Research ArticleLinear Programming Problem with Interval Type 2 FuzzyCoefficients and an Interpretation for Its Constraints
A Srinivasan and G Geetharamani
Department of Mathematics Anna University Bharathidasan Institute of Technology (BIT) Campus TiruchirappalliTamilnadu 620 024 India
Correspondence should be addressed to A Srinivasan yeacheenu4phdgmailcom
Received 16 August 2015 Accepted 3 November 2015
Academic Editor Frank Werner
Copyright copy 2016 A Srinivasan and G Geetharamani This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
Interval type 2 fuzzy numbers are a special kind of type 2 fuzzy numbers These numbers can be described by triangular andtrapezoidal shapes In this paper first perfectly normal interval type 2 trapezoidal fuzzy numbers with their left-hand and right-hand spreads and their core have been introduced which are normal and convex then a new type of fuzzy arithmetic operationsfor perfectly normal interval type 2 trapezoidal fuzzy numbers has been proposed based on the extension principle of normal type1 trapezoidal fuzzy numbers Moreover in this proposal linear programming problems with resources and technology coefficientsare perfectly normal interval type 2 fuzzy numbers To solve this kind of fuzzy linear programming problems a method basedon the degree of satisfaction (or possibility degree) of the constraints has been introduced In this method the fulfillment of theconstraints can be measured with the help of ranking method of fuzzy numbers Optimal solution is obtained at different degreeof satisfaction by using Barnes algorithm with the help of MATLAB Finally the optimal solution procedure is illustrated withnumerical example
1 Introduction
Linear programming and its applications are used in manyfields of operations research It is concerned with the opti-mization of a linear function while satisfying a set of linearequality andor inequality constraints or restriction In realworld situation a linear programmingmodel involves a lot ofparameters whose values are assigned by experts Howeverboth experts and decision makers often do not preciselyknow the value of those parameters Therefore it is useful toconsider the knowledge of experts about the parameters asfuzzy data To make this possible Zadeh [1] introduced fuzzyset theory The theory proposes a mathematical techniquefor dealing with imprecise concepts and problems that havemany possible solutions The concept of fuzzy mathematicalprogramming on general level was first proposed by Tanakaet al [2] in the framework of the fuzzy decision-making infuzzy environment given by Bellman and Zadeh [3]
The concept of fuzzy liner programming was first for-mulated by Zimmermann [4] After this motivation and
inspiration several authors such as Deng et al discussed thefact that the fuzzy bilevel linear programming with multiplefollowers model solved this complex problem by using thefuzzy structured element method [5] A new interior pointmethod has been presented to solve fuzzy number linearprogramming problems using linear ranking function byZhong et al [6] A fully fuzzy linear programming problemwith fuzzy equality constraints has been discussed and solvedusing a compromise programming approach by Cheng et al[7] An intuitionistic fuzzy chance constraints model (CCM)based on possibility and necessity measures has been devel-oped and proposed a newmethod for solving an intuitionisticfuzzy CCM using chance operators by Chakraborty et al[8] All the above authors and many others have consideredvarious types of fuzzy linear programming problems andproposed several approaches for solving these problemsAmong them Le and Gogne introduced a class of linear pro-gramming problems based on the possibility and necessityrelation [9] Figueroa-Garcıa and Hernandez proposed anextension of the fuzzy linear programming method which
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2016 Article ID 8496812 11 pageshttpdxdoiorg10115520168496812
2 Journal of Applied Mathematics
was proposed by Zimmermann to an interval type 2 fuzzy lin-ear programming problem with linear membership function[10] In another work Garcıa presented a general model forlinear programming where its technological coefficients areassumed as interval type 2 fuzzy sets and it is solved throughan 120572-cuts approach [11] Furthermore Garcıa [12] presenteda general model to handle uncertainty to right-hand sideparameters of a linear programmingmodel by interval type 2fuzzy sets and trapezoidal membership functions to optimizeby using classical algorithms Chen and Lee [13] proposed thedefinition of possibility degree of trapezoidal interval type 2fuzzy numbers and some arithmetic operations Hu et al [14]proposed a new approach based on possibility degree to solvemulticriteria decision-making problems in which the criteriavalue takes the form of interval type 2 fuzzy number
Based on these literaturesrsquo survey with our best knowl-edge however none of them introduced fuzzy linear pro-gramming model with both the right-hand side (resources)and the technological coefficients being perfectly normalinterval type 2 fuzzy numbers based on possibility In thispaper first we have presented a method to do Perfectlynormal Interval Type 2 Fuzzy Number (PnIT2FN) arith-metic operation using well-known arithmetical operationson type 1 fuzzy numbers Further the relation of possibilityfor PnIT2FN as well as its property are discussed Anda fuzzy linear programming model with both the right-hand side (resources) and the technological coefficients beingPnIT2FNs has been proposed By using Zadehrsquos extensionprinciple a pair of upper and lower linear problems are devel-oped to calculate the optimal solution of upper and lowerbounds of linear problems measure at different possibilitylevel 120572
The rest of this paper is organized as follows FirstlySection 2 deals with some preliminary definitions which aregiven In Section 3 the definitions of interval type 2 fuzzysets and trapezoidal interval type 2 fuzzy number have beenrecalled In Section 4 the new representation for PnIT2TrFNits properties and some arithmetic operations of PnIT2TrFNbased on type 1 fuzzy number are presented The possibilitydegree of PnIT2TrFN is discussed in Section 5 The fuzzylinear programming problem with the right-hand side andthe technological coefficientsmodel is presented in Section 6Section 7 has numerical illustrations of proposed fuzzy linearprogramming model with PnIT2TrFN finally Section 8 hasthe conclusion of the work
2 Preliminaries
Definition 1 (see [15]) Let119883 be a nonempty set A fuzzy set in 119883 is characterized by its membership function 120583
119860
119883 rarr
[0 1] and 120583
119860
(119909) is interpreted as the degree of membershipof element 119909 in fuzzy set for each 119909 isin 119883 It is clear that iscompletely determined by the set of tuples
= (119909 120583
119860
(119909)) | 119909 isin 119883 (1)
Definition 2 (see [15]) Let be a fuzzy subset of 119883 thesupport of denoted by Supp() is the crisp subset of
119883 whose elements all have nonzero membership grades in
Supp () = 119909 isin 119883 | 120583
119860
(119909) gt 0 (2)
Definition 3 (see [15]) A fuzzy subset of a classical set 119883is called normal if there exists 119909 isin 119883 such that 120583
119860
(119909) = 1Otherwise is subnormal
Definition 4 An 120572-level set (or 120572-cut) of a fuzzy set of119883 isa nonfuzzy set denoted by
120572
and defined by
120572
=
119909 isin 119883 | 120583
119860
(119909) ge 120572 if 120572 gt 0
cl (Supp ) if 120572 = 0
(3)
where cl(Supp) denotes the closure of the support of
Definition 5 (see [15]) A fuzzy set of 119883 is called convex if120572
is a convex subset of119883 for all 120572 isin [0 1]
Definition 6 (see [15 16]) A fuzzy number is a fuzzy set ofthe real lines with a normal (fuzzy) convex and continuousmembership function of bounded support Alternatively thefuzzy subset of R is called a fuzzy number if the followingconditions are satisfied
(i) is normal that is there exists 119909 isin R such that120583
119860
(119909) = 1(ii) the membership function 120583
119860
(119909) is quasiconcave thatis 120583
119860
(1205821199091
+ (1 minus 120582)1199092
) ge min120583
119860
(1199091
) 120583
119860
(1199092
) for all120572 isin [0 1]
(iii) the membership function 120583
119860
(119909) is upper semicontin-uous that is 119909 isin R 120583
119860
(119909) ge 120572 is a closed subset ofR for all 120572 isin [0 1]
(iv) the 0-level set 120572=0
is compact (closed and boundedin R)
We denote by119865(R) the set of all fuzzy numbers If is a fuzzynumber then from Zadeh [1] 120572-level set
120572
is a convex setfrom condition (ii) Combining this fact with condition (iii)the 120572-level set
120572
is a compact and convex set for all 120572 isin [0 1]
(since 120572=0
is bounded it says that 120572
sube 120572=0
is also boundedfor all 120572 isin (0 1]) Therefore we can write
120572
= [119886119871
120572
119886119880
120572
]
Definition 7 (see [9]) The trapezoidal fuzzy number is fullydetermined by quadruples (119886
119871
119886119880
120572 120573) of crisp numberssuch that 119886119871 le 119886
119880 120572 ge 0 and 120573 ge 0 whose membershipfunction can be denoted by
120583
119860
(119909) =
(119909 minus 119886119871
+ 120572)
120572
119886119871
minus 120572 le 119909 le 119886119871
1 119886119871
le 119909 le 119886119880
minus
(119909 minus 119886119880
minus 120573)
120573
119886119880
le 119909 le 119886119880
+ 120573
0 otherwise
(4)
Journal of Applied Mathematics 3
When 119886119871
= 119886119880 the trapezoidal fuzzy number becomes a
triangular fuzzy number If 120572 = 120573 the trapezoidal fuzzynumber becomes a symmetrical trapezoidal fuzzy number[119886119871
119886119880
] is the core of and 120572 ge 0 120573 ge 0 are the left-handand right-hand spreads
It can easily be shown that
120572
= [120572 (119886119871
minus (119886119871
minus 120572)) + (119886119871
minus 120572)
minus 120572 ((119886119880
+ 120573) minus 119886119880
) + (119886119880
+ 120573)]
(5)
and the support of is (119886119871 minus 120572 119886119880
+ 120573)
21 Arithmetic Operations In this subsection addition sub-traction and scalar multiplication operation of trapezoidalfuzzy numbers are reviewed [9 15]
Let = (119886119871
119886119880
120572 120573) and = (119887119871
119887119880
120579 120574) be twotrapezoidal fuzzy numbers then
= (119886119871
119886119880
120572 120573)
= (119887119871
119887119880
120574 120579)
+ = (119886119871
+ 119887119871
119886119880
+ 119887119880
120572 + 120574 120573 + 120579)
minus = (119886119871
minus 119887119880
119886119880
minus 119887119871
120572 + 120579 120573 + 120574)
120582 =
(120582119886119871
120582119886119880
120582120572 120582120573) 120582 ge 0
(120582119886119880
120582119886119871
|120582| 120573 |120582| 120572) 120582 lt 0
(6)
3 Interval Type 2 Fuzzy Sets
Interval type 2 fuzzy sets (IT2FSs) play a central role in fuzzysets as models for words and in engineering applications oftype 2 fuzzy sets These fuzzy sets are characterized by theirfootprints of uncertainty which in turn are characterized bytheir boundaries upper and lower membership functions
Definition 8 (see [17]) Type 2 fuzzy set in the universeof discourse 119883 can be represented by type 2 membershipfunction 120583
119860
(119909 119906) as follows
= ((119909 119906) 120583
119860
(119909 119906)) | forall119909 isin 119883 forall119906 isin 119869119909
sube [0 1] 0
le 120583
119860
(119909 119906) le 1
(7)
where 119869119909
sube [0 1] is the primary membership function at 119909and
int
119906isin119869
119909
120583
119860
(119909 119906)
119906
(8)
indicates the secondmembership at 119909 For discrete situationsint is replaced by sum
Definition 9 (see [17 18]) Let be type 2 fuzzy set in theuniverse of discourse 119883 represented by type 2 membershipfunction 120583
119860
(119909 119906) If all 120583
119860
(119909 119906) = 1 then is called IT2FS
IT2FS can be regarded as a special case of type 2 fuzzy setwhich is defined as
= int
119909isin119883
int
119906isin119869
119909
1
(119909 119906)
= int
119909isin119883
[int119906isin119869
119909
1119906]
119909
(9)
where 119909 is the primary variable 119869119909
sube [0 1] is the primarymembership of 119909 119906 is the secondary variable and int
119906isin119869
119909
1119906 isthe secondary membership function at 119909
It is obvious that IT2FS defined on 119883 is completelydetermined by the primary membership which is called thefootprint of uncertainty and the footprint of uncertainty canbe expressed as follows
FOU () = ⋃
119909isin119883
119869119909
= ⋃
119909isin119883
(119909 119906) | 119906 isin 119869119909
sube [0 1] (10)
Definition 10 (see [14 19]) Let be IT2FS uncertaintyin the primary membership of type 2 fuzzy set consistsof a bounded region called the footprint of uncertaintywhich is the union of all primary membership Footprint ofuncertainty is characterized by upper membership functionand lower membership function Both of the membershipfunctions are type 1 fuzzy sets Upper membership functionis denoted by 120583
119860
and lower membership function is denotedby 120583
119860
respectively
Definition 11 (see [14]) An interval type 2 fuzzy number iscalled trapezoidal interval type 2 fuzzy number where theuppermembership function and lowermembership functionare both trapezoidal fuzzy numbers that is
= (119860119871
119860119880
) = ((119886119871
1
119886119871
2
119886119871
3
119886119871
4
1198671
(119860119871
) 1198672
(119860119871
))
(119886119880
1
119886119880
2
119886119880
3
119886119880
4
1198671
(119860119880
) 1198672
(119860119880
)))
(11)
where 119867119895
(119860119871
) and 119867119895
(119860119880
) (119895 = 1 2) denote membershipvalues of the corresponding elements 119886119871
119895+1
and 119886119880119895+1
(119895 = 1 2)respectively
Definition 12 (see [13]) The upper membership function andlower membership function of IT2FSs are type 1 membershipfunction respectively
Definition 13 (see [20]) IT2FS is said to be perfectlynormal if both its upper and lower membership functions arenormal It is denoted by PnIT2FS that is
sup 120583
119860
(119909) = sup 120583
119860
= 1 (12)
4 Perfectly Normal IT2TrFN
In this section the concept of PnIT2TrFN is discussed andit is the extension work of Chiao [21] that proposed theconcept of trapezoidal interval type 2 fuzzy set in whichthe upper membership function and the lower membershipfunction are represented by trapezoidal fuzzy number whichis adopted for PnIT2TrFN PnIT2FN is a fuzzy subset of the
4 Journal of Applied Mathematics
real line which is both ldquonormalrdquo and ldquoconvexrdquo The opera-tions PnIT2FSs are very complex according to the decom-position theorem [22] and the IT2FSs are usually taken insome simplified formations in applications In subsectionthe arithmetic operation on PnIT2TrFNs is formulated byproposing the extension principle
Definition 14 Let = [119860119871
119860119880
] be PnIT2FS on 119883 Let 119871 =(119886119871
2
119886119871
3
120572119871
120573119871
) and
119880
= (119886119880
2
119886119880
3
120572119880
120573119880
) be the lower andupper trapezoidal fuzzy number respectively with respect to defined on the universe of discourse 119883 where 119886
119871
2
le 119886119871
3
119886119880
2
le 119886119880
3
120572119871
120572119880
ge 0 and 120573119871
120573119880
ge 0 [1198861198712
119886119871
3
] is the core of
119871 and 120572119871
120573119871
ge 0 are the left-hand and right-hand spreadsand [119886
119880
2
119886119880
3
] is the core of 119880 and 120572119880
120573119880
ge 0 are the left-hand and right-hand spreads The membership functions of119909 in
119871 and
119880 are expressed as follows
120583
119860
(119909) =
(119909 minus 119886119871
2
+ 120572119871
)
120572119871
119886119871
2
minus 120572119871
le 119909 le 119886119871
2
1 119886119871
2
le 119909 le 119886119871
3
minus
(119909 minus 119886119871
3
minus 120573119871
)
120573119871
119886119871
3
le 119909 le 119886119871
3
+ 120573119871
0 otherwise
120583
119860
(119909) =
(119909 minus 119886119880
2
+ 120572119880
)
120572119880
119886119880
2
minus 120572119880
le 119909 le 119886119880
2
1 119886119880
2
le 119909 le 119886119880
3
minus
(119909 minus 119886119880
3
minus 120573119880
)
120573119880
119886119880
3
le 119909 le 119886119880
3
+ 120573119880
0 otherwise
(13)
120583
119860
(119909) and 120583
119860
(119909) are lower and upper bounds respectivelyof (see Figure 1) Then is a PnIT2TrFN on 119883 and isrepresented by the following = [119860
119871
119860119880
] = [(119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)] Obviously if 1198861198712
= 119886119871
3
and 119886119880
2
= 119886119880
3
the PnIT2TrFN reduces to the perfectly normal interval type2 triangular fuzzy number (PnIT2TFN) If 119860119871 = 119860
119880 thenPnIT2TrFN becomes type 1 trapezoidal fuzzy number [1323]
Definition 15 (primary 120572-cut of PnIT2FS) The primary 120572-cutof PnIT2FS is 120572 = (119909 119906) | 119869
119909
ge 120572 119906 isin [0 1] which isbounded by two regions
120572
120583
119860
(119909) = (119909 120583
119860
(119909)) | 120583
119860
(119909) ge 120572 forall120572 isin [0 1]
120572
120583
119860
(119909) = (119909 120583
119860
(119909)) | 120583
119860
(119909) ge 120572 forall120572 isin [0 1]
(14)
Definition 16 (crisp bounds of PnIT2FN) The crisp boundaryof the primary 120572-cut of PnIT2FN = (119860
119871
119860119880
) is closedinterval 120572
119888119887
which will be obtained as follows 119860119871 and 119860119880
are the lower and upper interval valued bounds of Also
0
120583(x)
hU = hL = 1
aU2 minus 120572U aU2 minus 120572L aU2 aL2 aL3 aU3xaU3 + 120572L aU3 + 120572U
Figure 1 The lower trapezoidal membership function
119871 and theupper trapezoidal membership function
119880 of PnIT2FS
the boundary of 119860119871120572
and 119860119880
120572
can be defined as the boundaryof the 120572-cuts of each interval type 1 fuzzy set
119860119871
120572
= [inf119909
120572
120583
119860
(119909) sup119909
120572
120583
119860
(119909)] = [119886119871
119897
119886119871
119906
]
119860119880
120572
= [inf119909
120572
120583
119860
(119909) sup119909
120572
120583
119860
(119909)] = [119886119880
119897
119886119880
119906
]
120572
119888119887
= [inf119909
120572
120583
119860
(119909 119906) sup119909
120572
120583
119860
(119909 119906)]
= [[119886119880
119897
119886119871
119897
] [119886119871
119906
119886119880
119906
]]
= [[120572
119886119871
120572
119886119871
] [120572
119886119877
120572
119886119877
]]
(15)
which is equivalent to say
120572
119888119887
120583
119860
isin [[120572
119886119871
120572
119886119871
] [120572
119886119877
120572
119886119877
]] (16)
Evidently for PnIT2TrFN from Figure 2
120572
119886119871
le120572
119886119871
le120572
119886119877
le120572
119886119877
(17)
Definition 17 A fuzzy number = (119860119871
119860119880
) = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is said to be nonnegative PnIT2TrFNif 120583
119860
(119909) = 120583
119860
= 0 forall119909 gt 0
Definition 18 A fuzzy number = (119860119871
119860119880
) = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is said to be nonpositive PnIT2TrFN if120583
119860
(119909) = 120583
119860
= 0 forall119909 lt 0
Definition 19 A fuzzy number = (119860119871
119860119880
) = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is said to be zero PnIT2TrFN if 1198861198712
=
119887119871
2
= 0 1198861198713
= 119887119871
3
= 0 120572119871
= 120574119871
= 120573119871
= 120579119871
= 120572119880
= 120574119880
= 0 and120573119880
= 120579119880
= 0
Definition 20 PnIT2TrFNs = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are said to beidentically equal to = if and only if 119886119871
2
= 119887119871
2
1198861198713
= 119887119871
3
120572119871
= 120574119871
120573119871
= 120579119871
120572119880
= 120574119880
and 120573119880
= 120579119880
Journal of Applied Mathematics 5
0
120583(x)
hU = hL = 1
x120572aR120572aL 120572aR
120572aL
120572119888119887 A
Figure 2 Crisp bounds of PnIT2TrFN
41 Arithmetic Operations on PnIT2TrFN
Definition 21 If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are PnIT2TrFNs then = + is also PnIT2TrFN and defined by
= ((119886119871
2
+ 119887119871
2
119886119871
3
+ 119887119871
3
120572119871
+ 120574119871
120573119871
+ 120579119871
)
(119886119880
2
+ 119887119880
2
119886119880
3
+ 119887119880
3
120572119880
+ 120574119880
120573119880
+ 120579119880
))
(18)
Definition 22 If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are PnIT2TrFNs then = minus is also PnIT2TrFN and defined by
= ([119886119871
2
minus 119887119880
3
119886119871
3
minus 119887119880
2
120572119871
+ 120579119880
120573119871
+ 120574119880
]
[119886119880
2
minus 119887119871
3
119886119880
3
minus 119887119871
2
120572119880
+ 120579119871
120573119880
+ 120574119871
])
(19)
Definition 23 Let 120582 isin R If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is PnIT2TrFN then = 120582 is also PnIT2TrFN and isgiven by
= 120582 =
((120582119886119871
2
120582119886119871
3
120582120572119871
120582120573119871
) (120582119886119880
2
120582119886119880
3
120582120572119880
120582120573119880
)) if 120582 ge 0
((120582119886119880
3
120582119886119880
2
|120582| 120573119880
|120582| 120572119880
) (120582119886119871
3
120582119886119871
2
|120582| 120573119871
|120582| 120572119871
)) if 120582 lt 0
(20)
5 The Possibility Degree of PnIT2TrFN
Comparison of fuzzy numbers is considered one of the mostimportant topics in fuzzy logic theory The early and mostimportant work in the field of comparing fuzzy numbershas been presented by Dubois and Prade [24] On the otherhand the dominance possibility indices which have beenintroduced by Negi et al were utilized in the field of fuzzymathematical programming [25 26] The approach used inthese fields was based on formulating a possibility functionwhether in the case of trapezoidal fuzzy numbers or the caseof triangular fuzzy numbers In this paper we are going to
utilize the degree of possibility that the proposition statingthat ldquo119860 is less than or equal to 119861rdquo is true which is proposedby Chen and Lee [13] for calculating the ranking values ofperfectly normal trapezoidal interval type 2 fuzzy numberHere the height of the uppermembership function and lowermembership function is considered as 1 so the modifiedproposition can be as follows
Definition 24 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNsThen the possibility degrees of lower and upper membershipfunction are defined as follows
Pos (119860119871 le 119861119871
) = max(1
minusmax(max (120572119886119871 minus 120572119887119871 0) +max ((119886119871
2
minus 119887119871
2
) 0) +max ((1198861198713
minus 119887119871
3
) 0) +max (120572119886119877 minus 120572119887119877 0) + (120572
119886119877
minus120572
119887119871
)
10038161003816100381610038161003816
120572
119886119871
minus120572
119887119871
10038161003816100381610038161003816+1003816100381610038161003816119886119871
2
minus 119887119871
2
1003816100381610038161003816+1003816100381610038161003816119886119871
3
minus 119887119871
3
1003816100381610038161003816+
10038161003816100381610038161003816
120572
119886119877
minus120572
119887119877
10038161003816100381610038161003816+ (120572
119887119877
minus120572
119887119871
) + (120572
119886119877
minus120572
119886119871
)
0)
0)
6 Journal of Applied Mathematics
Pos (119860119880 le 119861119880
) = max(1
minusmax(max (120572119886119871 minus 120572119887
119871
0) +max ((1198861198802
minus 119887119880
2
) 0) +max ((1198861198803
minus 119887119880
3
) 0) +max (120572119886119877 minus 120572119887119877
0) + (120572
119886119877
minus120572
119887
119871
)
100381610038161003816100381610038161003816
120572
119886119871
minus120572
119887
119871100381610038161003816100381610038161003816
+1003816100381610038161003816119886119880
2
minus 119887119880
2
1003816100381610038161003816+1003816100381610038161003816119886119880
3
minus 119887119880
3
1003816100381610038161003816+
100381610038161003816100381610038161003816
120572
119886119877
minus120572
119887
119877100381610038161003816100381610038161003816
+ (120572
119887
119877
minus120572
119887
119871
) + (120572
119886119877
minus120572
119886119871
)
0)
0)
(21)
Proposition 25 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
))
and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2FNsPos(119860119871 ⪯ 119861
119871
) ge 120572 if and only if inf[119860119871]120572
le sup[119861119871]120572
that is
Pos (119860119871 ⪯ 119861119871
) ge 120572 iff 120572119886119871 le 120572119887119877 (22)
Alternatively
Pos (119860119871 ⪯ 119861119871
) = 120583Pos(119860119871119861119871) = (119860119871
⪯Pos
119861
119871
) (23)
An interpretation of the 120572minus relation associated with possibilitywhen comparing fuzzy numbers 119860119871 and 119861119871 is as follows For agiven level of satisfaction 120572 isin [0 1] a fuzzy number 119860119871 is notbetter than 119861119871 with respect to fuzzy relation ⪯Pos if the smallestvalue of119860119871 with the degree of satisfaction (or possibility degree)being greater than or equal to 120572minus is less than or equal to thelargest value of 119861119871 with the degree of satisfaction greater thanor equal to 120572 Similar to the above
Pos (119860119880 ⪯ 119861119880
) ge 120572 iff 120572119886119871 le 120572119887119877
(24)
Proposition 26 (see [15 23 24]) Let = ((119886119871
2
119886119871
3
120572119871
120573119871
)(119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) betwo PnIT2TrFNs then
Pos (119860119871 ⪯ 119861119871
)
=
1 119887119871
3
ge 119886119871
2
119887119871
3
minus 119886119871
2
+ 120579119871
+ 120572119871
120579119871
+ 120572119871
0 le 119886119871
2
minus 119887119871
3
le 120579119871
+ 120572119871
0 119886119871
2
minus 119887119871
3
gt 120579119871
+ 120572119871
(25)
see Figure 3
Pos (119860119880 ⪯ 119861119880
)
=
1 119887119880
3
ge 119886119880
2
119887119880
3
minus 119886119880
2
+ 120579119880
+ 120572119906
120579119880
+ 120572119880
0 le 119886119880
2
minus 119887119880
3
le 120579119880
+ 120572119880
0 119886119880
2
minus 119887119880
3
gt 120579119880
+ 120572119880
(26)
see Figure 4
Theorem 27 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNs 119901 isin
(0 1] Pos(119871 ⪯
119871
) ge 119901 if and only if 1198871198713
minus 119886119871
2
ge (119901 minus 1)(120579119871
+
120572119871
)
Proof If 119901 = 1 then from Pos(119871 ⪯
119871
) ge 1 one can getthat 1198871198713
ge 119886119871
2
and vice versa If 0 lt 119901 lt 1 then 119887119871
3
lt 119886119871
2
and119887119871
3
+ 120579119871
gt 119886119871
2
minus 120572119871
Pos(119871 ⪯
119871
) ge 119901 if and only if (1198871198713
minus 119886119871
2
+
120579119871
+120572119871
)(120579119871
+120572119871
) ge 119901 that is 1198871198713
minus119886119871
2
ge (119901minus1)(120579119871
+120572119871
)
Theorem 28 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNs 119901 isin
(0 1] Pos(119880 ⪯
119880
) ge 119901 if and only if 1198871198803
minus119886119880
2
ge (119901minus1)(120579119880
+
120572119880
)
Proof If 119901 = 1 then from Pos(119880 ⪯
119880
) ge 1 one can getthat 119887119880
3
ge 119886119880
2
and vice versa If 0 lt 119901 lt 1 then 119887119880
3
lt 119886119880
2
and 119887119880
3
+ 120579119880
gt 119886119880
2
minus 120572119880
Pos(119880 ⪯
119880
) ge 119901 if and onlyif (1198871198803
minus 119886119880
2
+ 120579119880
+ 120572119880
)(120579119880
+ 120572119880
) ge 119901 that is 1198871198803
minus 119886119880
2
ge
(119901 minus 1)(120579119880
+ 120572119880
)
6 Fuzzy Linear Programming
In this section we propose a fuzzy linear programmingmodel with the technological coefficients and right-hand side(resources) being PnIT2FN
MaxMin119909isin119883
119885 = 119888119909
St 119860119909 ⪯ 119887 119909 ge 0
(27)
where 119860 = (119886119894119895
)119898times119899
119887 = (1198871
1198872
119887119898
)119879 119888 = (119888
1
1198882
119888119899
)119909 = (119909
1
1199092
119909119899
)119879 and 119886
119894119895
119887119894
are PnIT2FN and 119909119895
119888119895
isin
R (119894 = 1 2 119898 119895 = 1 2 119899) ⪯ is type 2 fuzzy order
Definition 29 Consider a set of right-hand side (resources)parameters of a fuzzy linear programming problem definedas PnIT2FS 119887 defined on the closed interval
119887119894
isin [[120572
119887
119871
119894
120572
119887119871
119894
] [120572
119887119877
119894
120572
119887
119877
119894
]] isin R (28)
Journal of Applied Mathematics 7
10908070605040302010
Mem
bers
hip
func
tion
bL3 aL2
Pos[ALle B
L]
Figure 3 Pos[119871 le
119871
] lt 1 because 119871 gt
119871
10908070605040302010
Mem
bers
hip
func
tion
bU3 aU2
Pos[AUle B
U]
Figure 4 Pos[119880 le
119880
] lt 1 because 119880 gt
119880
and 119894 isin N119899
The membership function which represents thefuzzy space Supp(119887
119894
) is
119887119894
= int
119887
119894isinR
[int119906isin119869
119887119894
1119906]
119887119894
119894 isin N119899
119869119887
119894
sube [0 1] (29)
Here 119887 is bounded by both lower and upper primary mem-bership function namely
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (30)
with parameters 120572119887119871119894
and 120572119887119877119894
and
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (31)
with parameters 120572119887119871
119894
and 120572119887119877
119894
Definition 30 Consider a technological coefficient of fuzzylinear programming problem defined as PnIT2FS
119894119895
definedon the closed interval
119894119895
isin [inf119886
119894119895
120572
120583
119894119895
(119886119894119895
119906) sup119886
119894119895
120572
120583
119894119895
(119886119894119895
119906)]
= [[120572
119886119871
119894119895
120572
119886119871
119894119895
] [120572
119886119877
119894119895
120572
119886119877
119894119895
]] isin R 119894 isin N119899
119895 isin N119898
(32)
The membership function which represents the fuzzy spaceSupp(
119894119895
) is
119894119895
= int
119894119895isinR
[int119906isin119869
119894119895
1119906]
119894119895
119894 isin N119899
119895 isin N119898
119869
119894119895
sube [0 1]
(33)
Here 119894119895
is bounded by both lower and upper primarymembership functions namely
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (34)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
and
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (35)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
Proposition 31 Let 119894119895
isin 119865(R) 119909119895
ge 0 119894 isin N119899
119895 isin N119898
Then the fuzzy setsum119899
119895=1
119894119895
119909119895
defined by the extension principleis again a fuzzy number
Let = ⪯Pos be a fuzzy relation [27] fuzzy extension of the
usual binary relation le on R The fuzzy linear programmingproblem associated with a standard linear programmingproblem is denoted as
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St (
119899
sum
119895=1
119894119895
119909119895
) 119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(36)
In (36) the value sum119899119895=1
119894119895
119909119895
isin 119865(R) is compared with a fuzzynumber 119887
119894
isin 119865(R) by sum fuzzy relation The maximizationof the objective function is denoted by Max119885 = sum
119899
119895=1
119888119895
119909119895
Definition 32 Let 120583
119894119895
R rarr [0 1] and 120583
119887
119894
R rarr [0 1]119894 isin N
119899
119895 isin N119898
be membership function of fuzzy numbers119894119895
and 119887119894
respectively Let be a fuzzy extension of a binaryrelation le on R A fuzzy set whose membership function120583
119883
is defined as 119909 isin R119899 by
120583
119883
=
min 120583
119875
(11
1199091
+ sdot sdot sdot + 1119899
119909119899
1198871
) 120583
119875
(1198981
1199091
+ sdot sdot sdot + 119898119899
119909119899
119887119898
) if 119909119895
ge 0 119895 = 1 2 119899
0 otherwise(37)
8 Journal of Applied Mathematics
is called the fuzzy set of feasible regions of the fuzzy linearprogramming problem (36) For 120572 isin (0 1] a vector 119909 isin
[]120572
is called the 120572-feasible solution of the fuzzy linearprogramming problem
Notice that the feasible region of the fuzzy linearprogramming problem is a fuzzy set On the other hand 120572-feasible solution is a vector belonging to the 120572-cut of the fea-sible region If all the coefficients
119894119895
and 119887119894
are crisp fuzzynumber that is they are equivalent to the corresponding crispfuzzy number then the fuzzy feasible region is equivalent tothe set of all feasible solutions of the corresponding classicallinear programming
61 Type 2 Fuzzy Linear Programming Problem Let us con-sider the following fuzzy linear programming problem withright-hand side (resources) and technology coefficients arePNIT2TrFNs
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119894119895
119909119895
le119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(38)
where 119894119895
= ((119894119895
)119871
(119894119895
)119880
) = (((119886119894119895
)119871
2
(119886119894119895
)119871
3
(120572119894119895
)119871
(120573119894119895
)119871
)((119886119894119895
)119880
2
(119886119894119895
)119880
3
(120572119894119895
)119880
(120573119894119895
)119880
)) 119887119894
= ((119887119894
)119871
(119887119894
)119880
) = (((119887119894
)119871
2
(119887119894
)119871
3
(120574119894
)119871
(120579119894
)119871
) ((119887119894
)119880
2
(119887119894
)119880
3
(120574119894
)119880
(120579119894
)119880
)) are PnIT2TrFNs119888119894
are crisp coefficients of the objective and 119909119894
are the decisionvariable
From Definitions 15 21 22 and 23 and (38)
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119871
119894119895
119909119895
le119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119880
119894119895
119909119895
le119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(39)
Based on the relation of possibility we can rewrite (39) asfollows
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119871
1198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯Pos
119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119880
1198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯Pos
119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(40)
With respect to constraints of linear programming possibilitythe optimal solution is completely determined by degrees ofpossibility So we can obtain the optimal solution of linearprogramming possibility at 119901-cut levels by solving the twocrisp linear programming problems using Barnes algorithm
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198711198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯119887
119871
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198801198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯119887
119880
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(41)
FromTheorems 27 and 28 we have
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(42)
where 119860119871 = (119886119894119895
)119871
2
119860119871 = (119886119894119895
)119880
2
119887119880 = (119887119894
)119871
3
119887119880
= (119887119894
)119880
3
120572 = (120572
119894119895
)119871
120572 = (120572119894119895
)119880
120573 = (120573119894119895
)119871
120573 = (120573119894119895
)119880
120574 =
(120574119894
)119871
120574 = (120574119894
)119880
120579 = (120579119894
)119871
120579 = (120579119894
)119880
and 119909 = 119909119895
119895 = 1 2 119899 119894 = 1 2 119898
Journal of Applied Mathematics 9
The feasible regions of (119871119875119875119901
) and (119871119875119875119901
) are denoted by
119883119875
119901
= 119909 isin 119877119899
| 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119883
119875
119901
= 119909 isin 119877119899
| 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
(43)
Assume the optimal solutions of (119871119875119875119901
) and (119871119875119875119901
) are 119885119875119901
and 119885
119875
119901
respectively
Theorem 33 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883119875
119901
1
119883
119875
119901
2
sube
119883
119875
119901
1
and 119885119875119901
1
ge 119885119875
119901
2
119885
119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) ge
(1199011
minus1)(120572119909+120579) so 119909 isin 119883119875
119901
1
that is119883119875119901
2
sube 119883119875
119901
1
and119885119875119901
1
ge 119885119875
119901
2
In a similar way119883119875119901
2
sube 119883
119875
119901
1
and 119885
119875
119901
1
ge 119885
119875
119901
2
Theorem 34 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883
119875
119901
1
and
119885119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 +
120579) ge (1199011
minus 1)(120572119909 + 120579) so 119909 isin 119883
119875
119901
1
that is 119883119875119901
2
sube 119883
119875
119901
1
and
119885
119875
119901
1
ge 119885119875
119901
2
FromTheorems 33 and 34 we obtain that 119885119875119901
119885
119875
119901
consti-tute the lower and upper bound of fuzzy objective value atlevel 119901 Then the objective value of 119885119875
119901
at level 119901 is the mean
of [119885119875119901
119885
119875
119901
]
7 Numerical Illustration
An application of the proposed method is introduced withan example where all its technological coefficients and right-hand side (resources) are defined as PnIT2FNs The optimalsolution will be obtained in terms of crisp for the followingproblem
The company plans tomanufacture two types of productsThe selling prices for these products are as follows 119875
1
costs Rs 1750 per unit and 1198752
costs Rs 2000 per unitDaily production volume of each type of these products isconstrained by available man hours and available machinehours The production specifications for the given problemsituation are presented in as shown in Table 1
Max 119885 = 1198881
1199091
+ 1198882
1199092
St 11
1199091
+ 12
1199092
⪯1198871
Table 1
Resource requirementunitResource 119875
1
1198752
AvailabilityMan hours (in minutes) Around 4 Around 8 Around 32Machine hours (in minutes) Around 6 Around 4 Around 36
21
1199091
+ 22
1199092
⪯1198872
119909119895
ge 0 119895 = 1 2
(44)
where 11
= [(230 250 20 10) (230 250 30 20)] 12
=
[(460 500 40 20) (460 500 60 40)] 21
= [(345 375 30
15) (345 375 45 30)] 22
= [(230 250 20 10) (230 250
30 20)] 1198871
= [(1840 2000 160 80) (1840 2000 240 160)]1198872
= [(2070 2250 180 90) (2070 2250 270 180)] 1198881
=
1750 and 1198882
= 2000 From (42) we have the following crispoptimal problem based on the level 119901
Max 119885119875
119901
= 17501199091
+ 20001199092
St (210 + 20119901) 1199091
+ (420 + 40119901) 1199092
le 2080 minus 80119901
(315 + 30119901) 1199091
+ (210 + 20119901) 1199092
le 2340 minus 90119901
1199091
1199092
ge 0
(45)
Max 119885
119875
119901
= 17501199091
+ 20001199092
St (200 + 30119901) 1199091
+ (400 + 60119901) 1199092
le 2160 minus 160119901
(300 + 45119901) 1199091
+ (200 + 30119901) 1199092
le 2430 minus 180119901
1199091
1199092
ge 0
(46)
For different cut level 119901 we can get different optimal solutionand denote by 119885119875
119901
119885119875
119901
the optimal solution (see Figure 5 andTable 2) [28] of the crisp programming (45) and program-ming (46) respectively
8 Conclusion
In this paper a method to do perfectly normal interval type2 fuzzy arithmetic operation using extension principle is pre-sented and defined the possibility degrees of upper and lowermembership function to compare perfectly normal intervaltype 2 fuzzy numbers These functions are defined based onthe strength of upper and lower membership function ofperfectly normal interval type 2 fuzzy numbers Meanwhilesome properties and theorems were proved Then the lowerand upper fuzzy satisfaction based on possibility degrees of
10 Journal of Applied Mathematics
Table 2 Optimal solution at different possibility levels
119901-cut 01 02 03 04 05 06 07 08 09 1119885119875
119901
143549528 141658878 13980324 137981651 136193181 134436937 132712053 131017699 129353069 127717393
119885
119875
119901
155123153 151723301 148421051 145212264 142093022 139059633 136108597 133236607 130440529 127717393119885119875
119901
149336341 146691090 144112146 141596958 139143102 136748285 134410325 132127153 129896799 127717393
1090807060504030201012500 13000 13500 14000 14500 15000
p-c
ut
Z
Figure 5 Optimal solution at different possibility levels
membership function respectively are defined These fuzzysatisfactions create two nonstrict order relations on the setof overlapping intervals Also linear programming problemwas introduced with resources and technology coefficientsof perfectly normal interval type 2 fuzzy numbers Thenthe possibility degrees of membership function were appliedin order to interpret inequality constraints with intervalcoefficients According to the definition of possibility degreesof membership function and their properties the inequalityconstraints with interval coefficients were reduced in theirsatisfactory crisp equivalent forms Finally the decisionmaker can get the crisp optimal solution of the problem forevery grade 120572 isin [0 1] which was obtained by using Barnesalgorithm and then mean value is selected according to thedecision maker optimistic attitude
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] H Tanaka T Okuda and K Asai ldquoOn fuzzy-mathematicalprogrammingrdquo Journal of Cybernetics vol 3 no 4 pp 37ndash461973
[3] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 pp B141ndashB164197071
[4] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978
[5] S Deng L Zhou and X Wang ldquoSolving the fuzzy bilevellinear programmingwithmultiple followers through structured
element methodrdquo Mathematical Problems in Engineering vol2014 Article ID 418594 6 pages 2014
[6] Y-H Zhong Y-L Jia D Chen and Y Yang ldquoInterior pointmethod for solving fuzzy number linear programming prob-lems using linear ranking functionrdquo Journal of Applied Math-ematics vol 2013 Article ID 795098 9 pages 2013
[7] H Cheng W Huang and J Cai ldquoSolving a fully fuzzy linearprogramming problem through compromise programmingrdquoJournal of Applied Mathematics vol 2013 Article ID 726296 10pages 2013
[8] D Chakraborty D K Jana and T K Royl ldquoA new approach tosolve intuitionistic fuzzy optimization problem using possibil-ity necessity and credibility measuresrdquo International Journal ofEngineering Mathematics vol 2014 Article ID 593185 12 pages2014
[9] H Le and Z Gogne ldquoFuzzy linear programming with possibil-ity and necessity relationrdquo in Fuzzy Information and Engineering2010 vol 78 of Advances in Intelligent and Soft Computing pp305ndash311 Springer Berlin Germany 2010
[10] J C Figueroa-Garcıa and G Hernandez ldquoA method for solvinglinear programming models with interval type-2 fuzzy con-straintsrdquo Pesquisa Operacional vol 34 no 1 pp 73ndash89 2014
[11] J C F Garcıa ldquoA general model for linear programming withinterval type-2 fuzzy technological coefficientsrdquo in Proceedingsof the IEEE Annual Meeting of the North American FuzzyInformation Processing Society (NAFIPS rsquo12) pp 1ndash4 BerkeleyCalif USA August 2012
[12] J C F Garcıa ldquoLinear programming with interval type-2fuzzy right hand side parametersrdquo in Proceedings of the AnnualMeeting of the North American Fuzzy Information ProcessingSociety (NAFIPS rsquo08) pp 1ndash6 IEEE New York NY USA May2008
[13] S-M Chen and L-W Lee ldquoFuzzy multiple attributes groupdecision-making based on the ranking values and the arith-metic operations of interval type-2 fuzzy setsrdquo Expert Systemswith Applications vol 37 no 1 pp 824ndash833 2010
[14] J Hu Y Zhang X Chen and Y Liu ldquoMulti-criteria decisionmaking method based on possibility degree of interval type-2 fuzzy numberrdquo Knowledge-Based Systems vol 43 pp 21ndash292013
[15] R Fuller Fuzzy Reasoning and Fuzzy Optimization TurkuCentre for Computer Science 1998
[16] H-C Wu ldquoDuality theory in fuzzy linear programming prob-lems with fuzzy coefficientsrdquo Fuzzy Optimization and DecisionMaking vol 2 no 1 pp 61ndash73 2003
[17] J Qin and X Liu ldquoFrank aggregation operators for triangularinterval type-2 fuzzy set and its application inmultiple attributegroup decision makingrdquo Journal of Applied Mathematics vol2014 Article ID 923213 24 pages 2014
[18] Z M Zhang and S H Zhang ldquoA novel approach to multiattribute group decision making based on trapezoidal intervaltype-2 fuzzy soft setsrdquo Applied Mathematical Modelling vol 37no 7 pp 4948ndash4971 2013
Journal of Applied Mathematics 11
[19] Y Gong ldquoFuzzymulti-attribute group decisionmakingmethodbased on interval type-2 fuzzy sets and applications to globalsupplier selectionrdquo International Journal of Fuzzy Systems vol15 no 4 pp 392ndash400 2013
[20] D S Dinagar and A Anbalagan ldquoA new type-2 fuzzy numberarithmetic using extension principlerdquo in Proceedings of the 1stInternational Conference on Advances in Engineering Scienceand Management (ICAESM rsquo12) pp 113ndash118 March 2012
[21] K-P Chiao ldquoMultiple criteria group decision making withtriangular interval type-2 fuzzy setsrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems (FUZZ rsquo11) pp 2575ndash2582 IEEE Taipei Taiwan June 2011
[22] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006
[23] X Liu ldquoMeasuring the satisfaction of constraints in fuzzy linearprogrammingrdquo Fuzzy Sets and Systems vol 122 no 2 pp 263ndash275 2001
[24] D Dubois and H Prade ldquoRanking fuzzy numbers in the settingof possibility theoryrdquo Information Sciences vol 30 no 3 pp183ndash224 1983
[25] D S Negi and E S Lee ldquoPossibility programming by thecomparison of fuzzy numbersrdquo Computers ampMathematics withApplications vol 25 no 9 pp 43ndash50 1993
[26] MG Iskander ldquoComparison of fuzzy numbers using possibilityprogramming comments and new conceptsrdquo Computers ampMathematics with Applications vol 43 no 6-7 pp 833ndash8402002
[27] J Ramik ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006
[28] G R Lindfield and J E T Penny Numerical methods UsingMatlab Academic Press 3rd edition 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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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Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
was proposed by Zimmermann to an interval type 2 fuzzy lin-ear programming problem with linear membership function[10] In another work Garcıa presented a general model forlinear programming where its technological coefficients areassumed as interval type 2 fuzzy sets and it is solved throughan 120572-cuts approach [11] Furthermore Garcıa [12] presenteda general model to handle uncertainty to right-hand sideparameters of a linear programmingmodel by interval type 2fuzzy sets and trapezoidal membership functions to optimizeby using classical algorithms Chen and Lee [13] proposed thedefinition of possibility degree of trapezoidal interval type 2fuzzy numbers and some arithmetic operations Hu et al [14]proposed a new approach based on possibility degree to solvemulticriteria decision-making problems in which the criteriavalue takes the form of interval type 2 fuzzy number
Based on these literaturesrsquo survey with our best knowl-edge however none of them introduced fuzzy linear pro-gramming model with both the right-hand side (resources)and the technological coefficients being perfectly normalinterval type 2 fuzzy numbers based on possibility In thispaper first we have presented a method to do Perfectlynormal Interval Type 2 Fuzzy Number (PnIT2FN) arith-metic operation using well-known arithmetical operationson type 1 fuzzy numbers Further the relation of possibilityfor PnIT2FN as well as its property are discussed Anda fuzzy linear programming model with both the right-hand side (resources) and the technological coefficients beingPnIT2FNs has been proposed By using Zadehrsquos extensionprinciple a pair of upper and lower linear problems are devel-oped to calculate the optimal solution of upper and lowerbounds of linear problems measure at different possibilitylevel 120572
The rest of this paper is organized as follows FirstlySection 2 deals with some preliminary definitions which aregiven In Section 3 the definitions of interval type 2 fuzzysets and trapezoidal interval type 2 fuzzy number have beenrecalled In Section 4 the new representation for PnIT2TrFNits properties and some arithmetic operations of PnIT2TrFNbased on type 1 fuzzy number are presented The possibilitydegree of PnIT2TrFN is discussed in Section 5 The fuzzylinear programming problem with the right-hand side andthe technological coefficientsmodel is presented in Section 6Section 7 has numerical illustrations of proposed fuzzy linearprogramming model with PnIT2TrFN finally Section 8 hasthe conclusion of the work
2 Preliminaries
Definition 1 (see [15]) Let119883 be a nonempty set A fuzzy set in 119883 is characterized by its membership function 120583
119860
119883 rarr
[0 1] and 120583
119860
(119909) is interpreted as the degree of membershipof element 119909 in fuzzy set for each 119909 isin 119883 It is clear that iscompletely determined by the set of tuples
= (119909 120583
119860
(119909)) | 119909 isin 119883 (1)
Definition 2 (see [15]) Let be a fuzzy subset of 119883 thesupport of denoted by Supp() is the crisp subset of
119883 whose elements all have nonzero membership grades in
Supp () = 119909 isin 119883 | 120583
119860
(119909) gt 0 (2)
Definition 3 (see [15]) A fuzzy subset of a classical set 119883is called normal if there exists 119909 isin 119883 such that 120583
119860
(119909) = 1Otherwise is subnormal
Definition 4 An 120572-level set (or 120572-cut) of a fuzzy set of119883 isa nonfuzzy set denoted by
120572
and defined by
120572
=
119909 isin 119883 | 120583
119860
(119909) ge 120572 if 120572 gt 0
cl (Supp ) if 120572 = 0
(3)
where cl(Supp) denotes the closure of the support of
Definition 5 (see [15]) A fuzzy set of 119883 is called convex if120572
is a convex subset of119883 for all 120572 isin [0 1]
Definition 6 (see [15 16]) A fuzzy number is a fuzzy set ofthe real lines with a normal (fuzzy) convex and continuousmembership function of bounded support Alternatively thefuzzy subset of R is called a fuzzy number if the followingconditions are satisfied
(i) is normal that is there exists 119909 isin R such that120583
119860
(119909) = 1(ii) the membership function 120583
119860
(119909) is quasiconcave thatis 120583
119860
(1205821199091
+ (1 minus 120582)1199092
) ge min120583
119860
(1199091
) 120583
119860
(1199092
) for all120572 isin [0 1]
(iii) the membership function 120583
119860
(119909) is upper semicontin-uous that is 119909 isin R 120583
119860
(119909) ge 120572 is a closed subset ofR for all 120572 isin [0 1]
(iv) the 0-level set 120572=0
is compact (closed and boundedin R)
We denote by119865(R) the set of all fuzzy numbers If is a fuzzynumber then from Zadeh [1] 120572-level set
120572
is a convex setfrom condition (ii) Combining this fact with condition (iii)the 120572-level set
120572
is a compact and convex set for all 120572 isin [0 1]
(since 120572=0
is bounded it says that 120572
sube 120572=0
is also boundedfor all 120572 isin (0 1]) Therefore we can write
120572
= [119886119871
120572
119886119880
120572
]
Definition 7 (see [9]) The trapezoidal fuzzy number is fullydetermined by quadruples (119886
119871
119886119880
120572 120573) of crisp numberssuch that 119886119871 le 119886
119880 120572 ge 0 and 120573 ge 0 whose membershipfunction can be denoted by
120583
119860
(119909) =
(119909 minus 119886119871
+ 120572)
120572
119886119871
minus 120572 le 119909 le 119886119871
1 119886119871
le 119909 le 119886119880
minus
(119909 minus 119886119880
minus 120573)
120573
119886119880
le 119909 le 119886119880
+ 120573
0 otherwise
(4)
Journal of Applied Mathematics 3
When 119886119871
= 119886119880 the trapezoidal fuzzy number becomes a
triangular fuzzy number If 120572 = 120573 the trapezoidal fuzzynumber becomes a symmetrical trapezoidal fuzzy number[119886119871
119886119880
] is the core of and 120572 ge 0 120573 ge 0 are the left-handand right-hand spreads
It can easily be shown that
120572
= [120572 (119886119871
minus (119886119871
minus 120572)) + (119886119871
minus 120572)
minus 120572 ((119886119880
+ 120573) minus 119886119880
) + (119886119880
+ 120573)]
(5)
and the support of is (119886119871 minus 120572 119886119880
+ 120573)
21 Arithmetic Operations In this subsection addition sub-traction and scalar multiplication operation of trapezoidalfuzzy numbers are reviewed [9 15]
Let = (119886119871
119886119880
120572 120573) and = (119887119871
119887119880
120579 120574) be twotrapezoidal fuzzy numbers then
= (119886119871
119886119880
120572 120573)
= (119887119871
119887119880
120574 120579)
+ = (119886119871
+ 119887119871
119886119880
+ 119887119880
120572 + 120574 120573 + 120579)
minus = (119886119871
minus 119887119880
119886119880
minus 119887119871
120572 + 120579 120573 + 120574)
120582 =
(120582119886119871
120582119886119880
120582120572 120582120573) 120582 ge 0
(120582119886119880
120582119886119871
|120582| 120573 |120582| 120572) 120582 lt 0
(6)
3 Interval Type 2 Fuzzy Sets
Interval type 2 fuzzy sets (IT2FSs) play a central role in fuzzysets as models for words and in engineering applications oftype 2 fuzzy sets These fuzzy sets are characterized by theirfootprints of uncertainty which in turn are characterized bytheir boundaries upper and lower membership functions
Definition 8 (see [17]) Type 2 fuzzy set in the universeof discourse 119883 can be represented by type 2 membershipfunction 120583
119860
(119909 119906) as follows
= ((119909 119906) 120583
119860
(119909 119906)) | forall119909 isin 119883 forall119906 isin 119869119909
sube [0 1] 0
le 120583
119860
(119909 119906) le 1
(7)
where 119869119909
sube [0 1] is the primary membership function at 119909and
int
119906isin119869
119909
120583
119860
(119909 119906)
119906
(8)
indicates the secondmembership at 119909 For discrete situationsint is replaced by sum
Definition 9 (see [17 18]) Let be type 2 fuzzy set in theuniverse of discourse 119883 represented by type 2 membershipfunction 120583
119860
(119909 119906) If all 120583
119860
(119909 119906) = 1 then is called IT2FS
IT2FS can be regarded as a special case of type 2 fuzzy setwhich is defined as
= int
119909isin119883
int
119906isin119869
119909
1
(119909 119906)
= int
119909isin119883
[int119906isin119869
119909
1119906]
119909
(9)
where 119909 is the primary variable 119869119909
sube [0 1] is the primarymembership of 119909 119906 is the secondary variable and int
119906isin119869
119909
1119906 isthe secondary membership function at 119909
It is obvious that IT2FS defined on 119883 is completelydetermined by the primary membership which is called thefootprint of uncertainty and the footprint of uncertainty canbe expressed as follows
FOU () = ⋃
119909isin119883
119869119909
= ⋃
119909isin119883
(119909 119906) | 119906 isin 119869119909
sube [0 1] (10)
Definition 10 (see [14 19]) Let be IT2FS uncertaintyin the primary membership of type 2 fuzzy set consistsof a bounded region called the footprint of uncertaintywhich is the union of all primary membership Footprint ofuncertainty is characterized by upper membership functionand lower membership function Both of the membershipfunctions are type 1 fuzzy sets Upper membership functionis denoted by 120583
119860
and lower membership function is denotedby 120583
119860
respectively
Definition 11 (see [14]) An interval type 2 fuzzy number iscalled trapezoidal interval type 2 fuzzy number where theuppermembership function and lowermembership functionare both trapezoidal fuzzy numbers that is
= (119860119871
119860119880
) = ((119886119871
1
119886119871
2
119886119871
3
119886119871
4
1198671
(119860119871
) 1198672
(119860119871
))
(119886119880
1
119886119880
2
119886119880
3
119886119880
4
1198671
(119860119880
) 1198672
(119860119880
)))
(11)
where 119867119895
(119860119871
) and 119867119895
(119860119880
) (119895 = 1 2) denote membershipvalues of the corresponding elements 119886119871
119895+1
and 119886119880119895+1
(119895 = 1 2)respectively
Definition 12 (see [13]) The upper membership function andlower membership function of IT2FSs are type 1 membershipfunction respectively
Definition 13 (see [20]) IT2FS is said to be perfectlynormal if both its upper and lower membership functions arenormal It is denoted by PnIT2FS that is
sup 120583
119860
(119909) = sup 120583
119860
= 1 (12)
4 Perfectly Normal IT2TrFN
In this section the concept of PnIT2TrFN is discussed andit is the extension work of Chiao [21] that proposed theconcept of trapezoidal interval type 2 fuzzy set in whichthe upper membership function and the lower membershipfunction are represented by trapezoidal fuzzy number whichis adopted for PnIT2TrFN PnIT2FN is a fuzzy subset of the
4 Journal of Applied Mathematics
real line which is both ldquonormalrdquo and ldquoconvexrdquo The opera-tions PnIT2FSs are very complex according to the decom-position theorem [22] and the IT2FSs are usually taken insome simplified formations in applications In subsectionthe arithmetic operation on PnIT2TrFNs is formulated byproposing the extension principle
Definition 14 Let = [119860119871
119860119880
] be PnIT2FS on 119883 Let 119871 =(119886119871
2
119886119871
3
120572119871
120573119871
) and
119880
= (119886119880
2
119886119880
3
120572119880
120573119880
) be the lower andupper trapezoidal fuzzy number respectively with respect to defined on the universe of discourse 119883 where 119886
119871
2
le 119886119871
3
119886119880
2
le 119886119880
3
120572119871
120572119880
ge 0 and 120573119871
120573119880
ge 0 [1198861198712
119886119871
3
] is the core of
119871 and 120572119871
120573119871
ge 0 are the left-hand and right-hand spreadsand [119886
119880
2
119886119880
3
] is the core of 119880 and 120572119880
120573119880
ge 0 are the left-hand and right-hand spreads The membership functions of119909 in
119871 and
119880 are expressed as follows
120583
119860
(119909) =
(119909 minus 119886119871
2
+ 120572119871
)
120572119871
119886119871
2
minus 120572119871
le 119909 le 119886119871
2
1 119886119871
2
le 119909 le 119886119871
3
minus
(119909 minus 119886119871
3
minus 120573119871
)
120573119871
119886119871
3
le 119909 le 119886119871
3
+ 120573119871
0 otherwise
120583
119860
(119909) =
(119909 minus 119886119880
2
+ 120572119880
)
120572119880
119886119880
2
minus 120572119880
le 119909 le 119886119880
2
1 119886119880
2
le 119909 le 119886119880
3
minus
(119909 minus 119886119880
3
minus 120573119880
)
120573119880
119886119880
3
le 119909 le 119886119880
3
+ 120573119880
0 otherwise
(13)
120583
119860
(119909) and 120583
119860
(119909) are lower and upper bounds respectivelyof (see Figure 1) Then is a PnIT2TrFN on 119883 and isrepresented by the following = [119860
119871
119860119880
] = [(119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)] Obviously if 1198861198712
= 119886119871
3
and 119886119880
2
= 119886119880
3
the PnIT2TrFN reduces to the perfectly normal interval type2 triangular fuzzy number (PnIT2TFN) If 119860119871 = 119860
119880 thenPnIT2TrFN becomes type 1 trapezoidal fuzzy number [1323]
Definition 15 (primary 120572-cut of PnIT2FS) The primary 120572-cutof PnIT2FS is 120572 = (119909 119906) | 119869
119909
ge 120572 119906 isin [0 1] which isbounded by two regions
120572
120583
119860
(119909) = (119909 120583
119860
(119909)) | 120583
119860
(119909) ge 120572 forall120572 isin [0 1]
120572
120583
119860
(119909) = (119909 120583
119860
(119909)) | 120583
119860
(119909) ge 120572 forall120572 isin [0 1]
(14)
Definition 16 (crisp bounds of PnIT2FN) The crisp boundaryof the primary 120572-cut of PnIT2FN = (119860
119871
119860119880
) is closedinterval 120572
119888119887
which will be obtained as follows 119860119871 and 119860119880
are the lower and upper interval valued bounds of Also
0
120583(x)
hU = hL = 1
aU2 minus 120572U aU2 minus 120572L aU2 aL2 aL3 aU3xaU3 + 120572L aU3 + 120572U
Figure 1 The lower trapezoidal membership function
119871 and theupper trapezoidal membership function
119880 of PnIT2FS
the boundary of 119860119871120572
and 119860119880
120572
can be defined as the boundaryof the 120572-cuts of each interval type 1 fuzzy set
119860119871
120572
= [inf119909
120572
120583
119860
(119909) sup119909
120572
120583
119860
(119909)] = [119886119871
119897
119886119871
119906
]
119860119880
120572
= [inf119909
120572
120583
119860
(119909) sup119909
120572
120583
119860
(119909)] = [119886119880
119897
119886119880
119906
]
120572
119888119887
= [inf119909
120572
120583
119860
(119909 119906) sup119909
120572
120583
119860
(119909 119906)]
= [[119886119880
119897
119886119871
119897
] [119886119871
119906
119886119880
119906
]]
= [[120572
119886119871
120572
119886119871
] [120572
119886119877
120572
119886119877
]]
(15)
which is equivalent to say
120572
119888119887
120583
119860
isin [[120572
119886119871
120572
119886119871
] [120572
119886119877
120572
119886119877
]] (16)
Evidently for PnIT2TrFN from Figure 2
120572
119886119871
le120572
119886119871
le120572
119886119877
le120572
119886119877
(17)
Definition 17 A fuzzy number = (119860119871
119860119880
) = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is said to be nonnegative PnIT2TrFNif 120583
119860
(119909) = 120583
119860
= 0 forall119909 gt 0
Definition 18 A fuzzy number = (119860119871
119860119880
) = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is said to be nonpositive PnIT2TrFN if120583
119860
(119909) = 120583
119860
= 0 forall119909 lt 0
Definition 19 A fuzzy number = (119860119871
119860119880
) = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is said to be zero PnIT2TrFN if 1198861198712
=
119887119871
2
= 0 1198861198713
= 119887119871
3
= 0 120572119871
= 120574119871
= 120573119871
= 120579119871
= 120572119880
= 120574119880
= 0 and120573119880
= 120579119880
= 0
Definition 20 PnIT2TrFNs = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are said to beidentically equal to = if and only if 119886119871
2
= 119887119871
2
1198861198713
= 119887119871
3
120572119871
= 120574119871
120573119871
= 120579119871
120572119880
= 120574119880
and 120573119880
= 120579119880
Journal of Applied Mathematics 5
0
120583(x)
hU = hL = 1
x120572aR120572aL 120572aR
120572aL
120572119888119887 A
Figure 2 Crisp bounds of PnIT2TrFN
41 Arithmetic Operations on PnIT2TrFN
Definition 21 If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are PnIT2TrFNs then = + is also PnIT2TrFN and defined by
= ((119886119871
2
+ 119887119871
2
119886119871
3
+ 119887119871
3
120572119871
+ 120574119871
120573119871
+ 120579119871
)
(119886119880
2
+ 119887119880
2
119886119880
3
+ 119887119880
3
120572119880
+ 120574119880
120573119880
+ 120579119880
))
(18)
Definition 22 If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are PnIT2TrFNs then = minus is also PnIT2TrFN and defined by
= ([119886119871
2
minus 119887119880
3
119886119871
3
minus 119887119880
2
120572119871
+ 120579119880
120573119871
+ 120574119880
]
[119886119880
2
minus 119887119871
3
119886119880
3
minus 119887119871
2
120572119880
+ 120579119871
120573119880
+ 120574119871
])
(19)
Definition 23 Let 120582 isin R If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is PnIT2TrFN then = 120582 is also PnIT2TrFN and isgiven by
= 120582 =
((120582119886119871
2
120582119886119871
3
120582120572119871
120582120573119871
) (120582119886119880
2
120582119886119880
3
120582120572119880
120582120573119880
)) if 120582 ge 0
((120582119886119880
3
120582119886119880
2
|120582| 120573119880
|120582| 120572119880
) (120582119886119871
3
120582119886119871
2
|120582| 120573119871
|120582| 120572119871
)) if 120582 lt 0
(20)
5 The Possibility Degree of PnIT2TrFN
Comparison of fuzzy numbers is considered one of the mostimportant topics in fuzzy logic theory The early and mostimportant work in the field of comparing fuzzy numbershas been presented by Dubois and Prade [24] On the otherhand the dominance possibility indices which have beenintroduced by Negi et al were utilized in the field of fuzzymathematical programming [25 26] The approach used inthese fields was based on formulating a possibility functionwhether in the case of trapezoidal fuzzy numbers or the caseof triangular fuzzy numbers In this paper we are going to
utilize the degree of possibility that the proposition statingthat ldquo119860 is less than or equal to 119861rdquo is true which is proposedby Chen and Lee [13] for calculating the ranking values ofperfectly normal trapezoidal interval type 2 fuzzy numberHere the height of the uppermembership function and lowermembership function is considered as 1 so the modifiedproposition can be as follows
Definition 24 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNsThen the possibility degrees of lower and upper membershipfunction are defined as follows
Pos (119860119871 le 119861119871
) = max(1
minusmax(max (120572119886119871 minus 120572119887119871 0) +max ((119886119871
2
minus 119887119871
2
) 0) +max ((1198861198713
minus 119887119871
3
) 0) +max (120572119886119877 minus 120572119887119877 0) + (120572
119886119877
minus120572
119887119871
)
10038161003816100381610038161003816
120572
119886119871
minus120572
119887119871
10038161003816100381610038161003816+1003816100381610038161003816119886119871
2
minus 119887119871
2
1003816100381610038161003816+1003816100381610038161003816119886119871
3
minus 119887119871
3
1003816100381610038161003816+
10038161003816100381610038161003816
120572
119886119877
minus120572
119887119877
10038161003816100381610038161003816+ (120572
119887119877
minus120572
119887119871
) + (120572
119886119877
minus120572
119886119871
)
0)
0)
6 Journal of Applied Mathematics
Pos (119860119880 le 119861119880
) = max(1
minusmax(max (120572119886119871 minus 120572119887
119871
0) +max ((1198861198802
minus 119887119880
2
) 0) +max ((1198861198803
minus 119887119880
3
) 0) +max (120572119886119877 minus 120572119887119877
0) + (120572
119886119877
minus120572
119887
119871
)
100381610038161003816100381610038161003816
120572
119886119871
minus120572
119887
119871100381610038161003816100381610038161003816
+1003816100381610038161003816119886119880
2
minus 119887119880
2
1003816100381610038161003816+1003816100381610038161003816119886119880
3
minus 119887119880
3
1003816100381610038161003816+
100381610038161003816100381610038161003816
120572
119886119877
minus120572
119887
119877100381610038161003816100381610038161003816
+ (120572
119887
119877
minus120572
119887
119871
) + (120572
119886119877
minus120572
119886119871
)
0)
0)
(21)
Proposition 25 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
))
and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2FNsPos(119860119871 ⪯ 119861
119871
) ge 120572 if and only if inf[119860119871]120572
le sup[119861119871]120572
that is
Pos (119860119871 ⪯ 119861119871
) ge 120572 iff 120572119886119871 le 120572119887119877 (22)
Alternatively
Pos (119860119871 ⪯ 119861119871
) = 120583Pos(119860119871119861119871) = (119860119871
⪯Pos
119861
119871
) (23)
An interpretation of the 120572minus relation associated with possibilitywhen comparing fuzzy numbers 119860119871 and 119861119871 is as follows For agiven level of satisfaction 120572 isin [0 1] a fuzzy number 119860119871 is notbetter than 119861119871 with respect to fuzzy relation ⪯Pos if the smallestvalue of119860119871 with the degree of satisfaction (or possibility degree)being greater than or equal to 120572minus is less than or equal to thelargest value of 119861119871 with the degree of satisfaction greater thanor equal to 120572 Similar to the above
Pos (119860119880 ⪯ 119861119880
) ge 120572 iff 120572119886119871 le 120572119887119877
(24)
Proposition 26 (see [15 23 24]) Let = ((119886119871
2
119886119871
3
120572119871
120573119871
)(119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) betwo PnIT2TrFNs then
Pos (119860119871 ⪯ 119861119871
)
=
1 119887119871
3
ge 119886119871
2
119887119871
3
minus 119886119871
2
+ 120579119871
+ 120572119871
120579119871
+ 120572119871
0 le 119886119871
2
minus 119887119871
3
le 120579119871
+ 120572119871
0 119886119871
2
minus 119887119871
3
gt 120579119871
+ 120572119871
(25)
see Figure 3
Pos (119860119880 ⪯ 119861119880
)
=
1 119887119880
3
ge 119886119880
2
119887119880
3
minus 119886119880
2
+ 120579119880
+ 120572119906
120579119880
+ 120572119880
0 le 119886119880
2
minus 119887119880
3
le 120579119880
+ 120572119880
0 119886119880
2
minus 119887119880
3
gt 120579119880
+ 120572119880
(26)
see Figure 4
Theorem 27 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNs 119901 isin
(0 1] Pos(119871 ⪯
119871
) ge 119901 if and only if 1198871198713
minus 119886119871
2
ge (119901 minus 1)(120579119871
+
120572119871
)
Proof If 119901 = 1 then from Pos(119871 ⪯
119871
) ge 1 one can getthat 1198871198713
ge 119886119871
2
and vice versa If 0 lt 119901 lt 1 then 119887119871
3
lt 119886119871
2
and119887119871
3
+ 120579119871
gt 119886119871
2
minus 120572119871
Pos(119871 ⪯
119871
) ge 119901 if and only if (1198871198713
minus 119886119871
2
+
120579119871
+120572119871
)(120579119871
+120572119871
) ge 119901 that is 1198871198713
minus119886119871
2
ge (119901minus1)(120579119871
+120572119871
)
Theorem 28 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNs 119901 isin
(0 1] Pos(119880 ⪯
119880
) ge 119901 if and only if 1198871198803
minus119886119880
2
ge (119901minus1)(120579119880
+
120572119880
)
Proof If 119901 = 1 then from Pos(119880 ⪯
119880
) ge 1 one can getthat 119887119880
3
ge 119886119880
2
and vice versa If 0 lt 119901 lt 1 then 119887119880
3
lt 119886119880
2
and 119887119880
3
+ 120579119880
gt 119886119880
2
minus 120572119880
Pos(119880 ⪯
119880
) ge 119901 if and onlyif (1198871198803
minus 119886119880
2
+ 120579119880
+ 120572119880
)(120579119880
+ 120572119880
) ge 119901 that is 1198871198803
minus 119886119880
2
ge
(119901 minus 1)(120579119880
+ 120572119880
)
6 Fuzzy Linear Programming
In this section we propose a fuzzy linear programmingmodel with the technological coefficients and right-hand side(resources) being PnIT2FN
MaxMin119909isin119883
119885 = 119888119909
St 119860119909 ⪯ 119887 119909 ge 0
(27)
where 119860 = (119886119894119895
)119898times119899
119887 = (1198871
1198872
119887119898
)119879 119888 = (119888
1
1198882
119888119899
)119909 = (119909
1
1199092
119909119899
)119879 and 119886
119894119895
119887119894
are PnIT2FN and 119909119895
119888119895
isin
R (119894 = 1 2 119898 119895 = 1 2 119899) ⪯ is type 2 fuzzy order
Definition 29 Consider a set of right-hand side (resources)parameters of a fuzzy linear programming problem definedas PnIT2FS 119887 defined on the closed interval
119887119894
isin [[120572
119887
119871
119894
120572
119887119871
119894
] [120572
119887119877
119894
120572
119887
119877
119894
]] isin R (28)
Journal of Applied Mathematics 7
10908070605040302010
Mem
bers
hip
func
tion
bL3 aL2
Pos[ALle B
L]
Figure 3 Pos[119871 le
119871
] lt 1 because 119871 gt
119871
10908070605040302010
Mem
bers
hip
func
tion
bU3 aU2
Pos[AUle B
U]
Figure 4 Pos[119880 le
119880
] lt 1 because 119880 gt
119880
and 119894 isin N119899
The membership function which represents thefuzzy space Supp(119887
119894
) is
119887119894
= int
119887
119894isinR
[int119906isin119869
119887119894
1119906]
119887119894
119894 isin N119899
119869119887
119894
sube [0 1] (29)
Here 119887 is bounded by both lower and upper primary mem-bership function namely
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (30)
with parameters 120572119887119871119894
and 120572119887119877119894
and
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (31)
with parameters 120572119887119871
119894
and 120572119887119877
119894
Definition 30 Consider a technological coefficient of fuzzylinear programming problem defined as PnIT2FS
119894119895
definedon the closed interval
119894119895
isin [inf119886
119894119895
120572
120583
119894119895
(119886119894119895
119906) sup119886
119894119895
120572
120583
119894119895
(119886119894119895
119906)]
= [[120572
119886119871
119894119895
120572
119886119871
119894119895
] [120572
119886119877
119894119895
120572
119886119877
119894119895
]] isin R 119894 isin N119899
119895 isin N119898
(32)
The membership function which represents the fuzzy spaceSupp(
119894119895
) is
119894119895
= int
119894119895isinR
[int119906isin119869
119894119895
1119906]
119894119895
119894 isin N119899
119895 isin N119898
119869
119894119895
sube [0 1]
(33)
Here 119894119895
is bounded by both lower and upper primarymembership functions namely
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (34)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
and
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (35)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
Proposition 31 Let 119894119895
isin 119865(R) 119909119895
ge 0 119894 isin N119899
119895 isin N119898
Then the fuzzy setsum119899
119895=1
119894119895
119909119895
defined by the extension principleis again a fuzzy number
Let = ⪯Pos be a fuzzy relation [27] fuzzy extension of the
usual binary relation le on R The fuzzy linear programmingproblem associated with a standard linear programmingproblem is denoted as
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St (
119899
sum
119895=1
119894119895
119909119895
) 119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(36)
In (36) the value sum119899119895=1
119894119895
119909119895
isin 119865(R) is compared with a fuzzynumber 119887
119894
isin 119865(R) by sum fuzzy relation The maximizationof the objective function is denoted by Max119885 = sum
119899
119895=1
119888119895
119909119895
Definition 32 Let 120583
119894119895
R rarr [0 1] and 120583
119887
119894
R rarr [0 1]119894 isin N
119899
119895 isin N119898
be membership function of fuzzy numbers119894119895
and 119887119894
respectively Let be a fuzzy extension of a binaryrelation le on R A fuzzy set whose membership function120583
119883
is defined as 119909 isin R119899 by
120583
119883
=
min 120583
119875
(11
1199091
+ sdot sdot sdot + 1119899
119909119899
1198871
) 120583
119875
(1198981
1199091
+ sdot sdot sdot + 119898119899
119909119899
119887119898
) if 119909119895
ge 0 119895 = 1 2 119899
0 otherwise(37)
8 Journal of Applied Mathematics
is called the fuzzy set of feasible regions of the fuzzy linearprogramming problem (36) For 120572 isin (0 1] a vector 119909 isin
[]120572
is called the 120572-feasible solution of the fuzzy linearprogramming problem
Notice that the feasible region of the fuzzy linearprogramming problem is a fuzzy set On the other hand 120572-feasible solution is a vector belonging to the 120572-cut of the fea-sible region If all the coefficients
119894119895
and 119887119894
are crisp fuzzynumber that is they are equivalent to the corresponding crispfuzzy number then the fuzzy feasible region is equivalent tothe set of all feasible solutions of the corresponding classicallinear programming
61 Type 2 Fuzzy Linear Programming Problem Let us con-sider the following fuzzy linear programming problem withright-hand side (resources) and technology coefficients arePNIT2TrFNs
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119894119895
119909119895
le119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(38)
where 119894119895
= ((119894119895
)119871
(119894119895
)119880
) = (((119886119894119895
)119871
2
(119886119894119895
)119871
3
(120572119894119895
)119871
(120573119894119895
)119871
)((119886119894119895
)119880
2
(119886119894119895
)119880
3
(120572119894119895
)119880
(120573119894119895
)119880
)) 119887119894
= ((119887119894
)119871
(119887119894
)119880
) = (((119887119894
)119871
2
(119887119894
)119871
3
(120574119894
)119871
(120579119894
)119871
) ((119887119894
)119880
2
(119887119894
)119880
3
(120574119894
)119880
(120579119894
)119880
)) are PnIT2TrFNs119888119894
are crisp coefficients of the objective and 119909119894
are the decisionvariable
From Definitions 15 21 22 and 23 and (38)
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119871
119894119895
119909119895
le119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119880
119894119895
119909119895
le119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(39)
Based on the relation of possibility we can rewrite (39) asfollows
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119871
1198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯Pos
119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119880
1198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯Pos
119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(40)
With respect to constraints of linear programming possibilitythe optimal solution is completely determined by degrees ofpossibility So we can obtain the optimal solution of linearprogramming possibility at 119901-cut levels by solving the twocrisp linear programming problems using Barnes algorithm
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198711198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯119887
119871
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198801198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯119887
119880
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(41)
FromTheorems 27 and 28 we have
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(42)
where 119860119871 = (119886119894119895
)119871
2
119860119871 = (119886119894119895
)119880
2
119887119880 = (119887119894
)119871
3
119887119880
= (119887119894
)119880
3
120572 = (120572
119894119895
)119871
120572 = (120572119894119895
)119880
120573 = (120573119894119895
)119871
120573 = (120573119894119895
)119880
120574 =
(120574119894
)119871
120574 = (120574119894
)119880
120579 = (120579119894
)119871
120579 = (120579119894
)119880
and 119909 = 119909119895
119895 = 1 2 119899 119894 = 1 2 119898
Journal of Applied Mathematics 9
The feasible regions of (119871119875119875119901
) and (119871119875119875119901
) are denoted by
119883119875
119901
= 119909 isin 119877119899
| 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119883
119875
119901
= 119909 isin 119877119899
| 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
(43)
Assume the optimal solutions of (119871119875119875119901
) and (119871119875119875119901
) are 119885119875119901
and 119885
119875
119901
respectively
Theorem 33 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883119875
119901
1
119883
119875
119901
2
sube
119883
119875
119901
1
and 119885119875119901
1
ge 119885119875
119901
2
119885
119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) ge
(1199011
minus1)(120572119909+120579) so 119909 isin 119883119875
119901
1
that is119883119875119901
2
sube 119883119875
119901
1
and119885119875119901
1
ge 119885119875
119901
2
In a similar way119883119875119901
2
sube 119883
119875
119901
1
and 119885
119875
119901
1
ge 119885
119875
119901
2
Theorem 34 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883
119875
119901
1
and
119885119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 +
120579) ge (1199011
minus 1)(120572119909 + 120579) so 119909 isin 119883
119875
119901
1
that is 119883119875119901
2
sube 119883
119875
119901
1
and
119885
119875
119901
1
ge 119885119875
119901
2
FromTheorems 33 and 34 we obtain that 119885119875119901
119885
119875
119901
consti-tute the lower and upper bound of fuzzy objective value atlevel 119901 Then the objective value of 119885119875
119901
at level 119901 is the mean
of [119885119875119901
119885
119875
119901
]
7 Numerical Illustration
An application of the proposed method is introduced withan example where all its technological coefficients and right-hand side (resources) are defined as PnIT2FNs The optimalsolution will be obtained in terms of crisp for the followingproblem
The company plans tomanufacture two types of productsThe selling prices for these products are as follows 119875
1
costs Rs 1750 per unit and 1198752
costs Rs 2000 per unitDaily production volume of each type of these products isconstrained by available man hours and available machinehours The production specifications for the given problemsituation are presented in as shown in Table 1
Max 119885 = 1198881
1199091
+ 1198882
1199092
St 11
1199091
+ 12
1199092
⪯1198871
Table 1
Resource requirementunitResource 119875
1
1198752
AvailabilityMan hours (in minutes) Around 4 Around 8 Around 32Machine hours (in minutes) Around 6 Around 4 Around 36
21
1199091
+ 22
1199092
⪯1198872
119909119895
ge 0 119895 = 1 2
(44)
where 11
= [(230 250 20 10) (230 250 30 20)] 12
=
[(460 500 40 20) (460 500 60 40)] 21
= [(345 375 30
15) (345 375 45 30)] 22
= [(230 250 20 10) (230 250
30 20)] 1198871
= [(1840 2000 160 80) (1840 2000 240 160)]1198872
= [(2070 2250 180 90) (2070 2250 270 180)] 1198881
=
1750 and 1198882
= 2000 From (42) we have the following crispoptimal problem based on the level 119901
Max 119885119875
119901
= 17501199091
+ 20001199092
St (210 + 20119901) 1199091
+ (420 + 40119901) 1199092
le 2080 minus 80119901
(315 + 30119901) 1199091
+ (210 + 20119901) 1199092
le 2340 minus 90119901
1199091
1199092
ge 0
(45)
Max 119885
119875
119901
= 17501199091
+ 20001199092
St (200 + 30119901) 1199091
+ (400 + 60119901) 1199092
le 2160 minus 160119901
(300 + 45119901) 1199091
+ (200 + 30119901) 1199092
le 2430 minus 180119901
1199091
1199092
ge 0
(46)
For different cut level 119901 we can get different optimal solutionand denote by 119885119875
119901
119885119875
119901
the optimal solution (see Figure 5 andTable 2) [28] of the crisp programming (45) and program-ming (46) respectively
8 Conclusion
In this paper a method to do perfectly normal interval type2 fuzzy arithmetic operation using extension principle is pre-sented and defined the possibility degrees of upper and lowermembership function to compare perfectly normal intervaltype 2 fuzzy numbers These functions are defined based onthe strength of upper and lower membership function ofperfectly normal interval type 2 fuzzy numbers Meanwhilesome properties and theorems were proved Then the lowerand upper fuzzy satisfaction based on possibility degrees of
10 Journal of Applied Mathematics
Table 2 Optimal solution at different possibility levels
119901-cut 01 02 03 04 05 06 07 08 09 1119885119875
119901
143549528 141658878 13980324 137981651 136193181 134436937 132712053 131017699 129353069 127717393
119885
119875
119901
155123153 151723301 148421051 145212264 142093022 139059633 136108597 133236607 130440529 127717393119885119875
119901
149336341 146691090 144112146 141596958 139143102 136748285 134410325 132127153 129896799 127717393
1090807060504030201012500 13000 13500 14000 14500 15000
p-c
ut
Z
Figure 5 Optimal solution at different possibility levels
membership function respectively are defined These fuzzysatisfactions create two nonstrict order relations on the setof overlapping intervals Also linear programming problemwas introduced with resources and technology coefficientsof perfectly normal interval type 2 fuzzy numbers Thenthe possibility degrees of membership function were appliedin order to interpret inequality constraints with intervalcoefficients According to the definition of possibility degreesof membership function and their properties the inequalityconstraints with interval coefficients were reduced in theirsatisfactory crisp equivalent forms Finally the decisionmaker can get the crisp optimal solution of the problem forevery grade 120572 isin [0 1] which was obtained by using Barnesalgorithm and then mean value is selected according to thedecision maker optimistic attitude
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] H Tanaka T Okuda and K Asai ldquoOn fuzzy-mathematicalprogrammingrdquo Journal of Cybernetics vol 3 no 4 pp 37ndash461973
[3] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 pp B141ndashB164197071
[4] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978
[5] S Deng L Zhou and X Wang ldquoSolving the fuzzy bilevellinear programmingwithmultiple followers through structured
element methodrdquo Mathematical Problems in Engineering vol2014 Article ID 418594 6 pages 2014
[6] Y-H Zhong Y-L Jia D Chen and Y Yang ldquoInterior pointmethod for solving fuzzy number linear programming prob-lems using linear ranking functionrdquo Journal of Applied Math-ematics vol 2013 Article ID 795098 9 pages 2013
[7] H Cheng W Huang and J Cai ldquoSolving a fully fuzzy linearprogramming problem through compromise programmingrdquoJournal of Applied Mathematics vol 2013 Article ID 726296 10pages 2013
[8] D Chakraborty D K Jana and T K Royl ldquoA new approach tosolve intuitionistic fuzzy optimization problem using possibil-ity necessity and credibility measuresrdquo International Journal ofEngineering Mathematics vol 2014 Article ID 593185 12 pages2014
[9] H Le and Z Gogne ldquoFuzzy linear programming with possibil-ity and necessity relationrdquo in Fuzzy Information and Engineering2010 vol 78 of Advances in Intelligent and Soft Computing pp305ndash311 Springer Berlin Germany 2010
[10] J C Figueroa-Garcıa and G Hernandez ldquoA method for solvinglinear programming models with interval type-2 fuzzy con-straintsrdquo Pesquisa Operacional vol 34 no 1 pp 73ndash89 2014
[11] J C F Garcıa ldquoA general model for linear programming withinterval type-2 fuzzy technological coefficientsrdquo in Proceedingsof the IEEE Annual Meeting of the North American FuzzyInformation Processing Society (NAFIPS rsquo12) pp 1ndash4 BerkeleyCalif USA August 2012
[12] J C F Garcıa ldquoLinear programming with interval type-2fuzzy right hand side parametersrdquo in Proceedings of the AnnualMeeting of the North American Fuzzy Information ProcessingSociety (NAFIPS rsquo08) pp 1ndash6 IEEE New York NY USA May2008
[13] S-M Chen and L-W Lee ldquoFuzzy multiple attributes groupdecision-making based on the ranking values and the arith-metic operations of interval type-2 fuzzy setsrdquo Expert Systemswith Applications vol 37 no 1 pp 824ndash833 2010
[14] J Hu Y Zhang X Chen and Y Liu ldquoMulti-criteria decisionmaking method based on possibility degree of interval type-2 fuzzy numberrdquo Knowledge-Based Systems vol 43 pp 21ndash292013
[15] R Fuller Fuzzy Reasoning and Fuzzy Optimization TurkuCentre for Computer Science 1998
[16] H-C Wu ldquoDuality theory in fuzzy linear programming prob-lems with fuzzy coefficientsrdquo Fuzzy Optimization and DecisionMaking vol 2 no 1 pp 61ndash73 2003
[17] J Qin and X Liu ldquoFrank aggregation operators for triangularinterval type-2 fuzzy set and its application inmultiple attributegroup decision makingrdquo Journal of Applied Mathematics vol2014 Article ID 923213 24 pages 2014
[18] Z M Zhang and S H Zhang ldquoA novel approach to multiattribute group decision making based on trapezoidal intervaltype-2 fuzzy soft setsrdquo Applied Mathematical Modelling vol 37no 7 pp 4948ndash4971 2013
Journal of Applied Mathematics 11
[19] Y Gong ldquoFuzzymulti-attribute group decisionmakingmethodbased on interval type-2 fuzzy sets and applications to globalsupplier selectionrdquo International Journal of Fuzzy Systems vol15 no 4 pp 392ndash400 2013
[20] D S Dinagar and A Anbalagan ldquoA new type-2 fuzzy numberarithmetic using extension principlerdquo in Proceedings of the 1stInternational Conference on Advances in Engineering Scienceand Management (ICAESM rsquo12) pp 113ndash118 March 2012
[21] K-P Chiao ldquoMultiple criteria group decision making withtriangular interval type-2 fuzzy setsrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems (FUZZ rsquo11) pp 2575ndash2582 IEEE Taipei Taiwan June 2011
[22] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006
[23] X Liu ldquoMeasuring the satisfaction of constraints in fuzzy linearprogrammingrdquo Fuzzy Sets and Systems vol 122 no 2 pp 263ndash275 2001
[24] D Dubois and H Prade ldquoRanking fuzzy numbers in the settingof possibility theoryrdquo Information Sciences vol 30 no 3 pp183ndash224 1983
[25] D S Negi and E S Lee ldquoPossibility programming by thecomparison of fuzzy numbersrdquo Computers ampMathematics withApplications vol 25 no 9 pp 43ndash50 1993
[26] MG Iskander ldquoComparison of fuzzy numbers using possibilityprogramming comments and new conceptsrdquo Computers ampMathematics with Applications vol 43 no 6-7 pp 833ndash8402002
[27] J Ramik ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006
[28] G R Lindfield and J E T Penny Numerical methods UsingMatlab Academic Press 3rd edition 2013
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
Journal of Applied Mathematics 3
When 119886119871
= 119886119880 the trapezoidal fuzzy number becomes a
triangular fuzzy number If 120572 = 120573 the trapezoidal fuzzynumber becomes a symmetrical trapezoidal fuzzy number[119886119871
119886119880
] is the core of and 120572 ge 0 120573 ge 0 are the left-handand right-hand spreads
It can easily be shown that
120572
= [120572 (119886119871
minus (119886119871
minus 120572)) + (119886119871
minus 120572)
minus 120572 ((119886119880
+ 120573) minus 119886119880
) + (119886119880
+ 120573)]
(5)
and the support of is (119886119871 minus 120572 119886119880
+ 120573)
21 Arithmetic Operations In this subsection addition sub-traction and scalar multiplication operation of trapezoidalfuzzy numbers are reviewed [9 15]
Let = (119886119871
119886119880
120572 120573) and = (119887119871
119887119880
120579 120574) be twotrapezoidal fuzzy numbers then
= (119886119871
119886119880
120572 120573)
= (119887119871
119887119880
120574 120579)
+ = (119886119871
+ 119887119871
119886119880
+ 119887119880
120572 + 120574 120573 + 120579)
minus = (119886119871
minus 119887119880
119886119880
minus 119887119871
120572 + 120579 120573 + 120574)
120582 =
(120582119886119871
120582119886119880
120582120572 120582120573) 120582 ge 0
(120582119886119880
120582119886119871
|120582| 120573 |120582| 120572) 120582 lt 0
(6)
3 Interval Type 2 Fuzzy Sets
Interval type 2 fuzzy sets (IT2FSs) play a central role in fuzzysets as models for words and in engineering applications oftype 2 fuzzy sets These fuzzy sets are characterized by theirfootprints of uncertainty which in turn are characterized bytheir boundaries upper and lower membership functions
Definition 8 (see [17]) Type 2 fuzzy set in the universeof discourse 119883 can be represented by type 2 membershipfunction 120583
119860
(119909 119906) as follows
= ((119909 119906) 120583
119860
(119909 119906)) | forall119909 isin 119883 forall119906 isin 119869119909
sube [0 1] 0
le 120583
119860
(119909 119906) le 1
(7)
where 119869119909
sube [0 1] is the primary membership function at 119909and
int
119906isin119869
119909
120583
119860
(119909 119906)
119906
(8)
indicates the secondmembership at 119909 For discrete situationsint is replaced by sum
Definition 9 (see [17 18]) Let be type 2 fuzzy set in theuniverse of discourse 119883 represented by type 2 membershipfunction 120583
119860
(119909 119906) If all 120583
119860
(119909 119906) = 1 then is called IT2FS
IT2FS can be regarded as a special case of type 2 fuzzy setwhich is defined as
= int
119909isin119883
int
119906isin119869
119909
1
(119909 119906)
= int
119909isin119883
[int119906isin119869
119909
1119906]
119909
(9)
where 119909 is the primary variable 119869119909
sube [0 1] is the primarymembership of 119909 119906 is the secondary variable and int
119906isin119869
119909
1119906 isthe secondary membership function at 119909
It is obvious that IT2FS defined on 119883 is completelydetermined by the primary membership which is called thefootprint of uncertainty and the footprint of uncertainty canbe expressed as follows
FOU () = ⋃
119909isin119883
119869119909
= ⋃
119909isin119883
(119909 119906) | 119906 isin 119869119909
sube [0 1] (10)
Definition 10 (see [14 19]) Let be IT2FS uncertaintyin the primary membership of type 2 fuzzy set consistsof a bounded region called the footprint of uncertaintywhich is the union of all primary membership Footprint ofuncertainty is characterized by upper membership functionand lower membership function Both of the membershipfunctions are type 1 fuzzy sets Upper membership functionis denoted by 120583
119860
and lower membership function is denotedby 120583
119860
respectively
Definition 11 (see [14]) An interval type 2 fuzzy number iscalled trapezoidal interval type 2 fuzzy number where theuppermembership function and lowermembership functionare both trapezoidal fuzzy numbers that is
= (119860119871
119860119880
) = ((119886119871
1
119886119871
2
119886119871
3
119886119871
4
1198671
(119860119871
) 1198672
(119860119871
))
(119886119880
1
119886119880
2
119886119880
3
119886119880
4
1198671
(119860119880
) 1198672
(119860119880
)))
(11)
where 119867119895
(119860119871
) and 119867119895
(119860119880
) (119895 = 1 2) denote membershipvalues of the corresponding elements 119886119871
119895+1
and 119886119880119895+1
(119895 = 1 2)respectively
Definition 12 (see [13]) The upper membership function andlower membership function of IT2FSs are type 1 membershipfunction respectively
Definition 13 (see [20]) IT2FS is said to be perfectlynormal if both its upper and lower membership functions arenormal It is denoted by PnIT2FS that is
sup 120583
119860
(119909) = sup 120583
119860
= 1 (12)
4 Perfectly Normal IT2TrFN
In this section the concept of PnIT2TrFN is discussed andit is the extension work of Chiao [21] that proposed theconcept of trapezoidal interval type 2 fuzzy set in whichthe upper membership function and the lower membershipfunction are represented by trapezoidal fuzzy number whichis adopted for PnIT2TrFN PnIT2FN is a fuzzy subset of the
4 Journal of Applied Mathematics
real line which is both ldquonormalrdquo and ldquoconvexrdquo The opera-tions PnIT2FSs are very complex according to the decom-position theorem [22] and the IT2FSs are usually taken insome simplified formations in applications In subsectionthe arithmetic operation on PnIT2TrFNs is formulated byproposing the extension principle
Definition 14 Let = [119860119871
119860119880
] be PnIT2FS on 119883 Let 119871 =(119886119871
2
119886119871
3
120572119871
120573119871
) and
119880
= (119886119880
2
119886119880
3
120572119880
120573119880
) be the lower andupper trapezoidal fuzzy number respectively with respect to defined on the universe of discourse 119883 where 119886
119871
2
le 119886119871
3
119886119880
2
le 119886119880
3
120572119871
120572119880
ge 0 and 120573119871
120573119880
ge 0 [1198861198712
119886119871
3
] is the core of
119871 and 120572119871
120573119871
ge 0 are the left-hand and right-hand spreadsand [119886
119880
2
119886119880
3
] is the core of 119880 and 120572119880
120573119880
ge 0 are the left-hand and right-hand spreads The membership functions of119909 in
119871 and
119880 are expressed as follows
120583
119860
(119909) =
(119909 minus 119886119871
2
+ 120572119871
)
120572119871
119886119871
2
minus 120572119871
le 119909 le 119886119871
2
1 119886119871
2
le 119909 le 119886119871
3
minus
(119909 minus 119886119871
3
minus 120573119871
)
120573119871
119886119871
3
le 119909 le 119886119871
3
+ 120573119871
0 otherwise
120583
119860
(119909) =
(119909 minus 119886119880
2
+ 120572119880
)
120572119880
119886119880
2
minus 120572119880
le 119909 le 119886119880
2
1 119886119880
2
le 119909 le 119886119880
3
minus
(119909 minus 119886119880
3
minus 120573119880
)
120573119880
119886119880
3
le 119909 le 119886119880
3
+ 120573119880
0 otherwise
(13)
120583
119860
(119909) and 120583
119860
(119909) are lower and upper bounds respectivelyof (see Figure 1) Then is a PnIT2TrFN on 119883 and isrepresented by the following = [119860
119871
119860119880
] = [(119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)] Obviously if 1198861198712
= 119886119871
3
and 119886119880
2
= 119886119880
3
the PnIT2TrFN reduces to the perfectly normal interval type2 triangular fuzzy number (PnIT2TFN) If 119860119871 = 119860
119880 thenPnIT2TrFN becomes type 1 trapezoidal fuzzy number [1323]
Definition 15 (primary 120572-cut of PnIT2FS) The primary 120572-cutof PnIT2FS is 120572 = (119909 119906) | 119869
119909
ge 120572 119906 isin [0 1] which isbounded by two regions
120572
120583
119860
(119909) = (119909 120583
119860
(119909)) | 120583
119860
(119909) ge 120572 forall120572 isin [0 1]
120572
120583
119860
(119909) = (119909 120583
119860
(119909)) | 120583
119860
(119909) ge 120572 forall120572 isin [0 1]
(14)
Definition 16 (crisp bounds of PnIT2FN) The crisp boundaryof the primary 120572-cut of PnIT2FN = (119860
119871
119860119880
) is closedinterval 120572
119888119887
which will be obtained as follows 119860119871 and 119860119880
are the lower and upper interval valued bounds of Also
0
120583(x)
hU = hL = 1
aU2 minus 120572U aU2 minus 120572L aU2 aL2 aL3 aU3xaU3 + 120572L aU3 + 120572U
Figure 1 The lower trapezoidal membership function
119871 and theupper trapezoidal membership function
119880 of PnIT2FS
the boundary of 119860119871120572
and 119860119880
120572
can be defined as the boundaryof the 120572-cuts of each interval type 1 fuzzy set
119860119871
120572
= [inf119909
120572
120583
119860
(119909) sup119909
120572
120583
119860
(119909)] = [119886119871
119897
119886119871
119906
]
119860119880
120572
= [inf119909
120572
120583
119860
(119909) sup119909
120572
120583
119860
(119909)] = [119886119880
119897
119886119880
119906
]
120572
119888119887
= [inf119909
120572
120583
119860
(119909 119906) sup119909
120572
120583
119860
(119909 119906)]
= [[119886119880
119897
119886119871
119897
] [119886119871
119906
119886119880
119906
]]
= [[120572
119886119871
120572
119886119871
] [120572
119886119877
120572
119886119877
]]
(15)
which is equivalent to say
120572
119888119887
120583
119860
isin [[120572
119886119871
120572
119886119871
] [120572
119886119877
120572
119886119877
]] (16)
Evidently for PnIT2TrFN from Figure 2
120572
119886119871
le120572
119886119871
le120572
119886119877
le120572
119886119877
(17)
Definition 17 A fuzzy number = (119860119871
119860119880
) = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is said to be nonnegative PnIT2TrFNif 120583
119860
(119909) = 120583
119860
= 0 forall119909 gt 0
Definition 18 A fuzzy number = (119860119871
119860119880
) = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is said to be nonpositive PnIT2TrFN if120583
119860
(119909) = 120583
119860
= 0 forall119909 lt 0
Definition 19 A fuzzy number = (119860119871
119860119880
) = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is said to be zero PnIT2TrFN if 1198861198712
=
119887119871
2
= 0 1198861198713
= 119887119871
3
= 0 120572119871
= 120574119871
= 120573119871
= 120579119871
= 120572119880
= 120574119880
= 0 and120573119880
= 120579119880
= 0
Definition 20 PnIT2TrFNs = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are said to beidentically equal to = if and only if 119886119871
2
= 119887119871
2
1198861198713
= 119887119871
3
120572119871
= 120574119871
120573119871
= 120579119871
120572119880
= 120574119880
and 120573119880
= 120579119880
Journal of Applied Mathematics 5
0
120583(x)
hU = hL = 1
x120572aR120572aL 120572aR
120572aL
120572119888119887 A
Figure 2 Crisp bounds of PnIT2TrFN
41 Arithmetic Operations on PnIT2TrFN
Definition 21 If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are PnIT2TrFNs then = + is also PnIT2TrFN and defined by
= ((119886119871
2
+ 119887119871
2
119886119871
3
+ 119887119871
3
120572119871
+ 120574119871
120573119871
+ 120579119871
)
(119886119880
2
+ 119887119880
2
119886119880
3
+ 119887119880
3
120572119880
+ 120574119880
120573119880
+ 120579119880
))
(18)
Definition 22 If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are PnIT2TrFNs then = minus is also PnIT2TrFN and defined by
= ([119886119871
2
minus 119887119880
3
119886119871
3
minus 119887119880
2
120572119871
+ 120579119880
120573119871
+ 120574119880
]
[119886119880
2
minus 119887119871
3
119886119880
3
minus 119887119871
2
120572119880
+ 120579119871
120573119880
+ 120574119871
])
(19)
Definition 23 Let 120582 isin R If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is PnIT2TrFN then = 120582 is also PnIT2TrFN and isgiven by
= 120582 =
((120582119886119871
2
120582119886119871
3
120582120572119871
120582120573119871
) (120582119886119880
2
120582119886119880
3
120582120572119880
120582120573119880
)) if 120582 ge 0
((120582119886119880
3
120582119886119880
2
|120582| 120573119880
|120582| 120572119880
) (120582119886119871
3
120582119886119871
2
|120582| 120573119871
|120582| 120572119871
)) if 120582 lt 0
(20)
5 The Possibility Degree of PnIT2TrFN
Comparison of fuzzy numbers is considered one of the mostimportant topics in fuzzy logic theory The early and mostimportant work in the field of comparing fuzzy numbershas been presented by Dubois and Prade [24] On the otherhand the dominance possibility indices which have beenintroduced by Negi et al were utilized in the field of fuzzymathematical programming [25 26] The approach used inthese fields was based on formulating a possibility functionwhether in the case of trapezoidal fuzzy numbers or the caseof triangular fuzzy numbers In this paper we are going to
utilize the degree of possibility that the proposition statingthat ldquo119860 is less than or equal to 119861rdquo is true which is proposedby Chen and Lee [13] for calculating the ranking values ofperfectly normal trapezoidal interval type 2 fuzzy numberHere the height of the uppermembership function and lowermembership function is considered as 1 so the modifiedproposition can be as follows
Definition 24 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNsThen the possibility degrees of lower and upper membershipfunction are defined as follows
Pos (119860119871 le 119861119871
) = max(1
minusmax(max (120572119886119871 minus 120572119887119871 0) +max ((119886119871
2
minus 119887119871
2
) 0) +max ((1198861198713
minus 119887119871
3
) 0) +max (120572119886119877 minus 120572119887119877 0) + (120572
119886119877
minus120572
119887119871
)
10038161003816100381610038161003816
120572
119886119871
minus120572
119887119871
10038161003816100381610038161003816+1003816100381610038161003816119886119871
2
minus 119887119871
2
1003816100381610038161003816+1003816100381610038161003816119886119871
3
minus 119887119871
3
1003816100381610038161003816+
10038161003816100381610038161003816
120572
119886119877
minus120572
119887119877
10038161003816100381610038161003816+ (120572
119887119877
minus120572
119887119871
) + (120572
119886119877
minus120572
119886119871
)
0)
0)
6 Journal of Applied Mathematics
Pos (119860119880 le 119861119880
) = max(1
minusmax(max (120572119886119871 minus 120572119887
119871
0) +max ((1198861198802
minus 119887119880
2
) 0) +max ((1198861198803
minus 119887119880
3
) 0) +max (120572119886119877 minus 120572119887119877
0) + (120572
119886119877
minus120572
119887
119871
)
100381610038161003816100381610038161003816
120572
119886119871
minus120572
119887
119871100381610038161003816100381610038161003816
+1003816100381610038161003816119886119880
2
minus 119887119880
2
1003816100381610038161003816+1003816100381610038161003816119886119880
3
minus 119887119880
3
1003816100381610038161003816+
100381610038161003816100381610038161003816
120572
119886119877
minus120572
119887
119877100381610038161003816100381610038161003816
+ (120572
119887
119877
minus120572
119887
119871
) + (120572
119886119877
minus120572
119886119871
)
0)
0)
(21)
Proposition 25 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
))
and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2FNsPos(119860119871 ⪯ 119861
119871
) ge 120572 if and only if inf[119860119871]120572
le sup[119861119871]120572
that is
Pos (119860119871 ⪯ 119861119871
) ge 120572 iff 120572119886119871 le 120572119887119877 (22)
Alternatively
Pos (119860119871 ⪯ 119861119871
) = 120583Pos(119860119871119861119871) = (119860119871
⪯Pos
119861
119871
) (23)
An interpretation of the 120572minus relation associated with possibilitywhen comparing fuzzy numbers 119860119871 and 119861119871 is as follows For agiven level of satisfaction 120572 isin [0 1] a fuzzy number 119860119871 is notbetter than 119861119871 with respect to fuzzy relation ⪯Pos if the smallestvalue of119860119871 with the degree of satisfaction (or possibility degree)being greater than or equal to 120572minus is less than or equal to thelargest value of 119861119871 with the degree of satisfaction greater thanor equal to 120572 Similar to the above
Pos (119860119880 ⪯ 119861119880
) ge 120572 iff 120572119886119871 le 120572119887119877
(24)
Proposition 26 (see [15 23 24]) Let = ((119886119871
2
119886119871
3
120572119871
120573119871
)(119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) betwo PnIT2TrFNs then
Pos (119860119871 ⪯ 119861119871
)
=
1 119887119871
3
ge 119886119871
2
119887119871
3
minus 119886119871
2
+ 120579119871
+ 120572119871
120579119871
+ 120572119871
0 le 119886119871
2
minus 119887119871
3
le 120579119871
+ 120572119871
0 119886119871
2
minus 119887119871
3
gt 120579119871
+ 120572119871
(25)
see Figure 3
Pos (119860119880 ⪯ 119861119880
)
=
1 119887119880
3
ge 119886119880
2
119887119880
3
minus 119886119880
2
+ 120579119880
+ 120572119906
120579119880
+ 120572119880
0 le 119886119880
2
minus 119887119880
3
le 120579119880
+ 120572119880
0 119886119880
2
minus 119887119880
3
gt 120579119880
+ 120572119880
(26)
see Figure 4
Theorem 27 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNs 119901 isin
(0 1] Pos(119871 ⪯
119871
) ge 119901 if and only if 1198871198713
minus 119886119871
2
ge (119901 minus 1)(120579119871
+
120572119871
)
Proof If 119901 = 1 then from Pos(119871 ⪯
119871
) ge 1 one can getthat 1198871198713
ge 119886119871
2
and vice versa If 0 lt 119901 lt 1 then 119887119871
3
lt 119886119871
2
and119887119871
3
+ 120579119871
gt 119886119871
2
minus 120572119871
Pos(119871 ⪯
119871
) ge 119901 if and only if (1198871198713
minus 119886119871
2
+
120579119871
+120572119871
)(120579119871
+120572119871
) ge 119901 that is 1198871198713
minus119886119871
2
ge (119901minus1)(120579119871
+120572119871
)
Theorem 28 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNs 119901 isin
(0 1] Pos(119880 ⪯
119880
) ge 119901 if and only if 1198871198803
minus119886119880
2
ge (119901minus1)(120579119880
+
120572119880
)
Proof If 119901 = 1 then from Pos(119880 ⪯
119880
) ge 1 one can getthat 119887119880
3
ge 119886119880
2
and vice versa If 0 lt 119901 lt 1 then 119887119880
3
lt 119886119880
2
and 119887119880
3
+ 120579119880
gt 119886119880
2
minus 120572119880
Pos(119880 ⪯
119880
) ge 119901 if and onlyif (1198871198803
minus 119886119880
2
+ 120579119880
+ 120572119880
)(120579119880
+ 120572119880
) ge 119901 that is 1198871198803
minus 119886119880
2
ge
(119901 minus 1)(120579119880
+ 120572119880
)
6 Fuzzy Linear Programming
In this section we propose a fuzzy linear programmingmodel with the technological coefficients and right-hand side(resources) being PnIT2FN
MaxMin119909isin119883
119885 = 119888119909
St 119860119909 ⪯ 119887 119909 ge 0
(27)
where 119860 = (119886119894119895
)119898times119899
119887 = (1198871
1198872
119887119898
)119879 119888 = (119888
1
1198882
119888119899
)119909 = (119909
1
1199092
119909119899
)119879 and 119886
119894119895
119887119894
are PnIT2FN and 119909119895
119888119895
isin
R (119894 = 1 2 119898 119895 = 1 2 119899) ⪯ is type 2 fuzzy order
Definition 29 Consider a set of right-hand side (resources)parameters of a fuzzy linear programming problem definedas PnIT2FS 119887 defined on the closed interval
119887119894
isin [[120572
119887
119871
119894
120572
119887119871
119894
] [120572
119887119877
119894
120572
119887
119877
119894
]] isin R (28)
Journal of Applied Mathematics 7
10908070605040302010
Mem
bers
hip
func
tion
bL3 aL2
Pos[ALle B
L]
Figure 3 Pos[119871 le
119871
] lt 1 because 119871 gt
119871
10908070605040302010
Mem
bers
hip
func
tion
bU3 aU2
Pos[AUle B
U]
Figure 4 Pos[119880 le
119880
] lt 1 because 119880 gt
119880
and 119894 isin N119899
The membership function which represents thefuzzy space Supp(119887
119894
) is
119887119894
= int
119887
119894isinR
[int119906isin119869
119887119894
1119906]
119887119894
119894 isin N119899
119869119887
119894
sube [0 1] (29)
Here 119887 is bounded by both lower and upper primary mem-bership function namely
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (30)
with parameters 120572119887119871119894
and 120572119887119877119894
and
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (31)
with parameters 120572119887119871
119894
and 120572119887119877
119894
Definition 30 Consider a technological coefficient of fuzzylinear programming problem defined as PnIT2FS
119894119895
definedon the closed interval
119894119895
isin [inf119886
119894119895
120572
120583
119894119895
(119886119894119895
119906) sup119886
119894119895
120572
120583
119894119895
(119886119894119895
119906)]
= [[120572
119886119871
119894119895
120572
119886119871
119894119895
] [120572
119886119877
119894119895
120572
119886119877
119894119895
]] isin R 119894 isin N119899
119895 isin N119898
(32)
The membership function which represents the fuzzy spaceSupp(
119894119895
) is
119894119895
= int
119894119895isinR
[int119906isin119869
119894119895
1119906]
119894119895
119894 isin N119899
119895 isin N119898
119869
119894119895
sube [0 1]
(33)
Here 119894119895
is bounded by both lower and upper primarymembership functions namely
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (34)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
and
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (35)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
Proposition 31 Let 119894119895
isin 119865(R) 119909119895
ge 0 119894 isin N119899
119895 isin N119898
Then the fuzzy setsum119899
119895=1
119894119895
119909119895
defined by the extension principleis again a fuzzy number
Let = ⪯Pos be a fuzzy relation [27] fuzzy extension of the
usual binary relation le on R The fuzzy linear programmingproblem associated with a standard linear programmingproblem is denoted as
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St (
119899
sum
119895=1
119894119895
119909119895
) 119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(36)
In (36) the value sum119899119895=1
119894119895
119909119895
isin 119865(R) is compared with a fuzzynumber 119887
119894
isin 119865(R) by sum fuzzy relation The maximizationof the objective function is denoted by Max119885 = sum
119899
119895=1
119888119895
119909119895
Definition 32 Let 120583
119894119895
R rarr [0 1] and 120583
119887
119894
R rarr [0 1]119894 isin N
119899
119895 isin N119898
be membership function of fuzzy numbers119894119895
and 119887119894
respectively Let be a fuzzy extension of a binaryrelation le on R A fuzzy set whose membership function120583
119883
is defined as 119909 isin R119899 by
120583
119883
=
min 120583
119875
(11
1199091
+ sdot sdot sdot + 1119899
119909119899
1198871
) 120583
119875
(1198981
1199091
+ sdot sdot sdot + 119898119899
119909119899
119887119898
) if 119909119895
ge 0 119895 = 1 2 119899
0 otherwise(37)
8 Journal of Applied Mathematics
is called the fuzzy set of feasible regions of the fuzzy linearprogramming problem (36) For 120572 isin (0 1] a vector 119909 isin
[]120572
is called the 120572-feasible solution of the fuzzy linearprogramming problem
Notice that the feasible region of the fuzzy linearprogramming problem is a fuzzy set On the other hand 120572-feasible solution is a vector belonging to the 120572-cut of the fea-sible region If all the coefficients
119894119895
and 119887119894
are crisp fuzzynumber that is they are equivalent to the corresponding crispfuzzy number then the fuzzy feasible region is equivalent tothe set of all feasible solutions of the corresponding classicallinear programming
61 Type 2 Fuzzy Linear Programming Problem Let us con-sider the following fuzzy linear programming problem withright-hand side (resources) and technology coefficients arePNIT2TrFNs
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119894119895
119909119895
le119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(38)
where 119894119895
= ((119894119895
)119871
(119894119895
)119880
) = (((119886119894119895
)119871
2
(119886119894119895
)119871
3
(120572119894119895
)119871
(120573119894119895
)119871
)((119886119894119895
)119880
2
(119886119894119895
)119880
3
(120572119894119895
)119880
(120573119894119895
)119880
)) 119887119894
= ((119887119894
)119871
(119887119894
)119880
) = (((119887119894
)119871
2
(119887119894
)119871
3
(120574119894
)119871
(120579119894
)119871
) ((119887119894
)119880
2
(119887119894
)119880
3
(120574119894
)119880
(120579119894
)119880
)) are PnIT2TrFNs119888119894
are crisp coefficients of the objective and 119909119894
are the decisionvariable
From Definitions 15 21 22 and 23 and (38)
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119871
119894119895
119909119895
le119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119880
119894119895
119909119895
le119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(39)
Based on the relation of possibility we can rewrite (39) asfollows
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119871
1198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯Pos
119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119880
1198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯Pos
119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(40)
With respect to constraints of linear programming possibilitythe optimal solution is completely determined by degrees ofpossibility So we can obtain the optimal solution of linearprogramming possibility at 119901-cut levels by solving the twocrisp linear programming problems using Barnes algorithm
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198711198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯119887
119871
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198801198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯119887
119880
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(41)
FromTheorems 27 and 28 we have
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(42)
where 119860119871 = (119886119894119895
)119871
2
119860119871 = (119886119894119895
)119880
2
119887119880 = (119887119894
)119871
3
119887119880
= (119887119894
)119880
3
120572 = (120572
119894119895
)119871
120572 = (120572119894119895
)119880
120573 = (120573119894119895
)119871
120573 = (120573119894119895
)119880
120574 =
(120574119894
)119871
120574 = (120574119894
)119880
120579 = (120579119894
)119871
120579 = (120579119894
)119880
and 119909 = 119909119895
119895 = 1 2 119899 119894 = 1 2 119898
Journal of Applied Mathematics 9
The feasible regions of (119871119875119875119901
) and (119871119875119875119901
) are denoted by
119883119875
119901
= 119909 isin 119877119899
| 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119883
119875
119901
= 119909 isin 119877119899
| 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
(43)
Assume the optimal solutions of (119871119875119875119901
) and (119871119875119875119901
) are 119885119875119901
and 119885
119875
119901
respectively
Theorem 33 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883119875
119901
1
119883
119875
119901
2
sube
119883
119875
119901
1
and 119885119875119901
1
ge 119885119875
119901
2
119885
119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) ge
(1199011
minus1)(120572119909+120579) so 119909 isin 119883119875
119901
1
that is119883119875119901
2
sube 119883119875
119901
1
and119885119875119901
1
ge 119885119875
119901
2
In a similar way119883119875119901
2
sube 119883
119875
119901
1
and 119885
119875
119901
1
ge 119885
119875
119901
2
Theorem 34 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883
119875
119901
1
and
119885119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 +
120579) ge (1199011
minus 1)(120572119909 + 120579) so 119909 isin 119883
119875
119901
1
that is 119883119875119901
2
sube 119883
119875
119901
1
and
119885
119875
119901
1
ge 119885119875
119901
2
FromTheorems 33 and 34 we obtain that 119885119875119901
119885
119875
119901
consti-tute the lower and upper bound of fuzzy objective value atlevel 119901 Then the objective value of 119885119875
119901
at level 119901 is the mean
of [119885119875119901
119885
119875
119901
]
7 Numerical Illustration
An application of the proposed method is introduced withan example where all its technological coefficients and right-hand side (resources) are defined as PnIT2FNs The optimalsolution will be obtained in terms of crisp for the followingproblem
The company plans tomanufacture two types of productsThe selling prices for these products are as follows 119875
1
costs Rs 1750 per unit and 1198752
costs Rs 2000 per unitDaily production volume of each type of these products isconstrained by available man hours and available machinehours The production specifications for the given problemsituation are presented in as shown in Table 1
Max 119885 = 1198881
1199091
+ 1198882
1199092
St 11
1199091
+ 12
1199092
⪯1198871
Table 1
Resource requirementunitResource 119875
1
1198752
AvailabilityMan hours (in minutes) Around 4 Around 8 Around 32Machine hours (in minutes) Around 6 Around 4 Around 36
21
1199091
+ 22
1199092
⪯1198872
119909119895
ge 0 119895 = 1 2
(44)
where 11
= [(230 250 20 10) (230 250 30 20)] 12
=
[(460 500 40 20) (460 500 60 40)] 21
= [(345 375 30
15) (345 375 45 30)] 22
= [(230 250 20 10) (230 250
30 20)] 1198871
= [(1840 2000 160 80) (1840 2000 240 160)]1198872
= [(2070 2250 180 90) (2070 2250 270 180)] 1198881
=
1750 and 1198882
= 2000 From (42) we have the following crispoptimal problem based on the level 119901
Max 119885119875
119901
= 17501199091
+ 20001199092
St (210 + 20119901) 1199091
+ (420 + 40119901) 1199092
le 2080 minus 80119901
(315 + 30119901) 1199091
+ (210 + 20119901) 1199092
le 2340 minus 90119901
1199091
1199092
ge 0
(45)
Max 119885
119875
119901
= 17501199091
+ 20001199092
St (200 + 30119901) 1199091
+ (400 + 60119901) 1199092
le 2160 minus 160119901
(300 + 45119901) 1199091
+ (200 + 30119901) 1199092
le 2430 minus 180119901
1199091
1199092
ge 0
(46)
For different cut level 119901 we can get different optimal solutionand denote by 119885119875
119901
119885119875
119901
the optimal solution (see Figure 5 andTable 2) [28] of the crisp programming (45) and program-ming (46) respectively
8 Conclusion
In this paper a method to do perfectly normal interval type2 fuzzy arithmetic operation using extension principle is pre-sented and defined the possibility degrees of upper and lowermembership function to compare perfectly normal intervaltype 2 fuzzy numbers These functions are defined based onthe strength of upper and lower membership function ofperfectly normal interval type 2 fuzzy numbers Meanwhilesome properties and theorems were proved Then the lowerand upper fuzzy satisfaction based on possibility degrees of
10 Journal of Applied Mathematics
Table 2 Optimal solution at different possibility levels
119901-cut 01 02 03 04 05 06 07 08 09 1119885119875
119901
143549528 141658878 13980324 137981651 136193181 134436937 132712053 131017699 129353069 127717393
119885
119875
119901
155123153 151723301 148421051 145212264 142093022 139059633 136108597 133236607 130440529 127717393119885119875
119901
149336341 146691090 144112146 141596958 139143102 136748285 134410325 132127153 129896799 127717393
1090807060504030201012500 13000 13500 14000 14500 15000
p-c
ut
Z
Figure 5 Optimal solution at different possibility levels
membership function respectively are defined These fuzzysatisfactions create two nonstrict order relations on the setof overlapping intervals Also linear programming problemwas introduced with resources and technology coefficientsof perfectly normal interval type 2 fuzzy numbers Thenthe possibility degrees of membership function were appliedin order to interpret inequality constraints with intervalcoefficients According to the definition of possibility degreesof membership function and their properties the inequalityconstraints with interval coefficients were reduced in theirsatisfactory crisp equivalent forms Finally the decisionmaker can get the crisp optimal solution of the problem forevery grade 120572 isin [0 1] which was obtained by using Barnesalgorithm and then mean value is selected according to thedecision maker optimistic attitude
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] H Tanaka T Okuda and K Asai ldquoOn fuzzy-mathematicalprogrammingrdquo Journal of Cybernetics vol 3 no 4 pp 37ndash461973
[3] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 pp B141ndashB164197071
[4] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978
[5] S Deng L Zhou and X Wang ldquoSolving the fuzzy bilevellinear programmingwithmultiple followers through structured
element methodrdquo Mathematical Problems in Engineering vol2014 Article ID 418594 6 pages 2014
[6] Y-H Zhong Y-L Jia D Chen and Y Yang ldquoInterior pointmethod for solving fuzzy number linear programming prob-lems using linear ranking functionrdquo Journal of Applied Math-ematics vol 2013 Article ID 795098 9 pages 2013
[7] H Cheng W Huang and J Cai ldquoSolving a fully fuzzy linearprogramming problem through compromise programmingrdquoJournal of Applied Mathematics vol 2013 Article ID 726296 10pages 2013
[8] D Chakraborty D K Jana and T K Royl ldquoA new approach tosolve intuitionistic fuzzy optimization problem using possibil-ity necessity and credibility measuresrdquo International Journal ofEngineering Mathematics vol 2014 Article ID 593185 12 pages2014
[9] H Le and Z Gogne ldquoFuzzy linear programming with possibil-ity and necessity relationrdquo in Fuzzy Information and Engineering2010 vol 78 of Advances in Intelligent and Soft Computing pp305ndash311 Springer Berlin Germany 2010
[10] J C Figueroa-Garcıa and G Hernandez ldquoA method for solvinglinear programming models with interval type-2 fuzzy con-straintsrdquo Pesquisa Operacional vol 34 no 1 pp 73ndash89 2014
[11] J C F Garcıa ldquoA general model for linear programming withinterval type-2 fuzzy technological coefficientsrdquo in Proceedingsof the IEEE Annual Meeting of the North American FuzzyInformation Processing Society (NAFIPS rsquo12) pp 1ndash4 BerkeleyCalif USA August 2012
[12] J C F Garcıa ldquoLinear programming with interval type-2fuzzy right hand side parametersrdquo in Proceedings of the AnnualMeeting of the North American Fuzzy Information ProcessingSociety (NAFIPS rsquo08) pp 1ndash6 IEEE New York NY USA May2008
[13] S-M Chen and L-W Lee ldquoFuzzy multiple attributes groupdecision-making based on the ranking values and the arith-metic operations of interval type-2 fuzzy setsrdquo Expert Systemswith Applications vol 37 no 1 pp 824ndash833 2010
[14] J Hu Y Zhang X Chen and Y Liu ldquoMulti-criteria decisionmaking method based on possibility degree of interval type-2 fuzzy numberrdquo Knowledge-Based Systems vol 43 pp 21ndash292013
[15] R Fuller Fuzzy Reasoning and Fuzzy Optimization TurkuCentre for Computer Science 1998
[16] H-C Wu ldquoDuality theory in fuzzy linear programming prob-lems with fuzzy coefficientsrdquo Fuzzy Optimization and DecisionMaking vol 2 no 1 pp 61ndash73 2003
[17] J Qin and X Liu ldquoFrank aggregation operators for triangularinterval type-2 fuzzy set and its application inmultiple attributegroup decision makingrdquo Journal of Applied Mathematics vol2014 Article ID 923213 24 pages 2014
[18] Z M Zhang and S H Zhang ldquoA novel approach to multiattribute group decision making based on trapezoidal intervaltype-2 fuzzy soft setsrdquo Applied Mathematical Modelling vol 37no 7 pp 4948ndash4971 2013
Journal of Applied Mathematics 11
[19] Y Gong ldquoFuzzymulti-attribute group decisionmakingmethodbased on interval type-2 fuzzy sets and applications to globalsupplier selectionrdquo International Journal of Fuzzy Systems vol15 no 4 pp 392ndash400 2013
[20] D S Dinagar and A Anbalagan ldquoA new type-2 fuzzy numberarithmetic using extension principlerdquo in Proceedings of the 1stInternational Conference on Advances in Engineering Scienceand Management (ICAESM rsquo12) pp 113ndash118 March 2012
[21] K-P Chiao ldquoMultiple criteria group decision making withtriangular interval type-2 fuzzy setsrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems (FUZZ rsquo11) pp 2575ndash2582 IEEE Taipei Taiwan June 2011
[22] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006
[23] X Liu ldquoMeasuring the satisfaction of constraints in fuzzy linearprogrammingrdquo Fuzzy Sets and Systems vol 122 no 2 pp 263ndash275 2001
[24] D Dubois and H Prade ldquoRanking fuzzy numbers in the settingof possibility theoryrdquo Information Sciences vol 30 no 3 pp183ndash224 1983
[25] D S Negi and E S Lee ldquoPossibility programming by thecomparison of fuzzy numbersrdquo Computers ampMathematics withApplications vol 25 no 9 pp 43ndash50 1993
[26] MG Iskander ldquoComparison of fuzzy numbers using possibilityprogramming comments and new conceptsrdquo Computers ampMathematics with Applications vol 43 no 6-7 pp 833ndash8402002
[27] J Ramik ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006
[28] G R Lindfield and J E T Penny Numerical methods UsingMatlab Academic Press 3rd edition 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
real line which is both ldquonormalrdquo and ldquoconvexrdquo The opera-tions PnIT2FSs are very complex according to the decom-position theorem [22] and the IT2FSs are usually taken insome simplified formations in applications In subsectionthe arithmetic operation on PnIT2TrFNs is formulated byproposing the extension principle
Definition 14 Let = [119860119871
119860119880
] be PnIT2FS on 119883 Let 119871 =(119886119871
2
119886119871
3
120572119871
120573119871
) and
119880
= (119886119880
2
119886119880
3
120572119880
120573119880
) be the lower andupper trapezoidal fuzzy number respectively with respect to defined on the universe of discourse 119883 where 119886
119871
2
le 119886119871
3
119886119880
2
le 119886119880
3
120572119871
120572119880
ge 0 and 120573119871
120573119880
ge 0 [1198861198712
119886119871
3
] is the core of
119871 and 120572119871
120573119871
ge 0 are the left-hand and right-hand spreadsand [119886
119880
2
119886119880
3
] is the core of 119880 and 120572119880
120573119880
ge 0 are the left-hand and right-hand spreads The membership functions of119909 in
119871 and
119880 are expressed as follows
120583
119860
(119909) =
(119909 minus 119886119871
2
+ 120572119871
)
120572119871
119886119871
2
minus 120572119871
le 119909 le 119886119871
2
1 119886119871
2
le 119909 le 119886119871
3
minus
(119909 minus 119886119871
3
minus 120573119871
)
120573119871
119886119871
3
le 119909 le 119886119871
3
+ 120573119871
0 otherwise
120583
119860
(119909) =
(119909 minus 119886119880
2
+ 120572119880
)
120572119880
119886119880
2
minus 120572119880
le 119909 le 119886119880
2
1 119886119880
2
le 119909 le 119886119880
3
minus
(119909 minus 119886119880
3
minus 120573119880
)
120573119880
119886119880
3
le 119909 le 119886119880
3
+ 120573119880
0 otherwise
(13)
120583
119860
(119909) and 120583
119860
(119909) are lower and upper bounds respectivelyof (see Figure 1) Then is a PnIT2TrFN on 119883 and isrepresented by the following = [119860
119871
119860119880
] = [(119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)] Obviously if 1198861198712
= 119886119871
3
and 119886119880
2
= 119886119880
3
the PnIT2TrFN reduces to the perfectly normal interval type2 triangular fuzzy number (PnIT2TFN) If 119860119871 = 119860
119880 thenPnIT2TrFN becomes type 1 trapezoidal fuzzy number [1323]
Definition 15 (primary 120572-cut of PnIT2FS) The primary 120572-cutof PnIT2FS is 120572 = (119909 119906) | 119869
119909
ge 120572 119906 isin [0 1] which isbounded by two regions
120572
120583
119860
(119909) = (119909 120583
119860
(119909)) | 120583
119860
(119909) ge 120572 forall120572 isin [0 1]
120572
120583
119860
(119909) = (119909 120583
119860
(119909)) | 120583
119860
(119909) ge 120572 forall120572 isin [0 1]
(14)
Definition 16 (crisp bounds of PnIT2FN) The crisp boundaryof the primary 120572-cut of PnIT2FN = (119860
119871
119860119880
) is closedinterval 120572
119888119887
which will be obtained as follows 119860119871 and 119860119880
are the lower and upper interval valued bounds of Also
0
120583(x)
hU = hL = 1
aU2 minus 120572U aU2 minus 120572L aU2 aL2 aL3 aU3xaU3 + 120572L aU3 + 120572U
Figure 1 The lower trapezoidal membership function
119871 and theupper trapezoidal membership function
119880 of PnIT2FS
the boundary of 119860119871120572
and 119860119880
120572
can be defined as the boundaryof the 120572-cuts of each interval type 1 fuzzy set
119860119871
120572
= [inf119909
120572
120583
119860
(119909) sup119909
120572
120583
119860
(119909)] = [119886119871
119897
119886119871
119906
]
119860119880
120572
= [inf119909
120572
120583
119860
(119909) sup119909
120572
120583
119860
(119909)] = [119886119880
119897
119886119880
119906
]
120572
119888119887
= [inf119909
120572
120583
119860
(119909 119906) sup119909
120572
120583
119860
(119909 119906)]
= [[119886119880
119897
119886119871
119897
] [119886119871
119906
119886119880
119906
]]
= [[120572
119886119871
120572
119886119871
] [120572
119886119877
120572
119886119877
]]
(15)
which is equivalent to say
120572
119888119887
120583
119860
isin [[120572
119886119871
120572
119886119871
] [120572
119886119877
120572
119886119877
]] (16)
Evidently for PnIT2TrFN from Figure 2
120572
119886119871
le120572
119886119871
le120572
119886119877
le120572
119886119877
(17)
Definition 17 A fuzzy number = (119860119871
119860119880
) = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is said to be nonnegative PnIT2TrFNif 120583
119860
(119909) = 120583
119860
= 0 forall119909 gt 0
Definition 18 A fuzzy number = (119860119871
119860119880
) = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is said to be nonpositive PnIT2TrFN if120583
119860
(119909) = 120583
119860
= 0 forall119909 lt 0
Definition 19 A fuzzy number = (119860119871
119860119880
) = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is said to be zero PnIT2TrFN if 1198861198712
=
119887119871
2
= 0 1198861198713
= 119887119871
3
= 0 120572119871
= 120574119871
= 120573119871
= 120579119871
= 120572119880
= 120574119880
= 0 and120573119880
= 120579119880
= 0
Definition 20 PnIT2TrFNs = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are said to beidentically equal to = if and only if 119886119871
2
= 119887119871
2
1198861198713
= 119887119871
3
120572119871
= 120574119871
120573119871
= 120579119871
120572119880
= 120574119880
and 120573119880
= 120579119880
Journal of Applied Mathematics 5
0
120583(x)
hU = hL = 1
x120572aR120572aL 120572aR
120572aL
120572119888119887 A
Figure 2 Crisp bounds of PnIT2TrFN
41 Arithmetic Operations on PnIT2TrFN
Definition 21 If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are PnIT2TrFNs then = + is also PnIT2TrFN and defined by
= ((119886119871
2
+ 119887119871
2
119886119871
3
+ 119887119871
3
120572119871
+ 120574119871
120573119871
+ 120579119871
)
(119886119880
2
+ 119887119880
2
119886119880
3
+ 119887119880
3
120572119880
+ 120574119880
120573119880
+ 120579119880
))
(18)
Definition 22 If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are PnIT2TrFNs then = minus is also PnIT2TrFN and defined by
= ([119886119871
2
minus 119887119880
3
119886119871
3
minus 119887119880
2
120572119871
+ 120579119880
120573119871
+ 120574119880
]
[119886119880
2
minus 119887119871
3
119886119880
3
minus 119887119871
2
120572119880
+ 120579119871
120573119880
+ 120574119871
])
(19)
Definition 23 Let 120582 isin R If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is PnIT2TrFN then = 120582 is also PnIT2TrFN and isgiven by
= 120582 =
((120582119886119871
2
120582119886119871
3
120582120572119871
120582120573119871
) (120582119886119880
2
120582119886119880
3
120582120572119880
120582120573119880
)) if 120582 ge 0
((120582119886119880
3
120582119886119880
2
|120582| 120573119880
|120582| 120572119880
) (120582119886119871
3
120582119886119871
2
|120582| 120573119871
|120582| 120572119871
)) if 120582 lt 0
(20)
5 The Possibility Degree of PnIT2TrFN
Comparison of fuzzy numbers is considered one of the mostimportant topics in fuzzy logic theory The early and mostimportant work in the field of comparing fuzzy numbershas been presented by Dubois and Prade [24] On the otherhand the dominance possibility indices which have beenintroduced by Negi et al were utilized in the field of fuzzymathematical programming [25 26] The approach used inthese fields was based on formulating a possibility functionwhether in the case of trapezoidal fuzzy numbers or the caseof triangular fuzzy numbers In this paper we are going to
utilize the degree of possibility that the proposition statingthat ldquo119860 is less than or equal to 119861rdquo is true which is proposedby Chen and Lee [13] for calculating the ranking values ofperfectly normal trapezoidal interval type 2 fuzzy numberHere the height of the uppermembership function and lowermembership function is considered as 1 so the modifiedproposition can be as follows
Definition 24 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNsThen the possibility degrees of lower and upper membershipfunction are defined as follows
Pos (119860119871 le 119861119871
) = max(1
minusmax(max (120572119886119871 minus 120572119887119871 0) +max ((119886119871
2
minus 119887119871
2
) 0) +max ((1198861198713
minus 119887119871
3
) 0) +max (120572119886119877 minus 120572119887119877 0) + (120572
119886119877
minus120572
119887119871
)
10038161003816100381610038161003816
120572
119886119871
minus120572
119887119871
10038161003816100381610038161003816+1003816100381610038161003816119886119871
2
minus 119887119871
2
1003816100381610038161003816+1003816100381610038161003816119886119871
3
minus 119887119871
3
1003816100381610038161003816+
10038161003816100381610038161003816
120572
119886119877
minus120572
119887119877
10038161003816100381610038161003816+ (120572
119887119877
minus120572
119887119871
) + (120572
119886119877
minus120572
119886119871
)
0)
0)
6 Journal of Applied Mathematics
Pos (119860119880 le 119861119880
) = max(1
minusmax(max (120572119886119871 minus 120572119887
119871
0) +max ((1198861198802
minus 119887119880
2
) 0) +max ((1198861198803
minus 119887119880
3
) 0) +max (120572119886119877 minus 120572119887119877
0) + (120572
119886119877
minus120572
119887
119871
)
100381610038161003816100381610038161003816
120572
119886119871
minus120572
119887
119871100381610038161003816100381610038161003816
+1003816100381610038161003816119886119880
2
minus 119887119880
2
1003816100381610038161003816+1003816100381610038161003816119886119880
3
minus 119887119880
3
1003816100381610038161003816+
100381610038161003816100381610038161003816
120572
119886119877
minus120572
119887
119877100381610038161003816100381610038161003816
+ (120572
119887
119877
minus120572
119887
119871
) + (120572
119886119877
minus120572
119886119871
)
0)
0)
(21)
Proposition 25 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
))
and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2FNsPos(119860119871 ⪯ 119861
119871
) ge 120572 if and only if inf[119860119871]120572
le sup[119861119871]120572
that is
Pos (119860119871 ⪯ 119861119871
) ge 120572 iff 120572119886119871 le 120572119887119877 (22)
Alternatively
Pos (119860119871 ⪯ 119861119871
) = 120583Pos(119860119871119861119871) = (119860119871
⪯Pos
119861
119871
) (23)
An interpretation of the 120572minus relation associated with possibilitywhen comparing fuzzy numbers 119860119871 and 119861119871 is as follows For agiven level of satisfaction 120572 isin [0 1] a fuzzy number 119860119871 is notbetter than 119861119871 with respect to fuzzy relation ⪯Pos if the smallestvalue of119860119871 with the degree of satisfaction (or possibility degree)being greater than or equal to 120572minus is less than or equal to thelargest value of 119861119871 with the degree of satisfaction greater thanor equal to 120572 Similar to the above
Pos (119860119880 ⪯ 119861119880
) ge 120572 iff 120572119886119871 le 120572119887119877
(24)
Proposition 26 (see [15 23 24]) Let = ((119886119871
2
119886119871
3
120572119871
120573119871
)(119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) betwo PnIT2TrFNs then
Pos (119860119871 ⪯ 119861119871
)
=
1 119887119871
3
ge 119886119871
2
119887119871
3
minus 119886119871
2
+ 120579119871
+ 120572119871
120579119871
+ 120572119871
0 le 119886119871
2
minus 119887119871
3
le 120579119871
+ 120572119871
0 119886119871
2
minus 119887119871
3
gt 120579119871
+ 120572119871
(25)
see Figure 3
Pos (119860119880 ⪯ 119861119880
)
=
1 119887119880
3
ge 119886119880
2
119887119880
3
minus 119886119880
2
+ 120579119880
+ 120572119906
120579119880
+ 120572119880
0 le 119886119880
2
minus 119887119880
3
le 120579119880
+ 120572119880
0 119886119880
2
minus 119887119880
3
gt 120579119880
+ 120572119880
(26)
see Figure 4
Theorem 27 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNs 119901 isin
(0 1] Pos(119871 ⪯
119871
) ge 119901 if and only if 1198871198713
minus 119886119871
2
ge (119901 minus 1)(120579119871
+
120572119871
)
Proof If 119901 = 1 then from Pos(119871 ⪯
119871
) ge 1 one can getthat 1198871198713
ge 119886119871
2
and vice versa If 0 lt 119901 lt 1 then 119887119871
3
lt 119886119871
2
and119887119871
3
+ 120579119871
gt 119886119871
2
minus 120572119871
Pos(119871 ⪯
119871
) ge 119901 if and only if (1198871198713
minus 119886119871
2
+
120579119871
+120572119871
)(120579119871
+120572119871
) ge 119901 that is 1198871198713
minus119886119871
2
ge (119901minus1)(120579119871
+120572119871
)
Theorem 28 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNs 119901 isin
(0 1] Pos(119880 ⪯
119880
) ge 119901 if and only if 1198871198803
minus119886119880
2
ge (119901minus1)(120579119880
+
120572119880
)
Proof If 119901 = 1 then from Pos(119880 ⪯
119880
) ge 1 one can getthat 119887119880
3
ge 119886119880
2
and vice versa If 0 lt 119901 lt 1 then 119887119880
3
lt 119886119880
2
and 119887119880
3
+ 120579119880
gt 119886119880
2
minus 120572119880
Pos(119880 ⪯
119880
) ge 119901 if and onlyif (1198871198803
minus 119886119880
2
+ 120579119880
+ 120572119880
)(120579119880
+ 120572119880
) ge 119901 that is 1198871198803
minus 119886119880
2
ge
(119901 minus 1)(120579119880
+ 120572119880
)
6 Fuzzy Linear Programming
In this section we propose a fuzzy linear programmingmodel with the technological coefficients and right-hand side(resources) being PnIT2FN
MaxMin119909isin119883
119885 = 119888119909
St 119860119909 ⪯ 119887 119909 ge 0
(27)
where 119860 = (119886119894119895
)119898times119899
119887 = (1198871
1198872
119887119898
)119879 119888 = (119888
1
1198882
119888119899
)119909 = (119909
1
1199092
119909119899
)119879 and 119886
119894119895
119887119894
are PnIT2FN and 119909119895
119888119895
isin
R (119894 = 1 2 119898 119895 = 1 2 119899) ⪯ is type 2 fuzzy order
Definition 29 Consider a set of right-hand side (resources)parameters of a fuzzy linear programming problem definedas PnIT2FS 119887 defined on the closed interval
119887119894
isin [[120572
119887
119871
119894
120572
119887119871
119894
] [120572
119887119877
119894
120572
119887
119877
119894
]] isin R (28)
Journal of Applied Mathematics 7
10908070605040302010
Mem
bers
hip
func
tion
bL3 aL2
Pos[ALle B
L]
Figure 3 Pos[119871 le
119871
] lt 1 because 119871 gt
119871
10908070605040302010
Mem
bers
hip
func
tion
bU3 aU2
Pos[AUle B
U]
Figure 4 Pos[119880 le
119880
] lt 1 because 119880 gt
119880
and 119894 isin N119899
The membership function which represents thefuzzy space Supp(119887
119894
) is
119887119894
= int
119887
119894isinR
[int119906isin119869
119887119894
1119906]
119887119894
119894 isin N119899
119869119887
119894
sube [0 1] (29)
Here 119887 is bounded by both lower and upper primary mem-bership function namely
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (30)
with parameters 120572119887119871119894
and 120572119887119877119894
and
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (31)
with parameters 120572119887119871
119894
and 120572119887119877
119894
Definition 30 Consider a technological coefficient of fuzzylinear programming problem defined as PnIT2FS
119894119895
definedon the closed interval
119894119895
isin [inf119886
119894119895
120572
120583
119894119895
(119886119894119895
119906) sup119886
119894119895
120572
120583
119894119895
(119886119894119895
119906)]
= [[120572
119886119871
119894119895
120572
119886119871
119894119895
] [120572
119886119877
119894119895
120572
119886119877
119894119895
]] isin R 119894 isin N119899
119895 isin N119898
(32)
The membership function which represents the fuzzy spaceSupp(
119894119895
) is
119894119895
= int
119894119895isinR
[int119906isin119869
119894119895
1119906]
119894119895
119894 isin N119899
119895 isin N119898
119869
119894119895
sube [0 1]
(33)
Here 119894119895
is bounded by both lower and upper primarymembership functions namely
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (34)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
and
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (35)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
Proposition 31 Let 119894119895
isin 119865(R) 119909119895
ge 0 119894 isin N119899
119895 isin N119898
Then the fuzzy setsum119899
119895=1
119894119895
119909119895
defined by the extension principleis again a fuzzy number
Let = ⪯Pos be a fuzzy relation [27] fuzzy extension of the
usual binary relation le on R The fuzzy linear programmingproblem associated with a standard linear programmingproblem is denoted as
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St (
119899
sum
119895=1
119894119895
119909119895
) 119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(36)
In (36) the value sum119899119895=1
119894119895
119909119895
isin 119865(R) is compared with a fuzzynumber 119887
119894
isin 119865(R) by sum fuzzy relation The maximizationof the objective function is denoted by Max119885 = sum
119899
119895=1
119888119895
119909119895
Definition 32 Let 120583
119894119895
R rarr [0 1] and 120583
119887
119894
R rarr [0 1]119894 isin N
119899
119895 isin N119898
be membership function of fuzzy numbers119894119895
and 119887119894
respectively Let be a fuzzy extension of a binaryrelation le on R A fuzzy set whose membership function120583
119883
is defined as 119909 isin R119899 by
120583
119883
=
min 120583
119875
(11
1199091
+ sdot sdot sdot + 1119899
119909119899
1198871
) 120583
119875
(1198981
1199091
+ sdot sdot sdot + 119898119899
119909119899
119887119898
) if 119909119895
ge 0 119895 = 1 2 119899
0 otherwise(37)
8 Journal of Applied Mathematics
is called the fuzzy set of feasible regions of the fuzzy linearprogramming problem (36) For 120572 isin (0 1] a vector 119909 isin
[]120572
is called the 120572-feasible solution of the fuzzy linearprogramming problem
Notice that the feasible region of the fuzzy linearprogramming problem is a fuzzy set On the other hand 120572-feasible solution is a vector belonging to the 120572-cut of the fea-sible region If all the coefficients
119894119895
and 119887119894
are crisp fuzzynumber that is they are equivalent to the corresponding crispfuzzy number then the fuzzy feasible region is equivalent tothe set of all feasible solutions of the corresponding classicallinear programming
61 Type 2 Fuzzy Linear Programming Problem Let us con-sider the following fuzzy linear programming problem withright-hand side (resources) and technology coefficients arePNIT2TrFNs
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119894119895
119909119895
le119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(38)
where 119894119895
= ((119894119895
)119871
(119894119895
)119880
) = (((119886119894119895
)119871
2
(119886119894119895
)119871
3
(120572119894119895
)119871
(120573119894119895
)119871
)((119886119894119895
)119880
2
(119886119894119895
)119880
3
(120572119894119895
)119880
(120573119894119895
)119880
)) 119887119894
= ((119887119894
)119871
(119887119894
)119880
) = (((119887119894
)119871
2
(119887119894
)119871
3
(120574119894
)119871
(120579119894
)119871
) ((119887119894
)119880
2
(119887119894
)119880
3
(120574119894
)119880
(120579119894
)119880
)) are PnIT2TrFNs119888119894
are crisp coefficients of the objective and 119909119894
are the decisionvariable
From Definitions 15 21 22 and 23 and (38)
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119871
119894119895
119909119895
le119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119880
119894119895
119909119895
le119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(39)
Based on the relation of possibility we can rewrite (39) asfollows
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119871
1198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯Pos
119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119880
1198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯Pos
119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(40)
With respect to constraints of linear programming possibilitythe optimal solution is completely determined by degrees ofpossibility So we can obtain the optimal solution of linearprogramming possibility at 119901-cut levels by solving the twocrisp linear programming problems using Barnes algorithm
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198711198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯119887
119871
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198801198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯119887
119880
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(41)
FromTheorems 27 and 28 we have
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(42)
where 119860119871 = (119886119894119895
)119871
2
119860119871 = (119886119894119895
)119880
2
119887119880 = (119887119894
)119871
3
119887119880
= (119887119894
)119880
3
120572 = (120572
119894119895
)119871
120572 = (120572119894119895
)119880
120573 = (120573119894119895
)119871
120573 = (120573119894119895
)119880
120574 =
(120574119894
)119871
120574 = (120574119894
)119880
120579 = (120579119894
)119871
120579 = (120579119894
)119880
and 119909 = 119909119895
119895 = 1 2 119899 119894 = 1 2 119898
Journal of Applied Mathematics 9
The feasible regions of (119871119875119875119901
) and (119871119875119875119901
) are denoted by
119883119875
119901
= 119909 isin 119877119899
| 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119883
119875
119901
= 119909 isin 119877119899
| 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
(43)
Assume the optimal solutions of (119871119875119875119901
) and (119871119875119875119901
) are 119885119875119901
and 119885
119875
119901
respectively
Theorem 33 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883119875
119901
1
119883
119875
119901
2
sube
119883
119875
119901
1
and 119885119875119901
1
ge 119885119875
119901
2
119885
119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) ge
(1199011
minus1)(120572119909+120579) so 119909 isin 119883119875
119901
1
that is119883119875119901
2
sube 119883119875
119901
1
and119885119875119901
1
ge 119885119875
119901
2
In a similar way119883119875119901
2
sube 119883
119875
119901
1
and 119885
119875
119901
1
ge 119885
119875
119901
2
Theorem 34 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883
119875
119901
1
and
119885119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 +
120579) ge (1199011
minus 1)(120572119909 + 120579) so 119909 isin 119883
119875
119901
1
that is 119883119875119901
2
sube 119883
119875
119901
1
and
119885
119875
119901
1
ge 119885119875
119901
2
FromTheorems 33 and 34 we obtain that 119885119875119901
119885
119875
119901
consti-tute the lower and upper bound of fuzzy objective value atlevel 119901 Then the objective value of 119885119875
119901
at level 119901 is the mean
of [119885119875119901
119885
119875
119901
]
7 Numerical Illustration
An application of the proposed method is introduced withan example where all its technological coefficients and right-hand side (resources) are defined as PnIT2FNs The optimalsolution will be obtained in terms of crisp for the followingproblem
The company plans tomanufacture two types of productsThe selling prices for these products are as follows 119875
1
costs Rs 1750 per unit and 1198752
costs Rs 2000 per unitDaily production volume of each type of these products isconstrained by available man hours and available machinehours The production specifications for the given problemsituation are presented in as shown in Table 1
Max 119885 = 1198881
1199091
+ 1198882
1199092
St 11
1199091
+ 12
1199092
⪯1198871
Table 1
Resource requirementunitResource 119875
1
1198752
AvailabilityMan hours (in minutes) Around 4 Around 8 Around 32Machine hours (in minutes) Around 6 Around 4 Around 36
21
1199091
+ 22
1199092
⪯1198872
119909119895
ge 0 119895 = 1 2
(44)
where 11
= [(230 250 20 10) (230 250 30 20)] 12
=
[(460 500 40 20) (460 500 60 40)] 21
= [(345 375 30
15) (345 375 45 30)] 22
= [(230 250 20 10) (230 250
30 20)] 1198871
= [(1840 2000 160 80) (1840 2000 240 160)]1198872
= [(2070 2250 180 90) (2070 2250 270 180)] 1198881
=
1750 and 1198882
= 2000 From (42) we have the following crispoptimal problem based on the level 119901
Max 119885119875
119901
= 17501199091
+ 20001199092
St (210 + 20119901) 1199091
+ (420 + 40119901) 1199092
le 2080 minus 80119901
(315 + 30119901) 1199091
+ (210 + 20119901) 1199092
le 2340 minus 90119901
1199091
1199092
ge 0
(45)
Max 119885
119875
119901
= 17501199091
+ 20001199092
St (200 + 30119901) 1199091
+ (400 + 60119901) 1199092
le 2160 minus 160119901
(300 + 45119901) 1199091
+ (200 + 30119901) 1199092
le 2430 minus 180119901
1199091
1199092
ge 0
(46)
For different cut level 119901 we can get different optimal solutionand denote by 119885119875
119901
119885119875
119901
the optimal solution (see Figure 5 andTable 2) [28] of the crisp programming (45) and program-ming (46) respectively
8 Conclusion
In this paper a method to do perfectly normal interval type2 fuzzy arithmetic operation using extension principle is pre-sented and defined the possibility degrees of upper and lowermembership function to compare perfectly normal intervaltype 2 fuzzy numbers These functions are defined based onthe strength of upper and lower membership function ofperfectly normal interval type 2 fuzzy numbers Meanwhilesome properties and theorems were proved Then the lowerand upper fuzzy satisfaction based on possibility degrees of
10 Journal of Applied Mathematics
Table 2 Optimal solution at different possibility levels
119901-cut 01 02 03 04 05 06 07 08 09 1119885119875
119901
143549528 141658878 13980324 137981651 136193181 134436937 132712053 131017699 129353069 127717393
119885
119875
119901
155123153 151723301 148421051 145212264 142093022 139059633 136108597 133236607 130440529 127717393119885119875
119901
149336341 146691090 144112146 141596958 139143102 136748285 134410325 132127153 129896799 127717393
1090807060504030201012500 13000 13500 14000 14500 15000
p-c
ut
Z
Figure 5 Optimal solution at different possibility levels
membership function respectively are defined These fuzzysatisfactions create two nonstrict order relations on the setof overlapping intervals Also linear programming problemwas introduced with resources and technology coefficientsof perfectly normal interval type 2 fuzzy numbers Thenthe possibility degrees of membership function were appliedin order to interpret inequality constraints with intervalcoefficients According to the definition of possibility degreesof membership function and their properties the inequalityconstraints with interval coefficients were reduced in theirsatisfactory crisp equivalent forms Finally the decisionmaker can get the crisp optimal solution of the problem forevery grade 120572 isin [0 1] which was obtained by using Barnesalgorithm and then mean value is selected according to thedecision maker optimistic attitude
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] H Tanaka T Okuda and K Asai ldquoOn fuzzy-mathematicalprogrammingrdquo Journal of Cybernetics vol 3 no 4 pp 37ndash461973
[3] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 pp B141ndashB164197071
[4] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978
[5] S Deng L Zhou and X Wang ldquoSolving the fuzzy bilevellinear programmingwithmultiple followers through structured
element methodrdquo Mathematical Problems in Engineering vol2014 Article ID 418594 6 pages 2014
[6] Y-H Zhong Y-L Jia D Chen and Y Yang ldquoInterior pointmethod for solving fuzzy number linear programming prob-lems using linear ranking functionrdquo Journal of Applied Math-ematics vol 2013 Article ID 795098 9 pages 2013
[7] H Cheng W Huang and J Cai ldquoSolving a fully fuzzy linearprogramming problem through compromise programmingrdquoJournal of Applied Mathematics vol 2013 Article ID 726296 10pages 2013
[8] D Chakraborty D K Jana and T K Royl ldquoA new approach tosolve intuitionistic fuzzy optimization problem using possibil-ity necessity and credibility measuresrdquo International Journal ofEngineering Mathematics vol 2014 Article ID 593185 12 pages2014
[9] H Le and Z Gogne ldquoFuzzy linear programming with possibil-ity and necessity relationrdquo in Fuzzy Information and Engineering2010 vol 78 of Advances in Intelligent and Soft Computing pp305ndash311 Springer Berlin Germany 2010
[10] J C Figueroa-Garcıa and G Hernandez ldquoA method for solvinglinear programming models with interval type-2 fuzzy con-straintsrdquo Pesquisa Operacional vol 34 no 1 pp 73ndash89 2014
[11] J C F Garcıa ldquoA general model for linear programming withinterval type-2 fuzzy technological coefficientsrdquo in Proceedingsof the IEEE Annual Meeting of the North American FuzzyInformation Processing Society (NAFIPS rsquo12) pp 1ndash4 BerkeleyCalif USA August 2012
[12] J C F Garcıa ldquoLinear programming with interval type-2fuzzy right hand side parametersrdquo in Proceedings of the AnnualMeeting of the North American Fuzzy Information ProcessingSociety (NAFIPS rsquo08) pp 1ndash6 IEEE New York NY USA May2008
[13] S-M Chen and L-W Lee ldquoFuzzy multiple attributes groupdecision-making based on the ranking values and the arith-metic operations of interval type-2 fuzzy setsrdquo Expert Systemswith Applications vol 37 no 1 pp 824ndash833 2010
[14] J Hu Y Zhang X Chen and Y Liu ldquoMulti-criteria decisionmaking method based on possibility degree of interval type-2 fuzzy numberrdquo Knowledge-Based Systems vol 43 pp 21ndash292013
[15] R Fuller Fuzzy Reasoning and Fuzzy Optimization TurkuCentre for Computer Science 1998
[16] H-C Wu ldquoDuality theory in fuzzy linear programming prob-lems with fuzzy coefficientsrdquo Fuzzy Optimization and DecisionMaking vol 2 no 1 pp 61ndash73 2003
[17] J Qin and X Liu ldquoFrank aggregation operators for triangularinterval type-2 fuzzy set and its application inmultiple attributegroup decision makingrdquo Journal of Applied Mathematics vol2014 Article ID 923213 24 pages 2014
[18] Z M Zhang and S H Zhang ldquoA novel approach to multiattribute group decision making based on trapezoidal intervaltype-2 fuzzy soft setsrdquo Applied Mathematical Modelling vol 37no 7 pp 4948ndash4971 2013
Journal of Applied Mathematics 11
[19] Y Gong ldquoFuzzymulti-attribute group decisionmakingmethodbased on interval type-2 fuzzy sets and applications to globalsupplier selectionrdquo International Journal of Fuzzy Systems vol15 no 4 pp 392ndash400 2013
[20] D S Dinagar and A Anbalagan ldquoA new type-2 fuzzy numberarithmetic using extension principlerdquo in Proceedings of the 1stInternational Conference on Advances in Engineering Scienceand Management (ICAESM rsquo12) pp 113ndash118 March 2012
[21] K-P Chiao ldquoMultiple criteria group decision making withtriangular interval type-2 fuzzy setsrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems (FUZZ rsquo11) pp 2575ndash2582 IEEE Taipei Taiwan June 2011
[22] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006
[23] X Liu ldquoMeasuring the satisfaction of constraints in fuzzy linearprogrammingrdquo Fuzzy Sets and Systems vol 122 no 2 pp 263ndash275 2001
[24] D Dubois and H Prade ldquoRanking fuzzy numbers in the settingof possibility theoryrdquo Information Sciences vol 30 no 3 pp183ndash224 1983
[25] D S Negi and E S Lee ldquoPossibility programming by thecomparison of fuzzy numbersrdquo Computers ampMathematics withApplications vol 25 no 9 pp 43ndash50 1993
[26] MG Iskander ldquoComparison of fuzzy numbers using possibilityprogramming comments and new conceptsrdquo Computers ampMathematics with Applications vol 43 no 6-7 pp 833ndash8402002
[27] J Ramik ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006
[28] G R Lindfield and J E T Penny Numerical methods UsingMatlab Academic Press 3rd edition 2013
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
Journal of Applied Mathematics 5
0
120583(x)
hU = hL = 1
x120572aR120572aL 120572aR
120572aL
120572119888119887 A
Figure 2 Crisp bounds of PnIT2TrFN
41 Arithmetic Operations on PnIT2TrFN
Definition 21 If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are PnIT2TrFNs then = + is also PnIT2TrFN and defined by
= ((119886119871
2
+ 119887119871
2
119886119871
3
+ 119887119871
3
120572119871
+ 120574119871
120573119871
+ 120579119871
)
(119886119880
2
+ 119887119880
2
119886119880
3
+ 119887119880
3
120572119880
+ 120574119880
120573119880
+ 120579119880
))
(18)
Definition 22 If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) are PnIT2TrFNs then = minus is also PnIT2TrFN and defined by
= ([119886119871
2
minus 119887119880
3
119886119871
3
minus 119887119880
2
120572119871
+ 120579119880
120573119871
+ 120574119880
]
[119886119880
2
minus 119887119871
3
119886119880
3
minus 119887119871
2
120572119880
+ 120579119871
120573119880
+ 120574119871
])
(19)
Definition 23 Let 120582 isin R If = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) is PnIT2TrFN then = 120582 is also PnIT2TrFN and isgiven by
= 120582 =
((120582119886119871
2
120582119886119871
3
120582120572119871
120582120573119871
) (120582119886119880
2
120582119886119880
3
120582120572119880
120582120573119880
)) if 120582 ge 0
((120582119886119880
3
120582119886119880
2
|120582| 120573119880
|120582| 120572119880
) (120582119886119871
3
120582119886119871
2
|120582| 120573119871
|120582| 120572119871
)) if 120582 lt 0
(20)
5 The Possibility Degree of PnIT2TrFN
Comparison of fuzzy numbers is considered one of the mostimportant topics in fuzzy logic theory The early and mostimportant work in the field of comparing fuzzy numbershas been presented by Dubois and Prade [24] On the otherhand the dominance possibility indices which have beenintroduced by Negi et al were utilized in the field of fuzzymathematical programming [25 26] The approach used inthese fields was based on formulating a possibility functionwhether in the case of trapezoidal fuzzy numbers or the caseof triangular fuzzy numbers In this paper we are going to
utilize the degree of possibility that the proposition statingthat ldquo119860 is less than or equal to 119861rdquo is true which is proposedby Chen and Lee [13] for calculating the ranking values ofperfectly normal trapezoidal interval type 2 fuzzy numberHere the height of the uppermembership function and lowermembership function is considered as 1 so the modifiedproposition can be as follows
Definition 24 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNsThen the possibility degrees of lower and upper membershipfunction are defined as follows
Pos (119860119871 le 119861119871
) = max(1
minusmax(max (120572119886119871 minus 120572119887119871 0) +max ((119886119871
2
minus 119887119871
2
) 0) +max ((1198861198713
minus 119887119871
3
) 0) +max (120572119886119877 minus 120572119887119877 0) + (120572
119886119877
minus120572
119887119871
)
10038161003816100381610038161003816
120572
119886119871
minus120572
119887119871
10038161003816100381610038161003816+1003816100381610038161003816119886119871
2
minus 119887119871
2
1003816100381610038161003816+1003816100381610038161003816119886119871
3
minus 119887119871
3
1003816100381610038161003816+
10038161003816100381610038161003816
120572
119886119877
minus120572
119887119877
10038161003816100381610038161003816+ (120572
119887119877
minus120572
119887119871
) + (120572
119886119877
minus120572
119886119871
)
0)
0)
6 Journal of Applied Mathematics
Pos (119860119880 le 119861119880
) = max(1
minusmax(max (120572119886119871 minus 120572119887
119871
0) +max ((1198861198802
minus 119887119880
2
) 0) +max ((1198861198803
minus 119887119880
3
) 0) +max (120572119886119877 minus 120572119887119877
0) + (120572
119886119877
minus120572
119887
119871
)
100381610038161003816100381610038161003816
120572
119886119871
minus120572
119887
119871100381610038161003816100381610038161003816
+1003816100381610038161003816119886119880
2
minus 119887119880
2
1003816100381610038161003816+1003816100381610038161003816119886119880
3
minus 119887119880
3
1003816100381610038161003816+
100381610038161003816100381610038161003816
120572
119886119877
minus120572
119887
119877100381610038161003816100381610038161003816
+ (120572
119887
119877
minus120572
119887
119871
) + (120572
119886119877
minus120572
119886119871
)
0)
0)
(21)
Proposition 25 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
))
and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2FNsPos(119860119871 ⪯ 119861
119871
) ge 120572 if and only if inf[119860119871]120572
le sup[119861119871]120572
that is
Pos (119860119871 ⪯ 119861119871
) ge 120572 iff 120572119886119871 le 120572119887119877 (22)
Alternatively
Pos (119860119871 ⪯ 119861119871
) = 120583Pos(119860119871119861119871) = (119860119871
⪯Pos
119861
119871
) (23)
An interpretation of the 120572minus relation associated with possibilitywhen comparing fuzzy numbers 119860119871 and 119861119871 is as follows For agiven level of satisfaction 120572 isin [0 1] a fuzzy number 119860119871 is notbetter than 119861119871 with respect to fuzzy relation ⪯Pos if the smallestvalue of119860119871 with the degree of satisfaction (or possibility degree)being greater than or equal to 120572minus is less than or equal to thelargest value of 119861119871 with the degree of satisfaction greater thanor equal to 120572 Similar to the above
Pos (119860119880 ⪯ 119861119880
) ge 120572 iff 120572119886119871 le 120572119887119877
(24)
Proposition 26 (see [15 23 24]) Let = ((119886119871
2
119886119871
3
120572119871
120573119871
)(119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) betwo PnIT2TrFNs then
Pos (119860119871 ⪯ 119861119871
)
=
1 119887119871
3
ge 119886119871
2
119887119871
3
minus 119886119871
2
+ 120579119871
+ 120572119871
120579119871
+ 120572119871
0 le 119886119871
2
minus 119887119871
3
le 120579119871
+ 120572119871
0 119886119871
2
minus 119887119871
3
gt 120579119871
+ 120572119871
(25)
see Figure 3
Pos (119860119880 ⪯ 119861119880
)
=
1 119887119880
3
ge 119886119880
2
119887119880
3
minus 119886119880
2
+ 120579119880
+ 120572119906
120579119880
+ 120572119880
0 le 119886119880
2
minus 119887119880
3
le 120579119880
+ 120572119880
0 119886119880
2
minus 119887119880
3
gt 120579119880
+ 120572119880
(26)
see Figure 4
Theorem 27 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNs 119901 isin
(0 1] Pos(119871 ⪯
119871
) ge 119901 if and only if 1198871198713
minus 119886119871
2
ge (119901 minus 1)(120579119871
+
120572119871
)
Proof If 119901 = 1 then from Pos(119871 ⪯
119871
) ge 1 one can getthat 1198871198713
ge 119886119871
2
and vice versa If 0 lt 119901 lt 1 then 119887119871
3
lt 119886119871
2
and119887119871
3
+ 120579119871
gt 119886119871
2
minus 120572119871
Pos(119871 ⪯
119871
) ge 119901 if and only if (1198871198713
minus 119886119871
2
+
120579119871
+120572119871
)(120579119871
+120572119871
) ge 119901 that is 1198871198713
minus119886119871
2
ge (119901minus1)(120579119871
+120572119871
)
Theorem 28 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNs 119901 isin
(0 1] Pos(119880 ⪯
119880
) ge 119901 if and only if 1198871198803
minus119886119880
2
ge (119901minus1)(120579119880
+
120572119880
)
Proof If 119901 = 1 then from Pos(119880 ⪯
119880
) ge 1 one can getthat 119887119880
3
ge 119886119880
2
and vice versa If 0 lt 119901 lt 1 then 119887119880
3
lt 119886119880
2
and 119887119880
3
+ 120579119880
gt 119886119880
2
minus 120572119880
Pos(119880 ⪯
119880
) ge 119901 if and onlyif (1198871198803
minus 119886119880
2
+ 120579119880
+ 120572119880
)(120579119880
+ 120572119880
) ge 119901 that is 1198871198803
minus 119886119880
2
ge
(119901 minus 1)(120579119880
+ 120572119880
)
6 Fuzzy Linear Programming
In this section we propose a fuzzy linear programmingmodel with the technological coefficients and right-hand side(resources) being PnIT2FN
MaxMin119909isin119883
119885 = 119888119909
St 119860119909 ⪯ 119887 119909 ge 0
(27)
where 119860 = (119886119894119895
)119898times119899
119887 = (1198871
1198872
119887119898
)119879 119888 = (119888
1
1198882
119888119899
)119909 = (119909
1
1199092
119909119899
)119879 and 119886
119894119895
119887119894
are PnIT2FN and 119909119895
119888119895
isin
R (119894 = 1 2 119898 119895 = 1 2 119899) ⪯ is type 2 fuzzy order
Definition 29 Consider a set of right-hand side (resources)parameters of a fuzzy linear programming problem definedas PnIT2FS 119887 defined on the closed interval
119887119894
isin [[120572
119887
119871
119894
120572
119887119871
119894
] [120572
119887119877
119894
120572
119887
119877
119894
]] isin R (28)
Journal of Applied Mathematics 7
10908070605040302010
Mem
bers
hip
func
tion
bL3 aL2
Pos[ALle B
L]
Figure 3 Pos[119871 le
119871
] lt 1 because 119871 gt
119871
10908070605040302010
Mem
bers
hip
func
tion
bU3 aU2
Pos[AUle B
U]
Figure 4 Pos[119880 le
119880
] lt 1 because 119880 gt
119880
and 119894 isin N119899
The membership function which represents thefuzzy space Supp(119887
119894
) is
119887119894
= int
119887
119894isinR
[int119906isin119869
119887119894
1119906]
119887119894
119894 isin N119899
119869119887
119894
sube [0 1] (29)
Here 119887 is bounded by both lower and upper primary mem-bership function namely
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (30)
with parameters 120572119887119871119894
and 120572119887119877119894
and
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (31)
with parameters 120572119887119871
119894
and 120572119887119877
119894
Definition 30 Consider a technological coefficient of fuzzylinear programming problem defined as PnIT2FS
119894119895
definedon the closed interval
119894119895
isin [inf119886
119894119895
120572
120583
119894119895
(119886119894119895
119906) sup119886
119894119895
120572
120583
119894119895
(119886119894119895
119906)]
= [[120572
119886119871
119894119895
120572
119886119871
119894119895
] [120572
119886119877
119894119895
120572
119886119877
119894119895
]] isin R 119894 isin N119899
119895 isin N119898
(32)
The membership function which represents the fuzzy spaceSupp(
119894119895
) is
119894119895
= int
119894119895isinR
[int119906isin119869
119894119895
1119906]
119894119895
119894 isin N119899
119895 isin N119898
119869
119894119895
sube [0 1]
(33)
Here 119894119895
is bounded by both lower and upper primarymembership functions namely
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (34)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
and
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (35)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
Proposition 31 Let 119894119895
isin 119865(R) 119909119895
ge 0 119894 isin N119899
119895 isin N119898
Then the fuzzy setsum119899
119895=1
119894119895
119909119895
defined by the extension principleis again a fuzzy number
Let = ⪯Pos be a fuzzy relation [27] fuzzy extension of the
usual binary relation le on R The fuzzy linear programmingproblem associated with a standard linear programmingproblem is denoted as
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St (
119899
sum
119895=1
119894119895
119909119895
) 119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(36)
In (36) the value sum119899119895=1
119894119895
119909119895
isin 119865(R) is compared with a fuzzynumber 119887
119894
isin 119865(R) by sum fuzzy relation The maximizationof the objective function is denoted by Max119885 = sum
119899
119895=1
119888119895
119909119895
Definition 32 Let 120583
119894119895
R rarr [0 1] and 120583
119887
119894
R rarr [0 1]119894 isin N
119899
119895 isin N119898
be membership function of fuzzy numbers119894119895
and 119887119894
respectively Let be a fuzzy extension of a binaryrelation le on R A fuzzy set whose membership function120583
119883
is defined as 119909 isin R119899 by
120583
119883
=
min 120583
119875
(11
1199091
+ sdot sdot sdot + 1119899
119909119899
1198871
) 120583
119875
(1198981
1199091
+ sdot sdot sdot + 119898119899
119909119899
119887119898
) if 119909119895
ge 0 119895 = 1 2 119899
0 otherwise(37)
8 Journal of Applied Mathematics
is called the fuzzy set of feasible regions of the fuzzy linearprogramming problem (36) For 120572 isin (0 1] a vector 119909 isin
[]120572
is called the 120572-feasible solution of the fuzzy linearprogramming problem
Notice that the feasible region of the fuzzy linearprogramming problem is a fuzzy set On the other hand 120572-feasible solution is a vector belonging to the 120572-cut of the fea-sible region If all the coefficients
119894119895
and 119887119894
are crisp fuzzynumber that is they are equivalent to the corresponding crispfuzzy number then the fuzzy feasible region is equivalent tothe set of all feasible solutions of the corresponding classicallinear programming
61 Type 2 Fuzzy Linear Programming Problem Let us con-sider the following fuzzy linear programming problem withright-hand side (resources) and technology coefficients arePNIT2TrFNs
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119894119895
119909119895
le119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(38)
where 119894119895
= ((119894119895
)119871
(119894119895
)119880
) = (((119886119894119895
)119871
2
(119886119894119895
)119871
3
(120572119894119895
)119871
(120573119894119895
)119871
)((119886119894119895
)119880
2
(119886119894119895
)119880
3
(120572119894119895
)119880
(120573119894119895
)119880
)) 119887119894
= ((119887119894
)119871
(119887119894
)119880
) = (((119887119894
)119871
2
(119887119894
)119871
3
(120574119894
)119871
(120579119894
)119871
) ((119887119894
)119880
2
(119887119894
)119880
3
(120574119894
)119880
(120579119894
)119880
)) are PnIT2TrFNs119888119894
are crisp coefficients of the objective and 119909119894
are the decisionvariable
From Definitions 15 21 22 and 23 and (38)
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119871
119894119895
119909119895
le119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119880
119894119895
119909119895
le119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(39)
Based on the relation of possibility we can rewrite (39) asfollows
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119871
1198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯Pos
119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119880
1198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯Pos
119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(40)
With respect to constraints of linear programming possibilitythe optimal solution is completely determined by degrees ofpossibility So we can obtain the optimal solution of linearprogramming possibility at 119901-cut levels by solving the twocrisp linear programming problems using Barnes algorithm
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198711198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯119887
119871
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198801198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯119887
119880
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(41)
FromTheorems 27 and 28 we have
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(42)
where 119860119871 = (119886119894119895
)119871
2
119860119871 = (119886119894119895
)119880
2
119887119880 = (119887119894
)119871
3
119887119880
= (119887119894
)119880
3
120572 = (120572
119894119895
)119871
120572 = (120572119894119895
)119880
120573 = (120573119894119895
)119871
120573 = (120573119894119895
)119880
120574 =
(120574119894
)119871
120574 = (120574119894
)119880
120579 = (120579119894
)119871
120579 = (120579119894
)119880
and 119909 = 119909119895
119895 = 1 2 119899 119894 = 1 2 119898
Journal of Applied Mathematics 9
The feasible regions of (119871119875119875119901
) and (119871119875119875119901
) are denoted by
119883119875
119901
= 119909 isin 119877119899
| 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119883
119875
119901
= 119909 isin 119877119899
| 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
(43)
Assume the optimal solutions of (119871119875119875119901
) and (119871119875119875119901
) are 119885119875119901
and 119885
119875
119901
respectively
Theorem 33 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883119875
119901
1
119883
119875
119901
2
sube
119883
119875
119901
1
and 119885119875119901
1
ge 119885119875
119901
2
119885
119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) ge
(1199011
minus1)(120572119909+120579) so 119909 isin 119883119875
119901
1
that is119883119875119901
2
sube 119883119875
119901
1
and119885119875119901
1
ge 119885119875
119901
2
In a similar way119883119875119901
2
sube 119883
119875
119901
1
and 119885
119875
119901
1
ge 119885
119875
119901
2
Theorem 34 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883
119875
119901
1
and
119885119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 +
120579) ge (1199011
minus 1)(120572119909 + 120579) so 119909 isin 119883
119875
119901
1
that is 119883119875119901
2
sube 119883
119875
119901
1
and
119885
119875
119901
1
ge 119885119875
119901
2
FromTheorems 33 and 34 we obtain that 119885119875119901
119885
119875
119901
consti-tute the lower and upper bound of fuzzy objective value atlevel 119901 Then the objective value of 119885119875
119901
at level 119901 is the mean
of [119885119875119901
119885
119875
119901
]
7 Numerical Illustration
An application of the proposed method is introduced withan example where all its technological coefficients and right-hand side (resources) are defined as PnIT2FNs The optimalsolution will be obtained in terms of crisp for the followingproblem
The company plans tomanufacture two types of productsThe selling prices for these products are as follows 119875
1
costs Rs 1750 per unit and 1198752
costs Rs 2000 per unitDaily production volume of each type of these products isconstrained by available man hours and available machinehours The production specifications for the given problemsituation are presented in as shown in Table 1
Max 119885 = 1198881
1199091
+ 1198882
1199092
St 11
1199091
+ 12
1199092
⪯1198871
Table 1
Resource requirementunitResource 119875
1
1198752
AvailabilityMan hours (in minutes) Around 4 Around 8 Around 32Machine hours (in minutes) Around 6 Around 4 Around 36
21
1199091
+ 22
1199092
⪯1198872
119909119895
ge 0 119895 = 1 2
(44)
where 11
= [(230 250 20 10) (230 250 30 20)] 12
=
[(460 500 40 20) (460 500 60 40)] 21
= [(345 375 30
15) (345 375 45 30)] 22
= [(230 250 20 10) (230 250
30 20)] 1198871
= [(1840 2000 160 80) (1840 2000 240 160)]1198872
= [(2070 2250 180 90) (2070 2250 270 180)] 1198881
=
1750 and 1198882
= 2000 From (42) we have the following crispoptimal problem based on the level 119901
Max 119885119875
119901
= 17501199091
+ 20001199092
St (210 + 20119901) 1199091
+ (420 + 40119901) 1199092
le 2080 minus 80119901
(315 + 30119901) 1199091
+ (210 + 20119901) 1199092
le 2340 minus 90119901
1199091
1199092
ge 0
(45)
Max 119885
119875
119901
= 17501199091
+ 20001199092
St (200 + 30119901) 1199091
+ (400 + 60119901) 1199092
le 2160 minus 160119901
(300 + 45119901) 1199091
+ (200 + 30119901) 1199092
le 2430 minus 180119901
1199091
1199092
ge 0
(46)
For different cut level 119901 we can get different optimal solutionand denote by 119885119875
119901
119885119875
119901
the optimal solution (see Figure 5 andTable 2) [28] of the crisp programming (45) and program-ming (46) respectively
8 Conclusion
In this paper a method to do perfectly normal interval type2 fuzzy arithmetic operation using extension principle is pre-sented and defined the possibility degrees of upper and lowermembership function to compare perfectly normal intervaltype 2 fuzzy numbers These functions are defined based onthe strength of upper and lower membership function ofperfectly normal interval type 2 fuzzy numbers Meanwhilesome properties and theorems were proved Then the lowerand upper fuzzy satisfaction based on possibility degrees of
10 Journal of Applied Mathematics
Table 2 Optimal solution at different possibility levels
119901-cut 01 02 03 04 05 06 07 08 09 1119885119875
119901
143549528 141658878 13980324 137981651 136193181 134436937 132712053 131017699 129353069 127717393
119885
119875
119901
155123153 151723301 148421051 145212264 142093022 139059633 136108597 133236607 130440529 127717393119885119875
119901
149336341 146691090 144112146 141596958 139143102 136748285 134410325 132127153 129896799 127717393
1090807060504030201012500 13000 13500 14000 14500 15000
p-c
ut
Z
Figure 5 Optimal solution at different possibility levels
membership function respectively are defined These fuzzysatisfactions create two nonstrict order relations on the setof overlapping intervals Also linear programming problemwas introduced with resources and technology coefficientsof perfectly normal interval type 2 fuzzy numbers Thenthe possibility degrees of membership function were appliedin order to interpret inequality constraints with intervalcoefficients According to the definition of possibility degreesof membership function and their properties the inequalityconstraints with interval coefficients were reduced in theirsatisfactory crisp equivalent forms Finally the decisionmaker can get the crisp optimal solution of the problem forevery grade 120572 isin [0 1] which was obtained by using Barnesalgorithm and then mean value is selected according to thedecision maker optimistic attitude
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] H Tanaka T Okuda and K Asai ldquoOn fuzzy-mathematicalprogrammingrdquo Journal of Cybernetics vol 3 no 4 pp 37ndash461973
[3] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 pp B141ndashB164197071
[4] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978
[5] S Deng L Zhou and X Wang ldquoSolving the fuzzy bilevellinear programmingwithmultiple followers through structured
element methodrdquo Mathematical Problems in Engineering vol2014 Article ID 418594 6 pages 2014
[6] Y-H Zhong Y-L Jia D Chen and Y Yang ldquoInterior pointmethod for solving fuzzy number linear programming prob-lems using linear ranking functionrdquo Journal of Applied Math-ematics vol 2013 Article ID 795098 9 pages 2013
[7] H Cheng W Huang and J Cai ldquoSolving a fully fuzzy linearprogramming problem through compromise programmingrdquoJournal of Applied Mathematics vol 2013 Article ID 726296 10pages 2013
[8] D Chakraborty D K Jana and T K Royl ldquoA new approach tosolve intuitionistic fuzzy optimization problem using possibil-ity necessity and credibility measuresrdquo International Journal ofEngineering Mathematics vol 2014 Article ID 593185 12 pages2014
[9] H Le and Z Gogne ldquoFuzzy linear programming with possibil-ity and necessity relationrdquo in Fuzzy Information and Engineering2010 vol 78 of Advances in Intelligent and Soft Computing pp305ndash311 Springer Berlin Germany 2010
[10] J C Figueroa-Garcıa and G Hernandez ldquoA method for solvinglinear programming models with interval type-2 fuzzy con-straintsrdquo Pesquisa Operacional vol 34 no 1 pp 73ndash89 2014
[11] J C F Garcıa ldquoA general model for linear programming withinterval type-2 fuzzy technological coefficientsrdquo in Proceedingsof the IEEE Annual Meeting of the North American FuzzyInformation Processing Society (NAFIPS rsquo12) pp 1ndash4 BerkeleyCalif USA August 2012
[12] J C F Garcıa ldquoLinear programming with interval type-2fuzzy right hand side parametersrdquo in Proceedings of the AnnualMeeting of the North American Fuzzy Information ProcessingSociety (NAFIPS rsquo08) pp 1ndash6 IEEE New York NY USA May2008
[13] S-M Chen and L-W Lee ldquoFuzzy multiple attributes groupdecision-making based on the ranking values and the arith-metic operations of interval type-2 fuzzy setsrdquo Expert Systemswith Applications vol 37 no 1 pp 824ndash833 2010
[14] J Hu Y Zhang X Chen and Y Liu ldquoMulti-criteria decisionmaking method based on possibility degree of interval type-2 fuzzy numberrdquo Knowledge-Based Systems vol 43 pp 21ndash292013
[15] R Fuller Fuzzy Reasoning and Fuzzy Optimization TurkuCentre for Computer Science 1998
[16] H-C Wu ldquoDuality theory in fuzzy linear programming prob-lems with fuzzy coefficientsrdquo Fuzzy Optimization and DecisionMaking vol 2 no 1 pp 61ndash73 2003
[17] J Qin and X Liu ldquoFrank aggregation operators for triangularinterval type-2 fuzzy set and its application inmultiple attributegroup decision makingrdquo Journal of Applied Mathematics vol2014 Article ID 923213 24 pages 2014
[18] Z M Zhang and S H Zhang ldquoA novel approach to multiattribute group decision making based on trapezoidal intervaltype-2 fuzzy soft setsrdquo Applied Mathematical Modelling vol 37no 7 pp 4948ndash4971 2013
Journal of Applied Mathematics 11
[19] Y Gong ldquoFuzzymulti-attribute group decisionmakingmethodbased on interval type-2 fuzzy sets and applications to globalsupplier selectionrdquo International Journal of Fuzzy Systems vol15 no 4 pp 392ndash400 2013
[20] D S Dinagar and A Anbalagan ldquoA new type-2 fuzzy numberarithmetic using extension principlerdquo in Proceedings of the 1stInternational Conference on Advances in Engineering Scienceand Management (ICAESM rsquo12) pp 113ndash118 March 2012
[21] K-P Chiao ldquoMultiple criteria group decision making withtriangular interval type-2 fuzzy setsrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems (FUZZ rsquo11) pp 2575ndash2582 IEEE Taipei Taiwan June 2011
[22] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006
[23] X Liu ldquoMeasuring the satisfaction of constraints in fuzzy linearprogrammingrdquo Fuzzy Sets and Systems vol 122 no 2 pp 263ndash275 2001
[24] D Dubois and H Prade ldquoRanking fuzzy numbers in the settingof possibility theoryrdquo Information Sciences vol 30 no 3 pp183ndash224 1983
[25] D S Negi and E S Lee ldquoPossibility programming by thecomparison of fuzzy numbersrdquo Computers ampMathematics withApplications vol 25 no 9 pp 43ndash50 1993
[26] MG Iskander ldquoComparison of fuzzy numbers using possibilityprogramming comments and new conceptsrdquo Computers ampMathematics with Applications vol 43 no 6-7 pp 833ndash8402002
[27] J Ramik ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006
[28] G R Lindfield and J E T Penny Numerical methods UsingMatlab Academic Press 3rd edition 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
Pos (119860119880 le 119861119880
) = max(1
minusmax(max (120572119886119871 minus 120572119887
119871
0) +max ((1198861198802
minus 119887119880
2
) 0) +max ((1198861198803
minus 119887119880
3
) 0) +max (120572119886119877 minus 120572119887119877
0) + (120572
119886119877
minus120572
119887
119871
)
100381610038161003816100381610038161003816
120572
119886119871
minus120572
119887
119871100381610038161003816100381610038161003816
+1003816100381610038161003816119886119880
2
minus 119887119880
2
1003816100381610038161003816+1003816100381610038161003816119886119880
3
minus 119887119880
3
1003816100381610038161003816+
100381610038161003816100381610038161003816
120572
119886119877
minus120572
119887
119877100381610038161003816100381610038161003816
+ (120572
119887
119877
minus120572
119887
119871
) + (120572
119886119877
minus120572
119886119871
)
0)
0)
(21)
Proposition 25 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
))
and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2FNsPos(119860119871 ⪯ 119861
119871
) ge 120572 if and only if inf[119860119871]120572
le sup[119861119871]120572
that is
Pos (119860119871 ⪯ 119861119871
) ge 120572 iff 120572119886119871 le 120572119887119877 (22)
Alternatively
Pos (119860119871 ⪯ 119861119871
) = 120583Pos(119860119871119861119871) = (119860119871
⪯Pos
119861
119871
) (23)
An interpretation of the 120572minus relation associated with possibilitywhen comparing fuzzy numbers 119860119871 and 119861119871 is as follows For agiven level of satisfaction 120572 isin [0 1] a fuzzy number 119860119871 is notbetter than 119861119871 with respect to fuzzy relation ⪯Pos if the smallestvalue of119860119871 with the degree of satisfaction (or possibility degree)being greater than or equal to 120572minus is less than or equal to thelargest value of 119861119871 with the degree of satisfaction greater thanor equal to 120572 Similar to the above
Pos (119860119880 ⪯ 119861119880
) ge 120572 iff 120572119886119871 le 120572119887119877
(24)
Proposition 26 (see [15 23 24]) Let = ((119886119871
2
119886119871
3
120572119871
120573119871
)(119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) betwo PnIT2TrFNs then
Pos (119860119871 ⪯ 119861119871
)
=
1 119887119871
3
ge 119886119871
2
119887119871
3
minus 119886119871
2
+ 120579119871
+ 120572119871
120579119871
+ 120572119871
0 le 119886119871
2
minus 119887119871
3
le 120579119871
+ 120572119871
0 119886119871
2
minus 119887119871
3
gt 120579119871
+ 120572119871
(25)
see Figure 3
Pos (119860119880 ⪯ 119861119880
)
=
1 119887119880
3
ge 119886119880
2
119887119880
3
minus 119886119880
2
+ 120579119880
+ 120572119906
120579119880
+ 120572119880
0 le 119886119880
2
minus 119887119880
3
le 120579119880
+ 120572119880
0 119886119880
2
minus 119887119880
3
gt 120579119880
+ 120572119880
(26)
see Figure 4
Theorem 27 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNs 119901 isin
(0 1] Pos(119871 ⪯
119871
) ge 119901 if and only if 1198871198713
minus 119886119871
2
ge (119901 minus 1)(120579119871
+
120572119871
)
Proof If 119901 = 1 then from Pos(119871 ⪯
119871
) ge 1 one can getthat 1198871198713
ge 119886119871
2
and vice versa If 0 lt 119901 lt 1 then 119887119871
3
lt 119886119871
2
and119887119871
3
+ 120579119871
gt 119886119871
2
minus 120572119871
Pos(119871 ⪯
119871
) ge 119901 if and only if (1198871198713
minus 119886119871
2
+
120579119871
+120572119871
)(120579119871
+120572119871
) ge 119901 that is 1198871198713
minus119886119871
2
ge (119901minus1)(120579119871
+120572119871
)
Theorem 28 Let = ((119886119871
2
119886119871
3
120572119871
120573119871
) (119886119880
2
119886119880
3
120572119880
120573119880
)) and = ((119887
119871
2
119887119871
3
120574119871
120579119871
) (119887119880
2
119887119880
3
120574119880
120579119880
)) be two PnIT2TrFNs 119901 isin
(0 1] Pos(119880 ⪯
119880
) ge 119901 if and only if 1198871198803
minus119886119880
2
ge (119901minus1)(120579119880
+
120572119880
)
Proof If 119901 = 1 then from Pos(119880 ⪯
119880
) ge 1 one can getthat 119887119880
3
ge 119886119880
2
and vice versa If 0 lt 119901 lt 1 then 119887119880
3
lt 119886119880
2
and 119887119880
3
+ 120579119880
gt 119886119880
2
minus 120572119880
Pos(119880 ⪯
119880
) ge 119901 if and onlyif (1198871198803
minus 119886119880
2
+ 120579119880
+ 120572119880
)(120579119880
+ 120572119880
) ge 119901 that is 1198871198803
minus 119886119880
2
ge
(119901 minus 1)(120579119880
+ 120572119880
)
6 Fuzzy Linear Programming
In this section we propose a fuzzy linear programmingmodel with the technological coefficients and right-hand side(resources) being PnIT2FN
MaxMin119909isin119883
119885 = 119888119909
St 119860119909 ⪯ 119887 119909 ge 0
(27)
where 119860 = (119886119894119895
)119898times119899
119887 = (1198871
1198872
119887119898
)119879 119888 = (119888
1
1198882
119888119899
)119909 = (119909
1
1199092
119909119899
)119879 and 119886
119894119895
119887119894
are PnIT2FN and 119909119895
119888119895
isin
R (119894 = 1 2 119898 119895 = 1 2 119899) ⪯ is type 2 fuzzy order
Definition 29 Consider a set of right-hand side (resources)parameters of a fuzzy linear programming problem definedas PnIT2FS 119887 defined on the closed interval
119887119894
isin [[120572
119887
119871
119894
120572
119887119871
119894
] [120572
119887119877
119894
120572
119887
119877
119894
]] isin R (28)
Journal of Applied Mathematics 7
10908070605040302010
Mem
bers
hip
func
tion
bL3 aL2
Pos[ALle B
L]
Figure 3 Pos[119871 le
119871
] lt 1 because 119871 gt
119871
10908070605040302010
Mem
bers
hip
func
tion
bU3 aU2
Pos[AUle B
U]
Figure 4 Pos[119880 le
119880
] lt 1 because 119880 gt
119880
and 119894 isin N119899
The membership function which represents thefuzzy space Supp(119887
119894
) is
119887119894
= int
119887
119894isinR
[int119906isin119869
119887119894
1119906]
119887119894
119894 isin N119899
119869119887
119894
sube [0 1] (29)
Here 119887 is bounded by both lower and upper primary mem-bership function namely
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (30)
with parameters 120572119887119871119894
and 120572119887119877119894
and
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (31)
with parameters 120572119887119871
119894
and 120572119887119877
119894
Definition 30 Consider a technological coefficient of fuzzylinear programming problem defined as PnIT2FS
119894119895
definedon the closed interval
119894119895
isin [inf119886
119894119895
120572
120583
119894119895
(119886119894119895
119906) sup119886
119894119895
120572
120583
119894119895
(119886119894119895
119906)]
= [[120572
119886119871
119894119895
120572
119886119871
119894119895
] [120572
119886119877
119894119895
120572
119886119877
119894119895
]] isin R 119894 isin N119899
119895 isin N119898
(32)
The membership function which represents the fuzzy spaceSupp(
119894119895
) is
119894119895
= int
119894119895isinR
[int119906isin119869
119894119895
1119906]
119894119895
119894 isin N119899
119895 isin N119898
119869
119894119895
sube [0 1]
(33)
Here 119894119895
is bounded by both lower and upper primarymembership functions namely
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (34)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
and
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (35)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
Proposition 31 Let 119894119895
isin 119865(R) 119909119895
ge 0 119894 isin N119899
119895 isin N119898
Then the fuzzy setsum119899
119895=1
119894119895
119909119895
defined by the extension principleis again a fuzzy number
Let = ⪯Pos be a fuzzy relation [27] fuzzy extension of the
usual binary relation le on R The fuzzy linear programmingproblem associated with a standard linear programmingproblem is denoted as
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St (
119899
sum
119895=1
119894119895
119909119895
) 119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(36)
In (36) the value sum119899119895=1
119894119895
119909119895
isin 119865(R) is compared with a fuzzynumber 119887
119894
isin 119865(R) by sum fuzzy relation The maximizationof the objective function is denoted by Max119885 = sum
119899
119895=1
119888119895
119909119895
Definition 32 Let 120583
119894119895
R rarr [0 1] and 120583
119887
119894
R rarr [0 1]119894 isin N
119899
119895 isin N119898
be membership function of fuzzy numbers119894119895
and 119887119894
respectively Let be a fuzzy extension of a binaryrelation le on R A fuzzy set whose membership function120583
119883
is defined as 119909 isin R119899 by
120583
119883
=
min 120583
119875
(11
1199091
+ sdot sdot sdot + 1119899
119909119899
1198871
) 120583
119875
(1198981
1199091
+ sdot sdot sdot + 119898119899
119909119899
119887119898
) if 119909119895
ge 0 119895 = 1 2 119899
0 otherwise(37)
8 Journal of Applied Mathematics
is called the fuzzy set of feasible regions of the fuzzy linearprogramming problem (36) For 120572 isin (0 1] a vector 119909 isin
[]120572
is called the 120572-feasible solution of the fuzzy linearprogramming problem
Notice that the feasible region of the fuzzy linearprogramming problem is a fuzzy set On the other hand 120572-feasible solution is a vector belonging to the 120572-cut of the fea-sible region If all the coefficients
119894119895
and 119887119894
are crisp fuzzynumber that is they are equivalent to the corresponding crispfuzzy number then the fuzzy feasible region is equivalent tothe set of all feasible solutions of the corresponding classicallinear programming
61 Type 2 Fuzzy Linear Programming Problem Let us con-sider the following fuzzy linear programming problem withright-hand side (resources) and technology coefficients arePNIT2TrFNs
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119894119895
119909119895
le119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(38)
where 119894119895
= ((119894119895
)119871
(119894119895
)119880
) = (((119886119894119895
)119871
2
(119886119894119895
)119871
3
(120572119894119895
)119871
(120573119894119895
)119871
)((119886119894119895
)119880
2
(119886119894119895
)119880
3
(120572119894119895
)119880
(120573119894119895
)119880
)) 119887119894
= ((119887119894
)119871
(119887119894
)119880
) = (((119887119894
)119871
2
(119887119894
)119871
3
(120574119894
)119871
(120579119894
)119871
) ((119887119894
)119880
2
(119887119894
)119880
3
(120574119894
)119880
(120579119894
)119880
)) are PnIT2TrFNs119888119894
are crisp coefficients of the objective and 119909119894
are the decisionvariable
From Definitions 15 21 22 and 23 and (38)
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119871
119894119895
119909119895
le119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119880
119894119895
119909119895
le119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(39)
Based on the relation of possibility we can rewrite (39) asfollows
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119871
1198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯Pos
119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119880
1198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯Pos
119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(40)
With respect to constraints of linear programming possibilitythe optimal solution is completely determined by degrees ofpossibility So we can obtain the optimal solution of linearprogramming possibility at 119901-cut levels by solving the twocrisp linear programming problems using Barnes algorithm
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198711198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯119887
119871
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198801198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯119887
119880
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(41)
FromTheorems 27 and 28 we have
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(42)
where 119860119871 = (119886119894119895
)119871
2
119860119871 = (119886119894119895
)119880
2
119887119880 = (119887119894
)119871
3
119887119880
= (119887119894
)119880
3
120572 = (120572
119894119895
)119871
120572 = (120572119894119895
)119880
120573 = (120573119894119895
)119871
120573 = (120573119894119895
)119880
120574 =
(120574119894
)119871
120574 = (120574119894
)119880
120579 = (120579119894
)119871
120579 = (120579119894
)119880
and 119909 = 119909119895
119895 = 1 2 119899 119894 = 1 2 119898
Journal of Applied Mathematics 9
The feasible regions of (119871119875119875119901
) and (119871119875119875119901
) are denoted by
119883119875
119901
= 119909 isin 119877119899
| 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119883
119875
119901
= 119909 isin 119877119899
| 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
(43)
Assume the optimal solutions of (119871119875119875119901
) and (119871119875119875119901
) are 119885119875119901
and 119885
119875
119901
respectively
Theorem 33 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883119875
119901
1
119883
119875
119901
2
sube
119883
119875
119901
1
and 119885119875119901
1
ge 119885119875
119901
2
119885
119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) ge
(1199011
minus1)(120572119909+120579) so 119909 isin 119883119875
119901
1
that is119883119875119901
2
sube 119883119875
119901
1
and119885119875119901
1
ge 119885119875
119901
2
In a similar way119883119875119901
2
sube 119883
119875
119901
1
and 119885
119875
119901
1
ge 119885
119875
119901
2
Theorem 34 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883
119875
119901
1
and
119885119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 +
120579) ge (1199011
minus 1)(120572119909 + 120579) so 119909 isin 119883
119875
119901
1
that is 119883119875119901
2
sube 119883
119875
119901
1
and
119885
119875
119901
1
ge 119885119875
119901
2
FromTheorems 33 and 34 we obtain that 119885119875119901
119885
119875
119901
consti-tute the lower and upper bound of fuzzy objective value atlevel 119901 Then the objective value of 119885119875
119901
at level 119901 is the mean
of [119885119875119901
119885
119875
119901
]
7 Numerical Illustration
An application of the proposed method is introduced withan example where all its technological coefficients and right-hand side (resources) are defined as PnIT2FNs The optimalsolution will be obtained in terms of crisp for the followingproblem
The company plans tomanufacture two types of productsThe selling prices for these products are as follows 119875
1
costs Rs 1750 per unit and 1198752
costs Rs 2000 per unitDaily production volume of each type of these products isconstrained by available man hours and available machinehours The production specifications for the given problemsituation are presented in as shown in Table 1
Max 119885 = 1198881
1199091
+ 1198882
1199092
St 11
1199091
+ 12
1199092
⪯1198871
Table 1
Resource requirementunitResource 119875
1
1198752
AvailabilityMan hours (in minutes) Around 4 Around 8 Around 32Machine hours (in minutes) Around 6 Around 4 Around 36
21
1199091
+ 22
1199092
⪯1198872
119909119895
ge 0 119895 = 1 2
(44)
where 11
= [(230 250 20 10) (230 250 30 20)] 12
=
[(460 500 40 20) (460 500 60 40)] 21
= [(345 375 30
15) (345 375 45 30)] 22
= [(230 250 20 10) (230 250
30 20)] 1198871
= [(1840 2000 160 80) (1840 2000 240 160)]1198872
= [(2070 2250 180 90) (2070 2250 270 180)] 1198881
=
1750 and 1198882
= 2000 From (42) we have the following crispoptimal problem based on the level 119901
Max 119885119875
119901
= 17501199091
+ 20001199092
St (210 + 20119901) 1199091
+ (420 + 40119901) 1199092
le 2080 minus 80119901
(315 + 30119901) 1199091
+ (210 + 20119901) 1199092
le 2340 minus 90119901
1199091
1199092
ge 0
(45)
Max 119885
119875
119901
= 17501199091
+ 20001199092
St (200 + 30119901) 1199091
+ (400 + 60119901) 1199092
le 2160 minus 160119901
(300 + 45119901) 1199091
+ (200 + 30119901) 1199092
le 2430 minus 180119901
1199091
1199092
ge 0
(46)
For different cut level 119901 we can get different optimal solutionand denote by 119885119875
119901
119885119875
119901
the optimal solution (see Figure 5 andTable 2) [28] of the crisp programming (45) and program-ming (46) respectively
8 Conclusion
In this paper a method to do perfectly normal interval type2 fuzzy arithmetic operation using extension principle is pre-sented and defined the possibility degrees of upper and lowermembership function to compare perfectly normal intervaltype 2 fuzzy numbers These functions are defined based onthe strength of upper and lower membership function ofperfectly normal interval type 2 fuzzy numbers Meanwhilesome properties and theorems were proved Then the lowerand upper fuzzy satisfaction based on possibility degrees of
10 Journal of Applied Mathematics
Table 2 Optimal solution at different possibility levels
119901-cut 01 02 03 04 05 06 07 08 09 1119885119875
119901
143549528 141658878 13980324 137981651 136193181 134436937 132712053 131017699 129353069 127717393
119885
119875
119901
155123153 151723301 148421051 145212264 142093022 139059633 136108597 133236607 130440529 127717393119885119875
119901
149336341 146691090 144112146 141596958 139143102 136748285 134410325 132127153 129896799 127717393
1090807060504030201012500 13000 13500 14000 14500 15000
p-c
ut
Z
Figure 5 Optimal solution at different possibility levels
membership function respectively are defined These fuzzysatisfactions create two nonstrict order relations on the setof overlapping intervals Also linear programming problemwas introduced with resources and technology coefficientsof perfectly normal interval type 2 fuzzy numbers Thenthe possibility degrees of membership function were appliedin order to interpret inequality constraints with intervalcoefficients According to the definition of possibility degreesof membership function and their properties the inequalityconstraints with interval coefficients were reduced in theirsatisfactory crisp equivalent forms Finally the decisionmaker can get the crisp optimal solution of the problem forevery grade 120572 isin [0 1] which was obtained by using Barnesalgorithm and then mean value is selected according to thedecision maker optimistic attitude
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] H Tanaka T Okuda and K Asai ldquoOn fuzzy-mathematicalprogrammingrdquo Journal of Cybernetics vol 3 no 4 pp 37ndash461973
[3] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 pp B141ndashB164197071
[4] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978
[5] S Deng L Zhou and X Wang ldquoSolving the fuzzy bilevellinear programmingwithmultiple followers through structured
element methodrdquo Mathematical Problems in Engineering vol2014 Article ID 418594 6 pages 2014
[6] Y-H Zhong Y-L Jia D Chen and Y Yang ldquoInterior pointmethod for solving fuzzy number linear programming prob-lems using linear ranking functionrdquo Journal of Applied Math-ematics vol 2013 Article ID 795098 9 pages 2013
[7] H Cheng W Huang and J Cai ldquoSolving a fully fuzzy linearprogramming problem through compromise programmingrdquoJournal of Applied Mathematics vol 2013 Article ID 726296 10pages 2013
[8] D Chakraborty D K Jana and T K Royl ldquoA new approach tosolve intuitionistic fuzzy optimization problem using possibil-ity necessity and credibility measuresrdquo International Journal ofEngineering Mathematics vol 2014 Article ID 593185 12 pages2014
[9] H Le and Z Gogne ldquoFuzzy linear programming with possibil-ity and necessity relationrdquo in Fuzzy Information and Engineering2010 vol 78 of Advances in Intelligent and Soft Computing pp305ndash311 Springer Berlin Germany 2010
[10] J C Figueroa-Garcıa and G Hernandez ldquoA method for solvinglinear programming models with interval type-2 fuzzy con-straintsrdquo Pesquisa Operacional vol 34 no 1 pp 73ndash89 2014
[11] J C F Garcıa ldquoA general model for linear programming withinterval type-2 fuzzy technological coefficientsrdquo in Proceedingsof the IEEE Annual Meeting of the North American FuzzyInformation Processing Society (NAFIPS rsquo12) pp 1ndash4 BerkeleyCalif USA August 2012
[12] J C F Garcıa ldquoLinear programming with interval type-2fuzzy right hand side parametersrdquo in Proceedings of the AnnualMeeting of the North American Fuzzy Information ProcessingSociety (NAFIPS rsquo08) pp 1ndash6 IEEE New York NY USA May2008
[13] S-M Chen and L-W Lee ldquoFuzzy multiple attributes groupdecision-making based on the ranking values and the arith-metic operations of interval type-2 fuzzy setsrdquo Expert Systemswith Applications vol 37 no 1 pp 824ndash833 2010
[14] J Hu Y Zhang X Chen and Y Liu ldquoMulti-criteria decisionmaking method based on possibility degree of interval type-2 fuzzy numberrdquo Knowledge-Based Systems vol 43 pp 21ndash292013
[15] R Fuller Fuzzy Reasoning and Fuzzy Optimization TurkuCentre for Computer Science 1998
[16] H-C Wu ldquoDuality theory in fuzzy linear programming prob-lems with fuzzy coefficientsrdquo Fuzzy Optimization and DecisionMaking vol 2 no 1 pp 61ndash73 2003
[17] J Qin and X Liu ldquoFrank aggregation operators for triangularinterval type-2 fuzzy set and its application inmultiple attributegroup decision makingrdquo Journal of Applied Mathematics vol2014 Article ID 923213 24 pages 2014
[18] Z M Zhang and S H Zhang ldquoA novel approach to multiattribute group decision making based on trapezoidal intervaltype-2 fuzzy soft setsrdquo Applied Mathematical Modelling vol 37no 7 pp 4948ndash4971 2013
Journal of Applied Mathematics 11
[19] Y Gong ldquoFuzzymulti-attribute group decisionmakingmethodbased on interval type-2 fuzzy sets and applications to globalsupplier selectionrdquo International Journal of Fuzzy Systems vol15 no 4 pp 392ndash400 2013
[20] D S Dinagar and A Anbalagan ldquoA new type-2 fuzzy numberarithmetic using extension principlerdquo in Proceedings of the 1stInternational Conference on Advances in Engineering Scienceand Management (ICAESM rsquo12) pp 113ndash118 March 2012
[21] K-P Chiao ldquoMultiple criteria group decision making withtriangular interval type-2 fuzzy setsrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems (FUZZ rsquo11) pp 2575ndash2582 IEEE Taipei Taiwan June 2011
[22] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006
[23] X Liu ldquoMeasuring the satisfaction of constraints in fuzzy linearprogrammingrdquo Fuzzy Sets and Systems vol 122 no 2 pp 263ndash275 2001
[24] D Dubois and H Prade ldquoRanking fuzzy numbers in the settingof possibility theoryrdquo Information Sciences vol 30 no 3 pp183ndash224 1983
[25] D S Negi and E S Lee ldquoPossibility programming by thecomparison of fuzzy numbersrdquo Computers ampMathematics withApplications vol 25 no 9 pp 43ndash50 1993
[26] MG Iskander ldquoComparison of fuzzy numbers using possibilityprogramming comments and new conceptsrdquo Computers ampMathematics with Applications vol 43 no 6-7 pp 833ndash8402002
[27] J Ramik ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006
[28] G R Lindfield and J E T Penny Numerical methods UsingMatlab Academic Press 3rd edition 2013
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
Journal of Applied Mathematics 7
10908070605040302010
Mem
bers
hip
func
tion
bL3 aL2
Pos[ALle B
L]
Figure 3 Pos[119871 le
119871
] lt 1 because 119871 gt
119871
10908070605040302010
Mem
bers
hip
func
tion
bU3 aU2
Pos[AUle B
U]
Figure 4 Pos[119880 le
119880
] lt 1 because 119880 gt
119880
and 119894 isin N119899
The membership function which represents thefuzzy space Supp(119887
119894
) is
119887119894
= int
119887
119894isinR
[int119906isin119869
119887119894
1119906]
119887119894
119894 isin N119899
119869119887
119894
sube [0 1] (29)
Here 119887 is bounded by both lower and upper primary mem-bership function namely
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (30)
with parameters 120572119887119871119894
and 120572119887119877119894
and
120572
120583
119887
119894
= (119887119894
119906) | 120583
119887
119894
ge 120572 (31)
with parameters 120572119887119871
119894
and 120572119887119877
119894
Definition 30 Consider a technological coefficient of fuzzylinear programming problem defined as PnIT2FS
119894119895
definedon the closed interval
119894119895
isin [inf119886
119894119895
120572
120583
119894119895
(119886119894119895
119906) sup119886
119894119895
120572
120583
119894119895
(119886119894119895
119906)]
= [[120572
119886119871
119894119895
120572
119886119871
119894119895
] [120572
119886119877
119894119895
120572
119886119877
119894119895
]] isin R 119894 isin N119899
119895 isin N119898
(32)
The membership function which represents the fuzzy spaceSupp(
119894119895
) is
119894119895
= int
119894119895isinR
[int119906isin119869
119894119895
1119906]
119894119895
119894 isin N119899
119895 isin N119898
119869
119894119895
sube [0 1]
(33)
Here 119894119895
is bounded by both lower and upper primarymembership functions namely
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (34)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
and
120572
120583
119894119895
= (119886119894119895
119906) | 120583
119894119895
ge 120572 (35)
with parameters 120572119886119871119894119895
and 120572119886119877119894119895
Proposition 31 Let 119894119895
isin 119865(R) 119909119895
ge 0 119894 isin N119899
119895 isin N119898
Then the fuzzy setsum119899
119895=1
119894119895
119909119895
defined by the extension principleis again a fuzzy number
Let = ⪯Pos be a fuzzy relation [27] fuzzy extension of the
usual binary relation le on R The fuzzy linear programmingproblem associated with a standard linear programmingproblem is denoted as
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St (
119899
sum
119895=1
119894119895
119909119895
) 119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(36)
In (36) the value sum119899119895=1
119894119895
119909119895
isin 119865(R) is compared with a fuzzynumber 119887
119894
isin 119865(R) by sum fuzzy relation The maximizationof the objective function is denoted by Max119885 = sum
119899
119895=1
119888119895
119909119895
Definition 32 Let 120583
119894119895
R rarr [0 1] and 120583
119887
119894
R rarr [0 1]119894 isin N
119899
119895 isin N119898
be membership function of fuzzy numbers119894119895
and 119887119894
respectively Let be a fuzzy extension of a binaryrelation le on R A fuzzy set whose membership function120583
119883
is defined as 119909 isin R119899 by
120583
119883
=
min 120583
119875
(11
1199091
+ sdot sdot sdot + 1119899
119909119899
1198871
) 120583
119875
(1198981
1199091
+ sdot sdot sdot + 119898119899
119909119899
119887119898
) if 119909119895
ge 0 119895 = 1 2 119899
0 otherwise(37)
8 Journal of Applied Mathematics
is called the fuzzy set of feasible regions of the fuzzy linearprogramming problem (36) For 120572 isin (0 1] a vector 119909 isin
[]120572
is called the 120572-feasible solution of the fuzzy linearprogramming problem
Notice that the feasible region of the fuzzy linearprogramming problem is a fuzzy set On the other hand 120572-feasible solution is a vector belonging to the 120572-cut of the fea-sible region If all the coefficients
119894119895
and 119887119894
are crisp fuzzynumber that is they are equivalent to the corresponding crispfuzzy number then the fuzzy feasible region is equivalent tothe set of all feasible solutions of the corresponding classicallinear programming
61 Type 2 Fuzzy Linear Programming Problem Let us con-sider the following fuzzy linear programming problem withright-hand side (resources) and technology coefficients arePNIT2TrFNs
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119894119895
119909119895
le119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(38)
where 119894119895
= ((119894119895
)119871
(119894119895
)119880
) = (((119886119894119895
)119871
2
(119886119894119895
)119871
3
(120572119894119895
)119871
(120573119894119895
)119871
)((119886119894119895
)119880
2
(119886119894119895
)119880
3
(120572119894119895
)119880
(120573119894119895
)119880
)) 119887119894
= ((119887119894
)119871
(119887119894
)119880
) = (((119887119894
)119871
2
(119887119894
)119871
3
(120574119894
)119871
(120579119894
)119871
) ((119887119894
)119880
2
(119887119894
)119880
3
(120574119894
)119880
(120579119894
)119880
)) are PnIT2TrFNs119888119894
are crisp coefficients of the objective and 119909119894
are the decisionvariable
From Definitions 15 21 22 and 23 and (38)
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119871
119894119895
119909119895
le119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119880
119894119895
119909119895
le119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(39)
Based on the relation of possibility we can rewrite (39) asfollows
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119871
1198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯Pos
119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119880
1198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯Pos
119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(40)
With respect to constraints of linear programming possibilitythe optimal solution is completely determined by degrees ofpossibility So we can obtain the optimal solution of linearprogramming possibility at 119901-cut levels by solving the twocrisp linear programming problems using Barnes algorithm
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198711198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯119887
119871
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198801198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯119887
119880
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(41)
FromTheorems 27 and 28 we have
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(42)
where 119860119871 = (119886119894119895
)119871
2
119860119871 = (119886119894119895
)119880
2
119887119880 = (119887119894
)119871
3
119887119880
= (119887119894
)119880
3
120572 = (120572
119894119895
)119871
120572 = (120572119894119895
)119880
120573 = (120573119894119895
)119871
120573 = (120573119894119895
)119880
120574 =
(120574119894
)119871
120574 = (120574119894
)119880
120579 = (120579119894
)119871
120579 = (120579119894
)119880
and 119909 = 119909119895
119895 = 1 2 119899 119894 = 1 2 119898
Journal of Applied Mathematics 9
The feasible regions of (119871119875119875119901
) and (119871119875119875119901
) are denoted by
119883119875
119901
= 119909 isin 119877119899
| 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119883
119875
119901
= 119909 isin 119877119899
| 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
(43)
Assume the optimal solutions of (119871119875119875119901
) and (119871119875119875119901
) are 119885119875119901
and 119885
119875
119901
respectively
Theorem 33 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883119875
119901
1
119883
119875
119901
2
sube
119883
119875
119901
1
and 119885119875119901
1
ge 119885119875
119901
2
119885
119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) ge
(1199011
minus1)(120572119909+120579) so 119909 isin 119883119875
119901
1
that is119883119875119901
2
sube 119883119875
119901
1
and119885119875119901
1
ge 119885119875
119901
2
In a similar way119883119875119901
2
sube 119883
119875
119901
1
and 119885
119875
119901
1
ge 119885
119875
119901
2
Theorem 34 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883
119875
119901
1
and
119885119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 +
120579) ge (1199011
minus 1)(120572119909 + 120579) so 119909 isin 119883
119875
119901
1
that is 119883119875119901
2
sube 119883
119875
119901
1
and
119885
119875
119901
1
ge 119885119875
119901
2
FromTheorems 33 and 34 we obtain that 119885119875119901
119885
119875
119901
consti-tute the lower and upper bound of fuzzy objective value atlevel 119901 Then the objective value of 119885119875
119901
at level 119901 is the mean
of [119885119875119901
119885
119875
119901
]
7 Numerical Illustration
An application of the proposed method is introduced withan example where all its technological coefficients and right-hand side (resources) are defined as PnIT2FNs The optimalsolution will be obtained in terms of crisp for the followingproblem
The company plans tomanufacture two types of productsThe selling prices for these products are as follows 119875
1
costs Rs 1750 per unit and 1198752
costs Rs 2000 per unitDaily production volume of each type of these products isconstrained by available man hours and available machinehours The production specifications for the given problemsituation are presented in as shown in Table 1
Max 119885 = 1198881
1199091
+ 1198882
1199092
St 11
1199091
+ 12
1199092
⪯1198871
Table 1
Resource requirementunitResource 119875
1
1198752
AvailabilityMan hours (in minutes) Around 4 Around 8 Around 32Machine hours (in minutes) Around 6 Around 4 Around 36
21
1199091
+ 22
1199092
⪯1198872
119909119895
ge 0 119895 = 1 2
(44)
where 11
= [(230 250 20 10) (230 250 30 20)] 12
=
[(460 500 40 20) (460 500 60 40)] 21
= [(345 375 30
15) (345 375 45 30)] 22
= [(230 250 20 10) (230 250
30 20)] 1198871
= [(1840 2000 160 80) (1840 2000 240 160)]1198872
= [(2070 2250 180 90) (2070 2250 270 180)] 1198881
=
1750 and 1198882
= 2000 From (42) we have the following crispoptimal problem based on the level 119901
Max 119885119875
119901
= 17501199091
+ 20001199092
St (210 + 20119901) 1199091
+ (420 + 40119901) 1199092
le 2080 minus 80119901
(315 + 30119901) 1199091
+ (210 + 20119901) 1199092
le 2340 minus 90119901
1199091
1199092
ge 0
(45)
Max 119885
119875
119901
= 17501199091
+ 20001199092
St (200 + 30119901) 1199091
+ (400 + 60119901) 1199092
le 2160 minus 160119901
(300 + 45119901) 1199091
+ (200 + 30119901) 1199092
le 2430 minus 180119901
1199091
1199092
ge 0
(46)
For different cut level 119901 we can get different optimal solutionand denote by 119885119875
119901
119885119875
119901
the optimal solution (see Figure 5 andTable 2) [28] of the crisp programming (45) and program-ming (46) respectively
8 Conclusion
In this paper a method to do perfectly normal interval type2 fuzzy arithmetic operation using extension principle is pre-sented and defined the possibility degrees of upper and lowermembership function to compare perfectly normal intervaltype 2 fuzzy numbers These functions are defined based onthe strength of upper and lower membership function ofperfectly normal interval type 2 fuzzy numbers Meanwhilesome properties and theorems were proved Then the lowerand upper fuzzy satisfaction based on possibility degrees of
10 Journal of Applied Mathematics
Table 2 Optimal solution at different possibility levels
119901-cut 01 02 03 04 05 06 07 08 09 1119885119875
119901
143549528 141658878 13980324 137981651 136193181 134436937 132712053 131017699 129353069 127717393
119885
119875
119901
155123153 151723301 148421051 145212264 142093022 139059633 136108597 133236607 130440529 127717393119885119875
119901
149336341 146691090 144112146 141596958 139143102 136748285 134410325 132127153 129896799 127717393
1090807060504030201012500 13000 13500 14000 14500 15000
p-c
ut
Z
Figure 5 Optimal solution at different possibility levels
membership function respectively are defined These fuzzysatisfactions create two nonstrict order relations on the setof overlapping intervals Also linear programming problemwas introduced with resources and technology coefficientsof perfectly normal interval type 2 fuzzy numbers Thenthe possibility degrees of membership function were appliedin order to interpret inequality constraints with intervalcoefficients According to the definition of possibility degreesof membership function and their properties the inequalityconstraints with interval coefficients were reduced in theirsatisfactory crisp equivalent forms Finally the decisionmaker can get the crisp optimal solution of the problem forevery grade 120572 isin [0 1] which was obtained by using Barnesalgorithm and then mean value is selected according to thedecision maker optimistic attitude
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] H Tanaka T Okuda and K Asai ldquoOn fuzzy-mathematicalprogrammingrdquo Journal of Cybernetics vol 3 no 4 pp 37ndash461973
[3] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 pp B141ndashB164197071
[4] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978
[5] S Deng L Zhou and X Wang ldquoSolving the fuzzy bilevellinear programmingwithmultiple followers through structured
element methodrdquo Mathematical Problems in Engineering vol2014 Article ID 418594 6 pages 2014
[6] Y-H Zhong Y-L Jia D Chen and Y Yang ldquoInterior pointmethod for solving fuzzy number linear programming prob-lems using linear ranking functionrdquo Journal of Applied Math-ematics vol 2013 Article ID 795098 9 pages 2013
[7] H Cheng W Huang and J Cai ldquoSolving a fully fuzzy linearprogramming problem through compromise programmingrdquoJournal of Applied Mathematics vol 2013 Article ID 726296 10pages 2013
[8] D Chakraborty D K Jana and T K Royl ldquoA new approach tosolve intuitionistic fuzzy optimization problem using possibil-ity necessity and credibility measuresrdquo International Journal ofEngineering Mathematics vol 2014 Article ID 593185 12 pages2014
[9] H Le and Z Gogne ldquoFuzzy linear programming with possibil-ity and necessity relationrdquo in Fuzzy Information and Engineering2010 vol 78 of Advances in Intelligent and Soft Computing pp305ndash311 Springer Berlin Germany 2010
[10] J C Figueroa-Garcıa and G Hernandez ldquoA method for solvinglinear programming models with interval type-2 fuzzy con-straintsrdquo Pesquisa Operacional vol 34 no 1 pp 73ndash89 2014
[11] J C F Garcıa ldquoA general model for linear programming withinterval type-2 fuzzy technological coefficientsrdquo in Proceedingsof the IEEE Annual Meeting of the North American FuzzyInformation Processing Society (NAFIPS rsquo12) pp 1ndash4 BerkeleyCalif USA August 2012
[12] J C F Garcıa ldquoLinear programming with interval type-2fuzzy right hand side parametersrdquo in Proceedings of the AnnualMeeting of the North American Fuzzy Information ProcessingSociety (NAFIPS rsquo08) pp 1ndash6 IEEE New York NY USA May2008
[13] S-M Chen and L-W Lee ldquoFuzzy multiple attributes groupdecision-making based on the ranking values and the arith-metic operations of interval type-2 fuzzy setsrdquo Expert Systemswith Applications vol 37 no 1 pp 824ndash833 2010
[14] J Hu Y Zhang X Chen and Y Liu ldquoMulti-criteria decisionmaking method based on possibility degree of interval type-2 fuzzy numberrdquo Knowledge-Based Systems vol 43 pp 21ndash292013
[15] R Fuller Fuzzy Reasoning and Fuzzy Optimization TurkuCentre for Computer Science 1998
[16] H-C Wu ldquoDuality theory in fuzzy linear programming prob-lems with fuzzy coefficientsrdquo Fuzzy Optimization and DecisionMaking vol 2 no 1 pp 61ndash73 2003
[17] J Qin and X Liu ldquoFrank aggregation operators for triangularinterval type-2 fuzzy set and its application inmultiple attributegroup decision makingrdquo Journal of Applied Mathematics vol2014 Article ID 923213 24 pages 2014
[18] Z M Zhang and S H Zhang ldquoA novel approach to multiattribute group decision making based on trapezoidal intervaltype-2 fuzzy soft setsrdquo Applied Mathematical Modelling vol 37no 7 pp 4948ndash4971 2013
Journal of Applied Mathematics 11
[19] Y Gong ldquoFuzzymulti-attribute group decisionmakingmethodbased on interval type-2 fuzzy sets and applications to globalsupplier selectionrdquo International Journal of Fuzzy Systems vol15 no 4 pp 392ndash400 2013
[20] D S Dinagar and A Anbalagan ldquoA new type-2 fuzzy numberarithmetic using extension principlerdquo in Proceedings of the 1stInternational Conference on Advances in Engineering Scienceand Management (ICAESM rsquo12) pp 113ndash118 March 2012
[21] K-P Chiao ldquoMultiple criteria group decision making withtriangular interval type-2 fuzzy setsrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems (FUZZ rsquo11) pp 2575ndash2582 IEEE Taipei Taiwan June 2011
[22] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006
[23] X Liu ldquoMeasuring the satisfaction of constraints in fuzzy linearprogrammingrdquo Fuzzy Sets and Systems vol 122 no 2 pp 263ndash275 2001
[24] D Dubois and H Prade ldquoRanking fuzzy numbers in the settingof possibility theoryrdquo Information Sciences vol 30 no 3 pp183ndash224 1983
[25] D S Negi and E S Lee ldquoPossibility programming by thecomparison of fuzzy numbersrdquo Computers ampMathematics withApplications vol 25 no 9 pp 43ndash50 1993
[26] MG Iskander ldquoComparison of fuzzy numbers using possibilityprogramming comments and new conceptsrdquo Computers ampMathematics with Applications vol 43 no 6-7 pp 833ndash8402002
[27] J Ramik ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006
[28] G R Lindfield and J E T Penny Numerical methods UsingMatlab Academic Press 3rd edition 2013
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
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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
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
is called the fuzzy set of feasible regions of the fuzzy linearprogramming problem (36) For 120572 isin (0 1] a vector 119909 isin
[]120572
is called the 120572-feasible solution of the fuzzy linearprogramming problem
Notice that the feasible region of the fuzzy linearprogramming problem is a fuzzy set On the other hand 120572-feasible solution is a vector belonging to the 120572-cut of the fea-sible region If all the coefficients
119894119895
and 119887119894
are crisp fuzzynumber that is they are equivalent to the corresponding crispfuzzy number then the fuzzy feasible region is equivalent tothe set of all feasible solutions of the corresponding classicallinear programming
61 Type 2 Fuzzy Linear Programming Problem Let us con-sider the following fuzzy linear programming problem withright-hand side (resources) and technology coefficients arePNIT2TrFNs
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119894119895
119909119895
le119887119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(38)
where 119894119895
= ((119894119895
)119871
(119894119895
)119880
) = (((119886119894119895
)119871
2
(119886119894119895
)119871
3
(120572119894119895
)119871
(120573119894119895
)119871
)((119886119894119895
)119880
2
(119886119894119895
)119880
3
(120572119894119895
)119880
(120573119894119895
)119880
)) 119887119894
= ((119887119894
)119871
(119887119894
)119880
) = (((119887119894
)119871
2
(119887119894
)119871
3
(120574119894
)119871
(120579119894
)119871
) ((119887119894
)119880
2
(119887119894
)119880
3
(120574119894
)119880
(120579119894
)119880
)) are PnIT2TrFNs119888119894
are crisp coefficients of the objective and 119909119894
are the decisionvariable
From Definitions 15 21 22 and 23 and (38)
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119871
119894119895
119909119895
le119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St119899
sum
119895=1
119880
119894119895
119909119895
le119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(39)
Based on the relation of possibility we can rewrite (39) asfollows
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119871
1198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯Pos
119887
119871
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119880
1198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯Pos
119887
119880
119894
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(40)
With respect to constraints of linear programming possibilitythe optimal solution is completely determined by degrees ofpossibility So we can obtain the optimal solution of linearprogramming possibility at 119901-cut levels by solving the twocrisp linear programming problems using Barnes algorithm
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198711198941
1199091
+ 119871
1198942
1199092
+ sdot sdot sdot + 119871
119894119899
119909119899
⪯119887
119871
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St Pos (1198801198941
1199091
+ 119880
1198942
1199092
+ sdot sdot sdot + 119880
119894119899
119909119899
⪯119887
119880
119894
)
ge 119901
119894 = 1 2 119898
119909119895
ge 0 119895 = 1 2 119899
(41)
FromTheorems 27 and 28 we have
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(119871119875119875119901
) = MaxMin 119885 =
119899
sum
119895=1
119888119895
119909119895
St 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119909119895
ge 0 119895 = 1 2 119899
(42)
where 119860119871 = (119886119894119895
)119871
2
119860119871 = (119886119894119895
)119880
2
119887119880 = (119887119894
)119871
3
119887119880
= (119887119894
)119880
3
120572 = (120572
119894119895
)119871
120572 = (120572119894119895
)119880
120573 = (120573119894119895
)119871
120573 = (120573119894119895
)119880
120574 =
(120574119894
)119871
120574 = (120574119894
)119880
120579 = (120579119894
)119871
120579 = (120579119894
)119880
and 119909 = 119909119895
119895 = 1 2 119899 119894 = 1 2 119898
Journal of Applied Mathematics 9
The feasible regions of (119871119875119875119901
) and (119871119875119875119901
) are denoted by
119883119875
119901
= 119909 isin 119877119899
| 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119883
119875
119901
= 119909 isin 119877119899
| 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
(43)
Assume the optimal solutions of (119871119875119875119901
) and (119871119875119875119901
) are 119885119875119901
and 119885
119875
119901
respectively
Theorem 33 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883119875
119901
1
119883
119875
119901
2
sube
119883
119875
119901
1
and 119885119875119901
1
ge 119885119875
119901
2
119885
119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) ge
(1199011
minus1)(120572119909+120579) so 119909 isin 119883119875
119901
1
that is119883119875119901
2
sube 119883119875
119901
1
and119885119875119901
1
ge 119885119875
119901
2
In a similar way119883119875119901
2
sube 119883
119875
119901
1
and 119885
119875
119901
1
ge 119885
119875
119901
2
Theorem 34 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883
119875
119901
1
and
119885119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 +
120579) ge (1199011
minus 1)(120572119909 + 120579) so 119909 isin 119883
119875
119901
1
that is 119883119875119901
2
sube 119883
119875
119901
1
and
119885
119875
119901
1
ge 119885119875
119901
2
FromTheorems 33 and 34 we obtain that 119885119875119901
119885
119875
119901
consti-tute the lower and upper bound of fuzzy objective value atlevel 119901 Then the objective value of 119885119875
119901
at level 119901 is the mean
of [119885119875119901
119885
119875
119901
]
7 Numerical Illustration
An application of the proposed method is introduced withan example where all its technological coefficients and right-hand side (resources) are defined as PnIT2FNs The optimalsolution will be obtained in terms of crisp for the followingproblem
The company plans tomanufacture two types of productsThe selling prices for these products are as follows 119875
1
costs Rs 1750 per unit and 1198752
costs Rs 2000 per unitDaily production volume of each type of these products isconstrained by available man hours and available machinehours The production specifications for the given problemsituation are presented in as shown in Table 1
Max 119885 = 1198881
1199091
+ 1198882
1199092
St 11
1199091
+ 12
1199092
⪯1198871
Table 1
Resource requirementunitResource 119875
1
1198752
AvailabilityMan hours (in minutes) Around 4 Around 8 Around 32Machine hours (in minutes) Around 6 Around 4 Around 36
21
1199091
+ 22
1199092
⪯1198872
119909119895
ge 0 119895 = 1 2
(44)
where 11
= [(230 250 20 10) (230 250 30 20)] 12
=
[(460 500 40 20) (460 500 60 40)] 21
= [(345 375 30
15) (345 375 45 30)] 22
= [(230 250 20 10) (230 250
30 20)] 1198871
= [(1840 2000 160 80) (1840 2000 240 160)]1198872
= [(2070 2250 180 90) (2070 2250 270 180)] 1198881
=
1750 and 1198882
= 2000 From (42) we have the following crispoptimal problem based on the level 119901
Max 119885119875
119901
= 17501199091
+ 20001199092
St (210 + 20119901) 1199091
+ (420 + 40119901) 1199092
le 2080 minus 80119901
(315 + 30119901) 1199091
+ (210 + 20119901) 1199092
le 2340 minus 90119901
1199091
1199092
ge 0
(45)
Max 119885
119875
119901
= 17501199091
+ 20001199092
St (200 + 30119901) 1199091
+ (400 + 60119901) 1199092
le 2160 minus 160119901
(300 + 45119901) 1199091
+ (200 + 30119901) 1199092
le 2430 minus 180119901
1199091
1199092
ge 0
(46)
For different cut level 119901 we can get different optimal solutionand denote by 119885119875
119901
119885119875
119901
the optimal solution (see Figure 5 andTable 2) [28] of the crisp programming (45) and program-ming (46) respectively
8 Conclusion
In this paper a method to do perfectly normal interval type2 fuzzy arithmetic operation using extension principle is pre-sented and defined the possibility degrees of upper and lowermembership function to compare perfectly normal intervaltype 2 fuzzy numbers These functions are defined based onthe strength of upper and lower membership function ofperfectly normal interval type 2 fuzzy numbers Meanwhilesome properties and theorems were proved Then the lowerand upper fuzzy satisfaction based on possibility degrees of
10 Journal of Applied Mathematics
Table 2 Optimal solution at different possibility levels
119901-cut 01 02 03 04 05 06 07 08 09 1119885119875
119901
143549528 141658878 13980324 137981651 136193181 134436937 132712053 131017699 129353069 127717393
119885
119875
119901
155123153 151723301 148421051 145212264 142093022 139059633 136108597 133236607 130440529 127717393119885119875
119901
149336341 146691090 144112146 141596958 139143102 136748285 134410325 132127153 129896799 127717393
1090807060504030201012500 13000 13500 14000 14500 15000
p-c
ut
Z
Figure 5 Optimal solution at different possibility levels
membership function respectively are defined These fuzzysatisfactions create two nonstrict order relations on the setof overlapping intervals Also linear programming problemwas introduced with resources and technology coefficientsof perfectly normal interval type 2 fuzzy numbers Thenthe possibility degrees of membership function were appliedin order to interpret inequality constraints with intervalcoefficients According to the definition of possibility degreesof membership function and their properties the inequalityconstraints with interval coefficients were reduced in theirsatisfactory crisp equivalent forms Finally the decisionmaker can get the crisp optimal solution of the problem forevery grade 120572 isin [0 1] which was obtained by using Barnesalgorithm and then mean value is selected according to thedecision maker optimistic attitude
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] H Tanaka T Okuda and K Asai ldquoOn fuzzy-mathematicalprogrammingrdquo Journal of Cybernetics vol 3 no 4 pp 37ndash461973
[3] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 pp B141ndashB164197071
[4] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978
[5] S Deng L Zhou and X Wang ldquoSolving the fuzzy bilevellinear programmingwithmultiple followers through structured
element methodrdquo Mathematical Problems in Engineering vol2014 Article ID 418594 6 pages 2014
[6] Y-H Zhong Y-L Jia D Chen and Y Yang ldquoInterior pointmethod for solving fuzzy number linear programming prob-lems using linear ranking functionrdquo Journal of Applied Math-ematics vol 2013 Article ID 795098 9 pages 2013
[7] H Cheng W Huang and J Cai ldquoSolving a fully fuzzy linearprogramming problem through compromise programmingrdquoJournal of Applied Mathematics vol 2013 Article ID 726296 10pages 2013
[8] D Chakraborty D K Jana and T K Royl ldquoA new approach tosolve intuitionistic fuzzy optimization problem using possibil-ity necessity and credibility measuresrdquo International Journal ofEngineering Mathematics vol 2014 Article ID 593185 12 pages2014
[9] H Le and Z Gogne ldquoFuzzy linear programming with possibil-ity and necessity relationrdquo in Fuzzy Information and Engineering2010 vol 78 of Advances in Intelligent and Soft Computing pp305ndash311 Springer Berlin Germany 2010
[10] J C Figueroa-Garcıa and G Hernandez ldquoA method for solvinglinear programming models with interval type-2 fuzzy con-straintsrdquo Pesquisa Operacional vol 34 no 1 pp 73ndash89 2014
[11] J C F Garcıa ldquoA general model for linear programming withinterval type-2 fuzzy technological coefficientsrdquo in Proceedingsof the IEEE Annual Meeting of the North American FuzzyInformation Processing Society (NAFIPS rsquo12) pp 1ndash4 BerkeleyCalif USA August 2012
[12] J C F Garcıa ldquoLinear programming with interval type-2fuzzy right hand side parametersrdquo in Proceedings of the AnnualMeeting of the North American Fuzzy Information ProcessingSociety (NAFIPS rsquo08) pp 1ndash6 IEEE New York NY USA May2008
[13] S-M Chen and L-W Lee ldquoFuzzy multiple attributes groupdecision-making based on the ranking values and the arith-metic operations of interval type-2 fuzzy setsrdquo Expert Systemswith Applications vol 37 no 1 pp 824ndash833 2010
[14] J Hu Y Zhang X Chen and Y Liu ldquoMulti-criteria decisionmaking method based on possibility degree of interval type-2 fuzzy numberrdquo Knowledge-Based Systems vol 43 pp 21ndash292013
[15] R Fuller Fuzzy Reasoning and Fuzzy Optimization TurkuCentre for Computer Science 1998
[16] H-C Wu ldquoDuality theory in fuzzy linear programming prob-lems with fuzzy coefficientsrdquo Fuzzy Optimization and DecisionMaking vol 2 no 1 pp 61ndash73 2003
[17] J Qin and X Liu ldquoFrank aggregation operators for triangularinterval type-2 fuzzy set and its application inmultiple attributegroup decision makingrdquo Journal of Applied Mathematics vol2014 Article ID 923213 24 pages 2014
[18] Z M Zhang and S H Zhang ldquoA novel approach to multiattribute group decision making based on trapezoidal intervaltype-2 fuzzy soft setsrdquo Applied Mathematical Modelling vol 37no 7 pp 4948ndash4971 2013
Journal of Applied Mathematics 11
[19] Y Gong ldquoFuzzymulti-attribute group decisionmakingmethodbased on interval type-2 fuzzy sets and applications to globalsupplier selectionrdquo International Journal of Fuzzy Systems vol15 no 4 pp 392ndash400 2013
[20] D S Dinagar and A Anbalagan ldquoA new type-2 fuzzy numberarithmetic using extension principlerdquo in Proceedings of the 1stInternational Conference on Advances in Engineering Scienceand Management (ICAESM rsquo12) pp 113ndash118 March 2012
[21] K-P Chiao ldquoMultiple criteria group decision making withtriangular interval type-2 fuzzy setsrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems (FUZZ rsquo11) pp 2575ndash2582 IEEE Taipei Taiwan June 2011
[22] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006
[23] X Liu ldquoMeasuring the satisfaction of constraints in fuzzy linearprogrammingrdquo Fuzzy Sets and Systems vol 122 no 2 pp 263ndash275 2001
[24] D Dubois and H Prade ldquoRanking fuzzy numbers in the settingof possibility theoryrdquo Information Sciences vol 30 no 3 pp183ndash224 1983
[25] D S Negi and E S Lee ldquoPossibility programming by thecomparison of fuzzy numbersrdquo Computers ampMathematics withApplications vol 25 no 9 pp 43ndash50 1993
[26] MG Iskander ldquoComparison of fuzzy numbers using possibilityprogramming comments and new conceptsrdquo Computers ampMathematics with Applications vol 43 no 6-7 pp 833ndash8402002
[27] J Ramik ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006
[28] G R Lindfield and J E T Penny Numerical methods UsingMatlab Academic Press 3rd edition 2013
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
Journal of Applied Mathematics 9
The feasible regions of (119871119875119875119901
) and (119871119875119875119901
) are denoted by
119883119875
119901
= 119909 isin 119877119899
| 119887119880
minus 119860119871
119909 ge (119901 minus 1) (120572119909 + 120579)
119883
119875
119901
= 119909 isin 119877119899
| 119887
119880
minus 119860
119871
119909 ge (119901 minus 1) (120572119909 + 120579)
(43)
Assume the optimal solutions of (119871119875119875119901
) and (119871119875119875119901
) are 119885119875119901
and 119885
119875
119901
respectively
Theorem 33 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883119875
119901
1
119883
119875
119901
2
sube
119883
119875
119901
1
and 119885119875119901
1
ge 119885119875
119901
2
119885
119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) ge
(1199011
minus1)(120572119909+120579) so 119909 isin 119883119875
119901
1
that is119883119875119901
2
sube 119883119875
119901
1
and119885119875119901
1
ge 119885119875
119901
2
In a similar way119883119875119901
2
sube 119883
119875
119901
1
and 119885
119875
119901
1
ge 119885
119875
119901
2
Theorem 34 For linear programming (119871119875119875119901
) and program-
ming (119871119875119875119901
) 0 le 1199011
le 1199012
le 1 and then 119883119875
119901
2
sube 119883
119875
119901
1
and
119885119875
119901
1
ge 119885
119875
119901
2
Proof If 119909 isin 119883119875
119901
2
we have 119887119880 minus119860119871
119909 ge (1199012
minus 1)(120572119909 + 120579) from120573 gt 0 120574 gt 0 and 119909 gt 0 then 119887
119880
minus 119860119871
119909 ge (1199012
minus 1)(120572119909 +
120579) ge (1199011
minus 1)(120572119909 + 120579) so 119909 isin 119883
119875
119901
1
that is 119883119875119901
2
sube 119883
119875
119901
1
and
119885
119875
119901
1
ge 119885119875
119901
2
FromTheorems 33 and 34 we obtain that 119885119875119901
119885
119875
119901
consti-tute the lower and upper bound of fuzzy objective value atlevel 119901 Then the objective value of 119885119875
119901
at level 119901 is the mean
of [119885119875119901
119885
119875
119901
]
7 Numerical Illustration
An application of the proposed method is introduced withan example where all its technological coefficients and right-hand side (resources) are defined as PnIT2FNs The optimalsolution will be obtained in terms of crisp for the followingproblem
The company plans tomanufacture two types of productsThe selling prices for these products are as follows 119875
1
costs Rs 1750 per unit and 1198752
costs Rs 2000 per unitDaily production volume of each type of these products isconstrained by available man hours and available machinehours The production specifications for the given problemsituation are presented in as shown in Table 1
Max 119885 = 1198881
1199091
+ 1198882
1199092
St 11
1199091
+ 12
1199092
⪯1198871
Table 1
Resource requirementunitResource 119875
1
1198752
AvailabilityMan hours (in minutes) Around 4 Around 8 Around 32Machine hours (in minutes) Around 6 Around 4 Around 36
21
1199091
+ 22
1199092
⪯1198872
119909119895
ge 0 119895 = 1 2
(44)
where 11
= [(230 250 20 10) (230 250 30 20)] 12
=
[(460 500 40 20) (460 500 60 40)] 21
= [(345 375 30
15) (345 375 45 30)] 22
= [(230 250 20 10) (230 250
30 20)] 1198871
= [(1840 2000 160 80) (1840 2000 240 160)]1198872
= [(2070 2250 180 90) (2070 2250 270 180)] 1198881
=
1750 and 1198882
= 2000 From (42) we have the following crispoptimal problem based on the level 119901
Max 119885119875
119901
= 17501199091
+ 20001199092
St (210 + 20119901) 1199091
+ (420 + 40119901) 1199092
le 2080 minus 80119901
(315 + 30119901) 1199091
+ (210 + 20119901) 1199092
le 2340 minus 90119901
1199091
1199092
ge 0
(45)
Max 119885
119875
119901
= 17501199091
+ 20001199092
St (200 + 30119901) 1199091
+ (400 + 60119901) 1199092
le 2160 minus 160119901
(300 + 45119901) 1199091
+ (200 + 30119901) 1199092
le 2430 minus 180119901
1199091
1199092
ge 0
(46)
For different cut level 119901 we can get different optimal solutionand denote by 119885119875
119901
119885119875
119901
the optimal solution (see Figure 5 andTable 2) [28] of the crisp programming (45) and program-ming (46) respectively
8 Conclusion
In this paper a method to do perfectly normal interval type2 fuzzy arithmetic operation using extension principle is pre-sented and defined the possibility degrees of upper and lowermembership function to compare perfectly normal intervaltype 2 fuzzy numbers These functions are defined based onthe strength of upper and lower membership function ofperfectly normal interval type 2 fuzzy numbers Meanwhilesome properties and theorems were proved Then the lowerand upper fuzzy satisfaction based on possibility degrees of
10 Journal of Applied Mathematics
Table 2 Optimal solution at different possibility levels
119901-cut 01 02 03 04 05 06 07 08 09 1119885119875
119901
143549528 141658878 13980324 137981651 136193181 134436937 132712053 131017699 129353069 127717393
119885
119875
119901
155123153 151723301 148421051 145212264 142093022 139059633 136108597 133236607 130440529 127717393119885119875
119901
149336341 146691090 144112146 141596958 139143102 136748285 134410325 132127153 129896799 127717393
1090807060504030201012500 13000 13500 14000 14500 15000
p-c
ut
Z
Figure 5 Optimal solution at different possibility levels
membership function respectively are defined These fuzzysatisfactions create two nonstrict order relations on the setof overlapping intervals Also linear programming problemwas introduced with resources and technology coefficientsof perfectly normal interval type 2 fuzzy numbers Thenthe possibility degrees of membership function were appliedin order to interpret inequality constraints with intervalcoefficients According to the definition of possibility degreesof membership function and their properties the inequalityconstraints with interval coefficients were reduced in theirsatisfactory crisp equivalent forms Finally the decisionmaker can get the crisp optimal solution of the problem forevery grade 120572 isin [0 1] which was obtained by using Barnesalgorithm and then mean value is selected according to thedecision maker optimistic attitude
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] H Tanaka T Okuda and K Asai ldquoOn fuzzy-mathematicalprogrammingrdquo Journal of Cybernetics vol 3 no 4 pp 37ndash461973
[3] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 pp B141ndashB164197071
[4] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978
[5] S Deng L Zhou and X Wang ldquoSolving the fuzzy bilevellinear programmingwithmultiple followers through structured
element methodrdquo Mathematical Problems in Engineering vol2014 Article ID 418594 6 pages 2014
[6] Y-H Zhong Y-L Jia D Chen and Y Yang ldquoInterior pointmethod for solving fuzzy number linear programming prob-lems using linear ranking functionrdquo Journal of Applied Math-ematics vol 2013 Article ID 795098 9 pages 2013
[7] H Cheng W Huang and J Cai ldquoSolving a fully fuzzy linearprogramming problem through compromise programmingrdquoJournal of Applied Mathematics vol 2013 Article ID 726296 10pages 2013
[8] D Chakraborty D K Jana and T K Royl ldquoA new approach tosolve intuitionistic fuzzy optimization problem using possibil-ity necessity and credibility measuresrdquo International Journal ofEngineering Mathematics vol 2014 Article ID 593185 12 pages2014
[9] H Le and Z Gogne ldquoFuzzy linear programming with possibil-ity and necessity relationrdquo in Fuzzy Information and Engineering2010 vol 78 of Advances in Intelligent and Soft Computing pp305ndash311 Springer Berlin Germany 2010
[10] J C Figueroa-Garcıa and G Hernandez ldquoA method for solvinglinear programming models with interval type-2 fuzzy con-straintsrdquo Pesquisa Operacional vol 34 no 1 pp 73ndash89 2014
[11] J C F Garcıa ldquoA general model for linear programming withinterval type-2 fuzzy technological coefficientsrdquo in Proceedingsof the IEEE Annual Meeting of the North American FuzzyInformation Processing Society (NAFIPS rsquo12) pp 1ndash4 BerkeleyCalif USA August 2012
[12] J C F Garcıa ldquoLinear programming with interval type-2fuzzy right hand side parametersrdquo in Proceedings of the AnnualMeeting of the North American Fuzzy Information ProcessingSociety (NAFIPS rsquo08) pp 1ndash6 IEEE New York NY USA May2008
[13] S-M Chen and L-W Lee ldquoFuzzy multiple attributes groupdecision-making based on the ranking values and the arith-metic operations of interval type-2 fuzzy setsrdquo Expert Systemswith Applications vol 37 no 1 pp 824ndash833 2010
[14] J Hu Y Zhang X Chen and Y Liu ldquoMulti-criteria decisionmaking method based on possibility degree of interval type-2 fuzzy numberrdquo Knowledge-Based Systems vol 43 pp 21ndash292013
[15] R Fuller Fuzzy Reasoning and Fuzzy Optimization TurkuCentre for Computer Science 1998
[16] H-C Wu ldquoDuality theory in fuzzy linear programming prob-lems with fuzzy coefficientsrdquo Fuzzy Optimization and DecisionMaking vol 2 no 1 pp 61ndash73 2003
[17] J Qin and X Liu ldquoFrank aggregation operators for triangularinterval type-2 fuzzy set and its application inmultiple attributegroup decision makingrdquo Journal of Applied Mathematics vol2014 Article ID 923213 24 pages 2014
[18] Z M Zhang and S H Zhang ldquoA novel approach to multiattribute group decision making based on trapezoidal intervaltype-2 fuzzy soft setsrdquo Applied Mathematical Modelling vol 37no 7 pp 4948ndash4971 2013
Journal of Applied Mathematics 11
[19] Y Gong ldquoFuzzymulti-attribute group decisionmakingmethodbased on interval type-2 fuzzy sets and applications to globalsupplier selectionrdquo International Journal of Fuzzy Systems vol15 no 4 pp 392ndash400 2013
[20] D S Dinagar and A Anbalagan ldquoA new type-2 fuzzy numberarithmetic using extension principlerdquo in Proceedings of the 1stInternational Conference on Advances in Engineering Scienceand Management (ICAESM rsquo12) pp 113ndash118 March 2012
[21] K-P Chiao ldquoMultiple criteria group decision making withtriangular interval type-2 fuzzy setsrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems (FUZZ rsquo11) pp 2575ndash2582 IEEE Taipei Taiwan June 2011
[22] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006
[23] X Liu ldquoMeasuring the satisfaction of constraints in fuzzy linearprogrammingrdquo Fuzzy Sets and Systems vol 122 no 2 pp 263ndash275 2001
[24] D Dubois and H Prade ldquoRanking fuzzy numbers in the settingof possibility theoryrdquo Information Sciences vol 30 no 3 pp183ndash224 1983
[25] D S Negi and E S Lee ldquoPossibility programming by thecomparison of fuzzy numbersrdquo Computers ampMathematics withApplications vol 25 no 9 pp 43ndash50 1993
[26] MG Iskander ldquoComparison of fuzzy numbers using possibilityprogramming comments and new conceptsrdquo Computers ampMathematics with Applications vol 43 no 6-7 pp 833ndash8402002
[27] J Ramik ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006
[28] G R Lindfield and J E T Penny Numerical methods UsingMatlab Academic Press 3rd edition 2013
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 Journal of Applied Mathematics
Table 2 Optimal solution at different possibility levels
119901-cut 01 02 03 04 05 06 07 08 09 1119885119875
119901
143549528 141658878 13980324 137981651 136193181 134436937 132712053 131017699 129353069 127717393
119885
119875
119901
155123153 151723301 148421051 145212264 142093022 139059633 136108597 133236607 130440529 127717393119885119875
119901
149336341 146691090 144112146 141596958 139143102 136748285 134410325 132127153 129896799 127717393
1090807060504030201012500 13000 13500 14000 14500 15000
p-c
ut
Z
Figure 5 Optimal solution at different possibility levels
membership function respectively are defined These fuzzysatisfactions create two nonstrict order relations on the setof overlapping intervals Also linear programming problemwas introduced with resources and technology coefficientsof perfectly normal interval type 2 fuzzy numbers Thenthe possibility degrees of membership function were appliedin order to interpret inequality constraints with intervalcoefficients According to the definition of possibility degreesof membership function and their properties the inequalityconstraints with interval coefficients were reduced in theirsatisfactory crisp equivalent forms Finally the decisionmaker can get the crisp optimal solution of the problem forevery grade 120572 isin [0 1] which was obtained by using Barnesalgorithm and then mean value is selected according to thedecision maker optimistic attitude
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] H Tanaka T Okuda and K Asai ldquoOn fuzzy-mathematicalprogrammingrdquo Journal of Cybernetics vol 3 no 4 pp 37ndash461973
[3] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 pp B141ndashB164197071
[4] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978
[5] S Deng L Zhou and X Wang ldquoSolving the fuzzy bilevellinear programmingwithmultiple followers through structured
element methodrdquo Mathematical Problems in Engineering vol2014 Article ID 418594 6 pages 2014
[6] Y-H Zhong Y-L Jia D Chen and Y Yang ldquoInterior pointmethod for solving fuzzy number linear programming prob-lems using linear ranking functionrdquo Journal of Applied Math-ematics vol 2013 Article ID 795098 9 pages 2013
[7] H Cheng W Huang and J Cai ldquoSolving a fully fuzzy linearprogramming problem through compromise programmingrdquoJournal of Applied Mathematics vol 2013 Article ID 726296 10pages 2013
[8] D Chakraborty D K Jana and T K Royl ldquoA new approach tosolve intuitionistic fuzzy optimization problem using possibil-ity necessity and credibility measuresrdquo International Journal ofEngineering Mathematics vol 2014 Article ID 593185 12 pages2014
[9] H Le and Z Gogne ldquoFuzzy linear programming with possibil-ity and necessity relationrdquo in Fuzzy Information and Engineering2010 vol 78 of Advances in Intelligent and Soft Computing pp305ndash311 Springer Berlin Germany 2010
[10] J C Figueroa-Garcıa and G Hernandez ldquoA method for solvinglinear programming models with interval type-2 fuzzy con-straintsrdquo Pesquisa Operacional vol 34 no 1 pp 73ndash89 2014
[11] J C F Garcıa ldquoA general model for linear programming withinterval type-2 fuzzy technological coefficientsrdquo in Proceedingsof the IEEE Annual Meeting of the North American FuzzyInformation Processing Society (NAFIPS rsquo12) pp 1ndash4 BerkeleyCalif USA August 2012
[12] J C F Garcıa ldquoLinear programming with interval type-2fuzzy right hand side parametersrdquo in Proceedings of the AnnualMeeting of the North American Fuzzy Information ProcessingSociety (NAFIPS rsquo08) pp 1ndash6 IEEE New York NY USA May2008
[13] S-M Chen and L-W Lee ldquoFuzzy multiple attributes groupdecision-making based on the ranking values and the arith-metic operations of interval type-2 fuzzy setsrdquo Expert Systemswith Applications vol 37 no 1 pp 824ndash833 2010
[14] J Hu Y Zhang X Chen and Y Liu ldquoMulti-criteria decisionmaking method based on possibility degree of interval type-2 fuzzy numberrdquo Knowledge-Based Systems vol 43 pp 21ndash292013
[15] R Fuller Fuzzy Reasoning and Fuzzy Optimization TurkuCentre for Computer Science 1998
[16] H-C Wu ldquoDuality theory in fuzzy linear programming prob-lems with fuzzy coefficientsrdquo Fuzzy Optimization and DecisionMaking vol 2 no 1 pp 61ndash73 2003
[17] J Qin and X Liu ldquoFrank aggregation operators for triangularinterval type-2 fuzzy set and its application inmultiple attributegroup decision makingrdquo Journal of Applied Mathematics vol2014 Article ID 923213 24 pages 2014
[18] Z M Zhang and S H Zhang ldquoA novel approach to multiattribute group decision making based on trapezoidal intervaltype-2 fuzzy soft setsrdquo Applied Mathematical Modelling vol 37no 7 pp 4948ndash4971 2013
Journal of Applied Mathematics 11
[19] Y Gong ldquoFuzzymulti-attribute group decisionmakingmethodbased on interval type-2 fuzzy sets and applications to globalsupplier selectionrdquo International Journal of Fuzzy Systems vol15 no 4 pp 392ndash400 2013
[20] D S Dinagar and A Anbalagan ldquoA new type-2 fuzzy numberarithmetic using extension principlerdquo in Proceedings of the 1stInternational Conference on Advances in Engineering Scienceand Management (ICAESM rsquo12) pp 113ndash118 March 2012
[21] K-P Chiao ldquoMultiple criteria group decision making withtriangular interval type-2 fuzzy setsrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems (FUZZ rsquo11) pp 2575ndash2582 IEEE Taipei Taiwan June 2011
[22] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006
[23] X Liu ldquoMeasuring the satisfaction of constraints in fuzzy linearprogrammingrdquo Fuzzy Sets and Systems vol 122 no 2 pp 263ndash275 2001
[24] D Dubois and H Prade ldquoRanking fuzzy numbers in the settingof possibility theoryrdquo Information Sciences vol 30 no 3 pp183ndash224 1983
[25] D S Negi and E S Lee ldquoPossibility programming by thecomparison of fuzzy numbersrdquo Computers ampMathematics withApplications vol 25 no 9 pp 43ndash50 1993
[26] MG Iskander ldquoComparison of fuzzy numbers using possibilityprogramming comments and new conceptsrdquo Computers ampMathematics with Applications vol 43 no 6-7 pp 833ndash8402002
[27] J Ramik ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006
[28] G R Lindfield and J E T Penny Numerical methods UsingMatlab Academic Press 3rd edition 2013
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
Journal of Applied Mathematics 11
[19] Y Gong ldquoFuzzymulti-attribute group decisionmakingmethodbased on interval type-2 fuzzy sets and applications to globalsupplier selectionrdquo International Journal of Fuzzy Systems vol15 no 4 pp 392ndash400 2013
[20] D S Dinagar and A Anbalagan ldquoA new type-2 fuzzy numberarithmetic using extension principlerdquo in Proceedings of the 1stInternational Conference on Advances in Engineering Scienceand Management (ICAESM rsquo12) pp 113ndash118 March 2012
[21] K-P Chiao ldquoMultiple criteria group decision making withtriangular interval type-2 fuzzy setsrdquo in Proceedings of the IEEEInternational Conference on Fuzzy Systems (FUZZ rsquo11) pp 2575ndash2582 IEEE Taipei Taiwan June 2011
[22] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006
[23] X Liu ldquoMeasuring the satisfaction of constraints in fuzzy linearprogrammingrdquo Fuzzy Sets and Systems vol 122 no 2 pp 263ndash275 2001
[24] D Dubois and H Prade ldquoRanking fuzzy numbers in the settingof possibility theoryrdquo Information Sciences vol 30 no 3 pp183ndash224 1983
[25] D S Negi and E S Lee ldquoPossibility programming by thecomparison of fuzzy numbersrdquo Computers ampMathematics withApplications vol 25 no 9 pp 43ndash50 1993
[26] MG Iskander ldquoComparison of fuzzy numbers using possibilityprogramming comments and new conceptsrdquo Computers ampMathematics with Applications vol 43 no 6-7 pp 833ndash8402002
[27] J Ramik ldquoDuality in fuzzy linear programming with possibilityand necessity relationsrdquo Fuzzy Sets and Systems vol 157 no 10pp 1283ndash1302 2006
[28] G R Lindfield and J E T Penny Numerical methods UsingMatlab Academic Press 3rd edition 2013
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