research article linear programming problem with interval type...

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


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


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


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


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)


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


(0 1] Pos(119880 ⪯

119880

) ge 119901 if and only if 1198871198803

minus119886119880

2

ge (119901minus1)(120579119880

+

120572119880

)


119880


3

ge 119886119880

2


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)


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]


119880


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)


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


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


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


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


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


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


) 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


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


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


[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


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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)


When 119886119871



119886119880



120572

= [120572 (119886119871

minus (119886119871

minus 120572)) + (119886119871

minus 120572)

minus 120572 ((119886119880

+ 120573) minus 119886119880

) + (119886119880

+ 120573)]

(5)


+ 120573)


Let = (119886119871

119886119880

120572 120573) and = (119887119871

119887119880


= (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)




119860

(119909 119906) as follows

= ((119909 119906) 120583

119860


sube [0 1] 0

le 120583

119860

(119909 119906) le 1

(7)

where 119869119909


int

119906isin119869

119909

120583

119860

(119909 119906)

119906

(8)



119860

(119909 119906) If all 120583

119860



= int

119909isin119883

int

119906isin119869

119909

1

(119909 119906)

= int

119909isin119883

[int119906isin119869

119909

1119906]

119909

(9)



119906isin119869

119909



FOU () = ⋃

119909isin119883

119869119909

= ⋃

119909isin119883

(119909 119906) | 119906 isin 119869119909

sube [0 1] (10)


119860


119860

respectively


= (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

and 119886119880119895+1




sup 120583

119860

(119909) = sup 120583

119860

= 1 (12)





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


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


119880

2

119886119880

3

] is the core of 119880 and 120572119880

120573119880


119871 and


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


119871

119860119880

] = [(119886119871

2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880

)] Obviously if 1198861198712

= 119886119871

3

and 119886119880

2

= 119886119880

3




119909


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)


119871

119860119880


119888119887



0

120583(x)

hU = hL = 1




119880 of PnIT2FS

the boundary of 119860119871120572

and 119860119880

120572


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)


120572

119888119887

120583

119860

isin [[120572

119886119871

120572

119886119871

] [120572

119886119877

120572

119886119877

]] (16)


120572

119886119871

le120572

119886119871

le120572

119886119877

le120572

119886119877

(17)


119860119880

) = ((119886119871

2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880


119860

(119909) = 120583

119860

= 0 forall119909 gt 0


119860119880

) = ((119886119871

2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880


119860

(119909) = 120583

119860

= 0 forall119909 lt 0


119860119880

) = ((119886119871

2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880


=

119887119871

2

= 0 1198861198713

= 119887119871

3

= 0 120572119871

= 120574119871

= 120573119871

= 120579119871

= 120572119880

= 120574119880

= 0 and120573119880

= 120579119880

= 0


2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880

)) and = ((119887119871

2

119887119871

3

120574119871

120579119871

) (119887119880

2

119887119880

3

120574119880

120579119880


2

= 119887119871

2

1198861198713

= 119887119871

3

120572119871

= 120574119871

120573119871

= 120579119871

120572119880

= 120574119880

and 120573119880

= 120579119880


0

120583(x)

hU = hL = 1

x120572aR120572aL 120572aR

120572aL

120572119888119887 A



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


= ((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


= ([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)


2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880


= 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)




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


Pos (119860119871 le 119861119871

) = max(1


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)


Pos (119860119880 le 119861119880

) = max(1


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)


2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880

))

and = ((119887119871

2

119887119871

3

120574119871

120579119871

) (119887119880

2

119887119880

3

120574119880

120579119880


119871


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)


Pos (119860119880 ⪯ 119861119880

) ge 120572 iff 120572119886119871 le 120572119887119877

(24)


2

119886119871

3

120572119871

120573119871

)(119886119880

2

119886119880

3

120572119880

120573119880

)) and = ((119887119871

2

119887119871

3

120574119871

120579119871

) (119887119880

2

119887119880

3

120574119880

120579119880


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


(0 1] Pos(119871 ⪯

119871

) ge 119901 if and only if 1198871198713

minus 119886119871

2

ge (119901 minus 1)(120579119871

+

120572119871

)


119871


ge 119886119871

2


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


(0 1] Pos(119880 ⪯

119880

) ge 119901 if and only if 1198871198803

minus119886119880

2

ge (119901minus1)(120579119880

+

120572119880

)


119880


3

ge 119886119880

2


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

)




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


119888119895

isin



119887119894

isin [[120572

119887

119871

119894

120572

119887119871

119894

] [120572

119887119877

119894

120572

119887

119877

119894

]] isin R (28)


10908070605040302010

Mem

bers

hip

func

tion

bL3 aL2

Pos[ALle B

L]


119871


119871

10908070605040302010

Mem

bers

hip

func

tion

bU3 aU2

Pos[AUle B

U]


119880


119880

and 119894 isin N119899


119894

) is

119887119894

= int

119887

119894isinR

[int119906isin119869

119887119894

1119906]

119887119894

119894 isin N119899

119869119887

119894

sube [0 1] (29)


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)


119894

and 120572119887119877

119894


119894119895


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)


119894119895

) is

119894119895

= int

119894119895isinR

[int119906isin119869

119894119895

1119906]

119894119895

119894 isin N119899

119895 isin N119898

119869

119894119895

sube [0 1]

(33)

Here 119894119895


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


isin 119865(R) 119909119895

ge 0 119894 isin N119899

119895 isin N119898


119895=1

119894119895

119909119895




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


119894


119899

119895=1

119888119895

119909119895


119894119895

R rarr [0 1] and 120583

119887

119894


119899

119895 isin N119898


and 119887119894


119883


120583

119883

=

min 120583

119875

(11

1199091


119909119899

1198871

) 120583

119875

(1198981

1199091

+ sdot sdot sdot + 119898119899

119909119899

119887119898

) if 119909119895

ge 0 119895 = 1 2 119899

0 otherwise(37)



[]120572



119894119895

and 119887119894



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





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)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St 119871

1198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯Pos

119887

119880

119894

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(40)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St Pos (1198711198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯119887

119880

119894

)

ge 119901

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(41)


(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



) 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)


) and (119871119875119875119901

) are 119885119875119901

and 119885

119875

119901

respectively


) 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


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


) 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


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


119885

119875

119901


119901


of [119885119875119901

119885

119875

119901

]




1



Max 119885 = 1198881

1199091

+ 1198882

1199092

St 11

1199091

+ 12

1199092

⪯1198871

Table 1


1

1198752


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


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)


119901

119885119875

119901


8 Conclusion




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





References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










When 119886119871



119886119880



120572

= [120572 (119886119871

minus (119886119871

minus 120572)) + (119886119871

minus 120572)

minus 120572 ((119886119880

+ 120573) minus 119886119880

) + (119886119880

+ 120573)]

(5)


+ 120573)


Let = (119886119871

119886119880

120572 120573) and = (119887119871

119887119880


= (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)




119860

(119909 119906) as follows

= ((119909 119906) 120583

119860


sube [0 1] 0

le 120583

119860

(119909 119906) le 1

(7)

where 119869119909


int

119906isin119869

119909

120583

119860

(119909 119906)

119906

(8)



119860

(119909 119906) If all 120583

119860



= int

119909isin119883

int

119906isin119869

119909

1

(119909 119906)

= int

119909isin119883

[int119906isin119869

119909

1119906]

119909

(9)



119906isin119869

119909



FOU () = ⋃

119909isin119883

119869119909

= ⋃

119909isin119883

(119909 119906) | 119906 isin 119869119909

sube [0 1] (10)


119860


119860

respectively


= (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

and 119886119880119895+1




sup 120583

119860

(119909) = sup 120583

119860

= 1 (12)





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


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


119880

2

119886119880

3

] is the core of 119880 and 120572119880

120573119880


119871 and


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


119871

119860119880

] = [(119886119871

2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880

)] Obviously if 1198861198712

= 119886119871

3

and 119886119880

2

= 119886119880

3




119909


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)


119871

119860119880


119888119887



0

120583(x)

hU = hL = 1




119880 of PnIT2FS

the boundary of 119860119871120572

and 119860119880

120572


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)


120572

119888119887

120583

119860

isin [[120572

119886119871

120572

119886119871

] [120572

119886119877

120572

119886119877

]] (16)


120572

119886119871

le120572

119886119871

le120572

119886119877

le120572

119886119877

(17)


119860119880

) = ((119886119871

2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880


119860

(119909) = 120583

119860

= 0 forall119909 gt 0


119860119880

) = ((119886119871

2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880


119860

(119909) = 120583

119860

= 0 forall119909 lt 0


119860119880

) = ((119886119871

2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880


=

119887119871

2

= 0 1198861198713

= 119887119871

3

= 0 120572119871

= 120574119871

= 120573119871

= 120579119871

= 120572119880

= 120574119880

= 0 and120573119880

= 120579119880

= 0


2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880

)) and = ((119887119871

2

119887119871

3

120574119871

120579119871

) (119887119880

2

119887119880

3

120574119880

120579119880


2

= 119887119871

2

1198861198713

= 119887119871

3

120572119871

= 120574119871

120573119871

= 120579119871

120572119880

= 120574119880

and 120573119880

= 120579119880


0

120583(x)

hU = hL = 1

x120572aR120572aL 120572aR

120572aL

120572119888119887 A



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


= ((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


= ([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)


2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880


= 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)




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


Pos (119860119871 le 119861119871

) = max(1


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)


Pos (119860119880 le 119861119880

) = max(1


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)


2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880

))

and = ((119887119871

2

119887119871

3

120574119871

120579119871

) (119887119880

2

119887119880

3

120574119880

120579119880


119871


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)


Pos (119860119880 ⪯ 119861119880

) ge 120572 iff 120572119886119871 le 120572119887119877

(24)


2

119886119871

3

120572119871

120573119871

)(119886119880

2

119886119880

3

120572119880

120573119880

)) and = ((119887119871

2

119887119871

3

120574119871

120579119871

) (119887119880

2

119887119880

3

120574119880

120579119880


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


(0 1] Pos(119871 ⪯

119871

) ge 119901 if and only if 1198871198713

minus 119886119871

2

ge (119901 minus 1)(120579119871

+

120572119871

)


119871


ge 119886119871

2


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


(0 1] Pos(119880 ⪯

119880

) ge 119901 if and only if 1198871198803

minus119886119880

2

ge (119901minus1)(120579119880

+

120572119880

)


119880


3

ge 119886119880

2


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

)




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


119888119895

isin



119887119894

isin [[120572

119887

119871

119894

120572

119887119871

119894

] [120572

119887119877

119894

120572

119887

119877

119894

]] isin R (28)


10908070605040302010

Mem

bers

hip

func

tion

bL3 aL2

Pos[ALle B

L]


119871


119871

10908070605040302010

Mem

bers

hip

func

tion

bU3 aU2

Pos[AUle B

U]


119880


119880

and 119894 isin N119899


119894

) is

119887119894

= int

119887

119894isinR

[int119906isin119869

119887119894

1119906]

119887119894

119894 isin N119899

119869119887

119894

sube [0 1] (29)


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)


119894

and 120572119887119877

119894


119894119895


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)


119894119895

) is

119894119895

= int

119894119895isinR

[int119906isin119869

119894119895

1119906]

119894119895

119894 isin N119899

119895 isin N119898

119869

119894119895

sube [0 1]

(33)

Here 119894119895


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


isin 119865(R) 119909119895

ge 0 119894 isin N119899

119895 isin N119898


119895=1

119894119895

119909119895




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


119894


119899

119895=1

119888119895

119909119895


119894119895

R rarr [0 1] and 120583

119887

119894


119899

119895 isin N119898


and 119887119894


119883


120583

119883

=

min 120583

119875

(11

1199091


119909119899

1198871

) 120583

119875

(1198981

1199091

+ sdot sdot sdot + 119898119899

119909119899

119887119898

) if 119909119895

ge 0 119895 = 1 2 119899

0 otherwise(37)



[]120572



119894119895

and 119887119894



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





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)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St 119871

1198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯Pos

119887

119880

119894

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(40)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St Pos (1198711198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯119887

119880

119894

)

ge 119901

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(41)


(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



) 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)


) and (119871119875119875119901

) are 119885119875119901

and 119885

119875

119901

respectively


) 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


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


) 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


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


119885

119875

119901


119901


of [119885119875119901

119885

119875

119901

]




1



Max 119885 = 1198881

1199091

+ 1198882

1199092

St 11

1199091

+ 12

1199092

⪯1198871

Table 1


1

1198752


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


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)


119901

119885119875

119901


8 Conclusion




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





References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











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


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


119880

2

119886119880

3

] is the core of 119880 and 120572119880

120573119880


119871 and


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


119871

119860119880

] = [(119886119871

2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880

)] Obviously if 1198861198712

= 119886119871

3

and 119886119880

2

= 119886119880

3




119909


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)


119871

119860119880


119888119887



0

120583(x)

hU = hL = 1




119880 of PnIT2FS

the boundary of 119860119871120572

and 119860119880

120572


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)


120572

119888119887

120583

119860

isin [[120572

119886119871

120572

119886119871

] [120572

119886119877

120572

119886119877

]] (16)


120572

119886119871

le120572

119886119871

le120572

119886119877

le120572

119886119877

(17)


119860119880

) = ((119886119871

2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880


119860

(119909) = 120583

119860

= 0 forall119909 gt 0


119860119880

) = ((119886119871

2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880


119860

(119909) = 120583

119860

= 0 forall119909 lt 0


119860119880

) = ((119886119871

2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880


=

119887119871

2

= 0 1198861198713

= 119887119871

3

= 0 120572119871

= 120574119871

= 120573119871

= 120579119871

= 120572119880

= 120574119880

= 0 and120573119880

= 120579119880

= 0


2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880

)) and = ((119887119871

2

119887119871

3

120574119871

120579119871

) (119887119880

2

119887119880

3

120574119880

120579119880


2

= 119887119871

2

1198861198713

= 119887119871

3

120572119871

= 120574119871

120573119871

= 120579119871

120572119880

= 120574119880

and 120573119880

= 120579119880


0

120583(x)

hU = hL = 1

x120572aR120572aL 120572aR

120572aL

120572119888119887 A



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


= ((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


= ([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)


2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880


= 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)




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


Pos (119860119871 le 119861119871

) = max(1


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)


Pos (119860119880 le 119861119880

) = max(1


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)


2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880

))

and = ((119887119871

2

119887119871

3

120574119871

120579119871

) (119887119880

2

119887119880

3

120574119880

120579119880


119871


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)


Pos (119860119880 ⪯ 119861119880

) ge 120572 iff 120572119886119871 le 120572119887119877

(24)


2

119886119871

3

120572119871

120573119871

)(119886119880

2

119886119880

3

120572119880

120573119880

)) and = ((119887119871

2

119887119871

3

120574119871

120579119871

) (119887119880

2

119887119880

3

120574119880

120579119880


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


(0 1] Pos(119871 ⪯

119871

) ge 119901 if and only if 1198871198713

minus 119886119871

2

ge (119901 minus 1)(120579119871

+

120572119871

)


119871


ge 119886119871

2


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


(0 1] Pos(119880 ⪯

119880

) ge 119901 if and only if 1198871198803

minus119886119880

2

ge (119901minus1)(120579119880

+

120572119880

)


119880


3

ge 119886119880

2


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

)




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


119888119895

isin



119887119894

isin [[120572

119887

119871

119894

120572

119887119871

119894

] [120572

119887119877

119894

120572

119887

119877

119894

]] isin R (28)


10908070605040302010

Mem

bers

hip

func

tion

bL3 aL2

Pos[ALle B

L]


119871


119871

10908070605040302010

Mem

bers

hip

func

tion

bU3 aU2

Pos[AUle B

U]


119880


119880

and 119894 isin N119899


119894

) is

119887119894

= int

119887

119894isinR

[int119906isin119869

119887119894

1119906]

119887119894

119894 isin N119899

119869119887

119894

sube [0 1] (29)


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)


119894

and 120572119887119877

119894


119894119895


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)


119894119895

) is

119894119895

= int

119894119895isinR

[int119906isin119869

119894119895

1119906]

119894119895

119894 isin N119899

119895 isin N119898

119869

119894119895

sube [0 1]

(33)

Here 119894119895


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


isin 119865(R) 119909119895

ge 0 119894 isin N119899

119895 isin N119898


119895=1

119894119895

119909119895




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


119894


119899

119895=1

119888119895

119909119895


119894119895

R rarr [0 1] and 120583

119887

119894


119899

119895 isin N119898


and 119887119894


119883


120583

119883

=

min 120583

119875

(11

1199091


119909119899

1198871

) 120583

119875

(1198981

1199091

+ sdot sdot sdot + 119898119899

119909119899

119887119898

) if 119909119895

ge 0 119895 = 1 2 119899

0 otherwise(37)



[]120572



119894119895

and 119887119894



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





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)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St 119871

1198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯Pos

119887

119880

119894

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(40)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St Pos (1198711198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯119887

119880

119894

)

ge 119901

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(41)


(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



) 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)


) and (119871119875119875119901

) are 119885119875119901

and 119885

119875

119901

respectively


) 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


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


) 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


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


119885

119875

119901


119901


of [119885119875119901

119885

119875

119901

]




1



Max 119885 = 1198881

1199091

+ 1198882

1199092

St 11

1199091

+ 12

1199092

⪯1198871

Table 1


1

1198752


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


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)


119901

119885119875

119901


8 Conclusion




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





References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

120583(x)

hU = hL = 1

x120572aR120572aL 120572aR

120572aL

120572119888119887 A



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


= ((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


= ([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)


2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880


= 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)




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


Pos (119860119871 le 119861119871

) = max(1


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)


Pos (119860119880 le 119861119880

) = max(1


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)


2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880

))

and = ((119887119871

2

119887119871

3

120574119871

120579119871

) (119887119880

2

119887119880

3

120574119880

120579119880


119871


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)


Pos (119860119880 ⪯ 119861119880

) ge 120572 iff 120572119886119871 le 120572119887119877

(24)


2

119886119871

3

120572119871

120573119871

)(119886119880

2

119886119880

3

120572119880

120573119880

)) and = ((119887119871

2

119887119871

3

120574119871

120579119871

) (119887119880

2

119887119880

3

120574119880

120579119880


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


(0 1] Pos(119871 ⪯

119871

) ge 119901 if and only if 1198871198713

minus 119886119871

2

ge (119901 minus 1)(120579119871

+

120572119871

)


119871


ge 119886119871

2


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


(0 1] Pos(119880 ⪯

119880

) ge 119901 if and only if 1198871198803

minus119886119880

2

ge (119901minus1)(120579119880

+

120572119880

)


119880


3

ge 119886119880

2


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

)




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


119888119895

isin



119887119894

isin [[120572

119887

119871

119894

120572

119887119871

119894

] [120572

119887119877

119894

120572

119887

119877

119894

]] isin R (28)


10908070605040302010

Mem

bers

hip

func

tion

bL3 aL2

Pos[ALle B

L]


119871


119871

10908070605040302010

Mem

bers

hip

func

tion

bU3 aU2

Pos[AUle B

U]


119880


119880

and 119894 isin N119899


119894

) is

119887119894

= int

119887

119894isinR

[int119906isin119869

119887119894

1119906]

119887119894

119894 isin N119899

119869119887

119894

sube [0 1] (29)


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)


119894

and 120572119887119877

119894


119894119895


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)


119894119895

) is

119894119895

= int

119894119895isinR

[int119906isin119869

119894119895

1119906]

119894119895

119894 isin N119899

119895 isin N119898

119869

119894119895

sube [0 1]

(33)

Here 119894119895


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


isin 119865(R) 119909119895

ge 0 119894 isin N119899

119895 isin N119898


119895=1

119894119895

119909119895




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


119894


119899

119895=1

119888119895

119909119895


119894119895

R rarr [0 1] and 120583

119887

119894


119899

119895 isin N119898


and 119887119894


119883


120583

119883

=

min 120583

119875

(11

1199091


119909119899

1198871

) 120583

119875

(1198981

1199091

+ sdot sdot sdot + 119898119899

119909119899

119887119898

) if 119909119895

ge 0 119895 = 1 2 119899

0 otherwise(37)



[]120572



119894119895

and 119887119894



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





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)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St 119871

1198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯Pos

119887

119880

119894

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(40)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St Pos (1198711198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯119887

119880

119894

)

ge 119901

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(41)


(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



) 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)


) and (119871119875119875119901

) are 119885119875119901

and 119885

119875

119901

respectively


) 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


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


) 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


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


119885

119875

119901


119901


of [119885119875119901

119885

119875

119901

]




1



Max 119885 = 1198881

1199091

+ 1198882

1199092

St 11

1199091

+ 12

1199092

⪯1198871

Table 1


1

1198752


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


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)


119901

119885119875

119901


8 Conclusion




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





References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Pos (119860119880 le 119861119880

) = max(1


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)


2

119886119871

3

120572119871

120573119871

) (119886119880

2

119886119880

3

120572119880

120573119880

))

and = ((119887119871

2

119887119871

3

120574119871

120579119871

) (119887119880

2

119887119880

3

120574119880

120579119880


119871


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)


Pos (119860119880 ⪯ 119861119880

) ge 120572 iff 120572119886119871 le 120572119887119877

(24)


2

119886119871

3

120572119871

120573119871

)(119886119880

2

119886119880

3

120572119880

120573119880

)) and = ((119887119871

2

119887119871

3

120574119871

120579119871

) (119887119880

2

119887119880

3

120574119880

120579119880


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


(0 1] Pos(119871 ⪯

119871

) ge 119901 if and only if 1198871198713

minus 119886119871

2

ge (119901 minus 1)(120579119871

+

120572119871

)


119871


ge 119886119871

2


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


(0 1] Pos(119880 ⪯

119880

) ge 119901 if and only if 1198871198803

minus119886119880

2

ge (119901minus1)(120579119880

+

120572119880

)


119880


3

ge 119886119880

2


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

)




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


119888119895

isin



119887119894

isin [[120572

119887

119871

119894

120572

119887119871

119894

] [120572

119887119877

119894

120572

119887

119877

119894

]] isin R (28)


10908070605040302010

Mem

bers

hip

func

tion

bL3 aL2

Pos[ALle B

L]


119871


119871

10908070605040302010

Mem

bers

hip

func

tion

bU3 aU2

Pos[AUle B

U]


119880


119880

and 119894 isin N119899


119894

) is

119887119894

= int

119887

119894isinR

[int119906isin119869

119887119894

1119906]

119887119894

119894 isin N119899

119869119887

119894

sube [0 1] (29)


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)


119894

and 120572119887119877

119894


119894119895


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)


119894119895

) is

119894119895

= int

119894119895isinR

[int119906isin119869

119894119895

1119906]

119894119895

119894 isin N119899

119895 isin N119898

119869

119894119895

sube [0 1]

(33)

Here 119894119895


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


isin 119865(R) 119909119895

ge 0 119894 isin N119899

119895 isin N119898


119895=1

119894119895

119909119895




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


119894


119899

119895=1

119888119895

119909119895


119894119895

R rarr [0 1] and 120583

119887

119894


119899

119895 isin N119898


and 119887119894


119883


120583

119883

=

min 120583

119875

(11

1199091


119909119899

1198871

) 120583

119875

(1198981

1199091

+ sdot sdot sdot + 119898119899

119909119899

119887119898

) if 119909119895

ge 0 119895 = 1 2 119899

0 otherwise(37)



[]120572



119894119895

and 119887119894



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





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)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St 119871

1198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯Pos

119887

119880

119894

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(40)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St Pos (1198711198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯119887

119880

119894

)

ge 119901

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(41)


(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



) 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)


) and (119871119875119875119901

) are 119885119875119901

and 119885

119875

119901

respectively


) 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


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


) 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


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


119885

119875

119901


119901


of [119885119875119901

119885

119875

119901

]




1



Max 119885 = 1198881

1199091

+ 1198882

1199092

St 11

1199091

+ 12

1199092

⪯1198871

Table 1


1

1198752


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


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)


119901

119885119875

119901


8 Conclusion




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





References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10908070605040302010

Mem

bers

hip

func

tion

bL3 aL2

Pos[ALle B

L]


119871


119871

10908070605040302010

Mem

bers

hip

func

tion

bU3 aU2

Pos[AUle B

U]


119880


119880

and 119894 isin N119899


119894

) is

119887119894

= int

119887

119894isinR

[int119906isin119869

119887119894

1119906]

119887119894

119894 isin N119899

119869119887

119894

sube [0 1] (29)


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)


119894

and 120572119887119877

119894


119894119895


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)


119894119895

) is

119894119895

= int

119894119895isinR

[int119906isin119869

119894119895

1119906]

119894119895

119894 isin N119899

119895 isin N119898

119869

119894119895

sube [0 1]

(33)

Here 119894119895


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


isin 119865(R) 119909119895

ge 0 119894 isin N119899

119895 isin N119898


119895=1

119894119895

119909119895




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


119894


119899

119895=1

119888119895

119909119895


119894119895

R rarr [0 1] and 120583

119887

119894


119899

119895 isin N119898


and 119887119894


119883


120583

119883

=

min 120583

119875

(11

1199091


119909119899

1198871

) 120583

119875

(1198981

1199091

+ sdot sdot sdot + 119898119899

119909119899

119887119898

) if 119909119895

ge 0 119895 = 1 2 119899

0 otherwise(37)



[]120572



119894119895

and 119887119894



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





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)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St 119871

1198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯Pos

119887

119880

119894

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(40)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St Pos (1198711198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯119887

119880

119894

)

ge 119901

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(41)


(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



) 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)


) and (119871119875119875119901

) are 119885119875119901

and 119885

119875

119901

respectively


) 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


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


) 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


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


119885

119875

119901


119901


of [119885119875119901

119885

119875

119901

]




1



Max 119885 = 1198881

1199091

+ 1198882

1199092

St 11

1199091

+ 12

1199092

⪯1198871

Table 1


1

1198752


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


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)


119901

119885119875

119901


8 Conclusion




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





References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[]120572



119894119895

and 119887119894



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





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)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St 119871

1198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯Pos

119887

119880

119894

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(40)


MaxMin 119885 =

119899

sum

119895=1

119888119895

119909119895

St Pos (1198711198941

1199091

+ 119871

1198942

1199092


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


119894119899

119909119899

⪯119887

119880

119894

)

ge 119901

119894 = 1 2 119898

119909119895

ge 0 119895 = 1 2 119899

(41)


(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



) 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)


) and (119871119875119875119901

) are 119885119875119901

and 119885

119875

119901

respectively


) 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


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


) 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


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


119885

119875

119901


119901


of [119885119875119901

119885

119875

119901

]




1



Max 119885 = 1198881

1199091

+ 1198882

1199092

St 11

1199091

+ 12

1199092

⪯1198871

Table 1


1

1198752


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


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)


119901

119885119875

119901


8 Conclusion




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





References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











) 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)


) and (119871119875119875119901

) are 119885119875119901

and 119885

119875

119901

respectively


) 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


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


) 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


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


119885

119875

119901


119901


of [119885119875119901

119885

119875

119901

]




1



Max 119885 = 1198881

1199091

+ 1198882

1199092

St 11

1199091

+ 12

1199092

⪯1198871

Table 1


1

1198752


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


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)


119901

119885119875

119901


8 Conclusion




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





References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











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





References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article linear programming problem with interval type...

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