research article a new approach to solve intuitionistic...
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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
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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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Differential EquationsInternational Journal of
Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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 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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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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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
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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
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