 2 119899

119909 ge 0

(24)

where 119909 is the decision vector 120585119868 and 119868

119894are intuitionistic

fuzzy parameters 119891(119909 120585119868) is an imprecise objective functionand 119892

119894(119909 120585119868

) are constraints function for 119894 = 1 2 119899The general chance-constraints model for problem (24) is

as followsMax 119891

1+ 1198912

subject to Ch120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Ch] 119891 (119909 120585119868

) ge 1198912 le 120573

Ch120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Ch] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(25)

The abbreviations Ch120583and Ch] represent chance operator

(ie Pos or Nec measure) for membership and nonmem-bership functions 120572 120573 120582

119894 and 120595

119894are the predetermined

confidence levels such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

International Journal of Engineering Mathematics 7

41 Intuitionistic Fuzzy CCM Based on Possibility MeasureThe CCM based on possibility measure is as follows

Max 1198911+ 1198912

Subject to Pos120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Pos] 119891 (119909 120585119868

) ge 1198912 le 120573

Pos120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Pos] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(26)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 13 A solution 119909lowast of the problem (26) satisfies

Pos120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Pos](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution at (120582119894 120595119894) possibility

levels 119894 = 1 2 119899

Definition 14 A feasible solution at (120582119894 120595119894) possibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (26)

if and only if there exists no other feasible solution at(120582119894 120595119894) possibility levels such that Pos

120583119891(119909 120585

119868

) ge 120572 andPos]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

42 Intuitionistic Fuzzy CCM Based on Necessity MeasureThe CCM based on necessity measure is as follows

Max 1198911+ 1198912

Subject to Nes120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Nes] 119891 (119909 120585119868

) ge 1198912 le 120573

Nes120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Nes] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(27)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 15 A solution 119909lowast of the problem (27) satisfies

Nes120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Nes](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) necessity

levels 119894 = 1 2 119899

Definition 16 A feasible solution at (120582119894 120595119894) necessity levels

119909lowast is said to be (120572 120573) efficient solution for problem (27)

if and only if there exists no other feasible solution at(120582119894 120595119894) necessity levels such that Nes

120583119891(119909 120585

119868

) ge 120572 andNes]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

43 Intuitionistic Fuzzy CCM Based on Credibility MeasureThe CCM based on credibility measure is as follows

Max 1198911+ 1198912

Subject to Cr120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Cr] 119891 (119909 120585119868

) ge 1198912 le 120573

Cr120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Cr] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(28)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 17 A solution 119909lowast of the problem (28) satisfies

Cr120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Cr](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) credibility

levels 119894 = 1 2 119899

Definition 18 A feasible solution at (120582119894 120595119894) credibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (28)

if and only if there exists no other feasible solution at(120582119894 120595119894) credibility levels such that Cr

120583119891(119909 120585

119868

) ge 120572 andCr]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

5 Proposed Method to Solve IFLPPUsing Chance Operator

To solve intuitionistic fuzzy CCM based on possibility ornecessity or credibility measures we propose the followingmethod

Step 1 Apply chance operator possibilitynecessitycredi-bility in intuitionistic fuzzy programming (24) Problem (24)can be converted into following problem

Max 1198911+ 1198912

Subject to Pos120583119891 (119909I

119868

) ge 1198911 ge 120572

or Nes120583119891 (119909I

119868

) ge 1198911 ge 120572

or Cr120583119891 (119909I

119868

) ge 1198911 ge 120572

(29)

Pos] 119891 (119909I119868

) ge 1198912 le 120573

or Nes] 119891 (119909I119868

) ge 1198912 le 120573

or Cr] 119891 (119909I119868

) ge 1198912 le 120573

(30)

8 International Journal of Engineering Mathematics

Pos120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Nes120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Cr120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

(31)

Pos] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Nes] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Cr] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

(32)

119909 ge 0 for 119894 = 1 2 119899 (33)

0 le 120582119894+ 120595119894le 1 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

(34)

where (120582119894 120595119894) and (120572 120573) are the predefined confidence levels

Step 2 Using Lemmas 10 11 andor Lemma 12 the aboveproblem in Step 1 can also be written as

Max 1198911+ 1198912

Subject to 1198911+ 1198912ge 119885

(31) ndash (34)

(35)

where 119885 is obtained by applying Lemmas 10 11 andorLemma 12 in (30) and (29)

Step 3 The above problem is equivalent to

Max 119885

Subject to (31) ndash (34)

(36)

Step 4 Crisp programming problem obtained in Step 2 canbe solved using any well-known method to get the optimalsolution

6 Numerical Example

Let us consider the following intuitionistic fuzzy mathemati-cal programming problem as

Maximize 120585119868

11199091oplus 120585119868

21199092

subject to 120585119868

31199091oplus 120585119868

41199092⪯ 120585119868

5

120585119868

61199091oplus 120585119868

71199092⪯ 120585119868

8

1199091 1199092ge 0

(37)

where 120585119868

1= (5 6 7 8)(4 6 7 9) 120585119868

2= (4 5 6 7)(3 5 6 8)

120585119868

3= (1 2 3 4)(05 2 3 6) 120585119868

4= (2 3 4 5)(1 3 4 6) 120585119868

5=

(6 7 8 9)(5 7 8 10) 1205851198686= (3 4 5 6)(2 4 5 7) 120585119868

7= (1 2 3

4)(0 2 3 4) and 120585119868

8= (10 11 12 14)(9 11 12 16)

61 Intuitionistic Fuzzy CCM Based on Possibility MeasureNowby using Step 2 of themethod explained in Section 4 and

Start Applychance operator

Convertinto crisp

Writeequivalent Solve

Figure 7 Flow chart of the proposed algorithm

Lemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (37) problem (37) is con-verted into the following crisp programming problem

Maximize (15 minus 120572 + 2120573) 1199091+ (13 minus 120572 + 2120573) 119909

2

subject to (1 + 1205821) 1199091+ (2 + 120582

1) 1199092le 9 minus 120582

1

(2 minus 151205951) 1199091+ (3 minus 2120595

1) 1199092le 8 + 2120595

1

(3 + 1205822) 1199091+ (1 + 120582

2) 1199092le 14 minus 2120582

2

(4 minus 21205952) 1199091+ (2 minus 2120595

2) 1199092le 12 + 4120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(38)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and different possibility levels we get differentoptimal solutions Optimal solution of (38) at differentpossibility levels (in Figure 7) are presented in Table 1 FromTable 1 we can observe that maximum value (= 7309) can beobtained at (120582

1= 040 120595

1= 035) and (120582

2= 030 120595

2= 040)

possibility levels

62 Intuitionistic Fuzzy CCM Based on Necessity MeasureNowby using Step 2 of themethod explained in Section 4 andLemma 11 if we apply the necessity measure in (37) problem(37) is converted into following crisp programming problem

Maximize (10 minus 120572 + 2120573) 1199091+ (8 minus 120572 + 2120573) 119909

2

subject to (3 + 21205821) 1199091+ (4 + 2120582

1) 1199092le 6 + 120582

1

(5 minus 21205951) 1199091+ (6 minus 2120595

1) 1199092le 7 minus 2120595

1

(5 + 1205822) 1199091+ (3 + 120582

2) 1199092le 6 + 120582

2

(7 minus 21205952) 1199091+ (4 minus 120595

2) 1199092le 11 minus 2120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(39)

Solving the above crisp linear programming problem forefficient levels (120572 = 06 120573 = 04) and different necessitylevels we get different optimal solutions Optimal solutionsof (39) at different necessity levels (in Figures 8 and 9) arepresented in Table 2 From Table 2 we can observed that at(1205821= 035 120595

1= 045) and (120582

2= 035 120595

2= 045) the decision

maker will get the maximum value = 1298

International Journal of Engineering Mathematics 9

Table 1 Optimal solution of (38) at different possibility levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)030 030 035 035 119909

1= 341 119909

2= 137 7009

030 035 035 040 1199091= 329 119909

2= 166 7214

040 035 030 040 1199091= 343 119909

2= 157 7309

035 040 040 030 1199091= 310 119909

2= 19 722

040 045 040 045 1199091= 316 119909

2= 173 7105

050 045 045 05 1199091= 316 119909

2= 150 6793

045 050 050 040 1199091= 297 119909

2= 173 6802

060 045 045 06 1199091= 329 119909

2= 120 6593

050 050 050 050 1199091= 303 119909

2= 157 6700

055 040 055 040 1199091= 297 119909

2= 150 6510

Table 2 Optimal solution of (39) at different necessity levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)035 030 030 035 119909

1= 091 119909

2= 043 1293

045 030 040 035 1199091= 090 119909

2= 045 1289

050 035 050 035 1199091= 087 119909

2= 047 1286

060 040 060 040 1199091= 085 119909

2= 050 1284

070 020 070 020 1199091= 087 119909

2= 045 1271

060 030 060 020 1199091= 087 119909

2= 047 1279

065 035 075 020 1199091= 084 119909

2= 050 1275

070 010 070 010 1199091= 089 119909

2= 043 1267

050 050 050 050 1199091= 085 119909

2= 051 1294

035 045 035 045 1199091= 088 119909

2= 048 1298

Table 3 Input data for IFTP

1198631

119868

1198632

119868

1198633

119868 Availability (119886119894

119868

)

1198781

119868 (2 4 6 7) (1 4 6 9) (4 6 7 8) (3 6 7 9) (3 7 9 12) (2 7 9 13) (4 6 8 9) (2 6 8 10)1198782

119868 (1 3 4 6) (05 3 4 7) (3 5 6 7) (2 5 6 9) (2 6 7 11) (1 6 7 12) (0 05 1 2) (0 05 1 5)1198783

119868 (3 4 5 8) (2 4 5 10) (1 2 3 4) (05 2 3 5) (2 4 5 10) (1 4 5 11) (8 95 10 11) (65 95 10 11)Demand (119887

119895

119868

) (6 7 8 10) (5 7 8 12) (4 5 6 9) (3 5 6 11) (2 4 5 7) (05 4 5 8)

646668707274

(030 030)

(035

035

)

(035

040

)

(030

040

) (055 040)

(050 050)

(060 045)

(045 050)

(050 045)(0

40 0

30)

(040

045

)

(045

050

)

(050

040

)

(045

060

)

(045

045

)

(030

030

)(040 045)

(035 040)

(040 035)

(030 035)(1205821 1205951

)(1205822 120595

2 )

flowast

Figure 8 Optimal solution at different possibility levels

10 International Journal of Engineering Mathematics

1265127

1275128

1285129

129513

(035 030)

(050

035

)

(060

040

)

(070

020

)

(060

020

)

(075

020

)

(070

010

)

(050

050

)

(035

045

)

(035 045)

(050 050)

(070 010)

(065 035)

(060 030)

(070 020)

(060 040)

(050 035)

(045 030)

(040

035

)

(030

035

)

(1205821 1205951)

(1205822 120595

2 )

flowast

Figure 9 Optimal solution at different necessity levels

7 Intuitionistic Fuzzy TransportationProblem Based on Possibility Measure

Let us consider the following intuitionistic fuzzy transporta-tion problem (IFTP) (in Table 3)

Above transportation problem is a balanced transporta-tion problem as ⨁3

119894=1119886119868

119894= ⨁3

119895=1119868

119895 The above IFTP can be

written as

Minimize (2 4 6 7) (1 4 6 9) 11990911

oplus (4 6 7 8) (3 6 7 9) 11990912

oplus (3 7 9 12) (2 7 9 13) 11990913

oplus (1 3 4 6) (05 3 4 7) 11990921

oplus (3 5 6 7) (2 5 6 9) 11990922

oplus (2 6 7 11) (1 6 7 12) 11990923

oplus (3 4 5 8) (2 4 5 10) 11990931

oplus (1 2 3 4) (05 2 3 5) 11990932

oplus (2 4 5 10) (1 4 5 11) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (4 6 8 9) (2 6 8 10)

11990921

+ 11990922

+ 11990923

le (0 05 1 2) (0 05 1 5)

11990931

+ 11990932

+ 11990933

le (8 95 10 11) (65 95 10 11)

11990911

+ 11990921

+ 11990931

ge (6 7 8 10) (5 7 8 12)

11990912

+ 11990922

+ 11990932

ge (4 5 6 9) (3 5 6 11)

11990913

+ 11990923

+ 11990933

ge (2 4 5 7) (05 4 5 8)

119909119894119895ge 0 forall119894 119895

(40)

Nowby using Step 2 of themethod explained in Section 4 andLemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (40) problem (40) isconverted into following crisp programming problem

Minimize (6 + 2120572 minus 3120573) 11990911

+ (10 + 2120572 minus 2120573) 11990912

+ (10 + 4120572 minus 5120573) 11990913

+ (4 + 2120572 minus 25120573) 11990921

+ (8 + 2120572 minus 3120573) 11990922

+ (8 + 4120572 minus 5120573) 11990923

+ (7 + 120572 minus 5120573) 11990931

+ (3 + 120572 minus 15120573) 11990932

+ (6 + 2120572 minus 3120573) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (17 minus 1205821+ 21205951)

11990921

+ 11990922

+ 11990923

le (3 minus 1205822+ 41205952)

11990931

+ 11990932

+ 11990933

le (21 minus 1205823+ 1205953)

11990911

+ 11990921

+ 11990931

ge (13 + 1205824minus 21205954)

11990912

+ 11990922

+ 11990932

ge (9 + 1205825minus 21205955)

11990913

+ 11990923

+ 11990933

ge (6 + 21205826minus 35120595

6)

119909119894119895ge 0 for 119894 119895 = 1 2 3 0 le 120572 + 120573 le 1

0 le 120582119896+ 120595119896le 1 for 119896 = 1 2 6

(41)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and possibility levels (120582119894= 05 120595

119894= 05) for

119894 = 1 2 6 using Lingo-110 we get 11990911

= 075 11990912

= 011990913

= 0 11990921

= 45 11990922

= 0 11990923

= 0 11990931

= 725 11990932

= 85 and

International Journal of Engineering Mathematics 11

11990933

= 525 Now the minimum intuitionstic fuzzy optimalcost is

119888119868

= (7225 109 136 2035) (4562 109 136 24625) (42)

8 Discussion

Intuitionistic fuzzy sets being a generalization of fuzzy setsgive us an additional possibility to represent imperfect knowl-edge making it possible to describe many real problems ina more adequate way So in this paper we have developedthe possibility and necessity measures on intuitionistic fuzzyset Here we have presented first time the mathematicalrepresentation of different types of measures in intuitionisticfuzzy environments and some graphical representations ofthemare depictedWehave also developed the theoretical cal-culation on possibility necessity and credibility measures fordefuzzify intuitionistic fuzzy linear programming problemusing chance operators To validate the proposed methodwe have discussed three different approaches to defuzzifythe intuitionistic fuzzy relations using possibility necessityand credibility measures Using chance operator we canconvert a problem under imprecise models to correspondingcrisp models At different levels of possibility necessity andcredibility we have achieved different optimal solution Anumerical example is presented and solved using LINGO-110to illustrate the proposed approaches The proposed methodcan be applied for multiobjective multiitem transportationproblem This method can be also extended to be appliedinto different types of optimization problem namely optimalcontrol and solid transportation problems

Conflict of Interests

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

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov Intuitionistic Fuzzy Sets VII ITKRrsquos SessionSofia Bulgarian 1983

[3] S K De and S S Sana ldquoA multi-periods production-inventorymodel with capacity constraints for multi-manufacturersmdashaglobal optimality in intuitionistic fuzzy environmentrdquo AppliedMathematics andComputation vol 242 no 1 pp 825ndash841 2014

[4] HGargM Rani S P Sharma andYVishwakarma ldquoIntuition-istic fuzzy optimization technique for solving multi-objectivereliability optimization problems in interval environmentrdquoExpert Systems with Applications vol 41 no 7 pp 3157ndash31672014

[5] J Wu and Y Liu ldquoAn approach for multiple attribute groupdecision making problems with interval-valued intuitionistictrapezoidal fuzzy numbersrdquoComputers and Industrial Engineer-ing vol 66 no 2 pp 311ndash324 2013

[6] A Nagoorgani and K Ponnalagu ldquoA new approach on solv-ing intuitionistic fuzzy linear programming problemrdquo AppliedMathematical Sciences vol 6 no 70 pp 3467ndash3474 2012

[7] G S Mahapatra and T K Roy ldquoIntuitionistic fuzzy numberand its arithmetic operation with application on system failurerdquoJournal of Uncertain Systems vol 7 no 2 pp 92ndash107 2013

[8] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[9] S Pramanik D K Jana and M Maiti ldquoA multi objectivesolid transportation problem in fuzzy Bi-fuzzy environmentvia genetic algorithmrdquo International Journal of Advanced Oper-ations Management vol 6 no 1 pp 4ndash26 2014

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

[11] M Esmailzadeh and M Esmailzadeh ldquoNew distance betweentriangular intuitionistic fuzzy numbersrdquo Advances in Computa-tional Mathematics and Its Applications vol 2 no 3 pp 3ndash102013

[12] P P Angelov ldquoOptimization in an intuitionistic fuzzy environ-mentrdquo Fuzzy Sets and Systems vol 86 no 3 pp 299ndash306 1997

[13] D Dubey and A Mehra ldquoLinear programming with triangularintuitionistic fuzzy numberrdquo Advances in Intelligent SystemsResearch vol 1 no 1 pp 563ndash569 2011

[14] R Parvathi and C Malathi ldquoIntuitionistic fuzzy simplexmethodrdquo International Journal of Computer Applications vol48 no 6 pp 39ndash48 2012

[15] R J Hussain and S P Kumar ldquoAlgorithmic approach forsolving intuitionistic fuzzy transportation problemrdquo AppliedMathematical Sciences vol 6 no 77-80 pp 3981ndash3989 2012

[16] ANagoorGani and S Abbas ldquoSolving intuitionstic fuzzy trans-portation problem using zero suffix algorithmrdquo InternationalJournal of Mathematics Sciences amp Enggineering Applicationsvol 6 pp 73ndash82 2012

[17] J Ye ldquoExpected value method for intuitionistic trapezoidalfuzzy multicriteria decision-making problemsrdquo Expert Systemswith Applications vol 38 no 9 pp 11730ndash11734 2011

[18] 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 2013

[19] J J Buckley ldquoPossibility and necessity in optimizationrdquo FuzzySets and Systems vol 25 no 1 pp 1ndash13 1988

[20] K D Jamison and W A Lodwick ldquoThe construction ofconsistent possibility and necessity measuresrdquo Fuzzy Sets andSystems vol 132 no 1 pp 1ndash10 2002

[21] J Ramık ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006

[22] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[23] M Sakawa H Katagiri and T Matsui ldquoFuzzy random bilevellinear programming through expectation optimization usingpossibility and necessityrdquo International Journal of MachineLearning and Cybernetics vol 3 no 3 pp 183ndash192 2012

[24] S Pathak S Kar and S Sarkar ldquoFuzzy production inventorymodel for deteriorating items with shortages under the effect oftime dependent learning and forgetting a possibilitynecessityapproachrdquo OPSEARCH vol 50 no 2 pp 149ndash181 2013

[25] K Maity ldquoPossibility and necessity representations of fuzzyinequality and its application to two warehouse production-inventory problemrdquo Applied Mathematical Modelling vol 35no 3 pp 1252ndash1263 2011

12 International Journal of Engineering Mathematics

[26] H-C Wu ldquoFuzzy optimization problems based on the embed-ding theorem and possibility and necessity measuresrdquo Mathe-matical and Computer Modelling vol 40 no 3-4 pp 329ndash3362004

[27] J Xu and X Zhou Fuzzy-Like Multiple Objective DecesionMaking Springer 2010

[28] K Maity and M Maiti ldquoPossibility and necessity constraintsand their defuzzificationmdasha multi-item production-inventoryscenario via optimal control theoryrdquo European Journal of Oper-ational Research vol 177 no 2 pp 882ndash896 2007

[29] B Das K Maity and M Maiti ldquoA two warehouse supply-chain model under possibilitynecessitycredibility measuresrdquoMathematical andComputerModelling vol 46 no 3-4 pp 398ndash409 2007

[30] D Panda S Kar K Maity and M Maiti ldquoA single periodinventory model with imperfect production and stochasticdemand under chance and imprecise constraintsrdquo EuropeanJournal of Operational Research vol 188 no 1 pp 121ndash139 2008

[31] A I Ban ldquoIntuitionistic fuzzy-valued possibility and necessitymeasuresrdquo in Proceedings of the 8th International Conference onIntuitionistic Fuzzy Sets vol 10 pp 1ndash7 Varna Bulgaria 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Engineering Mathematics

0

1

b1 b2 b3 b4a2a1 a3 a4

Figure 3 Membership function of TIFN 119860119868 and 119861

119868 and Pos120583(119860119868

le

119861119868

)

0

1

a9984001 a9984004b2 b3 a2 a3b9984001 b9984004

Figure 4 Nonmembership function of TIFN 119860119868 and 119861

119868 andPos](119860

119868

le 119861119868

)

From the Definition 8 the possibilities of 119860119868 ge 119861119868 for

membership andnonmembership function (cf Figures 5 and6) are as follows

Pos120583(119860119868

ge 119861119868

) =

1 1198863ge 1198872

1198864minus 1198871

1198864minus 1198863+ 1198872minus 1198871

1198863lt 1198872 1198864gt 1198871

0 1198864le 1198871

Pos] (119860119868

ge 119861119868

) =

0 1198872le 1198863

1198872minus 1198863

1198872minus 1198871015840

1+ 1198861015840

4minus 1198863

1198872gt 1198863 1198861015840

4gt 1198871015840

1

1 1198871015840

1ge 1198861015840

4

(9)

1

0b1 b2 b3 b4a2 a4a1 a3

Figure 5 Membership function of TIFN 119860119868 and 119861

119868 and Pos120583(119860119868

ge

119861119868

)

0

1

a9984001 a9984004b2 b3a2 a3b9984001 b9984004

Figure 6 Nonmembership function of TIFN 119860119868 and 119861

119868 andPos](119860

119868

ge 119861119868

)

Now by Definition 8 necessity of the event 119860119868 le 119861119868 are

as follows

Nes120583(119860119868

le 119861119868

) = 1 minus Pos120583(119860119868

gt 119861119868

)

=

0 1198873ge 1198861

1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Nes] (119860119868

le 119861119868

) = 1 minus Pos] (119860119868

gt 119861119868

)

=

0 1198872ge 1198861015840

4

1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

1198872lt 1198861015840

4 1198863lt 1198871

1 1198863ge 1198871015840

1

(10)

International Journal of Engineering Mathematics 5

By Definition 8 necessity of the event 119860119868 ge 119861119868 are as follows

Nes120583(119860119868

ge 119861119868

) = 1 minus Pos120583(119860119868

lt 119861119868

)

=

0 1198862le 1198874

1198862minus 1198874

1198862minus 1198861minus 1198874+ 1198873

1198862gt 1198874 1198861lt 1198873

1 1198861ge 1198873

Nes] (119860119868

ge 119861119868

) = 1 minus Pos] (119860119868

lt 119861119868

)

=

0 1198873le 1198861015840

1

1198873minus 1198861015840

1

1198862minus 1198861015840

1minus 1198871015840

4+ 1198873

1198873gt 1198861015840

1 1198862gt 1198871015840

4

1 1198862le 1198871015840

4

(11)

By Definition 9 measures of the event 119860119868 le 119861119868 are as follows

Me120583(119860119868

le 119861119868

)

= 120582Pos120583(119860119868

le 119861119868

) + (1 minus 120582)Nes120583(119860119868

le 119861119868

)

=

0 1198874le 1198861

1205821198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

1198873gt 1198862 1198871lt 1198862

120582 1198873gt 1198862 1198871lt 1198863

120582 + (1 minus 120582)1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Me] (119860119868

le 119861119868

)

= 120582Pos] (119860119868

le 119861119868

) + (1 minus 120582)Nes] (119860119868

le 119861119868

)

=

1 1198871015840

4le 1198861015840

1

1205821198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

+ (1 minus 120582) 1198871015840

4gt 1198861015840

1 1198862gt 1198873

(1 minus 120582) 1198862lt 1198873 1198871015840

1lt 1198863

(1 minus 120582)1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

1198863lt 1198871015840

1 1198861015840

4gt 1198872

0 1198872ge 1198861015840

4

(12)

By Definition 9 measures of the event 119860119868 ge 119861119868 are as follows

Me120583(119860119868

ge 119861119868

)

= 120582Pos120583(119860119868

ge 119861119868

) + (1 minus 120582)Nes120583(119860119868

ge 119861119868

)

=

1 1198873le 1198862

(1 minus 120582)1198862minus 1198874

1198862minus 1198861minus 1198874+ 1198873

+ 120582 1198873gt 1198861 1198874lt 1198862

120582 1198874gt 1198862 1198872lt 1198863

1205821198864minus 1198871

1198864minus 1198863+ 1198872minus 1198871

1198863lt 1198862 1198864gt 1198871

0 1198871ge 1198864

Me] (119860119868

ge 119861119868

)

= 120582Pos] (119860119868

ge 119861119868

) + (1 minus 120582)Nes] (119860119868

ge 119861119868

)

=

0 1198873le 1198861015840

1

(1 minus 120582)1198873minus 1198861015840

1

1198862minus 1198861015840

1minus 1198871015840

4+ 1198873

1198873gt 1198861015840

1 1198862gt 1198871015840

4

(1 minus 120582) 1198862lt 1198871015840

4 1198872lt 1198863

(1 minus 120582) + 1205821198872minus 1198863

1198872minus 1198871015840

1+ 1198861015840

4minus 1198863

1198863lt 1198872 1198861015840

4gt 1198871015840

1

1 1198871015840

1ge 1198861015840

4

(13)

For 120582 = 05

Cr120583(119860119868

le119861119868

)=

0 1198871le 1198861

1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)

1198874gt 1198861 1198862gt 1198873

1

2

1198873gt 1198862 1198871lt 1198863

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Cr] (119860119868

le119861119868

)=

1 1198871015840

4le 1198861015840

1

21198862minus 21198873+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)

1198871015840

4gt 1198861015840

1 1198862gt 1198873

1

2

1198862lt 1198873 1198871015840

1gt 1198863

1198861015840

4minus 1198872

2 (1198861015840

4minus 1198863minus 1198872+ 1198871015840

1)

1198863lt 1198871015840

1 1198872lt 1198861015840

4

0 1198872ge 1198861015840

4

Cr120583(119860119868

ge119861119868

)=

1 1198873le 1198862

21198862minus 21198874+ 1198873minus 1198861

2 (1198862minus 1198861minus 1198874+ 1198873)

1198873gt 1198861 1198862gt 1198874

1

2

1198874gt 1198862 1198872lt 1198863

1198864minus 1198871

2 (1198864minus 1198863+ 1198872minus 1198871)

1198872gt 1198863 1198864gt 1198871

0 1198871ge 1198864

Cr] (119860119868

ge119861119868

)=

0 1198871015840

3le 1198861015840

1

1198873minus 1198861015840

1

2 (1198862minus 1198861015840

1minus 1198871015840

4+ 1198873)

1198873gt 1198861015840

1 1198862gt 1198871015840

4

1

2

1198862lt 1198871015840

4 1198872lt 1198863

21198872minus 21198863+ 1198861015840

4minus 1198871015840

1

2 (1198872minus 1198871015840

1+ 1198861015840

4minus 1198863)

1198872gt 1198863 1198871015840

1lt 1198861015840

4

1 1198871015840

1ge 1198861015840

4

(14)

Lemma 10 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Pos120583(119860119868

le 119861119868

) ge 120572 Pos] (119860119868

le 119861119868

) le 120573

lArrrArr1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

ge 120572

1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

le 120573

(15)

6 International Journal of Engineering Mathematics

Proof Let us consider

Pos120583(119860119868

le 119861119868

) ge 120572 Pos] (119860119868

ge 119861119868

) le 120573 (16)

Now from (8)

Pos120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

ge 120572

Pos] (119860119868

le 119861119868

) le 120573 lArrrArr1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

le 120573

(17)

Note Pos120583(119860119868

le 119909) ge 120572 and Pos](119860119868

le 119909) le 120573 hArr (119909 minus

1198861)(1198862minus 1198861) ge 120572 and (119886

2minus 119909)(119886

2minus 1198861015840

1) le 120573

Lemma 11 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Nes120583(119860119868

le 119861119868

) ge 120572 Nes] (119860119868

le 119861119868

) le 120573

lArrrArr1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

ge 120572

1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

le 120573

(18)

Proof Let us consider

Nes120583(119860119868

le 119861119868

) ge 120572 Nes] (119860119868

le 119861119868

) le 120573 (19)

Now from (10)

Nes120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

ge 120572

Nes] (119860119868

le 119861119868

) le 120573 lArrrArr1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

le 120573

(20)

Note Nes120583(119860119868

le 119909) ge 120572 and Nes](119860119868

le 119909) le 120573 hArr (119909 minus

1198863)(1198864minus 1198863) ge 120572 and (119886

1015840

4minus 119909)(119886

1015840

4minus 1198863) le 120573

Lemma 12 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Cr120583(119860119868

le 119861119868

) ge 120572 Cr] (119860119868

le 119861119868

) le 120573

lArrrArr1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)ge 120572

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)ge 120572

21198862minus 21198871015840

3+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)le 120573

1198861015840

4minus 1198872

2 (1198861015840

4minus 1198863minus 1198872+ 1198871015840

1)le 120573

(21)

Proof Let us consider

Cr120583(119860119868

le 119861119868

) ge 120572 Cr] (119860119868

le 119861119868

) le 120573 (22)

Now from (14)

Cr120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)ge 120572

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)ge 120572

Cr] (119860119868

le 119861119868

) le 120573 lArrrArr21198862minus 21198871015840

3+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)le 120573

1198872minus 1198861015840

4

2 (1198872minus 1198871015840

1minus 1198861015840

4+ 1198863)le 120573

(23)

Note Cr120583(119860119868

le 119909) ge 120572 Cr](119860119868

le 119909) ge 120573 hArr (119909 minus 1198861)2(1198862minus

1198861) ge 120572 (119886

4minus21198863+119909)2(119886

4minus1198863) ge 120572 and (2119886

2minus119909minus119886

1015840

1)2(1198862minus

1198861015840

1) le 120573 (1198861015840

4minus 119909)2(119886

1015840

4minus 1198863) le 120573

4 Intuitionistic Fuzzy CCM

The chance operator is actually taken as possibility or neces-sity or credibility measures We can use chance operatorto transform the intuitionistic fuzzy problem into crispproblem which is called as CCM [27] A general single-objective mathematical programming problem with intu-itionistic fuzzy parameter should have the following form

Max 119891 (119909 120585119868

)

subject to 119892119894(119909 120585119868

) le 119868

119894 119894 = 1 2 119899

119909 ge 0

(24)

where 119909 is the decision vector 120585119868 and 119868

119894are intuitionistic

fuzzy parameters 119891(119909 120585119868) is an imprecise objective functionand 119892

119894(119909 120585119868

) are constraints function for 119894 = 1 2 119899The general chance-constraints model for problem (24) is

as followsMax 119891

1+ 1198912

subject to Ch120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Ch] 119891 (119909 120585119868

) ge 1198912 le 120573

Ch120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Ch] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(25)

The abbreviations Ch120583and Ch] represent chance operator

(ie Pos or Nec measure) for membership and nonmem-bership functions 120572 120573 120582

119894 and 120595

119894are the predetermined

confidence levels such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

International Journal of Engineering Mathematics 7

41 Intuitionistic Fuzzy CCM Based on Possibility MeasureThe CCM based on possibility measure is as follows

Max 1198911+ 1198912

Subject to Pos120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Pos] 119891 (119909 120585119868

) ge 1198912 le 120573

Pos120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Pos] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(26)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 13 A solution 119909lowast of the problem (26) satisfies

Pos120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Pos](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution at (120582119894 120595119894) possibility

levels 119894 = 1 2 119899

Definition 14 A feasible solution at (120582119894 120595119894) possibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (26)

if and only if there exists no other feasible solution at(120582119894 120595119894) possibility levels such that Pos

120583119891(119909 120585

119868

) ge 120572 andPos]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

42 Intuitionistic Fuzzy CCM Based on Necessity MeasureThe CCM based on necessity measure is as follows

Max 1198911+ 1198912

Subject to Nes120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Nes] 119891 (119909 120585119868

) ge 1198912 le 120573

Nes120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Nes] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(27)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 15 A solution 119909lowast of the problem (27) satisfies

Nes120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Nes](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) necessity

levels 119894 = 1 2 119899

Definition 16 A feasible solution at (120582119894 120595119894) necessity levels

119909lowast is said to be (120572 120573) efficient solution for problem (27)

if and only if there exists no other feasible solution at(120582119894 120595119894) necessity levels such that Nes

120583119891(119909 120585

119868

) ge 120572 andNes]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

43 Intuitionistic Fuzzy CCM Based on Credibility MeasureThe CCM based on credibility measure is as follows

Max 1198911+ 1198912

Subject to Cr120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Cr] 119891 (119909 120585119868

) ge 1198912 le 120573

Cr120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Cr] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(28)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 17 A solution 119909lowast of the problem (28) satisfies

Cr120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Cr](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) credibility

levels 119894 = 1 2 119899

Definition 18 A feasible solution at (120582119894 120595119894) credibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (28)

if and only if there exists no other feasible solution at(120582119894 120595119894) credibility levels such that Cr

120583119891(119909 120585

119868

) ge 120572 andCr]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

5 Proposed Method to Solve IFLPPUsing Chance Operator

To solve intuitionistic fuzzy CCM based on possibility ornecessity or credibility measures we propose the followingmethod

Step 1 Apply chance operator possibilitynecessitycredi-bility in intuitionistic fuzzy programming (24) Problem (24)can be converted into following problem

Max 1198911+ 1198912

Subject to Pos120583119891 (119909I

119868

) ge 1198911 ge 120572

or Nes120583119891 (119909I

119868

) ge 1198911 ge 120572

or Cr120583119891 (119909I

119868

) ge 1198911 ge 120572

(29)

Pos] 119891 (119909I119868

) ge 1198912 le 120573

or Nes] 119891 (119909I119868

) ge 1198912 le 120573

or Cr] 119891 (119909I119868

) ge 1198912 le 120573

(30)

8 International Journal of Engineering Mathematics

Pos120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Nes120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Cr120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

(31)

Pos] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Nes] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Cr] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

(32)

119909 ge 0 for 119894 = 1 2 119899 (33)

0 le 120582119894+ 120595119894le 1 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

(34)

where (120582119894 120595119894) and (120572 120573) are the predefined confidence levels

Step 2 Using Lemmas 10 11 andor Lemma 12 the aboveproblem in Step 1 can also be written as

Max 1198911+ 1198912

Subject to 1198911+ 1198912ge 119885

(31) ndash (34)

(35)

where 119885 is obtained by applying Lemmas 10 11 andorLemma 12 in (30) and (29)

Step 3 The above problem is equivalent to

Max 119885

Subject to (31) ndash (34)

(36)

Step 4 Crisp programming problem obtained in Step 2 canbe solved using any well-known method to get the optimalsolution

6 Numerical Example

Let us consider the following intuitionistic fuzzy mathemati-cal programming problem as

Maximize 120585119868

11199091oplus 120585119868

21199092

subject to 120585119868

31199091oplus 120585119868

41199092⪯ 120585119868

5

120585119868

61199091oplus 120585119868

71199092⪯ 120585119868

8

1199091 1199092ge 0

(37)

where 120585119868

1= (5 6 7 8)(4 6 7 9) 120585119868

2= (4 5 6 7)(3 5 6 8)

120585119868

3= (1 2 3 4)(05 2 3 6) 120585119868

4= (2 3 4 5)(1 3 4 6) 120585119868

5=

(6 7 8 9)(5 7 8 10) 1205851198686= (3 4 5 6)(2 4 5 7) 120585119868

7= (1 2 3

4)(0 2 3 4) and 120585119868

8= (10 11 12 14)(9 11 12 16)

61 Intuitionistic Fuzzy CCM Based on Possibility MeasureNowby using Step 2 of themethod explained in Section 4 and

Start Applychance operator

Convertinto crisp

Writeequivalent Solve

Figure 7 Flow chart of the proposed algorithm

Lemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (37) problem (37) is con-verted into the following crisp programming problem

Maximize (15 minus 120572 + 2120573) 1199091+ (13 minus 120572 + 2120573) 119909

2

subject to (1 + 1205821) 1199091+ (2 + 120582

1) 1199092le 9 minus 120582

1

(2 minus 151205951) 1199091+ (3 minus 2120595

1) 1199092le 8 + 2120595

1

(3 + 1205822) 1199091+ (1 + 120582

2) 1199092le 14 minus 2120582

2

(4 minus 21205952) 1199091+ (2 minus 2120595

2) 1199092le 12 + 4120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(38)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and different possibility levels we get differentoptimal solutions Optimal solution of (38) at differentpossibility levels (in Figure 7) are presented in Table 1 FromTable 1 we can observe that maximum value (= 7309) can beobtained at (120582

1= 040 120595

1= 035) and (120582

2= 030 120595

2= 040)

possibility levels

62 Intuitionistic Fuzzy CCM Based on Necessity MeasureNowby using Step 2 of themethod explained in Section 4 andLemma 11 if we apply the necessity measure in (37) problem(37) is converted into following crisp programming problem

Maximize (10 minus 120572 + 2120573) 1199091+ (8 minus 120572 + 2120573) 119909

2

subject to (3 + 21205821) 1199091+ (4 + 2120582

1) 1199092le 6 + 120582

1

(5 minus 21205951) 1199091+ (6 minus 2120595

1) 1199092le 7 minus 2120595

1

(5 + 1205822) 1199091+ (3 + 120582

2) 1199092le 6 + 120582

2

(7 minus 21205952) 1199091+ (4 minus 120595

2) 1199092le 11 minus 2120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(39)

Solving the above crisp linear programming problem forefficient levels (120572 = 06 120573 = 04) and different necessitylevels we get different optimal solutions Optimal solutionsof (39) at different necessity levels (in Figures 8 and 9) arepresented in Table 2 From Table 2 we can observed that at(1205821= 035 120595

1= 045) and (120582

2= 035 120595

2= 045) the decision

maker will get the maximum value = 1298

International Journal of Engineering Mathematics 9

Table 1 Optimal solution of (38) at different possibility levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)030 030 035 035 119909

1= 341 119909

2= 137 7009

030 035 035 040 1199091= 329 119909

2= 166 7214

040 035 030 040 1199091= 343 119909

2= 157 7309

035 040 040 030 1199091= 310 119909

2= 19 722

040 045 040 045 1199091= 316 119909

2= 173 7105

050 045 045 05 1199091= 316 119909

2= 150 6793

045 050 050 040 1199091= 297 119909

2= 173 6802

060 045 045 06 1199091= 329 119909

2= 120 6593

050 050 050 050 1199091= 303 119909

2= 157 6700

055 040 055 040 1199091= 297 119909

2= 150 6510

Table 2 Optimal solution of (39) at different necessity levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)035 030 030 035 119909

1= 091 119909

2= 043 1293

045 030 040 035 1199091= 090 119909

2= 045 1289

050 035 050 035 1199091= 087 119909

2= 047 1286

060 040 060 040 1199091= 085 119909

2= 050 1284

070 020 070 020 1199091= 087 119909

2= 045 1271

060 030 060 020 1199091= 087 119909

2= 047 1279

065 035 075 020 1199091= 084 119909

2= 050 1275

070 010 070 010 1199091= 089 119909

2= 043 1267

050 050 050 050 1199091= 085 119909

2= 051 1294

035 045 035 045 1199091= 088 119909

2= 048 1298

Table 3 Input data for IFTP

1198631

119868

1198632

119868

1198633

119868 Availability (119886119894

119868

)

1198781

119868 (2 4 6 7) (1 4 6 9) (4 6 7 8) (3 6 7 9) (3 7 9 12) (2 7 9 13) (4 6 8 9) (2 6 8 10)1198782

119868 (1 3 4 6) (05 3 4 7) (3 5 6 7) (2 5 6 9) (2 6 7 11) (1 6 7 12) (0 05 1 2) (0 05 1 5)1198783

119868 (3 4 5 8) (2 4 5 10) (1 2 3 4) (05 2 3 5) (2 4 5 10) (1 4 5 11) (8 95 10 11) (65 95 10 11)Demand (119887

119895

119868

) (6 7 8 10) (5 7 8 12) (4 5 6 9) (3 5 6 11) (2 4 5 7) (05 4 5 8)

646668707274

(030 030)

(035

035

)

(035

040

)

(030

040

) (055 040)

(050 050)

(060 045)

(045 050)

(050 045)(0

40 0

30)

(040

045

)

(045

050

)

(050

040

)

(045

060

)

(045

045

)

(030

030

)(040 045)

(035 040)

(040 035)

(030 035)(1205821 1205951

)(1205822 120595

2 )

flowast

Figure 8 Optimal solution at different possibility levels

10 International Journal of Engineering Mathematics

1265127

1275128

1285129

129513

(035 030)

(050

035

)

(060

040

)

(070

020

)

(060

020

)

(075

020

)

(070

010

)

(050

050

)

(035

045

)

(035 045)

(050 050)

(070 010)

(065 035)

(060 030)

(070 020)

(060 040)

(050 035)

(045 030)

(040

035

)

(030

035

)

(1205821 1205951)

(1205822 120595

2 )

flowast

Figure 9 Optimal solution at different necessity levels

7 Intuitionistic Fuzzy TransportationProblem Based on Possibility Measure

Let us consider the following intuitionistic fuzzy transporta-tion problem (IFTP) (in Table 3)

Above transportation problem is a balanced transporta-tion problem as ⨁3

119894=1119886119868

119894= ⨁3

119895=1119868

119895 The above IFTP can be

written as

Minimize (2 4 6 7) (1 4 6 9) 11990911

oplus (4 6 7 8) (3 6 7 9) 11990912

oplus (3 7 9 12) (2 7 9 13) 11990913

oplus (1 3 4 6) (05 3 4 7) 11990921

oplus (3 5 6 7) (2 5 6 9) 11990922

oplus (2 6 7 11) (1 6 7 12) 11990923

oplus (3 4 5 8) (2 4 5 10) 11990931

oplus (1 2 3 4) (05 2 3 5) 11990932

oplus (2 4 5 10) (1 4 5 11) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (4 6 8 9) (2 6 8 10)

11990921

+ 11990922

+ 11990923

le (0 05 1 2) (0 05 1 5)

11990931

+ 11990932

+ 11990933

le (8 95 10 11) (65 95 10 11)

11990911

+ 11990921

+ 11990931

ge (6 7 8 10) (5 7 8 12)

11990912

+ 11990922

+ 11990932

ge (4 5 6 9) (3 5 6 11)

11990913

+ 11990923

+ 11990933

ge (2 4 5 7) (05 4 5 8)

119909119894119895ge 0 forall119894 119895

(40)

Nowby using Step 2 of themethod explained in Section 4 andLemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (40) problem (40) isconverted into following crisp programming problem

Minimize (6 + 2120572 minus 3120573) 11990911

+ (10 + 2120572 minus 2120573) 11990912

+ (10 + 4120572 minus 5120573) 11990913

+ (4 + 2120572 minus 25120573) 11990921

+ (8 + 2120572 minus 3120573) 11990922

+ (8 + 4120572 minus 5120573) 11990923

+ (7 + 120572 minus 5120573) 11990931

+ (3 + 120572 minus 15120573) 11990932

+ (6 + 2120572 minus 3120573) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (17 minus 1205821+ 21205951)

11990921

+ 11990922

+ 11990923

le (3 minus 1205822+ 41205952)

11990931

+ 11990932

+ 11990933

le (21 minus 1205823+ 1205953)

11990911

+ 11990921

+ 11990931

ge (13 + 1205824minus 21205954)

11990912

+ 11990922

+ 11990932

ge (9 + 1205825minus 21205955)

11990913

+ 11990923

+ 11990933

ge (6 + 21205826minus 35120595

6)

119909119894119895ge 0 for 119894 119895 = 1 2 3 0 le 120572 + 120573 le 1

0 le 120582119896+ 120595119896le 1 for 119896 = 1 2 6

(41)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and possibility levels (120582119894= 05 120595

119894= 05) for

119894 = 1 2 6 using Lingo-110 we get 11990911

= 075 11990912

= 011990913

= 0 11990921

= 45 11990922

= 0 11990923

= 0 11990931

= 725 11990932

= 85 and

International Journal of Engineering Mathematics 11

11990933

= 525 Now the minimum intuitionstic fuzzy optimalcost is

119888119868

= (7225 109 136 2035) (4562 109 136 24625) (42)

8 Discussion

Intuitionistic fuzzy sets being a generalization of fuzzy setsgive us an additional possibility to represent imperfect knowl-edge making it possible to describe many real problems ina more adequate way So in this paper we have developedthe possibility and necessity measures on intuitionistic fuzzyset Here we have presented first time the mathematicalrepresentation of different types of measures in intuitionisticfuzzy environments and some graphical representations ofthemare depictedWehave also developed the theoretical cal-culation on possibility necessity and credibility measures fordefuzzify intuitionistic fuzzy linear programming problemusing chance operators To validate the proposed methodwe have discussed three different approaches to defuzzifythe intuitionistic fuzzy relations using possibility necessityand credibility measures Using chance operator we canconvert a problem under imprecise models to correspondingcrisp models At different levels of possibility necessity andcredibility we have achieved different optimal solution Anumerical example is presented and solved using LINGO-110to illustrate the proposed approaches The proposed methodcan be applied for multiobjective multiitem transportationproblem This method can be also extended to be appliedinto different types of optimization problem namely optimalcontrol and solid transportation problems

Conflict of Interests

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

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov Intuitionistic Fuzzy Sets VII ITKRrsquos SessionSofia Bulgarian 1983

[3] S K De and S S Sana ldquoA multi-periods production-inventorymodel with capacity constraints for multi-manufacturersmdashaglobal optimality in intuitionistic fuzzy environmentrdquo AppliedMathematics andComputation vol 242 no 1 pp 825ndash841 2014

[4] HGargM Rani S P Sharma andYVishwakarma ldquoIntuition-istic fuzzy optimization technique for solving multi-objectivereliability optimization problems in interval environmentrdquoExpert Systems with Applications vol 41 no 7 pp 3157ndash31672014

[5] J Wu and Y Liu ldquoAn approach for multiple attribute groupdecision making problems with interval-valued intuitionistictrapezoidal fuzzy numbersrdquoComputers and Industrial Engineer-ing vol 66 no 2 pp 311ndash324 2013

[6] A Nagoorgani and K Ponnalagu ldquoA new approach on solv-ing intuitionistic fuzzy linear programming problemrdquo AppliedMathematical Sciences vol 6 no 70 pp 3467ndash3474 2012

[7] G S Mahapatra and T K Roy ldquoIntuitionistic fuzzy numberand its arithmetic operation with application on system failurerdquoJournal of Uncertain Systems vol 7 no 2 pp 92ndash107 2013

[8] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[9] S Pramanik D K Jana and M Maiti ldquoA multi objectivesolid transportation problem in fuzzy Bi-fuzzy environmentvia genetic algorithmrdquo International Journal of Advanced Oper-ations Management vol 6 no 1 pp 4ndash26 2014

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

[11] M Esmailzadeh and M Esmailzadeh ldquoNew distance betweentriangular intuitionistic fuzzy numbersrdquo Advances in Computa-tional Mathematics and Its Applications vol 2 no 3 pp 3ndash102013

[12] P P Angelov ldquoOptimization in an intuitionistic fuzzy environ-mentrdquo Fuzzy Sets and Systems vol 86 no 3 pp 299ndash306 1997

[13] D Dubey and A Mehra ldquoLinear programming with triangularintuitionistic fuzzy numberrdquo Advances in Intelligent SystemsResearch vol 1 no 1 pp 563ndash569 2011

[14] R Parvathi and C Malathi ldquoIntuitionistic fuzzy simplexmethodrdquo International Journal of Computer Applications vol48 no 6 pp 39ndash48 2012

[15] R J Hussain and S P Kumar ldquoAlgorithmic approach forsolving intuitionistic fuzzy transportation problemrdquo AppliedMathematical Sciences vol 6 no 77-80 pp 3981ndash3989 2012

[16] ANagoorGani and S Abbas ldquoSolving intuitionstic fuzzy trans-portation problem using zero suffix algorithmrdquo InternationalJournal of Mathematics Sciences amp Enggineering Applicationsvol 6 pp 73ndash82 2012

[17] J Ye ldquoExpected value method for intuitionistic trapezoidalfuzzy multicriteria decision-making problemsrdquo Expert Systemswith Applications vol 38 no 9 pp 11730ndash11734 2011

[18] 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 2013

[19] J J Buckley ldquoPossibility and necessity in optimizationrdquo FuzzySets and Systems vol 25 no 1 pp 1ndash13 1988

[20] K D Jamison and W A Lodwick ldquoThe construction ofconsistent possibility and necessity measuresrdquo Fuzzy Sets andSystems vol 132 no 1 pp 1ndash10 2002

[21] J Ramık ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006

[22] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[23] M Sakawa H Katagiri and T Matsui ldquoFuzzy random bilevellinear programming through expectation optimization usingpossibility and necessityrdquo International Journal of MachineLearning and Cybernetics vol 3 no 3 pp 183ndash192 2012

[24] S Pathak S Kar and S Sarkar ldquoFuzzy production inventorymodel for deteriorating items with shortages under the effect oftime dependent learning and forgetting a possibilitynecessityapproachrdquo OPSEARCH vol 50 no 2 pp 149ndash181 2013

[25] K Maity ldquoPossibility and necessity representations of fuzzyinequality and its application to two warehouse production-inventory problemrdquo Applied Mathematical Modelling vol 35no 3 pp 1252ndash1263 2011

12 International Journal of Engineering Mathematics

[26] H-C Wu ldquoFuzzy optimization problems based on the embed-ding theorem and possibility and necessity measuresrdquo Mathe-matical and Computer Modelling vol 40 no 3-4 pp 329ndash3362004

[27] J Xu and X Zhou Fuzzy-Like Multiple Objective DecesionMaking Springer 2010

[28] K Maity and M Maiti ldquoPossibility and necessity constraintsand their defuzzificationmdasha multi-item production-inventoryscenario via optimal control theoryrdquo European Journal of Oper-ational Research vol 177 no 2 pp 882ndash896 2007

[29] B Das K Maity and M Maiti ldquoA two warehouse supply-chain model under possibilitynecessitycredibility measuresrdquoMathematical andComputerModelling vol 46 no 3-4 pp 398ndash409 2007

[30] D Panda S Kar K Maity and M Maiti ldquoA single periodinventory model with imperfect production and stochasticdemand under chance and imprecise constraintsrdquo EuropeanJournal of Operational Research vol 188 no 1 pp 121ndash139 2008

[31] A I Ban ldquoIntuitionistic fuzzy-valued possibility and necessitymeasuresrdquo in Proceedings of the 8th International Conference onIntuitionistic Fuzzy Sets vol 10 pp 1ndash7 Varna Bulgaria 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Engineering Mathematics 5

By Definition 8 necessity of the event 119860119868 ge 119861119868 are as follows

Nes120583(119860119868

ge 119861119868

) = 1 minus Pos120583(119860119868

lt 119861119868

)

=

0 1198862le 1198874

1198862minus 1198874

1198862minus 1198861minus 1198874+ 1198873

1198862gt 1198874 1198861lt 1198873

1 1198861ge 1198873

Nes] (119860119868

ge 119861119868

) = 1 minus Pos] (119860119868

lt 119861119868

)

=

0 1198873le 1198861015840

1

1198873minus 1198861015840

1

1198862minus 1198861015840

1minus 1198871015840

4+ 1198873

1198873gt 1198861015840

1 1198862gt 1198871015840

4

1 1198862le 1198871015840

4

(11)

By Definition 9 measures of the event 119860119868 le 119861119868 are as follows

Me120583(119860119868

le 119861119868

)

= 120582Pos120583(119860119868

le 119861119868

) + (1 minus 120582)Nes120583(119860119868

le 119861119868

)

=

0 1198874le 1198861

1205821198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

1198873gt 1198862 1198871lt 1198862

120582 1198873gt 1198862 1198871lt 1198863

120582 + (1 minus 120582)1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Me] (119860119868

le 119861119868

)

= 120582Pos] (119860119868

le 119861119868

) + (1 minus 120582)Nes] (119860119868

le 119861119868

)

=

1 1198871015840

4le 1198861015840

1

1205821198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

+ (1 minus 120582) 1198871015840

4gt 1198861015840

1 1198862gt 1198873

(1 minus 120582) 1198862lt 1198873 1198871015840

1lt 1198863

(1 minus 120582)1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

1198863lt 1198871015840

1 1198861015840

4gt 1198872

0 1198872ge 1198861015840

4

(12)

By Definition 9 measures of the event 119860119868 ge 119861119868 are as follows

Me120583(119860119868

ge 119861119868

)

= 120582Pos120583(119860119868

ge 119861119868

) + (1 minus 120582)Nes120583(119860119868

ge 119861119868

)

=

1 1198873le 1198862

(1 minus 120582)1198862minus 1198874

1198862minus 1198861minus 1198874+ 1198873

+ 120582 1198873gt 1198861 1198874lt 1198862

120582 1198874gt 1198862 1198872lt 1198863

1205821198864minus 1198871

1198864minus 1198863+ 1198872minus 1198871

1198863lt 1198862 1198864gt 1198871

0 1198871ge 1198864

Me] (119860119868

ge 119861119868

)

= 120582Pos] (119860119868

ge 119861119868

) + (1 minus 120582)Nes] (119860119868

ge 119861119868

)

=

0 1198873le 1198861015840

1

(1 minus 120582)1198873minus 1198861015840

1

1198862minus 1198861015840

1minus 1198871015840

4+ 1198873

1198873gt 1198861015840

1 1198862gt 1198871015840

4

(1 minus 120582) 1198862lt 1198871015840

4 1198872lt 1198863

(1 minus 120582) + 1205821198872minus 1198863

1198872minus 1198871015840

1+ 1198861015840

4minus 1198863

1198863lt 1198872 1198861015840

4gt 1198871015840

1

1 1198871015840

1ge 1198861015840

4

(13)

For 120582 = 05

Cr120583(119860119868

le119861119868

)=

0 1198871le 1198861

1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)

1198874gt 1198861 1198862gt 1198873

1

2

1198873gt 1198862 1198871lt 1198863

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)

1198871gt 1198863 1198864gt 1198872

1 1198872ge 1198864

Cr] (119860119868

le119861119868

)=

1 1198871015840

4le 1198861015840

1

21198862minus 21198873+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)

1198871015840

4gt 1198861015840

1 1198862gt 1198873

1

2

1198862lt 1198873 1198871015840

1gt 1198863

1198861015840

4minus 1198872

2 (1198861015840

4minus 1198863minus 1198872+ 1198871015840

1)

1198863lt 1198871015840

1 1198872lt 1198861015840

4

0 1198872ge 1198861015840

4

Cr120583(119860119868

ge119861119868

)=

1 1198873le 1198862

21198862minus 21198874+ 1198873minus 1198861

2 (1198862minus 1198861minus 1198874+ 1198873)

1198873gt 1198861 1198862gt 1198874

1

2

1198874gt 1198862 1198872lt 1198863

1198864minus 1198871

2 (1198864minus 1198863+ 1198872minus 1198871)

1198872gt 1198863 1198864gt 1198871

0 1198871ge 1198864

Cr] (119860119868

ge119861119868

)=

0 1198871015840

3le 1198861015840

1

1198873minus 1198861015840

1

2 (1198862minus 1198861015840

1minus 1198871015840

4+ 1198873)

1198873gt 1198861015840

1 1198862gt 1198871015840

4

1

2

1198862lt 1198871015840

4 1198872lt 1198863

21198872minus 21198863+ 1198861015840

4minus 1198871015840

1

2 (1198872minus 1198871015840

1+ 1198861015840

4minus 1198863)

1198872gt 1198863 1198871015840

1lt 1198861015840

4

1 1198871015840

1ge 1198861015840

4

(14)

Lemma 10 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Pos120583(119860119868

le 119861119868

) ge 120572 Pos] (119860119868

le 119861119868

) le 120573

lArrrArr1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

ge 120572

1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

le 120573

(15)

6 International Journal of Engineering Mathematics

Proof Let us consider

Pos120583(119860119868

le 119861119868

) ge 120572 Pos] (119860119868

ge 119861119868

) le 120573 (16)

Now from (8)

Pos120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

ge 120572

Pos] (119860119868

le 119861119868

) le 120573 lArrrArr1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

le 120573

(17)

Note Pos120583(119860119868

le 119909) ge 120572 and Pos](119860119868

le 119909) le 120573 hArr (119909 minus

1198861)(1198862minus 1198861) ge 120572 and (119886

2minus 119909)(119886

2minus 1198861015840

1) le 120573

Lemma 11 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Nes120583(119860119868

le 119861119868

) ge 120572 Nes] (119860119868

le 119861119868

) le 120573

lArrrArr1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

ge 120572

1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

le 120573

(18)

Proof Let us consider

Nes120583(119860119868

le 119861119868

) ge 120572 Nes] (119860119868

le 119861119868

) le 120573 (19)

Now from (10)

Nes120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

ge 120572

Nes] (119860119868

le 119861119868

) le 120573 lArrrArr1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

le 120573

(20)

Note Nes120583(119860119868

le 119909) ge 120572 and Nes](119860119868

le 119909) le 120573 hArr (119909 minus

1198863)(1198864minus 1198863) ge 120572 and (119886

1015840

4minus 119909)(119886

1015840

4minus 1198863) le 120573

Lemma 12 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Cr120583(119860119868

le 119861119868

) ge 120572 Cr] (119860119868

le 119861119868

) le 120573

lArrrArr1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)ge 120572

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)ge 120572

21198862minus 21198871015840

3+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)le 120573

1198861015840

4minus 1198872

2 (1198861015840

4minus 1198863minus 1198872+ 1198871015840

1)le 120573

(21)

Proof Let us consider

Cr120583(119860119868

le 119861119868

) ge 120572 Cr] (119860119868

le 119861119868

) le 120573 (22)

Now from (14)

Cr120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)ge 120572

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)ge 120572

Cr] (119860119868

le 119861119868

) le 120573 lArrrArr21198862minus 21198871015840

3+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)le 120573

1198872minus 1198861015840

4

2 (1198872minus 1198871015840

1minus 1198861015840

4+ 1198863)le 120573

(23)

Note Cr120583(119860119868

le 119909) ge 120572 Cr](119860119868

le 119909) ge 120573 hArr (119909 minus 1198861)2(1198862minus

1198861) ge 120572 (119886

4minus21198863+119909)2(119886

4minus1198863) ge 120572 and (2119886

2minus119909minus119886

1015840

1)2(1198862minus

1198861015840

1) le 120573 (1198861015840

4minus 119909)2(119886

1015840

4minus 1198863) le 120573

4 Intuitionistic Fuzzy CCM

The chance operator is actually taken as possibility or neces-sity or credibility measures We can use chance operatorto transform the intuitionistic fuzzy problem into crispproblem which is called as CCM [27] A general single-objective mathematical programming problem with intu-itionistic fuzzy parameter should have the following form

Max 119891 (119909 120585119868

)

subject to 119892119894(119909 120585119868

) le 119868

119894 119894 = 1 2 119899

119909 ge 0

(24)

where 119909 is the decision vector 120585119868 and 119868

119894are intuitionistic

fuzzy parameters 119891(119909 120585119868) is an imprecise objective functionand 119892

119894(119909 120585119868

) are constraints function for 119894 = 1 2 119899The general chance-constraints model for problem (24) is

as followsMax 119891

1+ 1198912

subject to Ch120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Ch] 119891 (119909 120585119868

) ge 1198912 le 120573

Ch120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Ch] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(25)

The abbreviations Ch120583and Ch] represent chance operator

(ie Pos or Nec measure) for membership and nonmem-bership functions 120572 120573 120582

119894 and 120595

119894are the predetermined

confidence levels such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

International Journal of Engineering Mathematics 7

41 Intuitionistic Fuzzy CCM Based on Possibility MeasureThe CCM based on possibility measure is as follows

Max 1198911+ 1198912

Subject to Pos120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Pos] 119891 (119909 120585119868

) ge 1198912 le 120573

Pos120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Pos] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(26)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 13 A solution 119909lowast of the problem (26) satisfies

Pos120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Pos](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution at (120582119894 120595119894) possibility

levels 119894 = 1 2 119899

Definition 14 A feasible solution at (120582119894 120595119894) possibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (26)

if and only if there exists no other feasible solution at(120582119894 120595119894) possibility levels such that Pos

120583119891(119909 120585

119868

) ge 120572 andPos]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

42 Intuitionistic Fuzzy CCM Based on Necessity MeasureThe CCM based on necessity measure is as follows

Max 1198911+ 1198912

Subject to Nes120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Nes] 119891 (119909 120585119868

) ge 1198912 le 120573

Nes120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Nes] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(27)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 15 A solution 119909lowast of the problem (27) satisfies

Nes120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Nes](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) necessity

levels 119894 = 1 2 119899

Definition 16 A feasible solution at (120582119894 120595119894) necessity levels

119909lowast is said to be (120572 120573) efficient solution for problem (27)

if and only if there exists no other feasible solution at(120582119894 120595119894) necessity levels such that Nes

120583119891(119909 120585

119868

) ge 120572 andNes]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

43 Intuitionistic Fuzzy CCM Based on Credibility MeasureThe CCM based on credibility measure is as follows

Max 1198911+ 1198912

Subject to Cr120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Cr] 119891 (119909 120585119868

) ge 1198912 le 120573

Cr120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Cr] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(28)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 17 A solution 119909lowast of the problem (28) satisfies

Cr120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Cr](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) credibility

levels 119894 = 1 2 119899

Definition 18 A feasible solution at (120582119894 120595119894) credibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (28)

if and only if there exists no other feasible solution at(120582119894 120595119894) credibility levels such that Cr

120583119891(119909 120585

119868

) ge 120572 andCr]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

5 Proposed Method to Solve IFLPPUsing Chance Operator

To solve intuitionistic fuzzy CCM based on possibility ornecessity or credibility measures we propose the followingmethod

Step 1 Apply chance operator possibilitynecessitycredi-bility in intuitionistic fuzzy programming (24) Problem (24)can be converted into following problem

Max 1198911+ 1198912

Subject to Pos120583119891 (119909I

119868

) ge 1198911 ge 120572

or Nes120583119891 (119909I

119868

) ge 1198911 ge 120572

or Cr120583119891 (119909I

119868

) ge 1198911 ge 120572

(29)

Pos] 119891 (119909I119868

) ge 1198912 le 120573

or Nes] 119891 (119909I119868

) ge 1198912 le 120573

or Cr] 119891 (119909I119868

) ge 1198912 le 120573

(30)

8 International Journal of Engineering Mathematics

Pos120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Nes120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Cr120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

(31)

Pos] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Nes] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Cr] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

(32)

119909 ge 0 for 119894 = 1 2 119899 (33)

0 le 120582119894+ 120595119894le 1 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

(34)

where (120582119894 120595119894) and (120572 120573) are the predefined confidence levels

Step 2 Using Lemmas 10 11 andor Lemma 12 the aboveproblem in Step 1 can also be written as

Max 1198911+ 1198912

Subject to 1198911+ 1198912ge 119885

(31) ndash (34)

(35)

where 119885 is obtained by applying Lemmas 10 11 andorLemma 12 in (30) and (29)

Step 3 The above problem is equivalent to

Max 119885

Subject to (31) ndash (34)

(36)

Step 4 Crisp programming problem obtained in Step 2 canbe solved using any well-known method to get the optimalsolution

6 Numerical Example

Let us consider the following intuitionistic fuzzy mathemati-cal programming problem as

Maximize 120585119868

11199091oplus 120585119868

21199092

subject to 120585119868

31199091oplus 120585119868

41199092⪯ 120585119868

5

120585119868

61199091oplus 120585119868

71199092⪯ 120585119868

8

1199091 1199092ge 0

(37)

where 120585119868

1= (5 6 7 8)(4 6 7 9) 120585119868

2= (4 5 6 7)(3 5 6 8)

120585119868

3= (1 2 3 4)(05 2 3 6) 120585119868

4= (2 3 4 5)(1 3 4 6) 120585119868

5=

(6 7 8 9)(5 7 8 10) 1205851198686= (3 4 5 6)(2 4 5 7) 120585119868

7= (1 2 3

4)(0 2 3 4) and 120585119868

8= (10 11 12 14)(9 11 12 16)

61 Intuitionistic Fuzzy CCM Based on Possibility MeasureNowby using Step 2 of themethod explained in Section 4 and

Start Applychance operator

Convertinto crisp

Writeequivalent Solve

Figure 7 Flow chart of the proposed algorithm

Lemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (37) problem (37) is con-verted into the following crisp programming problem

Maximize (15 minus 120572 + 2120573) 1199091+ (13 minus 120572 + 2120573) 119909

2

subject to (1 + 1205821) 1199091+ (2 + 120582

1) 1199092le 9 minus 120582

1

(2 minus 151205951) 1199091+ (3 minus 2120595

1) 1199092le 8 + 2120595

1

(3 + 1205822) 1199091+ (1 + 120582

2) 1199092le 14 minus 2120582

2

(4 minus 21205952) 1199091+ (2 minus 2120595

2) 1199092le 12 + 4120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(38)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and different possibility levels we get differentoptimal solutions Optimal solution of (38) at differentpossibility levels (in Figure 7) are presented in Table 1 FromTable 1 we can observe that maximum value (= 7309) can beobtained at (120582

1= 040 120595

1= 035) and (120582

2= 030 120595

2= 040)

possibility levels

62 Intuitionistic Fuzzy CCM Based on Necessity MeasureNowby using Step 2 of themethod explained in Section 4 andLemma 11 if we apply the necessity measure in (37) problem(37) is converted into following crisp programming problem

Maximize (10 minus 120572 + 2120573) 1199091+ (8 minus 120572 + 2120573) 119909

2

subject to (3 + 21205821) 1199091+ (4 + 2120582

1) 1199092le 6 + 120582

1

(5 minus 21205951) 1199091+ (6 minus 2120595

1) 1199092le 7 minus 2120595

1

(5 + 1205822) 1199091+ (3 + 120582

2) 1199092le 6 + 120582

2

(7 minus 21205952) 1199091+ (4 minus 120595

2) 1199092le 11 minus 2120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(39)

Solving the above crisp linear programming problem forefficient levels (120572 = 06 120573 = 04) and different necessitylevels we get different optimal solutions Optimal solutionsof (39) at different necessity levels (in Figures 8 and 9) arepresented in Table 2 From Table 2 we can observed that at(1205821= 035 120595

1= 045) and (120582

2= 035 120595

2= 045) the decision

maker will get the maximum value = 1298

International Journal of Engineering Mathematics 9

Table 1 Optimal solution of (38) at different possibility levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)030 030 035 035 119909

1= 341 119909

2= 137 7009

030 035 035 040 1199091= 329 119909

2= 166 7214

040 035 030 040 1199091= 343 119909

2= 157 7309

035 040 040 030 1199091= 310 119909

2= 19 722

040 045 040 045 1199091= 316 119909

2= 173 7105

050 045 045 05 1199091= 316 119909

2= 150 6793

045 050 050 040 1199091= 297 119909

2= 173 6802

060 045 045 06 1199091= 329 119909

2= 120 6593

050 050 050 050 1199091= 303 119909

2= 157 6700

055 040 055 040 1199091= 297 119909

2= 150 6510

Table 2 Optimal solution of (39) at different necessity levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)035 030 030 035 119909

1= 091 119909

2= 043 1293

045 030 040 035 1199091= 090 119909

2= 045 1289

050 035 050 035 1199091= 087 119909

2= 047 1286

060 040 060 040 1199091= 085 119909

2= 050 1284

070 020 070 020 1199091= 087 119909

2= 045 1271

060 030 060 020 1199091= 087 119909

2= 047 1279

065 035 075 020 1199091= 084 119909

2= 050 1275

070 010 070 010 1199091= 089 119909

2= 043 1267

050 050 050 050 1199091= 085 119909

2= 051 1294

035 045 035 045 1199091= 088 119909

2= 048 1298

Table 3 Input data for IFTP

1198631

119868

1198632

119868

1198633

119868 Availability (119886119894

119868

)

1198781

119868 (2 4 6 7) (1 4 6 9) (4 6 7 8) (3 6 7 9) (3 7 9 12) (2 7 9 13) (4 6 8 9) (2 6 8 10)1198782

119868 (1 3 4 6) (05 3 4 7) (3 5 6 7) (2 5 6 9) (2 6 7 11) (1 6 7 12) (0 05 1 2) (0 05 1 5)1198783

119868 (3 4 5 8) (2 4 5 10) (1 2 3 4) (05 2 3 5) (2 4 5 10) (1 4 5 11) (8 95 10 11) (65 95 10 11)Demand (119887

119895

119868

) (6 7 8 10) (5 7 8 12) (4 5 6 9) (3 5 6 11) (2 4 5 7) (05 4 5 8)

646668707274

(030 030)

(035

035

)

(035

040

)

(030

040

) (055 040)

(050 050)

(060 045)

(045 050)

(050 045)(0

40 0

30)

(040

045

)

(045

050

)

(050

040

)

(045

060

)

(045

045

)

(030

030

)(040 045)

(035 040)

(040 035)

(030 035)(1205821 1205951

)(1205822 120595

2 )

flowast

Figure 8 Optimal solution at different possibility levels

10 International Journal of Engineering Mathematics

1265127

1275128

1285129

129513

(035 030)

(050

035

)

(060

040

)

(070

020

)

(060

020

)

(075

020

)

(070

010

)

(050

050

)

(035

045

)

(035 045)

(050 050)

(070 010)

(065 035)

(060 030)

(070 020)

(060 040)

(050 035)

(045 030)

(040

035

)

(030

035

)

(1205821 1205951)

(1205822 120595

2 )

flowast

Figure 9 Optimal solution at different necessity levels

7 Intuitionistic Fuzzy TransportationProblem Based on Possibility Measure

Let us consider the following intuitionistic fuzzy transporta-tion problem (IFTP) (in Table 3)

Above transportation problem is a balanced transporta-tion problem as ⨁3

119894=1119886119868

119894= ⨁3

119895=1119868

119895 The above IFTP can be

written as

Minimize (2 4 6 7) (1 4 6 9) 11990911

oplus (4 6 7 8) (3 6 7 9) 11990912

oplus (3 7 9 12) (2 7 9 13) 11990913

oplus (1 3 4 6) (05 3 4 7) 11990921

oplus (3 5 6 7) (2 5 6 9) 11990922

oplus (2 6 7 11) (1 6 7 12) 11990923

oplus (3 4 5 8) (2 4 5 10) 11990931

oplus (1 2 3 4) (05 2 3 5) 11990932

oplus (2 4 5 10) (1 4 5 11) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (4 6 8 9) (2 6 8 10)

11990921

+ 11990922

+ 11990923

le (0 05 1 2) (0 05 1 5)

11990931

+ 11990932

+ 11990933

le (8 95 10 11) (65 95 10 11)

11990911

+ 11990921

+ 11990931

ge (6 7 8 10) (5 7 8 12)

11990912

+ 11990922

+ 11990932

ge (4 5 6 9) (3 5 6 11)

11990913

+ 11990923

+ 11990933

ge (2 4 5 7) (05 4 5 8)

119909119894119895ge 0 forall119894 119895

(40)

Nowby using Step 2 of themethod explained in Section 4 andLemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (40) problem (40) isconverted into following crisp programming problem

Minimize (6 + 2120572 minus 3120573) 11990911

+ (10 + 2120572 minus 2120573) 11990912

+ (10 + 4120572 minus 5120573) 11990913

+ (4 + 2120572 minus 25120573) 11990921

+ (8 + 2120572 minus 3120573) 11990922

+ (8 + 4120572 minus 5120573) 11990923

+ (7 + 120572 minus 5120573) 11990931

+ (3 + 120572 minus 15120573) 11990932

+ (6 + 2120572 minus 3120573) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (17 minus 1205821+ 21205951)

11990921

+ 11990922

+ 11990923

le (3 minus 1205822+ 41205952)

11990931

+ 11990932

+ 11990933

le (21 minus 1205823+ 1205953)

11990911

+ 11990921

+ 11990931

ge (13 + 1205824minus 21205954)

11990912

+ 11990922

+ 11990932

ge (9 + 1205825minus 21205955)

11990913

+ 11990923

+ 11990933

ge (6 + 21205826minus 35120595

6)

119909119894119895ge 0 for 119894 119895 = 1 2 3 0 le 120572 + 120573 le 1

0 le 120582119896+ 120595119896le 1 for 119896 = 1 2 6

(41)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and possibility levels (120582119894= 05 120595

119894= 05) for

119894 = 1 2 6 using Lingo-110 we get 11990911

= 075 11990912

= 011990913

= 0 11990921

= 45 11990922

= 0 11990923

= 0 11990931

= 725 11990932

= 85 and

International Journal of Engineering Mathematics 11

11990933

= 525 Now the minimum intuitionstic fuzzy optimalcost is

119888119868

= (7225 109 136 2035) (4562 109 136 24625) (42)

8 Discussion

Intuitionistic fuzzy sets being a generalization of fuzzy setsgive us an additional possibility to represent imperfect knowl-edge making it possible to describe many real problems ina more adequate way So in this paper we have developedthe possibility and necessity measures on intuitionistic fuzzyset Here we have presented first time the mathematicalrepresentation of different types of measures in intuitionisticfuzzy environments and some graphical representations ofthemare depictedWehave also developed the theoretical cal-culation on possibility necessity and credibility measures fordefuzzify intuitionistic fuzzy linear programming problemusing chance operators To validate the proposed methodwe have discussed three different approaches to defuzzifythe intuitionistic fuzzy relations using possibility necessityand credibility measures Using chance operator we canconvert a problem under imprecise models to correspondingcrisp models At different levels of possibility necessity andcredibility we have achieved different optimal solution Anumerical example is presented and solved using LINGO-110to illustrate the proposed approaches The proposed methodcan be applied for multiobjective multiitem transportationproblem This method can be also extended to be appliedinto different types of optimization problem namely optimalcontrol and solid transportation problems

Conflict of Interests

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

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov Intuitionistic Fuzzy Sets VII ITKRrsquos SessionSofia Bulgarian 1983

[3] S K De and S S Sana ldquoA multi-periods production-inventorymodel with capacity constraints for multi-manufacturersmdashaglobal optimality in intuitionistic fuzzy environmentrdquo AppliedMathematics andComputation vol 242 no 1 pp 825ndash841 2014

[4] HGargM Rani S P Sharma andYVishwakarma ldquoIntuition-istic fuzzy optimization technique for solving multi-objectivereliability optimization problems in interval environmentrdquoExpert Systems with Applications vol 41 no 7 pp 3157ndash31672014

[5] J Wu and Y Liu ldquoAn approach for multiple attribute groupdecision making problems with interval-valued intuitionistictrapezoidal fuzzy numbersrdquoComputers and Industrial Engineer-ing vol 66 no 2 pp 311ndash324 2013

[6] A Nagoorgani and K Ponnalagu ldquoA new approach on solv-ing intuitionistic fuzzy linear programming problemrdquo AppliedMathematical Sciences vol 6 no 70 pp 3467ndash3474 2012

[7] G S Mahapatra and T K Roy ldquoIntuitionistic fuzzy numberand its arithmetic operation with application on system failurerdquoJournal of Uncertain Systems vol 7 no 2 pp 92ndash107 2013

[8] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[9] S Pramanik D K Jana and M Maiti ldquoA multi objectivesolid transportation problem in fuzzy Bi-fuzzy environmentvia genetic algorithmrdquo International Journal of Advanced Oper-ations Management vol 6 no 1 pp 4ndash26 2014

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

[11] M Esmailzadeh and M Esmailzadeh ldquoNew distance betweentriangular intuitionistic fuzzy numbersrdquo Advances in Computa-tional Mathematics and Its Applications vol 2 no 3 pp 3ndash102013

[12] P P Angelov ldquoOptimization in an intuitionistic fuzzy environ-mentrdquo Fuzzy Sets and Systems vol 86 no 3 pp 299ndash306 1997

[13] D Dubey and A Mehra ldquoLinear programming with triangularintuitionistic fuzzy numberrdquo Advances in Intelligent SystemsResearch vol 1 no 1 pp 563ndash569 2011

[14] R Parvathi and C Malathi ldquoIntuitionistic fuzzy simplexmethodrdquo International Journal of Computer Applications vol48 no 6 pp 39ndash48 2012

[15] R J Hussain and S P Kumar ldquoAlgorithmic approach forsolving intuitionistic fuzzy transportation problemrdquo AppliedMathematical Sciences vol 6 no 77-80 pp 3981ndash3989 2012

[16] ANagoorGani and S Abbas ldquoSolving intuitionstic fuzzy trans-portation problem using zero suffix algorithmrdquo InternationalJournal of Mathematics Sciences amp Enggineering Applicationsvol 6 pp 73ndash82 2012

[17] J Ye ldquoExpected value method for intuitionistic trapezoidalfuzzy multicriteria decision-making problemsrdquo Expert Systemswith Applications vol 38 no 9 pp 11730ndash11734 2011

[18] 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 2013

[19] J J Buckley ldquoPossibility and necessity in optimizationrdquo FuzzySets and Systems vol 25 no 1 pp 1ndash13 1988

[20] K D Jamison and W A Lodwick ldquoThe construction ofconsistent possibility and necessity measuresrdquo Fuzzy Sets andSystems vol 132 no 1 pp 1ndash10 2002

[21] J Ramık ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006

[22] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[23] M Sakawa H Katagiri and T Matsui ldquoFuzzy random bilevellinear programming through expectation optimization usingpossibility and necessityrdquo International Journal of MachineLearning and Cybernetics vol 3 no 3 pp 183ndash192 2012

[24] S Pathak S Kar and S Sarkar ldquoFuzzy production inventorymodel for deteriorating items with shortages under the effect oftime dependent learning and forgetting a possibilitynecessityapproachrdquo OPSEARCH vol 50 no 2 pp 149ndash181 2013

[25] K Maity ldquoPossibility and necessity representations of fuzzyinequality and its application to two warehouse production-inventory problemrdquo Applied Mathematical Modelling vol 35no 3 pp 1252ndash1263 2011

12 International Journal of Engineering Mathematics

[26] H-C Wu ldquoFuzzy optimization problems based on the embed-ding theorem and possibility and necessity measuresrdquo Mathe-matical and Computer Modelling vol 40 no 3-4 pp 329ndash3362004

[27] J Xu and X Zhou Fuzzy-Like Multiple Objective DecesionMaking Springer 2010

[28] K Maity and M Maiti ldquoPossibility and necessity constraintsand their defuzzificationmdasha multi-item production-inventoryscenario via optimal control theoryrdquo European Journal of Oper-ational Research vol 177 no 2 pp 882ndash896 2007

[29] B Das K Maity and M Maiti ldquoA two warehouse supply-chain model under possibilitynecessitycredibility measuresrdquoMathematical andComputerModelling vol 46 no 3-4 pp 398ndash409 2007

[30] D Panda S Kar K Maity and M Maiti ldquoA single periodinventory model with imperfect production and stochasticdemand under chance and imprecise constraintsrdquo EuropeanJournal of Operational Research vol 188 no 1 pp 121ndash139 2008

[31] A I Ban ldquoIntuitionistic fuzzy-valued possibility and necessitymeasuresrdquo in Proceedings of the 8th International Conference onIntuitionistic Fuzzy Sets vol 10 pp 1ndash7 Varna Bulgaria 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Engineering Mathematics

Proof Let us consider

Pos120583(119860119868

le 119861119868

) ge 120572 Pos] (119860119868

ge 119861119868

) le 120573 (16)

Now from (8)

Pos120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198874minus 1198861

1198874minus 1198873+ 1198862minus 1198861

ge 120572

Pos] (119860119868

le 119861119868

) le 120573 lArrrArr1198862minus 1198873

1198862minus 1198861015840

1+ 1198871015840

4minus 1198873

le 120573

(17)

Note Pos120583(119860119868

le 119909) ge 120572 and Pos](119860119868

le 119909) le 120573 hArr (119909 minus

1198861)(1198862minus 1198861) ge 120572 and (119886

2minus 119909)(119886

2minus 1198861015840

1) le 120573

Lemma 11 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Nes120583(119860119868

le 119861119868

) ge 120572 Nes] (119860119868

le 119861119868

) le 120573

lArrrArr1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

ge 120572

1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

le 120573

(18)

Proof Let us consider

Nes120583(119860119868

le 119861119868

) ge 120572 Nes] (119860119868

le 119861119868

) le 120573 (19)

Now from (10)

Nes120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198871minus 1198863

1198864minus 1198863minus 1198872+ 1198871

ge 120572

Nes] (119860119868

le 119861119868

) le 120573 lArrrArr1198861015840

4minus 1198872

1198861015840

4minus 1198863minus 1198872+ 1198871015840

1

le 120573

(20)

Note Nes120583(119860119868

le 119909) ge 120572 and Nes](119860119868

le 119909) le 120573 hArr (119909 minus

1198863)(1198864minus 1198863) ge 120572 and (119886

1015840

4minus 119909)(119886

1015840

4minus 1198863) le 120573

Lemma 12 If 119860119868 = (1198861 1198862 1198863 1198864)(1198861015840

1 1198862 1198863 1198861015840

4) and 119861

119868

=

(1198871 1198872 1198873 1198874)(1198871015840

1 1198872 1198873 1198871015840

4) then

Cr120583(119860119868

le 119861119868

) ge 120572 Cr] (119860119868

le 119861119868

) le 120573

lArrrArr1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)ge 120572

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)ge 120572

21198862minus 21198871015840

3+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)le 120573

1198861015840

4minus 1198872

2 (1198861015840

4minus 1198863minus 1198872+ 1198871015840

1)le 120573

(21)

Proof Let us consider

Cr120583(119860119868

le 119861119868

) ge 120572 Cr] (119860119868

le 119861119868

) le 120573 (22)

Now from (14)

Cr120583(119860119868

le 119861119868

) ge 120572 lArrrArr1198874minus 1198861

2 (1198874minus 1198873+ 1198862minus 1198861)ge 120572

1198864minus 21198863+ 21198871minus 1198872

2 (1198864minus 1198863minus 1198872+ 1198871)ge 120572

Cr] (119860119868

le 119861119868

) le 120573 lArrrArr21198862minus 21198871015840

3+ 1198871015840

4minus 1198861015840

1

2 (1198862minus 1198861015840

1+ 1198871015840

4minus 1198873)le 120573

1198872minus 1198861015840

4

2 (1198872minus 1198871015840

1minus 1198861015840

4+ 1198863)le 120573

(23)

Note Cr120583(119860119868

le 119909) ge 120572 Cr](119860119868

le 119909) ge 120573 hArr (119909 minus 1198861)2(1198862minus

1198861) ge 120572 (119886

4minus21198863+119909)2(119886

4minus1198863) ge 120572 and (2119886

2minus119909minus119886

1015840

1)2(1198862minus

1198861015840

1) le 120573 (1198861015840

4minus 119909)2(119886

1015840

4minus 1198863) le 120573

4 Intuitionistic Fuzzy CCM

The chance operator is actually taken as possibility or neces-sity or credibility measures We can use chance operatorto transform the intuitionistic fuzzy problem into crispproblem which is called as CCM [27] A general single-objective mathematical programming problem with intu-itionistic fuzzy parameter should have the following form

Max 119891 (119909 120585119868

)

subject to 119892119894(119909 120585119868

) le 119868

119894 119894 = 1 2 119899

119909 ge 0

(24)

where 119909 is the decision vector 120585119868 and 119868

119894are intuitionistic

fuzzy parameters 119891(119909 120585119868) is an imprecise objective functionand 119892

119894(119909 120585119868

) are constraints function for 119894 = 1 2 119899The general chance-constraints model for problem (24) is

as followsMax 119891

1+ 1198912

subject to Ch120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Ch] 119891 (119909 120585119868

) ge 1198912 le 120573

Ch120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Ch] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(25)

The abbreviations Ch120583and Ch] represent chance operator

(ie Pos or Nec measure) for membership and nonmem-bership functions 120572 120573 120582

119894 and 120595

119894are the predetermined

confidence levels such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

International Journal of Engineering Mathematics 7

41 Intuitionistic Fuzzy CCM Based on Possibility MeasureThe CCM based on possibility measure is as follows

Max 1198911+ 1198912

Subject to Pos120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Pos] 119891 (119909 120585119868

) ge 1198912 le 120573

Pos120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Pos] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(26)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 13 A solution 119909lowast of the problem (26) satisfies

Pos120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Pos](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution at (120582119894 120595119894) possibility

levels 119894 = 1 2 119899

Definition 14 A feasible solution at (120582119894 120595119894) possibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (26)

if and only if there exists no other feasible solution at(120582119894 120595119894) possibility levels such that Pos

120583119891(119909 120585

119868

) ge 120572 andPos]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

42 Intuitionistic Fuzzy CCM Based on Necessity MeasureThe CCM based on necessity measure is as follows

Max 1198911+ 1198912

Subject to Nes120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Nes] 119891 (119909 120585119868

) ge 1198912 le 120573

Nes120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Nes] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(27)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 15 A solution 119909lowast of the problem (27) satisfies

Nes120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Nes](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) necessity

levels 119894 = 1 2 119899

Definition 16 A feasible solution at (120582119894 120595119894) necessity levels

119909lowast is said to be (120572 120573) efficient solution for problem (27)

if and only if there exists no other feasible solution at(120582119894 120595119894) necessity levels such that Nes

120583119891(119909 120585

119868

) ge 120572 andNes]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

43 Intuitionistic Fuzzy CCM Based on Credibility MeasureThe CCM based on credibility measure is as follows

Max 1198911+ 1198912

Subject to Cr120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Cr] 119891 (119909 120585119868

) ge 1198912 le 120573

Cr120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Cr] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(28)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 17 A solution 119909lowast of the problem (28) satisfies

Cr120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Cr](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) credibility

levels 119894 = 1 2 119899

Definition 18 A feasible solution at (120582119894 120595119894) credibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (28)

if and only if there exists no other feasible solution at(120582119894 120595119894) credibility levels such that Cr

120583119891(119909 120585

119868

) ge 120572 andCr]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

5 Proposed Method to Solve IFLPPUsing Chance Operator

To solve intuitionistic fuzzy CCM based on possibility ornecessity or credibility measures we propose the followingmethod

Step 1 Apply chance operator possibilitynecessitycredi-bility in intuitionistic fuzzy programming (24) Problem (24)can be converted into following problem

Max 1198911+ 1198912

Subject to Pos120583119891 (119909I

119868

) ge 1198911 ge 120572

or Nes120583119891 (119909I

119868

) ge 1198911 ge 120572

or Cr120583119891 (119909I

119868

) ge 1198911 ge 120572

(29)

Pos] 119891 (119909I119868

) ge 1198912 le 120573

or Nes] 119891 (119909I119868

) ge 1198912 le 120573

or Cr] 119891 (119909I119868

) ge 1198912 le 120573

(30)

8 International Journal of Engineering Mathematics

Pos120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Nes120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Cr120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

(31)

Pos] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Nes] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Cr] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

(32)

119909 ge 0 for 119894 = 1 2 119899 (33)

0 le 120582119894+ 120595119894le 1 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

(34)

where (120582119894 120595119894) and (120572 120573) are the predefined confidence levels

Step 2 Using Lemmas 10 11 andor Lemma 12 the aboveproblem in Step 1 can also be written as

Max 1198911+ 1198912

Subject to 1198911+ 1198912ge 119885

(31) ndash (34)

(35)

where 119885 is obtained by applying Lemmas 10 11 andorLemma 12 in (30) and (29)

Step 3 The above problem is equivalent to

Max 119885

Subject to (31) ndash (34)

(36)

Step 4 Crisp programming problem obtained in Step 2 canbe solved using any well-known method to get the optimalsolution

6 Numerical Example

Let us consider the following intuitionistic fuzzy mathemati-cal programming problem as

Maximize 120585119868

11199091oplus 120585119868

21199092

subject to 120585119868

31199091oplus 120585119868

41199092⪯ 120585119868

5

120585119868

61199091oplus 120585119868

71199092⪯ 120585119868

8

1199091 1199092ge 0

(37)

where 120585119868

1= (5 6 7 8)(4 6 7 9) 120585119868

2= (4 5 6 7)(3 5 6 8)

120585119868

3= (1 2 3 4)(05 2 3 6) 120585119868

4= (2 3 4 5)(1 3 4 6) 120585119868

5=

(6 7 8 9)(5 7 8 10) 1205851198686= (3 4 5 6)(2 4 5 7) 120585119868

7= (1 2 3

4)(0 2 3 4) and 120585119868

8= (10 11 12 14)(9 11 12 16)

61 Intuitionistic Fuzzy CCM Based on Possibility MeasureNowby using Step 2 of themethod explained in Section 4 and

Start Applychance operator

Convertinto crisp

Writeequivalent Solve

Figure 7 Flow chart of the proposed algorithm

Lemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (37) problem (37) is con-verted into the following crisp programming problem

Maximize (15 minus 120572 + 2120573) 1199091+ (13 minus 120572 + 2120573) 119909

2

subject to (1 + 1205821) 1199091+ (2 + 120582

1) 1199092le 9 minus 120582

1

(2 minus 151205951) 1199091+ (3 minus 2120595

1) 1199092le 8 + 2120595

1

(3 + 1205822) 1199091+ (1 + 120582

2) 1199092le 14 minus 2120582

2

(4 minus 21205952) 1199091+ (2 minus 2120595

2) 1199092le 12 + 4120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(38)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and different possibility levels we get differentoptimal solutions Optimal solution of (38) at differentpossibility levels (in Figure 7) are presented in Table 1 FromTable 1 we can observe that maximum value (= 7309) can beobtained at (120582

1= 040 120595

1= 035) and (120582

2= 030 120595

2= 040)

possibility levels

62 Intuitionistic Fuzzy CCM Based on Necessity MeasureNowby using Step 2 of themethod explained in Section 4 andLemma 11 if we apply the necessity measure in (37) problem(37) is converted into following crisp programming problem

Maximize (10 minus 120572 + 2120573) 1199091+ (8 minus 120572 + 2120573) 119909

2

subject to (3 + 21205821) 1199091+ (4 + 2120582

1) 1199092le 6 + 120582

1

(5 minus 21205951) 1199091+ (6 minus 2120595

1) 1199092le 7 minus 2120595

1

(5 + 1205822) 1199091+ (3 + 120582

2) 1199092le 6 + 120582

2

(7 minus 21205952) 1199091+ (4 minus 120595

2) 1199092le 11 minus 2120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(39)

Solving the above crisp linear programming problem forefficient levels (120572 = 06 120573 = 04) and different necessitylevels we get different optimal solutions Optimal solutionsof (39) at different necessity levels (in Figures 8 and 9) arepresented in Table 2 From Table 2 we can observed that at(1205821= 035 120595

1= 045) and (120582

2= 035 120595

2= 045) the decision

maker will get the maximum value = 1298

International Journal of Engineering Mathematics 9

Table 1 Optimal solution of (38) at different possibility levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)030 030 035 035 119909

1= 341 119909

2= 137 7009

030 035 035 040 1199091= 329 119909

2= 166 7214

040 035 030 040 1199091= 343 119909

2= 157 7309

035 040 040 030 1199091= 310 119909

2= 19 722

040 045 040 045 1199091= 316 119909

2= 173 7105

050 045 045 05 1199091= 316 119909

2= 150 6793

045 050 050 040 1199091= 297 119909

2= 173 6802

060 045 045 06 1199091= 329 119909

2= 120 6593

050 050 050 050 1199091= 303 119909

2= 157 6700

055 040 055 040 1199091= 297 119909

2= 150 6510

Table 2 Optimal solution of (39) at different necessity levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)035 030 030 035 119909

1= 091 119909

2= 043 1293

045 030 040 035 1199091= 090 119909

2= 045 1289

050 035 050 035 1199091= 087 119909

2= 047 1286

060 040 060 040 1199091= 085 119909

2= 050 1284

070 020 070 020 1199091= 087 119909

2= 045 1271

060 030 060 020 1199091= 087 119909

2= 047 1279

065 035 075 020 1199091= 084 119909

2= 050 1275

070 010 070 010 1199091= 089 119909

2= 043 1267

050 050 050 050 1199091= 085 119909

2= 051 1294

035 045 035 045 1199091= 088 119909

2= 048 1298

Table 3 Input data for IFTP

1198631

119868

1198632

119868

1198633

119868 Availability (119886119894

119868

)

1198781

119868 (2 4 6 7) (1 4 6 9) (4 6 7 8) (3 6 7 9) (3 7 9 12) (2 7 9 13) (4 6 8 9) (2 6 8 10)1198782

119868 (1 3 4 6) (05 3 4 7) (3 5 6 7) (2 5 6 9) (2 6 7 11) (1 6 7 12) (0 05 1 2) (0 05 1 5)1198783

119868 (3 4 5 8) (2 4 5 10) (1 2 3 4) (05 2 3 5) (2 4 5 10) (1 4 5 11) (8 95 10 11) (65 95 10 11)Demand (119887

119895

119868

) (6 7 8 10) (5 7 8 12) (4 5 6 9) (3 5 6 11) (2 4 5 7) (05 4 5 8)

646668707274

(030 030)

(035

035

)

(035

040

)

(030

040

) (055 040)

(050 050)

(060 045)

(045 050)

(050 045)(0

40 0

30)

(040

045

)

(045

050

)

(050

040

)

(045

060

)

(045

045

)

(030

030

)(040 045)

(035 040)

(040 035)

(030 035)(1205821 1205951

)(1205822 120595

2 )

flowast

Figure 8 Optimal solution at different possibility levels

10 International Journal of Engineering Mathematics

1265127

1275128

1285129

129513

(035 030)

(050

035

)

(060

040

)

(070

020

)

(060

020

)

(075

020

)

(070

010

)

(050

050

)

(035

045

)

(035 045)

(050 050)

(070 010)

(065 035)

(060 030)

(070 020)

(060 040)

(050 035)

(045 030)

(040

035

)

(030

035

)

(1205821 1205951)

(1205822 120595

2 )

flowast

Figure 9 Optimal solution at different necessity levels

7 Intuitionistic Fuzzy TransportationProblem Based on Possibility Measure

Let us consider the following intuitionistic fuzzy transporta-tion problem (IFTP) (in Table 3)

Above transportation problem is a balanced transporta-tion problem as ⨁3

119894=1119886119868

119894= ⨁3

119895=1119868

119895 The above IFTP can be

written as

Minimize (2 4 6 7) (1 4 6 9) 11990911

oplus (4 6 7 8) (3 6 7 9) 11990912

oplus (3 7 9 12) (2 7 9 13) 11990913

oplus (1 3 4 6) (05 3 4 7) 11990921

oplus (3 5 6 7) (2 5 6 9) 11990922

oplus (2 6 7 11) (1 6 7 12) 11990923

oplus (3 4 5 8) (2 4 5 10) 11990931

oplus (1 2 3 4) (05 2 3 5) 11990932

oplus (2 4 5 10) (1 4 5 11) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (4 6 8 9) (2 6 8 10)

11990921

+ 11990922

+ 11990923

le (0 05 1 2) (0 05 1 5)

11990931

+ 11990932

+ 11990933

le (8 95 10 11) (65 95 10 11)

11990911

+ 11990921

+ 11990931

ge (6 7 8 10) (5 7 8 12)

11990912

+ 11990922

+ 11990932

ge (4 5 6 9) (3 5 6 11)

11990913

+ 11990923

+ 11990933

ge (2 4 5 7) (05 4 5 8)

119909119894119895ge 0 forall119894 119895

(40)

Nowby using Step 2 of themethod explained in Section 4 andLemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (40) problem (40) isconverted into following crisp programming problem

Minimize (6 + 2120572 minus 3120573) 11990911

+ (10 + 2120572 minus 2120573) 11990912

+ (10 + 4120572 minus 5120573) 11990913

+ (4 + 2120572 minus 25120573) 11990921

+ (8 + 2120572 minus 3120573) 11990922

+ (8 + 4120572 minus 5120573) 11990923

+ (7 + 120572 minus 5120573) 11990931

+ (3 + 120572 minus 15120573) 11990932

+ (6 + 2120572 minus 3120573) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (17 minus 1205821+ 21205951)

11990921

+ 11990922

+ 11990923

le (3 minus 1205822+ 41205952)

11990931

+ 11990932

+ 11990933

le (21 minus 1205823+ 1205953)

11990911

+ 11990921

+ 11990931

ge (13 + 1205824minus 21205954)

11990912

+ 11990922

+ 11990932

ge (9 + 1205825minus 21205955)

11990913

+ 11990923

+ 11990933

ge (6 + 21205826minus 35120595

6)

119909119894119895ge 0 for 119894 119895 = 1 2 3 0 le 120572 + 120573 le 1

0 le 120582119896+ 120595119896le 1 for 119896 = 1 2 6

(41)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and possibility levels (120582119894= 05 120595

119894= 05) for

119894 = 1 2 6 using Lingo-110 we get 11990911

= 075 11990912

= 011990913

= 0 11990921

= 45 11990922

= 0 11990923

= 0 11990931

= 725 11990932

= 85 and

International Journal of Engineering Mathematics 11

11990933

= 525 Now the minimum intuitionstic fuzzy optimalcost is

119888119868

= (7225 109 136 2035) (4562 109 136 24625) (42)

8 Discussion

Intuitionistic fuzzy sets being a generalization of fuzzy setsgive us an additional possibility to represent imperfect knowl-edge making it possible to describe many real problems ina more adequate way So in this paper we have developedthe possibility and necessity measures on intuitionistic fuzzyset Here we have presented first time the mathematicalrepresentation of different types of measures in intuitionisticfuzzy environments and some graphical representations ofthemare depictedWehave also developed the theoretical cal-culation on possibility necessity and credibility measures fordefuzzify intuitionistic fuzzy linear programming problemusing chance operators To validate the proposed methodwe have discussed three different approaches to defuzzifythe intuitionistic fuzzy relations using possibility necessityand credibility measures Using chance operator we canconvert a problem under imprecise models to correspondingcrisp models At different levels of possibility necessity andcredibility we have achieved different optimal solution Anumerical example is presented and solved using LINGO-110to illustrate the proposed approaches The proposed methodcan be applied for multiobjective multiitem transportationproblem This method can be also extended to be appliedinto different types of optimization problem namely optimalcontrol and solid transportation problems

Conflict of Interests

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

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov Intuitionistic Fuzzy Sets VII ITKRrsquos SessionSofia Bulgarian 1983

[3] S K De and S S Sana ldquoA multi-periods production-inventorymodel with capacity constraints for multi-manufacturersmdashaglobal optimality in intuitionistic fuzzy environmentrdquo AppliedMathematics andComputation vol 242 no 1 pp 825ndash841 2014

[4] HGargM Rani S P Sharma andYVishwakarma ldquoIntuition-istic fuzzy optimization technique for solving multi-objectivereliability optimization problems in interval environmentrdquoExpert Systems with Applications vol 41 no 7 pp 3157ndash31672014

[5] J Wu and Y Liu ldquoAn approach for multiple attribute groupdecision making problems with interval-valued intuitionistictrapezoidal fuzzy numbersrdquoComputers and Industrial Engineer-ing vol 66 no 2 pp 311ndash324 2013

[6] A Nagoorgani and K Ponnalagu ldquoA new approach on solv-ing intuitionistic fuzzy linear programming problemrdquo AppliedMathematical Sciences vol 6 no 70 pp 3467ndash3474 2012

[7] G S Mahapatra and T K Roy ldquoIntuitionistic fuzzy numberand its arithmetic operation with application on system failurerdquoJournal of Uncertain Systems vol 7 no 2 pp 92ndash107 2013

[8] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[9] S Pramanik D K Jana and M Maiti ldquoA multi objectivesolid transportation problem in fuzzy Bi-fuzzy environmentvia genetic algorithmrdquo International Journal of Advanced Oper-ations Management vol 6 no 1 pp 4ndash26 2014

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

[11] M Esmailzadeh and M Esmailzadeh ldquoNew distance betweentriangular intuitionistic fuzzy numbersrdquo Advances in Computa-tional Mathematics and Its Applications vol 2 no 3 pp 3ndash102013

[12] P P Angelov ldquoOptimization in an intuitionistic fuzzy environ-mentrdquo Fuzzy Sets and Systems vol 86 no 3 pp 299ndash306 1997

[13] D Dubey and A Mehra ldquoLinear programming with triangularintuitionistic fuzzy numberrdquo Advances in Intelligent SystemsResearch vol 1 no 1 pp 563ndash569 2011

[14] R Parvathi and C Malathi ldquoIntuitionistic fuzzy simplexmethodrdquo International Journal of Computer Applications vol48 no 6 pp 39ndash48 2012

[15] R J Hussain and S P Kumar ldquoAlgorithmic approach forsolving intuitionistic fuzzy transportation problemrdquo AppliedMathematical Sciences vol 6 no 77-80 pp 3981ndash3989 2012

[16] ANagoorGani and S Abbas ldquoSolving intuitionstic fuzzy trans-portation problem using zero suffix algorithmrdquo InternationalJournal of Mathematics Sciences amp Enggineering Applicationsvol 6 pp 73ndash82 2012

[17] J Ye ldquoExpected value method for intuitionistic trapezoidalfuzzy multicriteria decision-making problemsrdquo Expert Systemswith Applications vol 38 no 9 pp 11730ndash11734 2011

[18] 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 2013

[19] J J Buckley ldquoPossibility and necessity in optimizationrdquo FuzzySets and Systems vol 25 no 1 pp 1ndash13 1988

[20] K D Jamison and W A Lodwick ldquoThe construction ofconsistent possibility and necessity measuresrdquo Fuzzy Sets andSystems vol 132 no 1 pp 1ndash10 2002

[21] J Ramık ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006

[22] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[23] M Sakawa H Katagiri and T Matsui ldquoFuzzy random bilevellinear programming through expectation optimization usingpossibility and necessityrdquo International Journal of MachineLearning and Cybernetics vol 3 no 3 pp 183ndash192 2012

[24] S Pathak S Kar and S Sarkar ldquoFuzzy production inventorymodel for deteriorating items with shortages under the effect oftime dependent learning and forgetting a possibilitynecessityapproachrdquo OPSEARCH vol 50 no 2 pp 149ndash181 2013

[25] K Maity ldquoPossibility and necessity representations of fuzzyinequality and its application to two warehouse production-inventory problemrdquo Applied Mathematical Modelling vol 35no 3 pp 1252ndash1263 2011

12 International Journal of Engineering Mathematics

[26] H-C Wu ldquoFuzzy optimization problems based on the embed-ding theorem and possibility and necessity measuresrdquo Mathe-matical and Computer Modelling vol 40 no 3-4 pp 329ndash3362004

[27] J Xu and X Zhou Fuzzy-Like Multiple Objective DecesionMaking Springer 2010

[28] K Maity and M Maiti ldquoPossibility and necessity constraintsand their defuzzificationmdasha multi-item production-inventoryscenario via optimal control theoryrdquo European Journal of Oper-ational Research vol 177 no 2 pp 882ndash896 2007

[29] B Das K Maity and M Maiti ldquoA two warehouse supply-chain model under possibilitynecessitycredibility measuresrdquoMathematical andComputerModelling vol 46 no 3-4 pp 398ndash409 2007

[30] D Panda S Kar K Maity and M Maiti ldquoA single periodinventory model with imperfect production and stochasticdemand under chance and imprecise constraintsrdquo EuropeanJournal of Operational Research vol 188 no 1 pp 121ndash139 2008

[31] A I Ban ldquoIntuitionistic fuzzy-valued possibility and necessitymeasuresrdquo in Proceedings of the 8th International Conference onIntuitionistic Fuzzy Sets vol 10 pp 1ndash7 Varna Bulgaria 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Engineering Mathematics 7

41 Intuitionistic Fuzzy CCM Based on Possibility MeasureThe CCM based on possibility measure is as follows

Max 1198911+ 1198912

Subject to Pos120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Pos] 119891 (119909 120585119868

) ge 1198912 le 120573

Pos120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Pos] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(26)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 13 A solution 119909lowast of the problem (26) satisfies

Pos120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Pos](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution at (120582119894 120595119894) possibility

levels 119894 = 1 2 119899

Definition 14 A feasible solution at (120582119894 120595119894) possibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (26)

if and only if there exists no other feasible solution at(120582119894 120595119894) possibility levels such that Pos

120583119891(119909 120585

119868

) ge 120572 andPos]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

42 Intuitionistic Fuzzy CCM Based on Necessity MeasureThe CCM based on necessity measure is as follows

Max 1198911+ 1198912

Subject to Nes120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Nes] 119891 (119909 120585119868

) ge 1198912 le 120573

Nes120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Nes] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(27)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 15 A solution 119909lowast of the problem (27) satisfies

Nes120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Nes](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) necessity

levels 119894 = 1 2 119899

Definition 16 A feasible solution at (120582119894 120595119894) necessity levels

119909lowast is said to be (120572 120573) efficient solution for problem (27)

if and only if there exists no other feasible solution at(120582119894 120595119894) necessity levels such that Nes

120583119891(119909 120585

119868

) ge 120572 andNes]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

43 Intuitionistic Fuzzy CCM Based on Credibility MeasureThe CCM based on credibility measure is as follows

Max 1198911+ 1198912

Subject to Cr120583119891 (119909 120585

119868

) ge 1198911 ge 120572

Cr] 119891 (119909 120585119868

) ge 1198912 le 120573

Cr120583119892119894(119909 120585119868

) le 119868

119894 ge 120582119894

Cr] 119892119894 (119909 120585119868

) le 119868

119894 le 120595119894

119909 ge 0 119894 = 1 2 119899

(28)

where120572120573 120582119894 and120595

119894are the predetermined confidence levels

such that 0 le 120582119894+ 120595119894le 1 and 0 le 120572 + 120573 le 1 for 119894 = 1 2 119899

Definition 17 A solution 119909lowast of the problem (28) satisfies

Cr120583(119892119894(119909 120585119868

) le 119868

119894) ge 120582

119894and Cr](119892119894(119909 120585

119868

) le 119868

119894) le 120595

119894for

119894 = 1 2 119899 is called a feasible solution (120582119894 120595119894) credibility

levels 119894 = 1 2 119899

Definition 18 A feasible solution at (120582119894 120595119894) credibility levels

119909lowast is said to be (120572 120573) efficient solution for problem (28)

if and only if there exists no other feasible solution at(120582119894 120595119894) credibility levels such that Cr

120583119891(119909 120585

119868

) ge 120572 andCr]119891(119909 120585

119868

) le 120573 with 119891(119909) ge 1198911(119909lowast

) + 1198912(119909lowast

)

5 Proposed Method to Solve IFLPPUsing Chance Operator

To solve intuitionistic fuzzy CCM based on possibility ornecessity or credibility measures we propose the followingmethod

Step 1 Apply chance operator possibilitynecessitycredi-bility in intuitionistic fuzzy programming (24) Problem (24)can be converted into following problem

Max 1198911+ 1198912

Subject to Pos120583119891 (119909I

119868

) ge 1198911 ge 120572

or Nes120583119891 (119909I

119868

) ge 1198911 ge 120572

or Cr120583119891 (119909I

119868

) ge 1198911 ge 120572

(29)

Pos] 119891 (119909I119868

) ge 1198912 le 120573

or Nes] 119891 (119909I119868

) ge 1198912 le 120573

or Cr] 119891 (119909I119868

) ge 1198912 le 120573

(30)

8 International Journal of Engineering Mathematics

Pos120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Nes120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Cr120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

(31)

Pos] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Nes] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Cr] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

(32)

119909 ge 0 for 119894 = 1 2 119899 (33)

0 le 120582119894+ 120595119894le 1 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

(34)

where (120582119894 120595119894) and (120572 120573) are the predefined confidence levels

Step 2 Using Lemmas 10 11 andor Lemma 12 the aboveproblem in Step 1 can also be written as

Max 1198911+ 1198912

Subject to 1198911+ 1198912ge 119885

(31) ndash (34)

(35)

where 119885 is obtained by applying Lemmas 10 11 andorLemma 12 in (30) and (29)

Step 3 The above problem is equivalent to

Max 119885

Subject to (31) ndash (34)

(36)

Step 4 Crisp programming problem obtained in Step 2 canbe solved using any well-known method to get the optimalsolution

6 Numerical Example

Let us consider the following intuitionistic fuzzy mathemati-cal programming problem as

Maximize 120585119868

11199091oplus 120585119868

21199092

subject to 120585119868

31199091oplus 120585119868

41199092⪯ 120585119868

5

120585119868

61199091oplus 120585119868

71199092⪯ 120585119868

8

1199091 1199092ge 0

(37)

where 120585119868

1= (5 6 7 8)(4 6 7 9) 120585119868

2= (4 5 6 7)(3 5 6 8)

120585119868

3= (1 2 3 4)(05 2 3 6) 120585119868

4= (2 3 4 5)(1 3 4 6) 120585119868

5=

(6 7 8 9)(5 7 8 10) 1205851198686= (3 4 5 6)(2 4 5 7) 120585119868

7= (1 2 3

4)(0 2 3 4) and 120585119868

8= (10 11 12 14)(9 11 12 16)

61 Intuitionistic Fuzzy CCM Based on Possibility MeasureNowby using Step 2 of themethod explained in Section 4 and

Start Applychance operator

Convertinto crisp

Writeequivalent Solve

Figure 7 Flow chart of the proposed algorithm

Lemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (37) problem (37) is con-verted into the following crisp programming problem

Maximize (15 minus 120572 + 2120573) 1199091+ (13 minus 120572 + 2120573) 119909

2

subject to (1 + 1205821) 1199091+ (2 + 120582

1) 1199092le 9 minus 120582

1

(2 minus 151205951) 1199091+ (3 minus 2120595

1) 1199092le 8 + 2120595

1

(3 + 1205822) 1199091+ (1 + 120582

2) 1199092le 14 minus 2120582

2

(4 minus 21205952) 1199091+ (2 minus 2120595

2) 1199092le 12 + 4120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(38)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and different possibility levels we get differentoptimal solutions Optimal solution of (38) at differentpossibility levels (in Figure 7) are presented in Table 1 FromTable 1 we can observe that maximum value (= 7309) can beobtained at (120582

1= 040 120595

1= 035) and (120582

2= 030 120595

2= 040)

possibility levels

62 Intuitionistic Fuzzy CCM Based on Necessity MeasureNowby using Step 2 of themethod explained in Section 4 andLemma 11 if we apply the necessity measure in (37) problem(37) is converted into following crisp programming problem

Maximize (10 minus 120572 + 2120573) 1199091+ (8 minus 120572 + 2120573) 119909

2

subject to (3 + 21205821) 1199091+ (4 + 2120582

1) 1199092le 6 + 120582

1

(5 minus 21205951) 1199091+ (6 minus 2120595

1) 1199092le 7 minus 2120595

1

(5 + 1205822) 1199091+ (3 + 120582

2) 1199092le 6 + 120582

2

(7 minus 21205952) 1199091+ (4 minus 120595

2) 1199092le 11 minus 2120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(39)

Solving the above crisp linear programming problem forefficient levels (120572 = 06 120573 = 04) and different necessitylevels we get different optimal solutions Optimal solutionsof (39) at different necessity levels (in Figures 8 and 9) arepresented in Table 2 From Table 2 we can observed that at(1205821= 035 120595

1= 045) and (120582

2= 035 120595

2= 045) the decision

maker will get the maximum value = 1298

International Journal of Engineering Mathematics 9

Table 1 Optimal solution of (38) at different possibility levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)030 030 035 035 119909

1= 341 119909

2= 137 7009

030 035 035 040 1199091= 329 119909

2= 166 7214

040 035 030 040 1199091= 343 119909

2= 157 7309

035 040 040 030 1199091= 310 119909

2= 19 722

040 045 040 045 1199091= 316 119909

2= 173 7105

050 045 045 05 1199091= 316 119909

2= 150 6793

045 050 050 040 1199091= 297 119909

2= 173 6802

060 045 045 06 1199091= 329 119909

2= 120 6593

050 050 050 050 1199091= 303 119909

2= 157 6700

055 040 055 040 1199091= 297 119909

2= 150 6510

Table 2 Optimal solution of (39) at different necessity levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)035 030 030 035 119909

1= 091 119909

2= 043 1293

045 030 040 035 1199091= 090 119909

2= 045 1289

050 035 050 035 1199091= 087 119909

2= 047 1286

060 040 060 040 1199091= 085 119909

2= 050 1284

070 020 070 020 1199091= 087 119909

2= 045 1271

060 030 060 020 1199091= 087 119909

2= 047 1279

065 035 075 020 1199091= 084 119909

2= 050 1275

070 010 070 010 1199091= 089 119909

2= 043 1267

050 050 050 050 1199091= 085 119909

2= 051 1294

035 045 035 045 1199091= 088 119909

2= 048 1298

Table 3 Input data for IFTP

1198631

119868

1198632

119868

1198633

119868 Availability (119886119894

119868

)

1198781

119868 (2 4 6 7) (1 4 6 9) (4 6 7 8) (3 6 7 9) (3 7 9 12) (2 7 9 13) (4 6 8 9) (2 6 8 10)1198782

119868 (1 3 4 6) (05 3 4 7) (3 5 6 7) (2 5 6 9) (2 6 7 11) (1 6 7 12) (0 05 1 2) (0 05 1 5)1198783

119868 (3 4 5 8) (2 4 5 10) (1 2 3 4) (05 2 3 5) (2 4 5 10) (1 4 5 11) (8 95 10 11) (65 95 10 11)Demand (119887

119895

119868

) (6 7 8 10) (5 7 8 12) (4 5 6 9) (3 5 6 11) (2 4 5 7) (05 4 5 8)

646668707274

(030 030)

(035

035

)

(035

040

)

(030

040

) (055 040)

(050 050)

(060 045)

(045 050)

(050 045)(0

40 0

30)

(040

045

)

(045

050

)

(050

040

)

(045

060

)

(045

045

)

(030

030

)(040 045)

(035 040)

(040 035)

(030 035)(1205821 1205951

)(1205822 120595

2 )

flowast

Figure 8 Optimal solution at different possibility levels

10 International Journal of Engineering Mathematics

1265127

1275128

1285129

129513

(035 030)

(050

035

)

(060

040

)

(070

020

)

(060

020

)

(075

020

)

(070

010

)

(050

050

)

(035

045

)

(035 045)

(050 050)

(070 010)

(065 035)

(060 030)

(070 020)

(060 040)

(050 035)

(045 030)

(040

035

)

(030

035

)

(1205821 1205951)

(1205822 120595

2 )

flowast

Figure 9 Optimal solution at different necessity levels

7 Intuitionistic Fuzzy TransportationProblem Based on Possibility Measure

Let us consider the following intuitionistic fuzzy transporta-tion problem (IFTP) (in Table 3)

Above transportation problem is a balanced transporta-tion problem as ⨁3

119894=1119886119868

119894= ⨁3

119895=1119868

119895 The above IFTP can be

written as

Minimize (2 4 6 7) (1 4 6 9) 11990911

oplus (4 6 7 8) (3 6 7 9) 11990912

oplus (3 7 9 12) (2 7 9 13) 11990913

oplus (1 3 4 6) (05 3 4 7) 11990921

oplus (3 5 6 7) (2 5 6 9) 11990922

oplus (2 6 7 11) (1 6 7 12) 11990923

oplus (3 4 5 8) (2 4 5 10) 11990931

oplus (1 2 3 4) (05 2 3 5) 11990932

oplus (2 4 5 10) (1 4 5 11) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (4 6 8 9) (2 6 8 10)

11990921

+ 11990922

+ 11990923

le (0 05 1 2) (0 05 1 5)

11990931

+ 11990932

+ 11990933

le (8 95 10 11) (65 95 10 11)

11990911

+ 11990921

+ 11990931

ge (6 7 8 10) (5 7 8 12)

11990912

+ 11990922

+ 11990932

ge (4 5 6 9) (3 5 6 11)

11990913

+ 11990923

+ 11990933

ge (2 4 5 7) (05 4 5 8)

119909119894119895ge 0 forall119894 119895

(40)

Nowby using Step 2 of themethod explained in Section 4 andLemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (40) problem (40) isconverted into following crisp programming problem

Minimize (6 + 2120572 minus 3120573) 11990911

+ (10 + 2120572 minus 2120573) 11990912

+ (10 + 4120572 minus 5120573) 11990913

+ (4 + 2120572 minus 25120573) 11990921

+ (8 + 2120572 minus 3120573) 11990922

+ (8 + 4120572 minus 5120573) 11990923

+ (7 + 120572 minus 5120573) 11990931

+ (3 + 120572 minus 15120573) 11990932

+ (6 + 2120572 minus 3120573) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (17 minus 1205821+ 21205951)

11990921

+ 11990922

+ 11990923

le (3 minus 1205822+ 41205952)

11990931

+ 11990932

+ 11990933

le (21 minus 1205823+ 1205953)

11990911

+ 11990921

+ 11990931

ge (13 + 1205824minus 21205954)

11990912

+ 11990922

+ 11990932

ge (9 + 1205825minus 21205955)

11990913

+ 11990923

+ 11990933

ge (6 + 21205826minus 35120595

6)

119909119894119895ge 0 for 119894 119895 = 1 2 3 0 le 120572 + 120573 le 1

0 le 120582119896+ 120595119896le 1 for 119896 = 1 2 6

(41)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and possibility levels (120582119894= 05 120595

119894= 05) for

119894 = 1 2 6 using Lingo-110 we get 11990911

= 075 11990912

= 011990913

= 0 11990921

= 45 11990922

= 0 11990923

= 0 11990931

= 725 11990932

= 85 and

International Journal of Engineering Mathematics 11

11990933

= 525 Now the minimum intuitionstic fuzzy optimalcost is

119888119868

= (7225 109 136 2035) (4562 109 136 24625) (42)

8 Discussion

Intuitionistic fuzzy sets being a generalization of fuzzy setsgive us an additional possibility to represent imperfect knowl-edge making it possible to describe many real problems ina more adequate way So in this paper we have developedthe possibility and necessity measures on intuitionistic fuzzyset Here we have presented first time the mathematicalrepresentation of different types of measures in intuitionisticfuzzy environments and some graphical representations ofthemare depictedWehave also developed the theoretical cal-culation on possibility necessity and credibility measures fordefuzzify intuitionistic fuzzy linear programming problemusing chance operators To validate the proposed methodwe have discussed three different approaches to defuzzifythe intuitionistic fuzzy relations using possibility necessityand credibility measures Using chance operator we canconvert a problem under imprecise models to correspondingcrisp models At different levels of possibility necessity andcredibility we have achieved different optimal solution Anumerical example is presented and solved using LINGO-110to illustrate the proposed approaches The proposed methodcan be applied for multiobjective multiitem transportationproblem This method can be also extended to be appliedinto different types of optimization problem namely optimalcontrol and solid transportation problems

Conflict of Interests

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

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov Intuitionistic Fuzzy Sets VII ITKRrsquos SessionSofia Bulgarian 1983

[3] S K De and S S Sana ldquoA multi-periods production-inventorymodel with capacity constraints for multi-manufacturersmdashaglobal optimality in intuitionistic fuzzy environmentrdquo AppliedMathematics andComputation vol 242 no 1 pp 825ndash841 2014

[4] HGargM Rani S P Sharma andYVishwakarma ldquoIntuition-istic fuzzy optimization technique for solving multi-objectivereliability optimization problems in interval environmentrdquoExpert Systems with Applications vol 41 no 7 pp 3157ndash31672014

[5] J Wu and Y Liu ldquoAn approach for multiple attribute groupdecision making problems with interval-valued intuitionistictrapezoidal fuzzy numbersrdquoComputers and Industrial Engineer-ing vol 66 no 2 pp 311ndash324 2013

[6] A Nagoorgani and K Ponnalagu ldquoA new approach on solv-ing intuitionistic fuzzy linear programming problemrdquo AppliedMathematical Sciences vol 6 no 70 pp 3467ndash3474 2012

[7] G S Mahapatra and T K Roy ldquoIntuitionistic fuzzy numberand its arithmetic operation with application on system failurerdquoJournal of Uncertain Systems vol 7 no 2 pp 92ndash107 2013

[8] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[9] S Pramanik D K Jana and M Maiti ldquoA multi objectivesolid transportation problem in fuzzy Bi-fuzzy environmentvia genetic algorithmrdquo International Journal of Advanced Oper-ations Management vol 6 no 1 pp 4ndash26 2014

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

[11] M Esmailzadeh and M Esmailzadeh ldquoNew distance betweentriangular intuitionistic fuzzy numbersrdquo Advances in Computa-tional Mathematics and Its Applications vol 2 no 3 pp 3ndash102013

[12] P P Angelov ldquoOptimization in an intuitionistic fuzzy environ-mentrdquo Fuzzy Sets and Systems vol 86 no 3 pp 299ndash306 1997

[13] D Dubey and A Mehra ldquoLinear programming with triangularintuitionistic fuzzy numberrdquo Advances in Intelligent SystemsResearch vol 1 no 1 pp 563ndash569 2011

[14] R Parvathi and C Malathi ldquoIntuitionistic fuzzy simplexmethodrdquo International Journal of Computer Applications vol48 no 6 pp 39ndash48 2012

[15] R J Hussain and S P Kumar ldquoAlgorithmic approach forsolving intuitionistic fuzzy transportation problemrdquo AppliedMathematical Sciences vol 6 no 77-80 pp 3981ndash3989 2012

[16] ANagoorGani and S Abbas ldquoSolving intuitionstic fuzzy trans-portation problem using zero suffix algorithmrdquo InternationalJournal of Mathematics Sciences amp Enggineering Applicationsvol 6 pp 73ndash82 2012

[17] J Ye ldquoExpected value method for intuitionistic trapezoidalfuzzy multicriteria decision-making problemsrdquo Expert Systemswith Applications vol 38 no 9 pp 11730ndash11734 2011

[18] 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 2013

[19] J J Buckley ldquoPossibility and necessity in optimizationrdquo FuzzySets and Systems vol 25 no 1 pp 1ndash13 1988

[20] K D Jamison and W A Lodwick ldquoThe construction ofconsistent possibility and necessity measuresrdquo Fuzzy Sets andSystems vol 132 no 1 pp 1ndash10 2002

[21] J Ramık ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006

[22] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[23] M Sakawa H Katagiri and T Matsui ldquoFuzzy random bilevellinear programming through expectation optimization usingpossibility and necessityrdquo International Journal of MachineLearning and Cybernetics vol 3 no 3 pp 183ndash192 2012

[24] S Pathak S Kar and S Sarkar ldquoFuzzy production inventorymodel for deteriorating items with shortages under the effect oftime dependent learning and forgetting a possibilitynecessityapproachrdquo OPSEARCH vol 50 no 2 pp 149ndash181 2013

[25] K Maity ldquoPossibility and necessity representations of fuzzyinequality and its application to two warehouse production-inventory problemrdquo Applied Mathematical Modelling vol 35no 3 pp 1252ndash1263 2011

12 International Journal of Engineering Mathematics

[26] H-C Wu ldquoFuzzy optimization problems based on the embed-ding theorem and possibility and necessity measuresrdquo Mathe-matical and Computer Modelling vol 40 no 3-4 pp 329ndash3362004

[27] J Xu and X Zhou Fuzzy-Like Multiple Objective DecesionMaking Springer 2010

[28] K Maity and M Maiti ldquoPossibility and necessity constraintsand their defuzzificationmdasha multi-item production-inventoryscenario via optimal control theoryrdquo European Journal of Oper-ational Research vol 177 no 2 pp 882ndash896 2007

[29] B Das K Maity and M Maiti ldquoA two warehouse supply-chain model under possibilitynecessitycredibility measuresrdquoMathematical andComputerModelling vol 46 no 3-4 pp 398ndash409 2007

[30] D Panda S Kar K Maity and M Maiti ldquoA single periodinventory model with imperfect production and stochasticdemand under chance and imprecise constraintsrdquo EuropeanJournal of Operational Research vol 188 no 1 pp 121ndash139 2008

[31] A I Ban ldquoIntuitionistic fuzzy-valued possibility and necessitymeasuresrdquo in Proceedings of the 8th International Conference onIntuitionistic Fuzzy Sets vol 10 pp 1ndash7 Varna Bulgaria 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Engineering Mathematics

Pos120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Nes120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

or Cr120583119892119894(119909I119868

) le 119868

119894 ge 120582119894

(31)

Pos] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Nes] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

or Cr] 119892119894 (119909I119868

) le 119868

119894 le 120595119894

(32)

119909 ge 0 for 119894 = 1 2 119899 (33)

0 le 120582119894+ 120595119894le 1 0 le 120572 + 120573 le 1

for 119894 = 1 2 119899

(34)

where (120582119894 120595119894) and (120572 120573) are the predefined confidence levels

Step 2 Using Lemmas 10 11 andor Lemma 12 the aboveproblem in Step 1 can also be written as

Max 1198911+ 1198912

Subject to 1198911+ 1198912ge 119885

(31) ndash (34)

(35)

where 119885 is obtained by applying Lemmas 10 11 andorLemma 12 in (30) and (29)

Step 3 The above problem is equivalent to

Max 119885

Subject to (31) ndash (34)

(36)

Step 4 Crisp programming problem obtained in Step 2 canbe solved using any well-known method to get the optimalsolution

6 Numerical Example

Let us consider the following intuitionistic fuzzy mathemati-cal programming problem as

Maximize 120585119868

11199091oplus 120585119868

21199092

subject to 120585119868

31199091oplus 120585119868

41199092⪯ 120585119868

5

120585119868

61199091oplus 120585119868

71199092⪯ 120585119868

8

1199091 1199092ge 0

(37)

where 120585119868

1= (5 6 7 8)(4 6 7 9) 120585119868

2= (4 5 6 7)(3 5 6 8)

120585119868

3= (1 2 3 4)(05 2 3 6) 120585119868

4= (2 3 4 5)(1 3 4 6) 120585119868

5=

(6 7 8 9)(5 7 8 10) 1205851198686= (3 4 5 6)(2 4 5 7) 120585119868

7= (1 2 3

4)(0 2 3 4) and 120585119868

8= (10 11 12 14)(9 11 12 16)

61 Intuitionistic Fuzzy CCM Based on Possibility MeasureNowby using Step 2 of themethod explained in Section 4 and

Start Applychance operator

Convertinto crisp

Writeequivalent Solve

Figure 7 Flow chart of the proposed algorithm

Lemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (37) problem (37) is con-verted into the following crisp programming problem

Maximize (15 minus 120572 + 2120573) 1199091+ (13 minus 120572 + 2120573) 119909

2

subject to (1 + 1205821) 1199091+ (2 + 120582

1) 1199092le 9 minus 120582

1

(2 minus 151205951) 1199091+ (3 minus 2120595

1) 1199092le 8 + 2120595

1

(3 + 1205822) 1199091+ (1 + 120582

2) 1199092le 14 minus 2120582

2

(4 minus 21205952) 1199091+ (2 minus 2120595

2) 1199092le 12 + 4120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(38)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and different possibility levels we get differentoptimal solutions Optimal solution of (38) at differentpossibility levels (in Figure 7) are presented in Table 1 FromTable 1 we can observe that maximum value (= 7309) can beobtained at (120582

1= 040 120595

1= 035) and (120582

2= 030 120595

2= 040)

possibility levels

62 Intuitionistic Fuzzy CCM Based on Necessity MeasureNowby using Step 2 of themethod explained in Section 4 andLemma 11 if we apply the necessity measure in (37) problem(37) is converted into following crisp programming problem

Maximize (10 minus 120572 + 2120573) 1199091+ (8 minus 120572 + 2120573) 119909

2

subject to (3 + 21205821) 1199091+ (4 + 2120582

1) 1199092le 6 + 120582

1

(5 minus 21205951) 1199091+ (6 minus 2120595

1) 1199092le 7 minus 2120595

1

(5 + 1205822) 1199091+ (3 + 120582

2) 1199092le 6 + 120582

2

(7 minus 21205952) 1199091+ (4 minus 120595

2) 1199092le 11 minus 2120595

2

1199091 1199092ge 0 0 le 120572 + 120573 le 1

0 le 120582119894+ 120595119894le 1 for 119894 = 1 2

(39)

Solving the above crisp linear programming problem forefficient levels (120572 = 06 120573 = 04) and different necessitylevels we get different optimal solutions Optimal solutionsof (39) at different necessity levels (in Figures 8 and 9) arepresented in Table 2 From Table 2 we can observed that at(1205821= 035 120595

1= 045) and (120582

2= 035 120595

2= 045) the decision

maker will get the maximum value = 1298

International Journal of Engineering Mathematics 9

Table 1 Optimal solution of (38) at different possibility levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)030 030 035 035 119909

1= 341 119909

2= 137 7009

030 035 035 040 1199091= 329 119909

2= 166 7214

040 035 030 040 1199091= 343 119909

2= 157 7309

035 040 040 030 1199091= 310 119909

2= 19 722

040 045 040 045 1199091= 316 119909

2= 173 7105

050 045 045 05 1199091= 316 119909

2= 150 6793

045 050 050 040 1199091= 297 119909

2= 173 6802

060 045 045 06 1199091= 329 119909

2= 120 6593

050 050 050 050 1199091= 303 119909

2= 157 6700

055 040 055 040 1199091= 297 119909

2= 150 6510

Table 2 Optimal solution of (39) at different necessity levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)035 030 030 035 119909

1= 091 119909

2= 043 1293

045 030 040 035 1199091= 090 119909

2= 045 1289

050 035 050 035 1199091= 087 119909

2= 047 1286

060 040 060 040 1199091= 085 119909

2= 050 1284

070 020 070 020 1199091= 087 119909

2= 045 1271

060 030 060 020 1199091= 087 119909

2= 047 1279

065 035 075 020 1199091= 084 119909

2= 050 1275

070 010 070 010 1199091= 089 119909

2= 043 1267

050 050 050 050 1199091= 085 119909

2= 051 1294

035 045 035 045 1199091= 088 119909

2= 048 1298

Table 3 Input data for IFTP

1198631

119868

1198632

119868

1198633

119868 Availability (119886119894

119868

)

1198781

119868 (2 4 6 7) (1 4 6 9) (4 6 7 8) (3 6 7 9) (3 7 9 12) (2 7 9 13) (4 6 8 9) (2 6 8 10)1198782

119868 (1 3 4 6) (05 3 4 7) (3 5 6 7) (2 5 6 9) (2 6 7 11) (1 6 7 12) (0 05 1 2) (0 05 1 5)1198783

119868 (3 4 5 8) (2 4 5 10) (1 2 3 4) (05 2 3 5) (2 4 5 10) (1 4 5 11) (8 95 10 11) (65 95 10 11)Demand (119887

119895

119868

) (6 7 8 10) (5 7 8 12) (4 5 6 9) (3 5 6 11) (2 4 5 7) (05 4 5 8)

646668707274

(030 030)

(035

035

)

(035

040

)

(030

040

) (055 040)

(050 050)

(060 045)

(045 050)

(050 045)(0

40 0

30)

(040

045

)

(045

050

)

(050

040

)

(045

060

)

(045

045

)

(030

030

)(040 045)

(035 040)

(040 035)

(030 035)(1205821 1205951

)(1205822 120595

2 )

flowast

Figure 8 Optimal solution at different possibility levels

10 International Journal of Engineering Mathematics

1265127

1275128

1285129

129513

(035 030)

(050

035

)

(060

040

)

(070

020

)

(060

020

)

(075

020

)

(070

010

)

(050

050

)

(035

045

)

(035 045)

(050 050)

(070 010)

(065 035)

(060 030)

(070 020)

(060 040)

(050 035)

(045 030)

(040

035

)

(030

035

)

(1205821 1205951)

(1205822 120595

2 )

flowast

Figure 9 Optimal solution at different necessity levels

7 Intuitionistic Fuzzy TransportationProblem Based on Possibility Measure

Let us consider the following intuitionistic fuzzy transporta-tion problem (IFTP) (in Table 3)

Above transportation problem is a balanced transporta-tion problem as ⨁3

119894=1119886119868

119894= ⨁3

119895=1119868

119895 The above IFTP can be

written as

Minimize (2 4 6 7) (1 4 6 9) 11990911

oplus (4 6 7 8) (3 6 7 9) 11990912

oplus (3 7 9 12) (2 7 9 13) 11990913

oplus (1 3 4 6) (05 3 4 7) 11990921

oplus (3 5 6 7) (2 5 6 9) 11990922

oplus (2 6 7 11) (1 6 7 12) 11990923

oplus (3 4 5 8) (2 4 5 10) 11990931

oplus (1 2 3 4) (05 2 3 5) 11990932

oplus (2 4 5 10) (1 4 5 11) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (4 6 8 9) (2 6 8 10)

11990921

+ 11990922

+ 11990923

le (0 05 1 2) (0 05 1 5)

11990931

+ 11990932

+ 11990933

le (8 95 10 11) (65 95 10 11)

11990911

+ 11990921

+ 11990931

ge (6 7 8 10) (5 7 8 12)

11990912

+ 11990922

+ 11990932

ge (4 5 6 9) (3 5 6 11)

11990913

+ 11990923

+ 11990933

ge (2 4 5 7) (05 4 5 8)

119909119894119895ge 0 forall119894 119895

(40)

Nowby using Step 2 of themethod explained in Section 4 andLemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (40) problem (40) isconverted into following crisp programming problem

Minimize (6 + 2120572 minus 3120573) 11990911

+ (10 + 2120572 minus 2120573) 11990912

+ (10 + 4120572 minus 5120573) 11990913

+ (4 + 2120572 minus 25120573) 11990921

+ (8 + 2120572 minus 3120573) 11990922

+ (8 + 4120572 minus 5120573) 11990923

+ (7 + 120572 minus 5120573) 11990931

+ (3 + 120572 minus 15120573) 11990932

+ (6 + 2120572 minus 3120573) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (17 minus 1205821+ 21205951)

11990921

+ 11990922

+ 11990923

le (3 minus 1205822+ 41205952)

11990931

+ 11990932

+ 11990933

le (21 minus 1205823+ 1205953)

11990911

+ 11990921

+ 11990931

ge (13 + 1205824minus 21205954)

11990912

+ 11990922

+ 11990932

ge (9 + 1205825minus 21205955)

11990913

+ 11990923

+ 11990933

ge (6 + 21205826minus 35120595

6)

119909119894119895ge 0 for 119894 119895 = 1 2 3 0 le 120572 + 120573 le 1

0 le 120582119896+ 120595119896le 1 for 119896 = 1 2 6

(41)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and possibility levels (120582119894= 05 120595

119894= 05) for

119894 = 1 2 6 using Lingo-110 we get 11990911

= 075 11990912

= 011990913

= 0 11990921

= 45 11990922

= 0 11990923

= 0 11990931

= 725 11990932

= 85 and

International Journal of Engineering Mathematics 11

11990933

= 525 Now the minimum intuitionstic fuzzy optimalcost is

119888119868

= (7225 109 136 2035) (4562 109 136 24625) (42)

8 Discussion

Intuitionistic fuzzy sets being a generalization of fuzzy setsgive us an additional possibility to represent imperfect knowl-edge making it possible to describe many real problems ina more adequate way So in this paper we have developedthe possibility and necessity measures on intuitionistic fuzzyset Here we have presented first time the mathematicalrepresentation of different types of measures in intuitionisticfuzzy environments and some graphical representations ofthemare depictedWehave also developed the theoretical cal-culation on possibility necessity and credibility measures fordefuzzify intuitionistic fuzzy linear programming problemusing chance operators To validate the proposed methodwe have discussed three different approaches to defuzzifythe intuitionistic fuzzy relations using possibility necessityand credibility measures Using chance operator we canconvert a problem under imprecise models to correspondingcrisp models At different levels of possibility necessity andcredibility we have achieved different optimal solution Anumerical example is presented and solved using LINGO-110to illustrate the proposed approaches The proposed methodcan be applied for multiobjective multiitem transportationproblem This method can be also extended to be appliedinto different types of optimization problem namely optimalcontrol and solid transportation problems

Conflict of Interests

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

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov Intuitionistic Fuzzy Sets VII ITKRrsquos SessionSofia Bulgarian 1983

[3] S K De and S S Sana ldquoA multi-periods production-inventorymodel with capacity constraints for multi-manufacturersmdashaglobal optimality in intuitionistic fuzzy environmentrdquo AppliedMathematics andComputation vol 242 no 1 pp 825ndash841 2014

[4] HGargM Rani S P Sharma andYVishwakarma ldquoIntuition-istic fuzzy optimization technique for solving multi-objectivereliability optimization problems in interval environmentrdquoExpert Systems with Applications vol 41 no 7 pp 3157ndash31672014

[5] J Wu and Y Liu ldquoAn approach for multiple attribute groupdecision making problems with interval-valued intuitionistictrapezoidal fuzzy numbersrdquoComputers and Industrial Engineer-ing vol 66 no 2 pp 311ndash324 2013

[6] A Nagoorgani and K Ponnalagu ldquoA new approach on solv-ing intuitionistic fuzzy linear programming problemrdquo AppliedMathematical Sciences vol 6 no 70 pp 3467ndash3474 2012

[7] G S Mahapatra and T K Roy ldquoIntuitionistic fuzzy numberand its arithmetic operation with application on system failurerdquoJournal of Uncertain Systems vol 7 no 2 pp 92ndash107 2013

[8] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[9] S Pramanik D K Jana and M Maiti ldquoA multi objectivesolid transportation problem in fuzzy Bi-fuzzy environmentvia genetic algorithmrdquo International Journal of Advanced Oper-ations Management vol 6 no 1 pp 4ndash26 2014

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

[11] M Esmailzadeh and M Esmailzadeh ldquoNew distance betweentriangular intuitionistic fuzzy numbersrdquo Advances in Computa-tional Mathematics and Its Applications vol 2 no 3 pp 3ndash102013

[12] P P Angelov ldquoOptimization in an intuitionistic fuzzy environ-mentrdquo Fuzzy Sets and Systems vol 86 no 3 pp 299ndash306 1997

[13] D Dubey and A Mehra ldquoLinear programming with triangularintuitionistic fuzzy numberrdquo Advances in Intelligent SystemsResearch vol 1 no 1 pp 563ndash569 2011

[14] R Parvathi and C Malathi ldquoIntuitionistic fuzzy simplexmethodrdquo International Journal of Computer Applications vol48 no 6 pp 39ndash48 2012

[15] R J Hussain and S P Kumar ldquoAlgorithmic approach forsolving intuitionistic fuzzy transportation problemrdquo AppliedMathematical Sciences vol 6 no 77-80 pp 3981ndash3989 2012

[16] ANagoorGani and S Abbas ldquoSolving intuitionstic fuzzy trans-portation problem using zero suffix algorithmrdquo InternationalJournal of Mathematics Sciences amp Enggineering Applicationsvol 6 pp 73ndash82 2012

[17] J Ye ldquoExpected value method for intuitionistic trapezoidalfuzzy multicriteria decision-making problemsrdquo Expert Systemswith Applications vol 38 no 9 pp 11730ndash11734 2011

[18] 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 2013

[19] J J Buckley ldquoPossibility and necessity in optimizationrdquo FuzzySets and Systems vol 25 no 1 pp 1ndash13 1988

[20] K D Jamison and W A Lodwick ldquoThe construction ofconsistent possibility and necessity measuresrdquo Fuzzy Sets andSystems vol 132 no 1 pp 1ndash10 2002

[21] J Ramık ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006

[22] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[23] M Sakawa H Katagiri and T Matsui ldquoFuzzy random bilevellinear programming through expectation optimization usingpossibility and necessityrdquo International Journal of MachineLearning and Cybernetics vol 3 no 3 pp 183ndash192 2012

[24] S Pathak S Kar and S Sarkar ldquoFuzzy production inventorymodel for deteriorating items with shortages under the effect oftime dependent learning and forgetting a possibilitynecessityapproachrdquo OPSEARCH vol 50 no 2 pp 149ndash181 2013

[25] K Maity ldquoPossibility and necessity representations of fuzzyinequality and its application to two warehouse production-inventory problemrdquo Applied Mathematical Modelling vol 35no 3 pp 1252ndash1263 2011

12 International Journal of Engineering Mathematics

[26] H-C Wu ldquoFuzzy optimization problems based on the embed-ding theorem and possibility and necessity measuresrdquo Mathe-matical and Computer Modelling vol 40 no 3-4 pp 329ndash3362004

[27] J Xu and X Zhou Fuzzy-Like Multiple Objective DecesionMaking Springer 2010

[28] K Maity and M Maiti ldquoPossibility and necessity constraintsand their defuzzificationmdasha multi-item production-inventoryscenario via optimal control theoryrdquo European Journal of Oper-ational Research vol 177 no 2 pp 882ndash896 2007

[29] B Das K Maity and M Maiti ldquoA two warehouse supply-chain model under possibilitynecessitycredibility measuresrdquoMathematical andComputerModelling vol 46 no 3-4 pp 398ndash409 2007

[30] D Panda S Kar K Maity and M Maiti ldquoA single periodinventory model with imperfect production and stochasticdemand under chance and imprecise constraintsrdquo EuropeanJournal of Operational Research vol 188 no 1 pp 121ndash139 2008

[31] A I Ban ldquoIntuitionistic fuzzy-valued possibility and necessitymeasuresrdquo in Proceedings of the 8th International Conference onIntuitionistic Fuzzy Sets vol 10 pp 1ndash7 Varna Bulgaria 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Engineering Mathematics 9

Table 1 Optimal solution of (38) at different possibility levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)030 030 035 035 119909

1= 341 119909

2= 137 7009

030 035 035 040 1199091= 329 119909

2= 166 7214

040 035 030 040 1199091= 343 119909

2= 157 7309

035 040 040 030 1199091= 310 119909

2= 19 722

040 045 040 045 1199091= 316 119909

2= 173 7105

050 045 045 05 1199091= 316 119909

2= 150 6793

045 050 050 040 1199091= 297 119909

2= 173 6802

060 045 045 06 1199091= 329 119909

2= 120 6593

050 050 050 050 1199091= 303 119909

2= 157 6700

055 040 055 040 1199091= 297 119909

2= 150 6510

Table 2 Optimal solution of (39) at different necessity levels

1205821

1205951

1205822

1205952

Optimal solution Optimal value (119891lowast)035 030 030 035 119909

1= 091 119909

2= 043 1293

045 030 040 035 1199091= 090 119909

2= 045 1289

050 035 050 035 1199091= 087 119909

2= 047 1286

060 040 060 040 1199091= 085 119909

2= 050 1284

070 020 070 020 1199091= 087 119909

2= 045 1271

060 030 060 020 1199091= 087 119909

2= 047 1279

065 035 075 020 1199091= 084 119909

2= 050 1275

070 010 070 010 1199091= 089 119909

2= 043 1267

050 050 050 050 1199091= 085 119909

2= 051 1294

035 045 035 045 1199091= 088 119909

2= 048 1298

Table 3 Input data for IFTP

1198631

119868

1198632

119868

1198633

119868 Availability (119886119894

119868

)

1198781

119868 (2 4 6 7) (1 4 6 9) (4 6 7 8) (3 6 7 9) (3 7 9 12) (2 7 9 13) (4 6 8 9) (2 6 8 10)1198782

119868 (1 3 4 6) (05 3 4 7) (3 5 6 7) (2 5 6 9) (2 6 7 11) (1 6 7 12) (0 05 1 2) (0 05 1 5)1198783

119868 (3 4 5 8) (2 4 5 10) (1 2 3 4) (05 2 3 5) (2 4 5 10) (1 4 5 11) (8 95 10 11) (65 95 10 11)Demand (119887

119895

119868

) (6 7 8 10) (5 7 8 12) (4 5 6 9) (3 5 6 11) (2 4 5 7) (05 4 5 8)

646668707274

(030 030)

(035

035

)

(035

040

)

(030

040

) (055 040)

(050 050)

(060 045)

(045 050)

(050 045)(0

40 0

30)

(040

045

)

(045

050

)

(050

040

)

(045

060

)

(045

045

)

(030

030

)(040 045)

(035 040)

(040 035)

(030 035)(1205821 1205951

)(1205822 120595

2 )

flowast

Figure 8 Optimal solution at different possibility levels

10 International Journal of Engineering Mathematics

1265127

1275128

1285129

129513

(035 030)

(050

035

)

(060

040

)

(070

020

)

(060

020

)

(075

020

)

(070

010

)

(050

050

)

(035

045

)

(035 045)

(050 050)

(070 010)

(065 035)

(060 030)

(070 020)

(060 040)

(050 035)

(045 030)

(040

035

)

(030

035

)

(1205821 1205951)

(1205822 120595

2 )

flowast

Figure 9 Optimal solution at different necessity levels

7 Intuitionistic Fuzzy TransportationProblem Based on Possibility Measure

Let us consider the following intuitionistic fuzzy transporta-tion problem (IFTP) (in Table 3)

Above transportation problem is a balanced transporta-tion problem as ⨁3

119894=1119886119868

119894= ⨁3

119895=1119868

119895 The above IFTP can be

written as

Minimize (2 4 6 7) (1 4 6 9) 11990911

oplus (4 6 7 8) (3 6 7 9) 11990912

oplus (3 7 9 12) (2 7 9 13) 11990913

oplus (1 3 4 6) (05 3 4 7) 11990921

oplus (3 5 6 7) (2 5 6 9) 11990922

oplus (2 6 7 11) (1 6 7 12) 11990923

oplus (3 4 5 8) (2 4 5 10) 11990931

oplus (1 2 3 4) (05 2 3 5) 11990932

oplus (2 4 5 10) (1 4 5 11) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (4 6 8 9) (2 6 8 10)

11990921

+ 11990922

+ 11990923

le (0 05 1 2) (0 05 1 5)

11990931

+ 11990932

+ 11990933

le (8 95 10 11) (65 95 10 11)

11990911

+ 11990921

+ 11990931

ge (6 7 8 10) (5 7 8 12)

11990912

+ 11990922

+ 11990932

ge (4 5 6 9) (3 5 6 11)

11990913

+ 11990923

+ 11990933

ge (2 4 5 7) (05 4 5 8)

119909119894119895ge 0 forall119894 119895

(40)

Nowby using Step 2 of themethod explained in Section 4 andLemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (40) problem (40) isconverted into following crisp programming problem

Minimize (6 + 2120572 minus 3120573) 11990911

+ (10 + 2120572 minus 2120573) 11990912

+ (10 + 4120572 minus 5120573) 11990913

+ (4 + 2120572 minus 25120573) 11990921

+ (8 + 2120572 minus 3120573) 11990922

+ (8 + 4120572 minus 5120573) 11990923

+ (7 + 120572 minus 5120573) 11990931

+ (3 + 120572 minus 15120573) 11990932

+ (6 + 2120572 minus 3120573) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (17 minus 1205821+ 21205951)

11990921

+ 11990922

+ 11990923

le (3 minus 1205822+ 41205952)

11990931

+ 11990932

+ 11990933

le (21 minus 1205823+ 1205953)

11990911

+ 11990921

+ 11990931

ge (13 + 1205824minus 21205954)

11990912

+ 11990922

+ 11990932

ge (9 + 1205825minus 21205955)

11990913

+ 11990923

+ 11990933

ge (6 + 21205826minus 35120595

6)

119909119894119895ge 0 for 119894 119895 = 1 2 3 0 le 120572 + 120573 le 1

0 le 120582119896+ 120595119896le 1 for 119896 = 1 2 6

(41)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and possibility levels (120582119894= 05 120595

119894= 05) for

119894 = 1 2 6 using Lingo-110 we get 11990911

= 075 11990912

= 011990913

= 0 11990921

= 45 11990922

= 0 11990923

= 0 11990931

= 725 11990932

= 85 and

International Journal of Engineering Mathematics 11

11990933

= 525 Now the minimum intuitionstic fuzzy optimalcost is

119888119868

= (7225 109 136 2035) (4562 109 136 24625) (42)

8 Discussion

Intuitionistic fuzzy sets being a generalization of fuzzy setsgive us an additional possibility to represent imperfect knowl-edge making it possible to describe many real problems ina more adequate way So in this paper we have developedthe possibility and necessity measures on intuitionistic fuzzyset Here we have presented first time the mathematicalrepresentation of different types of measures in intuitionisticfuzzy environments and some graphical representations ofthemare depictedWehave also developed the theoretical cal-culation on possibility necessity and credibility measures fordefuzzify intuitionistic fuzzy linear programming problemusing chance operators To validate the proposed methodwe have discussed three different approaches to defuzzifythe intuitionistic fuzzy relations using possibility necessityand credibility measures Using chance operator we canconvert a problem under imprecise models to correspondingcrisp models At different levels of possibility necessity andcredibility we have achieved different optimal solution Anumerical example is presented and solved using LINGO-110to illustrate the proposed approaches The proposed methodcan be applied for multiobjective multiitem transportationproblem This method can be also extended to be appliedinto different types of optimization problem namely optimalcontrol and solid transportation problems

Conflict of Interests

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

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov Intuitionistic Fuzzy Sets VII ITKRrsquos SessionSofia Bulgarian 1983

[3] S K De and S S Sana ldquoA multi-periods production-inventorymodel with capacity constraints for multi-manufacturersmdashaglobal optimality in intuitionistic fuzzy environmentrdquo AppliedMathematics andComputation vol 242 no 1 pp 825ndash841 2014

[4] HGargM Rani S P Sharma andYVishwakarma ldquoIntuition-istic fuzzy optimization technique for solving multi-objectivereliability optimization problems in interval environmentrdquoExpert Systems with Applications vol 41 no 7 pp 3157ndash31672014

[5] J Wu and Y Liu ldquoAn approach for multiple attribute groupdecision making problems with interval-valued intuitionistictrapezoidal fuzzy numbersrdquoComputers and Industrial Engineer-ing vol 66 no 2 pp 311ndash324 2013

[6] A Nagoorgani and K Ponnalagu ldquoA new approach on solv-ing intuitionistic fuzzy linear programming problemrdquo AppliedMathematical Sciences vol 6 no 70 pp 3467ndash3474 2012

[7] G S Mahapatra and T K Roy ldquoIntuitionistic fuzzy numberand its arithmetic operation with application on system failurerdquoJournal of Uncertain Systems vol 7 no 2 pp 92ndash107 2013

[8] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[9] S Pramanik D K Jana and M Maiti ldquoA multi objectivesolid transportation problem in fuzzy Bi-fuzzy environmentvia genetic algorithmrdquo International Journal of Advanced Oper-ations Management vol 6 no 1 pp 4ndash26 2014

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

[11] M Esmailzadeh and M Esmailzadeh ldquoNew distance betweentriangular intuitionistic fuzzy numbersrdquo Advances in Computa-tional Mathematics and Its Applications vol 2 no 3 pp 3ndash102013

[12] P P Angelov ldquoOptimization in an intuitionistic fuzzy environ-mentrdquo Fuzzy Sets and Systems vol 86 no 3 pp 299ndash306 1997

[13] D Dubey and A Mehra ldquoLinear programming with triangularintuitionistic fuzzy numberrdquo Advances in Intelligent SystemsResearch vol 1 no 1 pp 563ndash569 2011

[14] R Parvathi and C Malathi ldquoIntuitionistic fuzzy simplexmethodrdquo International Journal of Computer Applications vol48 no 6 pp 39ndash48 2012

[15] R J Hussain and S P Kumar ldquoAlgorithmic approach forsolving intuitionistic fuzzy transportation problemrdquo AppliedMathematical Sciences vol 6 no 77-80 pp 3981ndash3989 2012

[16] ANagoorGani and S Abbas ldquoSolving intuitionstic fuzzy trans-portation problem using zero suffix algorithmrdquo InternationalJournal of Mathematics Sciences amp Enggineering Applicationsvol 6 pp 73ndash82 2012

[17] J Ye ldquoExpected value method for intuitionistic trapezoidalfuzzy multicriteria decision-making problemsrdquo Expert Systemswith Applications vol 38 no 9 pp 11730ndash11734 2011

[18] 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 2013

[19] J J Buckley ldquoPossibility and necessity in optimizationrdquo FuzzySets and Systems vol 25 no 1 pp 1ndash13 1988

[20] K D Jamison and W A Lodwick ldquoThe construction ofconsistent possibility and necessity measuresrdquo Fuzzy Sets andSystems vol 132 no 1 pp 1ndash10 2002

[21] J Ramık ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006

[22] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[23] M Sakawa H Katagiri and T Matsui ldquoFuzzy random bilevellinear programming through expectation optimization usingpossibility and necessityrdquo International Journal of MachineLearning and Cybernetics vol 3 no 3 pp 183ndash192 2012

[24] S Pathak S Kar and S Sarkar ldquoFuzzy production inventorymodel for deteriorating items with shortages under the effect oftime dependent learning and forgetting a possibilitynecessityapproachrdquo OPSEARCH vol 50 no 2 pp 149ndash181 2013

[25] K Maity ldquoPossibility and necessity representations of fuzzyinequality and its application to two warehouse production-inventory problemrdquo Applied Mathematical Modelling vol 35no 3 pp 1252ndash1263 2011

12 International Journal of Engineering Mathematics

[26] H-C Wu ldquoFuzzy optimization problems based on the embed-ding theorem and possibility and necessity measuresrdquo Mathe-matical and Computer Modelling vol 40 no 3-4 pp 329ndash3362004

[27] J Xu and X Zhou Fuzzy-Like Multiple Objective DecesionMaking Springer 2010

[28] K Maity and M Maiti ldquoPossibility and necessity constraintsand their defuzzificationmdasha multi-item production-inventoryscenario via optimal control theoryrdquo European Journal of Oper-ational Research vol 177 no 2 pp 882ndash896 2007

[29] B Das K Maity and M Maiti ldquoA two warehouse supply-chain model under possibilitynecessitycredibility measuresrdquoMathematical andComputerModelling vol 46 no 3-4 pp 398ndash409 2007

[30] D Panda S Kar K Maity and M Maiti ldquoA single periodinventory model with imperfect production and stochasticdemand under chance and imprecise constraintsrdquo EuropeanJournal of Operational Research vol 188 no 1 pp 121ndash139 2008

[31] A I Ban ldquoIntuitionistic fuzzy-valued possibility and necessitymeasuresrdquo in Proceedings of the 8th International Conference onIntuitionistic Fuzzy Sets vol 10 pp 1ndash7 Varna Bulgaria 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Journal of Engineering Mathematics

1265127

1275128

1285129

129513

(035 030)

(050

035

)

(060

040

)

(070

020

)

(060

020

)

(075

020

)

(070

010

)

(050

050

)

(035

045

)

(035 045)

(050 050)

(070 010)

(065 035)

(060 030)

(070 020)

(060 040)

(050 035)

(045 030)

(040

035

)

(030

035

)

(1205821 1205951)

(1205822 120595

2 )

flowast

Figure 9 Optimal solution at different necessity levels

7 Intuitionistic Fuzzy TransportationProblem Based on Possibility Measure

Let us consider the following intuitionistic fuzzy transporta-tion problem (IFTP) (in Table 3)

Above transportation problem is a balanced transporta-tion problem as ⨁3

119894=1119886119868

119894= ⨁3

119895=1119868

119895 The above IFTP can be

written as

Minimize (2 4 6 7) (1 4 6 9) 11990911

oplus (4 6 7 8) (3 6 7 9) 11990912

oplus (3 7 9 12) (2 7 9 13) 11990913

oplus (1 3 4 6) (05 3 4 7) 11990921

oplus (3 5 6 7) (2 5 6 9) 11990922

oplus (2 6 7 11) (1 6 7 12) 11990923

oplus (3 4 5 8) (2 4 5 10) 11990931

oplus (1 2 3 4) (05 2 3 5) 11990932

oplus (2 4 5 10) (1 4 5 11) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (4 6 8 9) (2 6 8 10)

11990921

+ 11990922

+ 11990923

le (0 05 1 2) (0 05 1 5)

11990931

+ 11990932

+ 11990933

le (8 95 10 11) (65 95 10 11)

11990911

+ 11990921

+ 11990931

ge (6 7 8 10) (5 7 8 12)

11990912

+ 11990922

+ 11990932

ge (4 5 6 9) (3 5 6 11)

11990913

+ 11990923

+ 11990933

ge (2 4 5 7) (05 4 5 8)

119909119894119895ge 0 forall119894 119895

(40)

Nowby using Step 2 of themethod explained in Section 4 andLemma 10 if we apply the possibilitymeasure in intuitionisticfuzzy mathematical programming (40) problem (40) isconverted into following crisp programming problem

Minimize (6 + 2120572 minus 3120573) 11990911

+ (10 + 2120572 minus 2120573) 11990912

+ (10 + 4120572 minus 5120573) 11990913

+ (4 + 2120572 minus 25120573) 11990921

+ (8 + 2120572 minus 3120573) 11990922

+ (8 + 4120572 minus 5120573) 11990923

+ (7 + 120572 minus 5120573) 11990931

+ (3 + 120572 minus 15120573) 11990932

+ (6 + 2120572 minus 3120573) 11990933

subject to 11990911

+ 11990912

+ 11990913

le (17 minus 1205821+ 21205951)

11990921

+ 11990922

+ 11990923

le (3 minus 1205822+ 41205952)

11990931

+ 11990932

+ 11990933

le (21 minus 1205823+ 1205953)

11990911

+ 11990921

+ 11990931

ge (13 + 1205824minus 21205954)

11990912

+ 11990922

+ 11990932

ge (9 + 1205825minus 21205955)

11990913

+ 11990923

+ 11990933

ge (6 + 21205826minus 35120595

6)

119909119894119895ge 0 for 119894 119895 = 1 2 3 0 le 120572 + 120573 le 1

0 le 120582119896+ 120595119896le 1 for 119896 = 1 2 6

(41)

Solving the above crisp problem for efficient levels (120572 =

06 120573 = 04) and possibility levels (120582119894= 05 120595

119894= 05) for

119894 = 1 2 6 using Lingo-110 we get 11990911

= 075 11990912

= 011990913

= 0 11990921

= 45 11990922

= 0 11990923

= 0 11990931

= 725 11990932

= 85 and

International Journal of Engineering Mathematics 11

11990933

= 525 Now the minimum intuitionstic fuzzy optimalcost is

119888119868

= (7225 109 136 2035) (4562 109 136 24625) (42)

8 Discussion

Intuitionistic fuzzy sets being a generalization of fuzzy setsgive us an additional possibility to represent imperfect knowl-edge making it possible to describe many real problems ina more adequate way So in this paper we have developedthe possibility and necessity measures on intuitionistic fuzzyset Here we have presented first time the mathematicalrepresentation of different types of measures in intuitionisticfuzzy environments and some graphical representations ofthemare depictedWehave also developed the theoretical cal-culation on possibility necessity and credibility measures fordefuzzify intuitionistic fuzzy linear programming problemusing chance operators To validate the proposed methodwe have discussed three different approaches to defuzzifythe intuitionistic fuzzy relations using possibility necessityand credibility measures Using chance operator we canconvert a problem under imprecise models to correspondingcrisp models At different levels of possibility necessity andcredibility we have achieved different optimal solution Anumerical example is presented and solved using LINGO-110to illustrate the proposed approaches The proposed methodcan be applied for multiobjective multiitem transportationproblem This method can be also extended to be appliedinto different types of optimization problem namely optimalcontrol and solid transportation problems

Conflict of Interests

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

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov Intuitionistic Fuzzy Sets VII ITKRrsquos SessionSofia Bulgarian 1983

[3] S K De and S S Sana ldquoA multi-periods production-inventorymodel with capacity constraints for multi-manufacturersmdashaglobal optimality in intuitionistic fuzzy environmentrdquo AppliedMathematics andComputation vol 242 no 1 pp 825ndash841 2014

[4] HGargM Rani S P Sharma andYVishwakarma ldquoIntuition-istic fuzzy optimization technique for solving multi-objectivereliability optimization problems in interval environmentrdquoExpert Systems with Applications vol 41 no 7 pp 3157ndash31672014

[5] J Wu and Y Liu ldquoAn approach for multiple attribute groupdecision making problems with interval-valued intuitionistictrapezoidal fuzzy numbersrdquoComputers and Industrial Engineer-ing vol 66 no 2 pp 311ndash324 2013

[6] A Nagoorgani and K Ponnalagu ldquoA new approach on solv-ing intuitionistic fuzzy linear programming problemrdquo AppliedMathematical Sciences vol 6 no 70 pp 3467ndash3474 2012

[7] G S Mahapatra and T K Roy ldquoIntuitionistic fuzzy numberand its arithmetic operation with application on system failurerdquoJournal of Uncertain Systems vol 7 no 2 pp 92ndash107 2013

[8] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[9] S Pramanik D K Jana and M Maiti ldquoA multi objectivesolid transportation problem in fuzzy Bi-fuzzy environmentvia genetic algorithmrdquo International Journal of Advanced Oper-ations Management vol 6 no 1 pp 4ndash26 2014

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

[11] M Esmailzadeh and M Esmailzadeh ldquoNew distance betweentriangular intuitionistic fuzzy numbersrdquo Advances in Computa-tional Mathematics and Its Applications vol 2 no 3 pp 3ndash102013

[12] P P Angelov ldquoOptimization in an intuitionistic fuzzy environ-mentrdquo Fuzzy Sets and Systems vol 86 no 3 pp 299ndash306 1997

[13] D Dubey and A Mehra ldquoLinear programming with triangularintuitionistic fuzzy numberrdquo Advances in Intelligent SystemsResearch vol 1 no 1 pp 563ndash569 2011

[14] R Parvathi and C Malathi ldquoIntuitionistic fuzzy simplexmethodrdquo International Journal of Computer Applications vol48 no 6 pp 39ndash48 2012

[15] R J Hussain and S P Kumar ldquoAlgorithmic approach forsolving intuitionistic fuzzy transportation problemrdquo AppliedMathematical Sciences vol 6 no 77-80 pp 3981ndash3989 2012

[16] ANagoorGani and S Abbas ldquoSolving intuitionstic fuzzy trans-portation problem using zero suffix algorithmrdquo InternationalJournal of Mathematics Sciences amp Enggineering Applicationsvol 6 pp 73ndash82 2012

[17] J Ye ldquoExpected value method for intuitionistic trapezoidalfuzzy multicriteria decision-making problemsrdquo Expert Systemswith Applications vol 38 no 9 pp 11730ndash11734 2011

[18] 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 2013

[19] J J Buckley ldquoPossibility and necessity in optimizationrdquo FuzzySets and Systems vol 25 no 1 pp 1ndash13 1988

[20] K D Jamison and W A Lodwick ldquoThe construction ofconsistent possibility and necessity measuresrdquo Fuzzy Sets andSystems vol 132 no 1 pp 1ndash10 2002

[21] J Ramık ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006

[22] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[23] M Sakawa H Katagiri and T Matsui ldquoFuzzy random bilevellinear programming through expectation optimization usingpossibility and necessityrdquo International Journal of MachineLearning and Cybernetics vol 3 no 3 pp 183ndash192 2012

[24] S Pathak S Kar and S Sarkar ldquoFuzzy production inventorymodel for deteriorating items with shortages under the effect oftime dependent learning and forgetting a possibilitynecessityapproachrdquo OPSEARCH vol 50 no 2 pp 149ndash181 2013

[25] K Maity ldquoPossibility and necessity representations of fuzzyinequality and its application to two warehouse production-inventory problemrdquo Applied Mathematical Modelling vol 35no 3 pp 1252ndash1263 2011

12 International Journal of Engineering Mathematics

[26] H-C Wu ldquoFuzzy optimization problems based on the embed-ding theorem and possibility and necessity measuresrdquo Mathe-matical and Computer Modelling vol 40 no 3-4 pp 329ndash3362004

[27] J Xu and X Zhou Fuzzy-Like Multiple Objective DecesionMaking Springer 2010

[28] K Maity and M Maiti ldquoPossibility and necessity constraintsand their defuzzificationmdasha multi-item production-inventoryscenario via optimal control theoryrdquo European Journal of Oper-ational Research vol 177 no 2 pp 882ndash896 2007

[29] B Das K Maity and M Maiti ldquoA two warehouse supply-chain model under possibilitynecessitycredibility measuresrdquoMathematical andComputerModelling vol 46 no 3-4 pp 398ndash409 2007

[30] D Panda S Kar K Maity and M Maiti ldquoA single periodinventory model with imperfect production and stochasticdemand under chance and imprecise constraintsrdquo EuropeanJournal of Operational Research vol 188 no 1 pp 121ndash139 2008

[31] A I Ban ldquoIntuitionistic fuzzy-valued possibility and necessitymeasuresrdquo in Proceedings of the 8th International Conference onIntuitionistic Fuzzy Sets vol 10 pp 1ndash7 Varna Bulgaria 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Engineering Mathematics 11

11990933

= 525 Now the minimum intuitionstic fuzzy optimalcost is

119888119868

= (7225 109 136 2035) (4562 109 136 24625) (42)

8 Discussion

Intuitionistic fuzzy sets being a generalization of fuzzy setsgive us an additional possibility to represent imperfect knowl-edge making it possible to describe many real problems ina more adequate way So in this paper we have developedthe possibility and necessity measures on intuitionistic fuzzyset Here we have presented first time the mathematicalrepresentation of different types of measures in intuitionisticfuzzy environments and some graphical representations ofthemare depictedWehave also developed the theoretical cal-culation on possibility necessity and credibility measures fordefuzzify intuitionistic fuzzy linear programming problemusing chance operators To validate the proposed methodwe have discussed three different approaches to defuzzifythe intuitionistic fuzzy relations using possibility necessityand credibility measures Using chance operator we canconvert a problem under imprecise models to correspondingcrisp models At different levels of possibility necessity andcredibility we have achieved different optimal solution Anumerical example is presented and solved using LINGO-110to illustrate the proposed approaches The proposed methodcan be applied for multiobjective multiitem transportationproblem This method can be also extended to be appliedinto different types of optimization problem namely optimalcontrol and solid transportation problems

Conflict of Interests

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

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov Intuitionistic Fuzzy Sets VII ITKRrsquos SessionSofia Bulgarian 1983

[3] S K De and S S Sana ldquoA multi-periods production-inventorymodel with capacity constraints for multi-manufacturersmdashaglobal optimality in intuitionistic fuzzy environmentrdquo AppliedMathematics andComputation vol 242 no 1 pp 825ndash841 2014

[4] HGargM Rani S P Sharma andYVishwakarma ldquoIntuition-istic fuzzy optimization technique for solving multi-objectivereliability optimization problems in interval environmentrdquoExpert Systems with Applications vol 41 no 7 pp 3157ndash31672014

[5] J Wu and Y Liu ldquoAn approach for multiple attribute groupdecision making problems with interval-valued intuitionistictrapezoidal fuzzy numbersrdquoComputers and Industrial Engineer-ing vol 66 no 2 pp 311ndash324 2013

[6] A Nagoorgani and K Ponnalagu ldquoA new approach on solv-ing intuitionistic fuzzy linear programming problemrdquo AppliedMathematical Sciences vol 6 no 70 pp 3467ndash3474 2012

[7] G S Mahapatra and T K Roy ldquoIntuitionistic fuzzy numberand its arithmetic operation with application on system failurerdquoJournal of Uncertain Systems vol 7 no 2 pp 92ndash107 2013

[8] S Pramanik D K Jana and M Maiti ldquoMulti-objective solidtransportation problem in imprecise environmentsrdquo Journal ofTransportation Security vol 6 no 2 pp 131ndash150 2013

[9] S Pramanik D K Jana and M Maiti ldquoA multi objectivesolid transportation problem in fuzzy Bi-fuzzy environmentvia genetic algorithmrdquo International Journal of Advanced Oper-ations Management vol 6 no 1 pp 4ndash26 2014

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

[11] M Esmailzadeh and M Esmailzadeh ldquoNew distance betweentriangular intuitionistic fuzzy numbersrdquo Advances in Computa-tional Mathematics and Its Applications vol 2 no 3 pp 3ndash102013

[12] P P Angelov ldquoOptimization in an intuitionistic fuzzy environ-mentrdquo Fuzzy Sets and Systems vol 86 no 3 pp 299ndash306 1997

[13] D Dubey and A Mehra ldquoLinear programming with triangularintuitionistic fuzzy numberrdquo Advances in Intelligent SystemsResearch vol 1 no 1 pp 563ndash569 2011

[14] R Parvathi and C Malathi ldquoIntuitionistic fuzzy simplexmethodrdquo International Journal of Computer Applications vol48 no 6 pp 39ndash48 2012

[15] R J Hussain and S P Kumar ldquoAlgorithmic approach forsolving intuitionistic fuzzy transportation problemrdquo AppliedMathematical Sciences vol 6 no 77-80 pp 3981ndash3989 2012

[16] ANagoorGani and S Abbas ldquoSolving intuitionstic fuzzy trans-portation problem using zero suffix algorithmrdquo InternationalJournal of Mathematics Sciences amp Enggineering Applicationsvol 6 pp 73ndash82 2012

[17] J Ye ldquoExpected value method for intuitionistic trapezoidalfuzzy multicriteria decision-making problemsrdquo Expert Systemswith Applications vol 38 no 9 pp 11730ndash11734 2011

[18] 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 2013

[19] J J Buckley ldquoPossibility and necessity in optimizationrdquo FuzzySets and Systems vol 25 no 1 pp 1ndash13 1988

[20] K D Jamison and W A Lodwick ldquoThe construction ofconsistent possibility and necessity measuresrdquo Fuzzy Sets andSystems vol 132 no 1 pp 1ndash10 2002

[21] J Ramık ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006

[22] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[23] M Sakawa H Katagiri and T Matsui ldquoFuzzy random bilevellinear programming through expectation optimization usingpossibility and necessityrdquo International Journal of MachineLearning and Cybernetics vol 3 no 3 pp 183ndash192 2012

[24] S Pathak S Kar and S Sarkar ldquoFuzzy production inventorymodel for deteriorating items with shortages under the effect oftime dependent learning and forgetting a possibilitynecessityapproachrdquo OPSEARCH vol 50 no 2 pp 149ndash181 2013

[25] K Maity ldquoPossibility and necessity representations of fuzzyinequality and its application to two warehouse production-inventory problemrdquo Applied Mathematical Modelling vol 35no 3 pp 1252ndash1263 2011

12 International Journal of Engineering Mathematics

[26] H-C Wu ldquoFuzzy optimization problems based on the embed-ding theorem and possibility and necessity measuresrdquo Mathe-matical and Computer Modelling vol 40 no 3-4 pp 329ndash3362004

[27] J Xu and X Zhou Fuzzy-Like Multiple Objective DecesionMaking Springer 2010

[28] K Maity and M Maiti ldquoPossibility and necessity constraintsand their defuzzificationmdasha multi-item production-inventoryscenario via optimal control theoryrdquo European Journal of Oper-ational Research vol 177 no 2 pp 882ndash896 2007

[29] B Das K Maity and M Maiti ldquoA two warehouse supply-chain model under possibilitynecessitycredibility measuresrdquoMathematical andComputerModelling vol 46 no 3-4 pp 398ndash409 2007

[30] D Panda S Kar K Maity and M Maiti ldquoA single periodinventory model with imperfect production and stochasticdemand under chance and imprecise constraintsrdquo EuropeanJournal of Operational Research vol 188 no 1 pp 121ndash139 2008

[31] A I Ban ldquoIntuitionistic fuzzy-valued possibility and necessitymeasuresrdquo in Proceedings of the 8th International Conference onIntuitionistic Fuzzy Sets vol 10 pp 1ndash7 Varna Bulgaria 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 International Journal of Engineering Mathematics

[26] H-C Wu ldquoFuzzy optimization problems based on the embed-ding theorem and possibility and necessity measuresrdquo Mathe-matical and Computer Modelling vol 40 no 3-4 pp 329ndash3362004

[27] J Xu and X Zhou Fuzzy-Like Multiple Objective DecesionMaking Springer 2010

[28] K Maity and M Maiti ldquoPossibility and necessity constraintsand their defuzzificationmdasha multi-item production-inventoryscenario via optimal control theoryrdquo European Journal of Oper-ational Research vol 177 no 2 pp 882ndash896 2007

[29] B Das K Maity and M Maiti ldquoA two warehouse supply-chain model under possibilitynecessitycredibility measuresrdquoMathematical andComputerModelling vol 46 no 3-4 pp 398ndash409 2007

[30] D Panda S Kar K Maity and M Maiti ldquoA single periodinventory model with imperfect production and stochasticdemand under chance and imprecise constraintsrdquo EuropeanJournal of Operational Research vol 188 no 1 pp 121ndash139 2008

[31] A I Ban ldquoIntuitionistic fuzzy-valued possibility and necessitymeasuresrdquo in Proceedings of the 8th International Conference onIntuitionistic Fuzzy Sets vol 10 pp 1ndash7 Varna Bulgaria 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of