research article the knowledge of expert opinion in

Research ArticleThe Knowledge of Expert Opinion in Intuitionistic FuzzyLinear Programming Problem

A Nagoorgani1 J Kavikumar2 and K Ponnalagu3

1PG amp Research Department of Mathematics Jamal Mohamed College (Autonomous) Tiruchirappalli Tamil Nadu 620 020 India2Department of Mathematics Faculty of Science Technology and Human Development Universiti Tun Hussein Onn Malaysia86400 Johor Malaysia3Department of Mathematics Sri Krishna Arts and Science College (Autonomous) Coimbatore Tamil Nadu 641 008 India

Correspondence should be addressed to A Nagoorgani ganijmcyahoocoin

Received 26 March 2015 Revised 30 June 2015 Accepted 2 July 2015

Academic Editor Yan-Jun Liu

Copyright copy 2015 A Nagoorgani et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In real life information available for certain situations is vague and such uncertainty is unavoidable One possible solution is toconsider the knowledge of experts on the parameters involved as intuitionistic fuzzy data We examine a linear programmingproblem in which all the coefficients are intuitionistic in nature An approach is presented to solve an intuitionistic fuzzy linearprogramming problem In this proposed approach a procedure for allocating limited resources effectively among competingdemands is developed An example is given to highlight the illustrated study

1 Introduction

A linear programming problem is a mathematical programin which the objective function is linear and the constraintsconsist of linear equalities and linear inequalities The firstand most fruitful industrial applications of linear program-ming can be found in the petroleum industry including oilextraction refining blending and distribution The compu-tational task is then to devise an algorithm for these systemsto choose the best schedule of actions from among thepossible alternatives Some people learn to make such selec-tions via intuitive processes In some cases making the rightselection is mainly a problem of organizing and interpret-ing facts Linear programming has proved to be extremelyuseful for solving certain types of industrial problems forit provides a precise way of using statements of limitationssuch as ldquonot more thanrdquo and ldquonot less thanrdquo in mathematicalcomputations When applying OR methods to industrialproblems for instance the problems to be modelled andsolved are normally quite clear cut well described and crispThey can generally be modelled and solved by using classicallinear programmingmethods If uncertainty occurs it can beproperly modelled using fuzzy theory

The linear programming problems in which at least onecoefficient is a fuzzy number when one or more coefficientsof linear programming problems have uncertain values areknown as fuzzy linear programming problems (FLPP) Itis regarded to treat uncertainty of optimization problemssuch as fuzzy data envelopment analysis and fuzzy networkoptimization [1 2] The fuzzy linear programming problemsin which all the parameters as well as the variables arerepresented by fuzzy numbers are known as fully fuzzy linearprogramming problems The main advantage of fuzzy linearprogramming problems compared to the crisp problem for-mulation is the fact that the decision maker is not forced intoa precise formulation Over the past decades solving fuzzylinear programming has become one of the fundamentalresearch subjects in the field of fuzzy sets and systems Fuzzylinear programming applications in real world situations arenumerous and diverse

Though fuzzy optimization formulations are more flex-ible one of the poorly studied problems in this field is thedefinition of membership degrees Fuzzy set theory has beenwidely developed and various modifications have been doneOut of several higher order fuzzy sets intuitionistic fuzzy sets(IFS) have been found to be highly useful in dealing with

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 875460 8 pageshttpdxdoiorg1011552015875460

2 Mathematical Problems in Engineering

vagueness Here the degrees of satisfaction and rejection areconsidered so that the sum of both the values is always lessthan or equal to one The concept of IFS was viewed as analternative approach for imprecise data Therefore consider-ing nonmembership function as the complement ofmember-ship function developed Intuitionistic Fuzzy Optimization(IFO) problems The main advantage of IFO problems isthat they are given the richest apparatus for the formulationof optimization problems and the solution of IFO problemssatisfies the objective with a higher degree of determinacycompared to the fuzzy and crisp cases In order to avoidunrealistic modelling the use of intuitionistic fuzzy linearprogramming problem (IFLPP) can be recommended Theirapplication implies that the problems will be solved in aninteractive way In this paper we consider a problem in whichall coefficients and variables are intuitionistic fuzzy triangularnumbers in nature In this manner we want to propose a newmatrix-analysis method to improve the efficiency of solvinglarge-scale IFLPP which will reduce the number of steps inthe classical simplex method The basic idea of this methodis to arrange IFLPP data in matrix form and solve for variousdeterminants to obtain the optimum solution of IFLPP Toillustrate the proposed method numerical example is solvedand the obtained result is discussed

The paper is organized as follows We present the worksrelated to finding the optimal solution of an intuitionisticfuzzy linear programming problem (IFLPP) in Section 2Section 3 provides preliminary background on intuitionisticfuzzy sets (IFS) intuitionistic fuzzy numbers (IFN) andIFLPP The procedure for the proposed method is describedin Section 4 An illustrative example is explained briefly inSection 5 Finally conclusions are presented in Section 6

2 Related Works

The research towards uncertain systems has attracted a lotof attention [3] especially the adaptive control of linear andnonlinear systems with completely unknown functions Thefuzzy logic systems (FLS) the neural networks (NN) andthe fuzzy-neural networks (FNN) are very effective tools forcontrolling uncertainty systems [4] As an application theFLS NN and FNN have been widely used in the area of sys-tem modeling [5] fuzzy control [6] and fuzzy optimizationproblems [7] Fuzzy control which directly uses fuzzy rules isthe most important application in fuzzy theory In a practicalsituation sometimes it is quite difficult to obtain an optimalsolution for fuzzy optimization problems the use of fuzzycontrols helps to find better solutions by the decision makerin order to terminatemathematical programming algorithms[8] The intuitionistic fuzzy set theory is an extension ofthe fuzzy set theory by Atanassov [9] and intuitionisticfuzzy linear programming problem (IFLPP) is a specialtype of fuzzy linear programming problem (FLPP) Thereare lots of articles in this area which cannot be reviewedcompletely and only a few of them are reviewed hereInterval valued intuitionistic fuzzy sets were first introducedby Atanassov and Gargov [10] since then there has beenmany types of intuitionistic fuzzy numbers (IFN) addressedsuch as interval valued intuitionistic fuzzy numbers (IVIFN)

triangular intuitionistic fuzzy numbers (TIFNs) and trape-zoidal intuitionistic fuzzy numbers Mahapatra and Roy[11] discussed briefly intuitionistic fuzzy numbers and theirarithmetic operations The arithmetic operations and logicoperations of triangular intuitionistic fuzzy numbers havebeen addressed by Wang et al [12] Ranking of intuitionisticfuzzy numbers plays a vital role in practical problems andso Li [13] developed a new ranking method based on theconcept of a ratio of the index of the ambiguity index Thearticle by Wu and Chiclana [14] describes new score andaccuracy functions for interval valued intuitionistic fuzzynumbers A ranking procedure for triangular intuitionisticfuzzy numbers was developed by Wan and Dong [15] andits applications to multiattribute decision making was alsogiven Evaluation and ranking of fuzzy quantities were dealtwith by Anzilli et al [16] The concept of the FLP was firstproposed by Tanaka et al [17] which were based on theconcept of decision analysis in fuzzy environment by Bellmanand Zadeh [18] Zimmermann [19 20] introduced fuzzy setsin operations research and presented a fuzzy approach tomultiobjective linear programming problems A new conceptof the optimization problem under uncertainty was proposedand treated in [21] On the other hand Zhu and Xu [22]developed a fuzzy linear programming method to deal withgroup decision-making problems The optimal solution forseveral degrees of feasibility of fuzzy linear and nonlinearprogramming problems was given by Mohtashami [23] Areal life multiobjective linear programming problem wastaken into an intuitionistic fuzzy environment and solved byNishad and Singh [24] Moreover Ye [25] proposed a linearprogramming model to solve interval valued intuitionisticmulticriteria decision-making problems Li [26] used intervalvalued intuitionistic fuzzy sets to capture fuzziness in linearprogramming

Motivated by these articles we proposed a study on thesolutions of intuitionistic fuzzy linear programming prob-lem (IFLPP) The classical simplex method requires muchiteration to solve IFLPP To overcome this limitation a newmatrix-analysis method is proposed in this paper The IFLPPis represented inmatrix format and variousmatrix operationsare performed to obtain the optimum solution

3 Preliminaries

In this section the basic notations and definitions are pre-sented We start by defining an intuitionistic fuzzy set

31 Intuitionistic Fuzzy Set (IFS) Given a fixed set 119883 = 11990911199092 1199093 119909119899 an intuitionistic fuzzy set is defined as 119860 =

(⟨119909119894 120583119860(119909119894) ]119860(119909119894)⟩ | 119909119894isin 119883) which assigns to each element

119909119894amembership degree 120583

119860(119909119894) and a nonmembership degree

]119860(119909119894) under the condition 0 le 120583

119860(119909119894) + ]119860(119909119894) le 1 for all

119909119894isin 119883

32 Intuitionistic Fuzzy Number (IFN) An intuitionisticfuzzy number 119860119868 is

(i) an intuitionistic fuzzy subset of the real line

Mathematical Problems in Engineering 3

(ii) normal that is there is some 1199090 isin R such that120583119860119868(1199090) = 1 ]

119860119868(1199090) = 0

(iii) convex for the membership function 120583119860119868(119909) that is

120583119860119868(1205821199091+(1minus120582)1199092) ge min(120583

119860119868(1199091) 120583119860119868(1199092)) for every

1199091 1199092 isin R 120582 isin [0 1](iv) concave for the nonmembership function ]

119860119868(119909) that

is ]119860119868(1205821199091 + (1 minus 120582)1199092) ge max(]

119860119868(1199091) ]119860119868(1199092)) for

every 1199091 1199092 isin R 120582 isin [0 1]

33 Triangular Intuitionistic Fuzzy Number (TIFN) A trian-gular intuitionistic fuzzy number119860119868 is an intuitionistic fuzzyset in R with the following membership function 120583

119860119868(119909) and

nonmembership function ]119860119868(119909) (Figure 1)

120583119860119868 =

119909 minus 11988611198862 minus 1198861

1198861 le 119909 le 1198862

119909 minus 11988631198862 minus 1198863

1198862 le 119909 le 1198863

0 otherwise

]119860119868 =

1198862 minus 119909

1198862 minus 1198861015840

1

1198861015840

1le 119909 le 1198862

119909 minus 11988621198861015840

3minus 1198862

1198862 le 119909 le 1198861015840

3

1 otherwise

(1)

where 1198861015840

1le 1198861 le 1198862 le 1198863 le 119886

1015840

3and 120583

119860119868(119909) + ]

119860119868 le 1 or

120583119860119868(119909) = ]

119860119868(119909) forall119909 isin R This TIFN is denoted by

119860119868

= (1198861 1198862 1198863 1198861015840

1 1198862 1198861015840

3)

= (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840

3)

(2)

34 Arithmetic Operations Arithmetic operations of tri-angular intuitionistic fuzzy number based on (120572 120573)-cutsmethod are as follows

(i) If 119860119868 = (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840

3) and 119861

119868

= (1198871 1198872 1198873)

(1198871015840

1 1198872 1198871015840

3) are two TIFNs then their sum is119860119868 +119861

119868

=

(1198861+1198871 1198862+1198872 1198863+1198873) (1198861015840

1+1198871015840

1 1198862+1198872 119886

1015840

3+1198871015840

3)which

is also a TIFN(ii) If 119860119868 = (1198861 1198862 1198863) (119886

1015840

1 1198862 1198861015840

3) and 119861

119868

= (1198871 1198872 1198873)

(1198871015840

1 1198872 1198871015840

3) are two TIFNs then their difference is119860119868minus

119861119868

= (1198861 minus1198873 1198862 minus1198872 1198863 minus1198871) (1198861015840

1minus1198871015840

3 1198862 minus1198872 119886

1015840

3minus1198871015840

1)

which is also a TIFN(iii) If 119860119868 = (1198861 1198862 1198863) (119886

1015840

1 1198862 1198861015840

3) and 119861

119868

= (1198871 1198872 1198873)

(1198871015840

1 1198872 1198871015840

3) are two TIFNs then their product is 119860119868 times

119861119868

= (11988611198871 11988621198872 11988631198873) (1198861015840

11198871015840

1 11988621198872 119886

1015840

31198871015840

3)which is also

a TIFN(iv) If 119860119868 = (1198861 1198862 1198863) (119886

1015840

1 1198862 1198861015840

3) is a TIFN and 119910 =

119896119886 (with 119896 gt 0) then 119910119868

= 119896119860119868

= (1198961198861 1198961198862 1198961198863)(1198961198861015840

1 1198961198862 119896119886

1015840

3) is also a TIFN

(v) If 119860119868 = (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840


119896119886 (with 119896 lt 0) then 119910119868

= 119896119860119868

= (1198961198863 1198961198862 1198961198861)(1198961198861015840

3 1198961198862 119896119886

1015840

1) is also a TIFN

0

05

1

120583

a998400

1a1 a2 a3 a

998400

3 x

120583A119868

A119868

Figure 1 Membership and nonmembership functions of TIFN

(vi) If 119860119868 = (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840

3) and 119861

119868

= (1198871 1198872 1198873)

(1198871015840

1 1198872 1198871015840

3) are two positive TIFNs then 119860

119868

119861119868 is

also a TIFN where 119860119868

119861119868

= (11988611198873 11988621198872 11988631198871)(1198861015840

11198871015840

3 11988621198872 119886

1015840

31198871015840

1)

35 Score Function and Accuracy Function Let 119860119868 = (11988611198862 1198863) (119886

1015840

1 1198862 1198861015840

3) be a TIFN then we define a score function

for membership and nonmembership as follows

119878 (119860119868120572

) =1198861 + 21198862 + 1198863

4

119878 (119861119868120573

) =1198861015840

1 + 21198862 + 1198861015840

34

(3)

Let 119860119868 = (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840

3) be a TIFN then

119867(119860119868

) =

(1198861 + 21198862 + 1198863) + (1198861015840

1 + 21198862 + 1198861015840

3)

8(4)

is an accuracy function of 119860119868 which is used to defuzzify thegiven number

36 Ranking Using Score Function Let 119860119868 = (1198861 1198862 1198863) (1198861015840

1

1198862 1198861015840

3) and 119861119868

= (1198871 1198872 1198873) (1198871015840

1 1198872 1198871015840

3) be two TIFNs Let(119878(119860119868120572

) 119878(119860119868120573

)) and (119878(119861119868120572

) 119878(119861119868120573

)) be the scores of 119860119868 and119861119868 respectively Then consider the following

(i) If 119878(119860119868120572) le 119878(119861119868120572

) and 119878(119860119868120573

) le 119878(119861119868120573

) then 119860119868

lt 119861119868

(ii) If 119878(119860119868120572) ge 119878(119861119868120572

) and 119878(119860119868120573

) ge 119878(119861119868120573

) then 119860119868

gt 119861119868

(iii) If 119878(119860119868120572) = 119878(119861119868120572

) and 119878(119860119868120573

) = 119878(119861119868120573

) then119860119868

= 119861119868

37 Intuitionistic Fuzzy Linear Programming Problem (IFLPP)A linear programming with triangular intuitionistic fuzzyvariables is defined as

(IFLP)max 119885119868

=

119899

sum

119895=1119888119868

119895119909119868

119895 (5)

Subject to119899

sum

119895=1119886119868

119894119895119909119868

119895le 119868

119894 (6)

119909119868

119895ge 0 119894 = 1 2 119898 (7)


where119860119868 = (119886119868

119894119895) 119888119868119895 119868119894 and 119909119868

119895are (119898times119899) (1times119899) (119898times1) and

(119899 times 1) intuitionistic fuzzy matrices consisting of triangularintuitionistic fuzzy numbers (TIFNs)

38 Representation of IFLPP in Matrix Form Consider theIFLPP defined in Section 37 The matrix notation of IFLPPis defined as follows

119909119868

1 119909119868

2 119909119868

3 sdot sdot sdot 119909119868

119899

((((

(

119888119868

1

119886119868

11

119886119868

21

119886119868

1198981

119888119868

2

119886119868

12

119886119868

22

119886119868

1198982

119888119868

3

119886119868

13

119886119868

23

119886119868

1198983

sdot sdot sdot

sdot sdot sdot

sdot sdot sdot

d

sdot sdot sdot

119888119868

119899

1198861119899

1198862119899

119886119898119899

119868

119894

119868

1

119868

2

119868

119898

)))))

)

(8)

39 Intuitionistic Fuzzy Basic Solution A basic solution to(6) is a solution obtained by setting any 119899 intuitionistic fuzzyvariables (among 119898 + 119899 variables) equal to zero and solv-ing remaining 119898 intuitionistic fuzzy variables provided thedeterminant of the coefficient of these 119898 intuitionistic fuzzyvariables is nonzero Such 119898 intuitionistic fuzzy variables(any of them may be zero) are called basic variables andremaining 119899 zero variables are called nonbasic variables Thenumber of intuitionistic fuzzy basic solutions obtainedwill beat most (119898 + 119899)119862

119898= (119898 + 119899)119899119898 which is the number of

combinations of 119899 + 119898 things taken119898 at a time

310 Intuitionistic Fuzzy Basic Feasible Solution An intu-itionistic fuzzy basic feasible solution is an intuitionistic fuzzybasic solution which also satisfies (7) that is all the basicvariables are nonnegative

311 Intuitionistic Fuzzy OptimumFeasible Solution Let119883119868 =(119909119868

1 119909119868

2 119909119868

3 119909119868

119899) be the set of all intuitionistic fuzzy feasible

solutions of (5) An intuitionistic fuzzy feasible solution 119909119868

0 isin

119883119868 is said to be an intuitionistic fuzzy optimum solution to

(5) if

119888119868

119909119868

0 ge 119888119868

119909119868

forall119909119868

isin 119883119868

(9)

where 119888119868 = (119888119868

1 119888119868

2 119888119868

3 119888119868

119899) and 119888

119868

119909119868

= 119888119868

1119909119868

1 + 119888119868

2119909119868

2 + 119888119868

3119909119868

3 +

sdot sdot sdot + 119888119868

119899119909119868

119899

Theorem 1 The set of all feasible solutions to the intuitionisticfuzzy linear programming problem is a convex set

Proof Weneed to show that every convex combination of anytwo feasible solutions is also a feasible solution The theoremis true if the set has only one element Assume that there areat least two solutions such as 1198831198681 = (119909

119868

1 119909119868

2 119909119868

3 119909119868

119899) and

119883119868

2 = (119910119868

1 119910119868

2 119910119868

3 119910119868

119899) Consider the system defined in (6)

as 119860119868119883119868 le 119868

Since1198831198681 and119883119868

2 are solutions we have 119860119868

119883119868

1 =

119868 for 1198831198681 ge 0 and 119860

119868

119883119868

2 = 119868 for 1198831198682 ge 0 For 0 le 120572 le 1

let 119883119868 = 120572119883119868

1 + (1 minus 120572)119883119868

2 be any convex combination of 1198831198681

and 119883119868

2 We note that all the elements of the vector 119883119868 arenonnegative Now 119860

119868

119883119868

= 119860119868

(120572119883119868

1 + (1 minus 120572)119883119868

2) = 119860119868

120572119883119868

1 +

119860119868

119883119868

2 minus 119860119868

120572119883119868

2 = 120572119868

+ 119868

minus 120572119868

= 119868 This completes the

proof

4 Methodology

This matrix-analysis method is the development of an ideaor an approach to programming It changes the form of theproblem information in such a way so as to make possibledirect logical comparisons of alternatives This greatly sim-plifies calculation

41 Steps for Setting Up and Solving a Problem The use ofmatrix-analysis method involves a number of steps whichfollow a definite patternThe steps in the order in which theyshould be taken are given below

411 Step 1 Set Up the Unknowns The unknowns are theamounts of each product to be made from the availableamount of productive capacity to maximize profits Theunknowns represent the answers that are wanted They willhave either positive or zero values

412 Step 2 Form a Matrix for Analysis The first step is toarrange the problem data in a table or matrix to facilitatesolution which is a matrix consisting of the coefficients ofa family of related equations and inequations The matrix isconstructed as defined in Section 38

413 Step 3 Unitize the Rows of the Matrix The purpose ofthis step is to set up each resource as unity This is done bydividing the numbers in each row by their constant value atthe end of the row When this process is completed the timeto make one unit of a product is expressed as a fraction of thecapacity This calculation does not affect the profit per unitfor each product These computations in no way change thevalidity of the inequations in the first matrix They state thesame fact in different ways The unitized matrix provides away of comparing the requirements for each product in termsof the capacity of each department

414 Step 4 Equalize the Columns of the Matrix Since theobjective is to earn as much as profit as possible the profitpossibilities of each product must be set up and comparedThe process of setting up and converting the information to acomparable basis is termed as equalizing The comparisonsof profit are accomplished by establishing groups of eachproduct selected so that the profit on each group is the sameas that of a group of any other productTherefore wemultiplyeach column by the least common multiple of the profitsand the matrix is now said to be a unitized equalized matrixHowever now that direct comparison is possible facts can befound and compared easily compared to the first matrix

415 Step 5 Find the First Key Number The next stepis to determine whether it is more profitable to make acombination of products instead of just one product First


select the largest number in each column Secondly selectthe smallest of these largest numbers Hereafter the smallestof the largest number is known as the first key number Therow and the column in which this number is found will beknown respectively as the first key row and the first keycolumnThere may be other key numbers in the matrix If sothey will be called the second key number and the third keynumber The rows and columns of the second and the thirdkey numbers if any will be called the second and the thirdkey rows and key columns respectively

416 Step 6 Rearrange the Matrix Based on Key NumberThe sequence of the rows and columns are changed buteach number is kept in its row and column The first keynumber is put in the upper left hand corner of the matrixThis immediately establishes the restriction for the first rowand for the first columnThe second key number is placed onerowdown and one column to the rightThere is a requirementto be met in this arrangement that is the numbers to theimmediate right of each key numbermust be smaller than thekey number The next step in the procedure is to see whetherany other key number can be added to this group of figuresThis can be done as it was done before

417 Step 7 Check for Profit Potential If 119898 constraints and119899 unknowns are given in Step 1 consider 119899 equations with 119899

unknowns from the rearranged matrix Solve this system of119899 times 119899 equations and the solution for this system of equationsgives the optimum result

418 Step 8 Set Up a Profit Table This step is the interpreta-tion of the results of the previous steps The equalized groupscan be converted to actual number of units by multiplyingtheir respective factors of least commonmultiples which wasconsidered in Step 4 The development of a profit table wouldbe used as a guide for management decisions

5 Numerical Example

A construction site engineer has been assigned the task ofmoving huge debris onto a field from the site The siteengineer has been informed that only a maximum of 100119868

vehicles are allowed each to be used for a single round tripThere are two types of vehicles available to the site engineera Caterpillar articulated truck that can carry 24119868 cubic metersand a Howo dump truck that can carry 16119868 cubic meters Thearticulated truck is estimated to use 50119868 L of fuel on a roundtrip whereas the dump truck will use only 25119868 L of fuel perround trip The project is granted with a total of 4000119868 L offuel The vehicles will require maintenance after each roundtrip each dump truck requires 3119868 hrs and each articulatedtruck requires 9119868 hrs The maintenance wing has 720119868 hrsavailable Tomove amaximum amount of cubic meters whatmix of articulated trucks and dump trucks should the siteengineer choose

The data is given as triangular intuitionistic fuzzy num-bers as follows

100119868 = (99 100 101) (95 100 102)

24119868 = (23 24 25) (22 24 26)

16119868 = (15 16 17) (14 16 18)

50119868 = (48 50 51) (47 50 52)

25119868 = (24 25 26) (23 25 28)

3119868 = (2 3 4) (1 3 5)

720119868 = (715 720 723) (711 720 725)

9119868 = (8 9 10) (7 9 11)

1119868 = (05 1 15) (04 1 18)

1119868 = (09 1 11) (08 1 12)

4000119868 = (3995 4000 4010) (3990 4000 4015)

(10)

SolutionLet 1199091198681 be the number of articulated trucks to be usedLet 1199091198682 be the number of dump trucks to be usedThe objective function is to maximize the volume trans-

portedMathematical formulation of the given problem is as

follows

Max 119868 = 241198681199091198681 + 161198681199091198682 (11)

subject to the following constraints

(119860) fuel limit 501198681199091198681 + 251198681199091198682 le 4000119868

(119861) maintenance hour limit 31198681199091198681 + 91198681199091198682 le 720119868

(119862) number limit on vehicles 11198681199091198681 + 11198681199091198682 le 100119868

non-negativity of variables 1199091198681 119909119868

2 ge 0

(12)

where

119888119868

1 = 24119868 = (23 24 25) (22 24 26)

119888119868

2 = 16119868 = (15 16 17) (14 16 18)

119886119868

11 = 50119868 = (48 50 51) (47 50 52)

119886119868

12 = 25119868 = (24 25 26) (23 25 28)


119886119868

21 = 3119868 = (2 3 4) (1 3 5)

119886119868

22 = 9119868 = (8 9 10) (7 9 11)

119886119868

31 = 1119868 = (05 1 15) (04 1 18)

119886119868

32 = 1119868 = (09 1 11) (08 1 12)

119868

1 = 4000119868

= (3995 4000 4010) (3990 4000 4015)

119868

2 = 720119868 = (715 720 723) (711 720 725)

119868

3 = 100119868 = (99 100 101) (95 100 102) (13)

Step 1 Arrange the elements in a matrix form as follows

119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

50119868

3119868

1119868

16119868

25119868

9119868

1119868

4000119868

720119868

100119868

)

)

(14)

Step 2 Unitize the rows of the matrixDivide the numbers in each row by their constant value

at the end of the row

119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

00125119868

00042119868

001119868

16119868

00063119868

00125119868

001119868

1119868

1119868

1119868

)

)

(15)

The elements of first row are

119886119868

11 =50119868

4000119868= 00125

119868

= (00119 00125 00127)

(00117 00125 00130)

119886119868

12 =25119868

4000119868= 00063

119868

= (00059 00063 00065)

(00057 00063 00070)

119886119868

13 =4000119868

4000119868= 1119868 = (09963 1 10037)

(09937 1 10062)

(16)

The elements of second row are

119886119868

21 =3119868

720119868= 00042

119868

= (00027 00042 00055)

(00013 00042 00070)

119886119868

22 =9119868

720119868= 00125

119868

= (00110 00125 00139)

(00096 00125 00154)

119886119868

23 =720119868

720119868= 1119868 = (09889 1 10111)

(09806 1 10196)

(17)

The elements of third row are

119886119868

31 =1119868

100119868= 001119868

= (00049 001 00151) (00039 001 00189)

119886119868

32 =1119868

100119868= 001119868

= (00089 001 00111) (00078 001 00126)

119886119868

33 =100119868

100119868= 1119868

= (09801 1 10202) (09313 1 10736)

(18)

Step 3 Equalize the columns of the matrix

Multiply second column by 15

119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

00125119868

00042119868

001119868

24119868

00094119868

00187119868

0015119868

1119868

1119868

1119868

)

)

(19)

The elements of the second column are

24119868 = (225 24 255) (21 24 27)

119886119868

12 = 00094119868

= (00088 00094 00097)

(00085 00094 00105)

119886119868

22 = 00187119868

= (00165 00187 00208)

(00144 00187 00231)

119886119868

32 = 0015119868

= (00133 0015 00166)

(00117 0015 00189)

(20)


Table 1

Departments Availability Maximum quantity utilized by vehiclesArticulated truck 119909

119868

1 Dump truck 119909119868

2 TotalFuel 4000

119868

L 3000119868

L 1000119868

L 4000119868

LMaintenance hours 720

119868

hrs 180119868

hrs 360119868

hrs 540119868

hrsNumber of vehicles 100

119868

60119868 40119868 100119868

Equalized profit units 5968119868

2688119868

Actual units produced 60119868 40119868 2080119868

Maximum amount of load transferred (in cubic meters)

Step 4 Find the key element

Min (max element)

= Minimum of maximum element in each column

= min (00125119868

00187119868

) = 00125119868

(21)

The numbers are compared using score function given in thispaper

Now 00125119868

is the key element or pivot elementThe row corresponding to the key element (first row) is

the key row and the column corresponding to the key element(first column) is the key column

Step 5 Rearrange the system based on pivot element

119909119868

1 119909119868

2

119888119868

119895

119860

119862

119861

(

(

24119868

00125119868

001119868

00042119868

24119868

00094119868

0015119868

00187119868

1119868

1119868

1119868

)

)

(22)

Step 6 Consider the 2 times 2 system of equations

00125119868

119909119868

1119890 + 00094119868

119909119868

2119890 = 1119868

0011198681199091198681119890 + 0015119868

119909119868

2119890 = 1119868(23)

Solving these two equations we get the followingEqualized profit units

119909119868

1119890 = 5968119868

119909119868

2119890 = 2688119868

(24)

Actual units produced

119909119868

1119890 = 5968119868

119909119868

2119890 = 15times 2688119868

= 4032119868

(25)

where 5968119868

= (2047 5968 15814) (minus3567 596824392) and 4032

119868

= (2767 4032 6063) (1568 403212742)

This indicates that approximately 5968119868

asymp 60119868 articulatedtrucks and 4032

119868

asymp 40119868 dump trucks can be chosen by the siteengineer to move a maximum amount of 2080119868 cubic meters

Step 7 Set up a profit tableThis step interprets the results of the previous steps in

table form (Table 1)

6 Discussion and Conclusion

This algorithm can be an aid in a wide area of decisionmaking Its usefulness covers a broad range of problems Aspecific problem in which a number of different demandscompete for limited amounts of resources will demonstratethis technique The objective is to meet as many of thesedemands as possible so as to be most beneficial over all

The matrix-analysis method will quickly provide an opti-mum solution and greatly reduce the number of steps in theclassical simplex methodThis method starts with the benefitthat will bring the largest overall returns the simplex methodstarts with the benefit that brings the largest unit profitTherewill be cases where it is impractical or perhaps impossible tosolve the matrix using matrix-analysis method As shown bythe sample problem the matrix-analysis method of IFLPPcan be used to aid management decisions The solutionobtained through this method gives complete operating andprofit information for management use It is a formal logicalapproach for making the best decision when alternatives andchoices exist In future work the efficiency of this process canbe greatly improved by detecting and removing redundantinequalities

Conflict of Interests

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

Acknowledgments

The authors would like to thank to the editor and anonymousreferees for various suggestionswhich have led to an improve-ment in both the quality and clarity of the paper


References

[1] M Wen and H Li ldquoFuzzy data envelopment analysis (DEA)model and ranking methodrdquo Journal of Computational andApplied Mathematics vol 223 no 2 pp 872ndash878 2009

[2] C Kahraman T Ertay and G Buyukozkan ldquoA fuzzy optimiza-tion model for QFD planning process using analytic networkapproachrdquo European Journal of Operational Research vol 171no 2 pp 390ndash411 2006

[3] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014

[4] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013

[5] W-J Wang and W-W Lin ldquoDecentralized PDC for large-scaleT-S fuzzy systemsrdquo IEEE Transactions on Fuzzy Systems vol 13no 6 pp 779ndash786 2005

[6] W-J Wang Y-J Chen and C-H Sun ldquoRelaxed stabilizationcriteria for discrete-time TndashS fuzzy control systems based ona switching fuzzy model and piecewise Lyapunov functionrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 37 no 3 pp 551ndash559 2007

[7] S Effati M Pakdaman and M Ranjbar ldquoA new fuzzy neuralnetwork model for solving fuzzy linear programming problemsand its applicationsrdquo Neural Computing and Applications vol20 no 8 pp 1285ndash1294 2011

[8] M Delgado J L Verdegay and M A Vila ldquoFuzzy linearprogramming from classical methods to new applicationsrdquo inFuzzy Optimization Recent Advances M Delgado J KacprzykJ L Verdegay and M A Vila Eds pp 111ndash134 Physica 1994

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

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

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

[12] J-Q Wang R Nie H-Y Zhang and X-H Chen ldquoNew oper-ators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[13] D-F Li ldquoA ratio ranking method of triangular intuitionisticfuzzy numbers and its application to MADM problemsrdquo Com-puters and Mathematics with Applications vol 60 no 6 pp1557ndash1570 2010

[14] J Wu and F Chiclana ldquoA risk attitudinal ranking method forinterval-valued intuitionistic fuzzy numbers based on novelattitudinal expected score and accuracy functionsrdquoApplied SoftComputing vol 22 pp 272ndash286 2014

[15] S-P Wan and J-Y Dong ldquoPossibility method for triangularintuitionistic fuzzy multi-attribute group decision making withincomplete weight informationrdquo International Journal of Com-putational Intelligence Systems vol 7 no 1 pp 65ndash79 2014

[16] L Anzilli G Facchinetti and G Mastroleo ldquoEvaluation ofranking of intuitionistic fuzzy quantitiesrdquo in Fuzzy Logic ampApplications vol 8256 pp 139ndash149 Springer 2013

[17] H Tanaka T Okuda and K Asai ldquoOn fuzzy-mathematicalprogrammingrdquo Journal of Cybernetics vol 3 no 4 pp 37ndash461973

[18] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 no 4 pp B141ndashB1641970

[19] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978

[20] H-J Zimmerman ldquoUsing fuzzy sets in operational researchrdquoEuropean Journal of Operational Research vol 13 no 3 pp 201ndash216 1983

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

[22] B Zhu and Z Xu ldquoA fuzzy linear programming method forgroup decision making with additive reciprocal fuzzy prefer-ence relationsrdquo Fuzzy Sets and Systems vol 246 pp 19ndash33 2014

[23] A Mohtashami ldquoThe optimal solution for several differentdegrees of feasibility for fuzzy linear and non-linear program-ming problemsrdquo Journal of Intelligent amp Fuzzy Systems vol 27no 5 pp 2611ndash2622 2014

[24] A K Nishad and S R Singh ldquoSolving multi-objective decisionmaking problem in intuitionistic fuzzy environmentrdquo Interna-tional Journal of System Assurance Engineering and Manage-ment vol 6 no 2 pp 206ndash215 2015

[25] J Ye ldquoA linear programming method based on an improvedscore function for interval-valued intuitionistic fuzzy multicri-teria decision makingrdquo The Engineering Economist vol 58 no3 pp 179ndash188 2013

[26] D-F Li ldquoLinear programming method for MADM withinterval-valued intuitionistic fuzzy setsrdquo Expert Systems withApplications vol 37 no 8 pp 5939ndash5945 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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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


vagueness Here the degrees of satisfaction and rejection areconsidered so that the sum of both the values is always lessthan or equal to one The concept of IFS was viewed as analternative approach for imprecise data Therefore consider-ing nonmembership function as the complement ofmember-ship function developed Intuitionistic Fuzzy Optimization(IFO) problems The main advantage of IFO problems isthat they are given the richest apparatus for the formulationof optimization problems and the solution of IFO problemssatisfies the objective with a higher degree of determinacycompared to the fuzzy and crisp cases In order to avoidunrealistic modelling the use of intuitionistic fuzzy linearprogramming problem (IFLPP) can be recommended Theirapplication implies that the problems will be solved in aninteractive way In this paper we consider a problem in whichall coefficients and variables are intuitionistic fuzzy triangularnumbers in nature In this manner we want to propose a newmatrix-analysis method to improve the efficiency of solvinglarge-scale IFLPP which will reduce the number of steps inthe classical simplex method The basic idea of this methodis to arrange IFLPP data in matrix form and solve for variousdeterminants to obtain the optimum solution of IFLPP Toillustrate the proposed method numerical example is solvedand the obtained result is discussed

The paper is organized as follows We present the worksrelated to finding the optimal solution of an intuitionisticfuzzy linear programming problem (IFLPP) in Section 2Section 3 provides preliminary background on intuitionisticfuzzy sets (IFS) intuitionistic fuzzy numbers (IFN) andIFLPP The procedure for the proposed method is describedin Section 4 An illustrative example is explained briefly inSection 5 Finally conclusions are presented in Section 6

2 Related Works

The research towards uncertain systems has attracted a lotof attention [3] especially the adaptive control of linear andnonlinear systems with completely unknown functions Thefuzzy logic systems (FLS) the neural networks (NN) andthe fuzzy-neural networks (FNN) are very effective tools forcontrolling uncertainty systems [4] As an application theFLS NN and FNN have been widely used in the area of sys-tem modeling [5] fuzzy control [6] and fuzzy optimizationproblems [7] Fuzzy control which directly uses fuzzy rules isthe most important application in fuzzy theory In a practicalsituation sometimes it is quite difficult to obtain an optimalsolution for fuzzy optimization problems the use of fuzzycontrols helps to find better solutions by the decision makerin order to terminatemathematical programming algorithms[8] The intuitionistic fuzzy set theory is an extension ofthe fuzzy set theory by Atanassov [9] and intuitionisticfuzzy linear programming problem (IFLPP) is a specialtype of fuzzy linear programming problem (FLPP) Thereare lots of articles in this area which cannot be reviewedcompletely and only a few of them are reviewed hereInterval valued intuitionistic fuzzy sets were first introducedby Atanassov and Gargov [10] since then there has beenmany types of intuitionistic fuzzy numbers (IFN) addressedsuch as interval valued intuitionistic fuzzy numbers (IVIFN)

triangular intuitionistic fuzzy numbers (TIFNs) and trape-zoidal intuitionistic fuzzy numbers Mahapatra and Roy[11] discussed briefly intuitionistic fuzzy numbers and theirarithmetic operations The arithmetic operations and logicoperations of triangular intuitionistic fuzzy numbers havebeen addressed by Wang et al [12] Ranking of intuitionisticfuzzy numbers plays a vital role in practical problems andso Li [13] developed a new ranking method based on theconcept of a ratio of the index of the ambiguity index Thearticle by Wu and Chiclana [14] describes new score andaccuracy functions for interval valued intuitionistic fuzzynumbers A ranking procedure for triangular intuitionisticfuzzy numbers was developed by Wan and Dong [15] andits applications to multiattribute decision making was alsogiven Evaluation and ranking of fuzzy quantities were dealtwith by Anzilli et al [16] The concept of the FLP was firstproposed by Tanaka et al [17] which were based on theconcept of decision analysis in fuzzy environment by Bellmanand Zadeh [18] Zimmermann [19 20] introduced fuzzy setsin operations research and presented a fuzzy approach tomultiobjective linear programming problems A new conceptof the optimization problem under uncertainty was proposedand treated in [21] On the other hand Zhu and Xu [22]developed a fuzzy linear programming method to deal withgroup decision-making problems The optimal solution forseveral degrees of feasibility of fuzzy linear and nonlinearprogramming problems was given by Mohtashami [23] Areal life multiobjective linear programming problem wastaken into an intuitionistic fuzzy environment and solved byNishad and Singh [24] Moreover Ye [25] proposed a linearprogramming model to solve interval valued intuitionisticmulticriteria decision-making problems Li [26] used intervalvalued intuitionistic fuzzy sets to capture fuzziness in linearprogramming

Motivated by these articles we proposed a study on thesolutions of intuitionistic fuzzy linear programming prob-lem (IFLPP) The classical simplex method requires muchiteration to solve IFLPP To overcome this limitation a newmatrix-analysis method is proposed in this paper The IFLPPis represented inmatrix format and variousmatrix operationsare performed to obtain the optimum solution

3 Preliminaries

In this section the basic notations and definitions are pre-sented We start by defining an intuitionistic fuzzy set

31 Intuitionistic Fuzzy Set (IFS) Given a fixed set 119883 = 11990911199092 1199093 119909119899 an intuitionistic fuzzy set is defined as 119860 =

(⟨119909119894 120583119860(119909119894) ]119860(119909119894)⟩ | 119909119894isin 119883) which assigns to each element

119909119894amembership degree 120583

119860(119909119894) and a nonmembership degree

]119860(119909119894) under the condition 0 le 120583

119860(119909119894) + ]119860(119909119894) le 1 for all

119909119894isin 119883

32 Intuitionistic Fuzzy Number (IFN) An intuitionisticfuzzy number 119860119868 is

(i) an intuitionistic fuzzy subset of the real line



119860119868(1199090) = 0


120583119860119868(1205821199091+(1minus120582)1199092) ge min(120583

119860119868(1199091) 120583119860119868(1199092)) for every


119860119868(119909) that

is ]119860119868(1205821199091 + (1 minus 120582)1199092) ge max(]

119860119868(1199091) ]119860119868(1199092)) for

every 1199091 1199092 isin R 120582 isin [0 1]


119860119868(119909) and


120583119860119868 =

119909 minus 11988611198862 minus 1198861

1198861 le 119909 le 1198862

119909 minus 11988631198862 minus 1198863

1198862 le 119909 le 1198863

0 otherwise

]119860119868 =

1198862 minus 119909

1198862 minus 1198861015840

1

1198861015840

1le 119909 le 1198862

119909 minus 11988621198861015840

3minus 1198862

1198862 le 119909 le 1198861015840

3

1 otherwise

(1)

where 1198861015840

1le 1198861 le 1198862 le 1198863 le 119886

1015840

3and 120583

119860119868(119909) + ]

119860119868 le 1 or

120583119860119868(119909) = ]


119860119868

= (1198861 1198862 1198863 1198861015840

1 1198862 1198861015840

3)

= (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840

3)

(2)


(i) If 119860119868 = (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840

3) and 119861

119868

= (1198871 1198872 1198873)

(1198871015840

1 1198872 1198871015840


119868

=

(1198861+1198871 1198862+1198872 1198863+1198873) (1198861015840

1+1198871015840

1 1198862+1198872 119886

1015840

3+1198871015840

3)which


1015840

1 1198862 1198861015840

3) and 119861

119868

= (1198871 1198872 1198873)

(1198871015840

1 1198872 1198871015840


119861119868


1minus1198871015840

3 1198862 minus1198872 119886

1015840

3minus1198871015840

1)


1015840

1 1198862 1198861015840

3) and 119861

119868

= (1198871 1198872 1198873)

(1198871015840

1 1198872 1198871015840


119861119868

= (11988611198871 11988621198872 11988631198873) (1198861015840

11198871015840

1 11988621198872 119886

1015840

31198871015840

3)which is also

a TIFN(iv) If 119860119868 = (1198861 1198862 1198863) (119886

1015840

1 1198862 1198861015840


119896119886 (with 119896 gt 0) then 119910119868

= 119896119860119868

= (1198961198861 1198961198862 1198961198863)(1198961198861015840

1 1198961198862 119896119886

1015840

3) is also a TIFN

(v) If 119860119868 = (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840


119896119886 (with 119896 lt 0) then 119910119868

= 119896119860119868

= (1198961198863 1198961198862 1198961198861)(1198961198861015840

3 1198961198862 119896119886

1015840

1) is also a TIFN

0

05

1

120583

a998400

1a1 a2 a3 a

998400

3 x

120583A119868

A119868


(vi) If 119860119868 = (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840

3) and 119861

119868

= (1198871 1198872 1198873)

(1198871015840

1 1198872 1198871015840


119868

119861119868 is


119861119868

= (11988611198873 11988621198872 11988631198871)(1198861015840

11198871015840

3 11988621198872 119886

1015840

31198871015840

1)


1015840

1 1198862 1198861015840



119878 (119860119868120572

) =1198861 + 21198862 + 1198863

4

119878 (119861119868120573

) =1198861015840

1 + 21198862 + 1198861015840

34

(3)

Let 119860119868 = (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840

3) be a TIFN then

119867(119860119868

) =

(1198861 + 21198862 + 1198863) + (1198861015840

1 + 21198862 + 1198861015840

3)

8(4)



1

1198862 1198861015840

3) and 119861119868

= (1198871 1198872 1198873) (1198871015840

1 1198872 1198871015840

3) be two TIFNs Let(119878(119860119868120572

) 119878(119860119868120573

)) and (119878(119861119868120572

) 119878(119861119868120573


(i) If 119878(119860119868120572) le 119878(119861119868120572

) and 119878(119860119868120573

) le 119878(119861119868120573

) then 119860119868

lt 119861119868

(ii) If 119878(119860119868120572) ge 119878(119861119868120572

) and 119878(119860119868120573

) ge 119878(119861119868120573

) then 119860119868

gt 119861119868

(iii) If 119878(119860119868120572) = 119878(119861119868120572

) and 119878(119860119868120573

) = 119878(119861119868120573

) then119860119868

= 119861119868


(IFLP)max 119885119868

=

119899

sum

119895=1119888119868

119895119909119868

119895 (5)

Subject to119899

sum

119895=1119886119868

119894119895119909119868

119895le 119868

119894 (6)

119909119868

119895ge 0 119894 = 1 2 119898 (7)


where119860119868 = (119886119868

119894119895) 119888119868119895 119868119894 and 119909119868




119909119868

1 119909119868

2 119909119868


119899

((((

(

119888119868

1

119886119868

11

119886119868

21

119886119868

1198981

119888119868

2

119886119868

12

119886119868

22

119886119868

1198982

119888119868

3

119886119868

13

119886119868

23

119886119868

1198983

sdot sdot sdot

sdot sdot sdot

sdot sdot sdot

d

sdot sdot sdot

119888119868

119899

1198861119899

1198862119899

119886119898119899

119868

119894

119868

1

119868

2

119868

119898

)))))

)

(8)






1 119909119868

2 119909119868

3 119909119868



0 isin


(5) if

119888119868

119909119868

0 ge 119888119868

119909119868

forall119909119868

isin 119883119868

(9)

where 119888119868 = (119888119868

1 119888119868

2 119888119868

3 119888119868

119899) and 119888

119868

119909119868

= 119888119868

1119909119868

1 + 119888119868

2119909119868

2 + 119888119868

3119909119868

3 +


119899119909119868

119899



119868

1 119909119868

2 119909119868

3 119909119868

119899) and

119883119868

2 = (119910119868

1 119910119868

2 119910119868

3 119910119868


as 119860119868119883119868 le 119868

Since1198831198681 and119883119868


119883119868

1 =

119868 for 1198831198681 ge 0 and 119860

119868

119883119868

2 = 119868 for 1198831198682 ge 0 For 0 le 120572 le 1

let 119883119868 = 120572119883119868

1 + (1 minus 120572)119883119868


and 119883119868


119868

119883119868

= 119860119868

(120572119883119868

1 + (1 minus 120572)119883119868

2) = 119860119868

120572119883119868

1 +

119860119868

119883119868

2 minus 119860119868

120572119883119868

2 = 120572119868

+ 119868

minus 120572119868


proof

4 Methodology














5 Numerical Example




100119868 = (99 100 101) (95 100 102)

24119868 = (23 24 25) (22 24 26)

16119868 = (15 16 17) (14 16 18)

50119868 = (48 50 51) (47 50 52)

25119868 = (24 25 26) (23 25 28)

3119868 = (2 3 4) (1 3 5)

720119868 = (715 720 723) (711 720 725)

9119868 = (8 9 10) (7 9 11)

1119868 = (05 1 15) (04 1 18)

1119868 = (09 1 11) (08 1 12)

4000119868 = (3995 4000 4010) (3990 4000 4015)

(10)



follows

Max 119868 = 241198681199091198681 + 161198681199091198682 (11)


(119860) fuel limit 501198681199091198681 + 251198681199091198682 le 4000119868




2 ge 0

(12)

where

119888119868

1 = 24119868 = (23 24 25) (22 24 26)

119888119868

2 = 16119868 = (15 16 17) (14 16 18)

119886119868

11 = 50119868 = (48 50 51) (47 50 52)

119886119868

12 = 25119868 = (24 25 26) (23 25 28)


119886119868

21 = 3119868 = (2 3 4) (1 3 5)

119886119868

22 = 9119868 = (8 9 10) (7 9 11)

119886119868

31 = 1119868 = (05 1 15) (04 1 18)

119886119868

32 = 1119868 = (09 1 11) (08 1 12)

119868

1 = 4000119868

= (3995 4000 4010) (3990 4000 4015)

119868

2 = 720119868 = (715 720 723) (711 720 725)

119868

3 = 100119868 = (99 100 101) (95 100 102) (13)


119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

50119868

3119868

1119868

16119868

25119868

9119868

1119868

4000119868

720119868

100119868

)

)

(14)



119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

00125119868

00042119868

001119868

16119868

00063119868

00125119868

001119868

1119868

1119868

1119868

)

)

(15)


119886119868

11 =50119868

4000119868= 00125

119868

= (00119 00125 00127)

(00117 00125 00130)

119886119868

12 =25119868

4000119868= 00063

119868

= (00059 00063 00065)

(00057 00063 00070)

119886119868

13 =4000119868

4000119868= 1119868 = (09963 1 10037)

(09937 1 10062)

(16)


119886119868

21 =3119868

720119868= 00042

119868

= (00027 00042 00055)

(00013 00042 00070)

119886119868

22 =9119868

720119868= 00125

119868

= (00110 00125 00139)

(00096 00125 00154)

119886119868

23 =720119868

720119868= 1119868 = (09889 1 10111)

(09806 1 10196)

(17)


119886119868

31 =1119868

100119868= 001119868

= (00049 001 00151) (00039 001 00189)

119886119868

32 =1119868

100119868= 001119868

= (00089 001 00111) (00078 001 00126)

119886119868

33 =100119868

100119868= 1119868

= (09801 1 10202) (09313 1 10736)

(18)



119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

00125119868

00042119868

001119868

24119868

00094119868

00187119868

0015119868

1119868

1119868

1119868

)

)

(19)


24119868 = (225 24 255) (21 24 27)

119886119868

12 = 00094119868

= (00088 00094 00097)

(00085 00094 00105)

119886119868

22 = 00187119868

= (00165 00187 00208)

(00144 00187 00231)

119886119868

32 = 0015119868

= (00133 0015 00166)

(00117 0015 00189)

(20)


Table 1


119868

1 Dump truck 119909119868

2 TotalFuel 4000

119868

L 3000119868

L 1000119868

L 4000119868


119868

hrs 180119868

hrs 360119868

hrs 540119868


119868

60119868 40119868 100119868


2688119868




Min (max element)


= min (00125119868

00187119868

) = 00125119868

(21)


Now 00125119868




119909119868

1 119909119868

2

119888119868

119895

119860

119862

119861

(

(

24119868

00125119868

001119868

00042119868

24119868

00094119868

0015119868

00187119868

1119868

1119868

1119868

)

)

(22)


00125119868

119909119868

1119890 + 00094119868

119909119868

2119890 = 1119868

0011198681199091198681119890 + 0015119868

119909119868

2119890 = 1119868(23)


119909119868

1119890 = 5968119868

119909119868

2119890 = 2688119868

(24)


119909119868

1119890 = 5968119868

119909119868

2119890 = 15times 2688119868

= 4032119868

(25)

where 5968119868

= (2047 5968 15814) (minus3567 596824392) and 4032

119868

= (2767 4032 6063) (1568 403212742)



119868









Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119860119868(1199090) = 0


120583119860119868(1205821199091+(1minus120582)1199092) ge min(120583

119860119868(1199091) 120583119860119868(1199092)) for every


119860119868(119909) that

is ]119860119868(1205821199091 + (1 minus 120582)1199092) ge max(]

119860119868(1199091) ]119860119868(1199092)) for

every 1199091 1199092 isin R 120582 isin [0 1]


119860119868(119909) and


120583119860119868 =

119909 minus 11988611198862 minus 1198861

1198861 le 119909 le 1198862

119909 minus 11988631198862 minus 1198863

1198862 le 119909 le 1198863

0 otherwise

]119860119868 =

1198862 minus 119909

1198862 minus 1198861015840

1

1198861015840

1le 119909 le 1198862

119909 minus 11988621198861015840

3minus 1198862

1198862 le 119909 le 1198861015840

3

1 otherwise

(1)

where 1198861015840

1le 1198861 le 1198862 le 1198863 le 119886

1015840

3and 120583

119860119868(119909) + ]

119860119868 le 1 or

120583119860119868(119909) = ]


119860119868

= (1198861 1198862 1198863 1198861015840

1 1198862 1198861015840

3)

= (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840

3)

(2)


(i) If 119860119868 = (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840

3) and 119861

119868

= (1198871 1198872 1198873)

(1198871015840

1 1198872 1198871015840


119868

=

(1198861+1198871 1198862+1198872 1198863+1198873) (1198861015840

1+1198871015840

1 1198862+1198872 119886

1015840

3+1198871015840

3)which


1015840

1 1198862 1198861015840

3) and 119861

119868

= (1198871 1198872 1198873)

(1198871015840

1 1198872 1198871015840


119861119868


1minus1198871015840

3 1198862 minus1198872 119886

1015840

3minus1198871015840

1)


1015840

1 1198862 1198861015840

3) and 119861

119868

= (1198871 1198872 1198873)

(1198871015840

1 1198872 1198871015840


119861119868

= (11988611198871 11988621198872 11988631198873) (1198861015840

11198871015840

1 11988621198872 119886

1015840

31198871015840

3)which is also

a TIFN(iv) If 119860119868 = (1198861 1198862 1198863) (119886

1015840

1 1198862 1198861015840


119896119886 (with 119896 gt 0) then 119910119868

= 119896119860119868

= (1198961198861 1198961198862 1198961198863)(1198961198861015840

1 1198961198862 119896119886

1015840

3) is also a TIFN

(v) If 119860119868 = (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840


119896119886 (with 119896 lt 0) then 119910119868

= 119896119860119868

= (1198961198863 1198961198862 1198961198861)(1198961198861015840

3 1198961198862 119896119886

1015840

1) is also a TIFN

0

05

1

120583

a998400

1a1 a2 a3 a

998400

3 x

120583A119868

A119868


(vi) If 119860119868 = (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840

3) and 119861

119868

= (1198871 1198872 1198873)

(1198871015840

1 1198872 1198871015840


119868

119861119868 is


119861119868

= (11988611198873 11988621198872 11988631198871)(1198861015840

11198871015840

3 11988621198872 119886

1015840

31198871015840

1)


1015840

1 1198862 1198861015840



119878 (119860119868120572

) =1198861 + 21198862 + 1198863

4

119878 (119861119868120573

) =1198861015840

1 + 21198862 + 1198861015840

34

(3)

Let 119860119868 = (1198861 1198862 1198863) (1198861015840

1 1198862 1198861015840

3) be a TIFN then

119867(119860119868

) =

(1198861 + 21198862 + 1198863) + (1198861015840

1 + 21198862 + 1198861015840

3)

8(4)



1

1198862 1198861015840

3) and 119861119868

= (1198871 1198872 1198873) (1198871015840

1 1198872 1198871015840

3) be two TIFNs Let(119878(119860119868120572

) 119878(119860119868120573

)) and (119878(119861119868120572

) 119878(119861119868120573


(i) If 119878(119860119868120572) le 119878(119861119868120572

) and 119878(119860119868120573

) le 119878(119861119868120573

) then 119860119868

lt 119861119868

(ii) If 119878(119860119868120572) ge 119878(119861119868120572

) and 119878(119860119868120573

) ge 119878(119861119868120573

) then 119860119868

gt 119861119868

(iii) If 119878(119860119868120572) = 119878(119861119868120572

) and 119878(119860119868120573

) = 119878(119861119868120573

) then119860119868

= 119861119868


(IFLP)max 119885119868

=

119899

sum

119895=1119888119868

119895119909119868

119895 (5)

Subject to119899

sum

119895=1119886119868

119894119895119909119868

119895le 119868

119894 (6)

119909119868

119895ge 0 119894 = 1 2 119898 (7)


where119860119868 = (119886119868

119894119895) 119888119868119895 119868119894 and 119909119868




119909119868

1 119909119868

2 119909119868


119899

((((

(

119888119868

1

119886119868

11

119886119868

21

119886119868

1198981

119888119868

2

119886119868

12

119886119868

22

119886119868

1198982

119888119868

3

119886119868

13

119886119868

23

119886119868

1198983

sdot sdot sdot

sdot sdot sdot

sdot sdot sdot

d

sdot sdot sdot

119888119868

119899

1198861119899

1198862119899

119886119898119899

119868

119894

119868

1

119868

2

119868

119898

)))))

)

(8)






1 119909119868

2 119909119868

3 119909119868



0 isin


(5) if

119888119868

119909119868

0 ge 119888119868

119909119868

forall119909119868

isin 119883119868

(9)

where 119888119868 = (119888119868

1 119888119868

2 119888119868

3 119888119868

119899) and 119888

119868

119909119868

= 119888119868

1119909119868

1 + 119888119868

2119909119868

2 + 119888119868

3119909119868

3 +


119899119909119868

119899



119868

1 119909119868

2 119909119868

3 119909119868

119899) and

119883119868

2 = (119910119868

1 119910119868

2 119910119868

3 119910119868


as 119860119868119883119868 le 119868

Since1198831198681 and119883119868


119883119868

1 =

119868 for 1198831198681 ge 0 and 119860

119868

119883119868

2 = 119868 for 1198831198682 ge 0 For 0 le 120572 le 1

let 119883119868 = 120572119883119868

1 + (1 minus 120572)119883119868


and 119883119868


119868

119883119868

= 119860119868

(120572119883119868

1 + (1 minus 120572)119883119868

2) = 119860119868

120572119883119868

1 +

119860119868

119883119868

2 minus 119860119868

120572119883119868

2 = 120572119868

+ 119868

minus 120572119868


proof

4 Methodology














5 Numerical Example




100119868 = (99 100 101) (95 100 102)

24119868 = (23 24 25) (22 24 26)

16119868 = (15 16 17) (14 16 18)

50119868 = (48 50 51) (47 50 52)

25119868 = (24 25 26) (23 25 28)

3119868 = (2 3 4) (1 3 5)

720119868 = (715 720 723) (711 720 725)

9119868 = (8 9 10) (7 9 11)

1119868 = (05 1 15) (04 1 18)

1119868 = (09 1 11) (08 1 12)

4000119868 = (3995 4000 4010) (3990 4000 4015)

(10)



follows

Max 119868 = 241198681199091198681 + 161198681199091198682 (11)


(119860) fuel limit 501198681199091198681 + 251198681199091198682 le 4000119868




2 ge 0

(12)

where

119888119868

1 = 24119868 = (23 24 25) (22 24 26)

119888119868

2 = 16119868 = (15 16 17) (14 16 18)

119886119868

11 = 50119868 = (48 50 51) (47 50 52)

119886119868

12 = 25119868 = (24 25 26) (23 25 28)


119886119868

21 = 3119868 = (2 3 4) (1 3 5)

119886119868

22 = 9119868 = (8 9 10) (7 9 11)

119886119868

31 = 1119868 = (05 1 15) (04 1 18)

119886119868

32 = 1119868 = (09 1 11) (08 1 12)

119868

1 = 4000119868

= (3995 4000 4010) (3990 4000 4015)

119868

2 = 720119868 = (715 720 723) (711 720 725)

119868

3 = 100119868 = (99 100 101) (95 100 102) (13)


119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

50119868

3119868

1119868

16119868

25119868

9119868

1119868

4000119868

720119868

100119868

)

)

(14)



119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

00125119868

00042119868

001119868

16119868

00063119868

00125119868

001119868

1119868

1119868

1119868

)

)

(15)


119886119868

11 =50119868

4000119868= 00125

119868

= (00119 00125 00127)

(00117 00125 00130)

119886119868

12 =25119868

4000119868= 00063

119868

= (00059 00063 00065)

(00057 00063 00070)

119886119868

13 =4000119868

4000119868= 1119868 = (09963 1 10037)

(09937 1 10062)

(16)


119886119868

21 =3119868

720119868= 00042

119868

= (00027 00042 00055)

(00013 00042 00070)

119886119868

22 =9119868

720119868= 00125

119868

= (00110 00125 00139)

(00096 00125 00154)

119886119868

23 =720119868

720119868= 1119868 = (09889 1 10111)

(09806 1 10196)

(17)


119886119868

31 =1119868

100119868= 001119868

= (00049 001 00151) (00039 001 00189)

119886119868

32 =1119868

100119868= 001119868

= (00089 001 00111) (00078 001 00126)

119886119868

33 =100119868

100119868= 1119868

= (09801 1 10202) (09313 1 10736)

(18)



119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

00125119868

00042119868

001119868

24119868

00094119868

00187119868

0015119868

1119868

1119868

1119868

)

)

(19)


24119868 = (225 24 255) (21 24 27)

119886119868

12 = 00094119868

= (00088 00094 00097)

(00085 00094 00105)

119886119868

22 = 00187119868

= (00165 00187 00208)

(00144 00187 00231)

119886119868

32 = 0015119868

= (00133 0015 00166)

(00117 0015 00189)

(20)


Table 1


119868

1 Dump truck 119909119868

2 TotalFuel 4000

119868

L 3000119868

L 1000119868

L 4000119868


119868

hrs 180119868

hrs 360119868

hrs 540119868


119868

60119868 40119868 100119868


2688119868




Min (max element)


= min (00125119868

00187119868

) = 00125119868

(21)


Now 00125119868




119909119868

1 119909119868

2

119888119868

119895

119860

119862

119861

(

(

24119868

00125119868

001119868

00042119868

24119868

00094119868

0015119868

00187119868

1119868

1119868

1119868

)

)

(22)


00125119868

119909119868

1119890 + 00094119868

119909119868

2119890 = 1119868

0011198681199091198681119890 + 0015119868

119909119868

2119890 = 1119868(23)


119909119868

1119890 = 5968119868

119909119868

2119890 = 2688119868

(24)


119909119868

1119890 = 5968119868

119909119868

2119890 = 15times 2688119868

= 4032119868

(25)

where 5968119868

= (2047 5968 15814) (minus3567 596824392) and 4032

119868

= (2767 4032 6063) (1568 403212742)



119868









Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where119860119868 = (119886119868

119894119895) 119888119868119895 119868119894 and 119909119868




119909119868

1 119909119868

2 119909119868


119899

((((

(

119888119868

1

119886119868

11

119886119868

21

119886119868

1198981

119888119868

2

119886119868

12

119886119868

22

119886119868

1198982

119888119868

3

119886119868

13

119886119868

23

119886119868

1198983

sdot sdot sdot

sdot sdot sdot

sdot sdot sdot

d

sdot sdot sdot

119888119868

119899

1198861119899

1198862119899

119886119898119899

119868

119894

119868

1

119868

2

119868

119898

)))))

)

(8)






1 119909119868

2 119909119868

3 119909119868



0 isin


(5) if

119888119868

119909119868

0 ge 119888119868

119909119868

forall119909119868

isin 119883119868

(9)

where 119888119868 = (119888119868

1 119888119868

2 119888119868

3 119888119868

119899) and 119888

119868

119909119868

= 119888119868

1119909119868

1 + 119888119868

2119909119868

2 + 119888119868

3119909119868

3 +


119899119909119868

119899



119868

1 119909119868

2 119909119868

3 119909119868

119899) and

119883119868

2 = (119910119868

1 119910119868

2 119910119868

3 119910119868


as 119860119868119883119868 le 119868

Since1198831198681 and119883119868


119883119868

1 =

119868 for 1198831198681 ge 0 and 119860

119868

119883119868

2 = 119868 for 1198831198682 ge 0 For 0 le 120572 le 1

let 119883119868 = 120572119883119868

1 + (1 minus 120572)119883119868


and 119883119868


119868

119883119868

= 119860119868

(120572119883119868

1 + (1 minus 120572)119883119868

2) = 119860119868

120572119883119868

1 +

119860119868

119883119868

2 minus 119860119868

120572119883119868

2 = 120572119868

+ 119868

minus 120572119868


proof

4 Methodology














5 Numerical Example




100119868 = (99 100 101) (95 100 102)

24119868 = (23 24 25) (22 24 26)

16119868 = (15 16 17) (14 16 18)

50119868 = (48 50 51) (47 50 52)

25119868 = (24 25 26) (23 25 28)

3119868 = (2 3 4) (1 3 5)

720119868 = (715 720 723) (711 720 725)

9119868 = (8 9 10) (7 9 11)

1119868 = (05 1 15) (04 1 18)

1119868 = (09 1 11) (08 1 12)

4000119868 = (3995 4000 4010) (3990 4000 4015)

(10)



follows

Max 119868 = 241198681199091198681 + 161198681199091198682 (11)


(119860) fuel limit 501198681199091198681 + 251198681199091198682 le 4000119868




2 ge 0

(12)

where

119888119868

1 = 24119868 = (23 24 25) (22 24 26)

119888119868

2 = 16119868 = (15 16 17) (14 16 18)

119886119868

11 = 50119868 = (48 50 51) (47 50 52)

119886119868

12 = 25119868 = (24 25 26) (23 25 28)


119886119868

21 = 3119868 = (2 3 4) (1 3 5)

119886119868

22 = 9119868 = (8 9 10) (7 9 11)

119886119868

31 = 1119868 = (05 1 15) (04 1 18)

119886119868

32 = 1119868 = (09 1 11) (08 1 12)

119868

1 = 4000119868

= (3995 4000 4010) (3990 4000 4015)

119868

2 = 720119868 = (715 720 723) (711 720 725)

119868

3 = 100119868 = (99 100 101) (95 100 102) (13)


119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

50119868

3119868

1119868

16119868

25119868

9119868

1119868

4000119868

720119868

100119868

)

)

(14)



119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

00125119868

00042119868

001119868

16119868

00063119868

00125119868

001119868

1119868

1119868

1119868

)

)

(15)


119886119868

11 =50119868

4000119868= 00125

119868

= (00119 00125 00127)

(00117 00125 00130)

119886119868

12 =25119868

4000119868= 00063

119868

= (00059 00063 00065)

(00057 00063 00070)

119886119868

13 =4000119868

4000119868= 1119868 = (09963 1 10037)

(09937 1 10062)

(16)


119886119868

21 =3119868

720119868= 00042

119868

= (00027 00042 00055)

(00013 00042 00070)

119886119868

22 =9119868

720119868= 00125

119868

= (00110 00125 00139)

(00096 00125 00154)

119886119868

23 =720119868

720119868= 1119868 = (09889 1 10111)

(09806 1 10196)

(17)


119886119868

31 =1119868

100119868= 001119868

= (00049 001 00151) (00039 001 00189)

119886119868

32 =1119868

100119868= 001119868

= (00089 001 00111) (00078 001 00126)

119886119868

33 =100119868

100119868= 1119868

= (09801 1 10202) (09313 1 10736)

(18)



119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

00125119868

00042119868

001119868

24119868

00094119868

00187119868

0015119868

1119868

1119868

1119868

)

)

(19)


24119868 = (225 24 255) (21 24 27)

119886119868

12 = 00094119868

= (00088 00094 00097)

(00085 00094 00105)

119886119868

22 = 00187119868

= (00165 00187 00208)

(00144 00187 00231)

119886119868

32 = 0015119868

= (00133 0015 00166)

(00117 0015 00189)

(20)


Table 1


119868

1 Dump truck 119909119868

2 TotalFuel 4000

119868

L 3000119868

L 1000119868

L 4000119868


119868

hrs 180119868

hrs 360119868

hrs 540119868


119868

60119868 40119868 100119868


2688119868




Min (max element)


= min (00125119868

00187119868

) = 00125119868

(21)


Now 00125119868




119909119868

1 119909119868

2

119888119868

119895

119860

119862

119861

(

(

24119868

00125119868

001119868

00042119868

24119868

00094119868

0015119868

00187119868

1119868

1119868

1119868

)

)

(22)


00125119868

119909119868

1119890 + 00094119868

119909119868

2119890 = 1119868

0011198681199091198681119890 + 0015119868

119909119868

2119890 = 1119868(23)


119909119868

1119890 = 5968119868

119909119868

2119890 = 2688119868

(24)


119909119868

1119890 = 5968119868

119909119868

2119890 = 15times 2688119868

= 4032119868

(25)

where 5968119868

= (2047 5968 15814) (minus3567 596824392) and 4032

119868

= (2767 4032 6063) (1568 403212742)



119868









Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















5 Numerical Example




100119868 = (99 100 101) (95 100 102)

24119868 = (23 24 25) (22 24 26)

16119868 = (15 16 17) (14 16 18)

50119868 = (48 50 51) (47 50 52)

25119868 = (24 25 26) (23 25 28)

3119868 = (2 3 4) (1 3 5)

720119868 = (715 720 723) (711 720 725)

9119868 = (8 9 10) (7 9 11)

1119868 = (05 1 15) (04 1 18)

1119868 = (09 1 11) (08 1 12)

4000119868 = (3995 4000 4010) (3990 4000 4015)

(10)



follows

Max 119868 = 241198681199091198681 + 161198681199091198682 (11)


(119860) fuel limit 501198681199091198681 + 251198681199091198682 le 4000119868




2 ge 0

(12)

where

119888119868

1 = 24119868 = (23 24 25) (22 24 26)

119888119868

2 = 16119868 = (15 16 17) (14 16 18)

119886119868

11 = 50119868 = (48 50 51) (47 50 52)

119886119868

12 = 25119868 = (24 25 26) (23 25 28)


119886119868

21 = 3119868 = (2 3 4) (1 3 5)

119886119868

22 = 9119868 = (8 9 10) (7 9 11)

119886119868

31 = 1119868 = (05 1 15) (04 1 18)

119886119868

32 = 1119868 = (09 1 11) (08 1 12)

119868

1 = 4000119868

= (3995 4000 4010) (3990 4000 4015)

119868

2 = 720119868 = (715 720 723) (711 720 725)

119868

3 = 100119868 = (99 100 101) (95 100 102) (13)


119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

50119868

3119868

1119868

16119868

25119868

9119868

1119868

4000119868

720119868

100119868

)

)

(14)



119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

00125119868

00042119868

001119868

16119868

00063119868

00125119868

001119868

1119868

1119868

1119868

)

)

(15)


119886119868

11 =50119868

4000119868= 00125

119868

= (00119 00125 00127)

(00117 00125 00130)

119886119868

12 =25119868

4000119868= 00063

119868

= (00059 00063 00065)

(00057 00063 00070)

119886119868

13 =4000119868

4000119868= 1119868 = (09963 1 10037)

(09937 1 10062)

(16)


119886119868

21 =3119868

720119868= 00042

119868

= (00027 00042 00055)

(00013 00042 00070)

119886119868

22 =9119868

720119868= 00125

119868

= (00110 00125 00139)

(00096 00125 00154)

119886119868

23 =720119868

720119868= 1119868 = (09889 1 10111)

(09806 1 10196)

(17)


119886119868

31 =1119868

100119868= 001119868

= (00049 001 00151) (00039 001 00189)

119886119868

32 =1119868

100119868= 001119868

= (00089 001 00111) (00078 001 00126)

119886119868

33 =100119868

100119868= 1119868

= (09801 1 10202) (09313 1 10736)

(18)



119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

00125119868

00042119868

001119868

24119868

00094119868

00187119868

0015119868

1119868

1119868

1119868

)

)

(19)


24119868 = (225 24 255) (21 24 27)

119886119868

12 = 00094119868

= (00088 00094 00097)

(00085 00094 00105)

119886119868

22 = 00187119868

= (00165 00187 00208)

(00144 00187 00231)

119886119868

32 = 0015119868

= (00133 0015 00166)

(00117 0015 00189)

(20)


Table 1


119868

1 Dump truck 119909119868

2 TotalFuel 4000

119868

L 3000119868

L 1000119868

L 4000119868


119868

hrs 180119868

hrs 360119868

hrs 540119868


119868

60119868 40119868 100119868


2688119868




Min (max element)


= min (00125119868

00187119868

) = 00125119868

(21)


Now 00125119868




119909119868

1 119909119868

2

119888119868

119895

119860

119862

119861

(

(

24119868

00125119868

001119868

00042119868

24119868

00094119868

0015119868

00187119868

1119868

1119868

1119868

)

)

(22)


00125119868

119909119868

1119890 + 00094119868

119909119868

2119890 = 1119868

0011198681199091198681119890 + 0015119868

119909119868

2119890 = 1119868(23)


119909119868

1119890 = 5968119868

119909119868

2119890 = 2688119868

(24)


119909119868

1119890 = 5968119868

119909119868

2119890 = 15times 2688119868

= 4032119868

(25)

where 5968119868

= (2047 5968 15814) (minus3567 596824392) and 4032

119868

= (2767 4032 6063) (1568 403212742)



119868









Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119886119868

21 = 3119868 = (2 3 4) (1 3 5)

119886119868

22 = 9119868 = (8 9 10) (7 9 11)

119886119868

31 = 1119868 = (05 1 15) (04 1 18)

119886119868

32 = 1119868 = (09 1 11) (08 1 12)

119868

1 = 4000119868

= (3995 4000 4010) (3990 4000 4015)

119868

2 = 720119868 = (715 720 723) (711 720 725)

119868

3 = 100119868 = (99 100 101) (95 100 102) (13)


119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

50119868

3119868

1119868

16119868

25119868

9119868

1119868

4000119868

720119868

100119868

)

)

(14)



119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

00125119868

00042119868

001119868

16119868

00063119868

00125119868

001119868

1119868

1119868

1119868

)

)

(15)


119886119868

11 =50119868

4000119868= 00125

119868

= (00119 00125 00127)

(00117 00125 00130)

119886119868

12 =25119868

4000119868= 00063

119868

= (00059 00063 00065)

(00057 00063 00070)

119886119868

13 =4000119868

4000119868= 1119868 = (09963 1 10037)

(09937 1 10062)

(16)


119886119868

21 =3119868

720119868= 00042

119868

= (00027 00042 00055)

(00013 00042 00070)

119886119868

22 =9119868

720119868= 00125

119868

= (00110 00125 00139)

(00096 00125 00154)

119886119868

23 =720119868

720119868= 1119868 = (09889 1 10111)

(09806 1 10196)

(17)


119886119868

31 =1119868

100119868= 001119868

= (00049 001 00151) (00039 001 00189)

119886119868

32 =1119868

100119868= 001119868

= (00089 001 00111) (00078 001 00126)

119886119868

33 =100119868

100119868= 1119868

= (09801 1 10202) (09313 1 10736)

(18)



119909119868

1 119909119868

2

119888119868

119895

119860

119861

119862

(

(

24119868

00125119868

00042119868

001119868

24119868

00094119868

00187119868

0015119868

1119868

1119868

1119868

)

)

(19)


24119868 = (225 24 255) (21 24 27)

119886119868

12 = 00094119868

= (00088 00094 00097)

(00085 00094 00105)

119886119868

22 = 00187119868

= (00165 00187 00208)

(00144 00187 00231)

119886119868

32 = 0015119868

= (00133 0015 00166)

(00117 0015 00189)

(20)


Table 1


119868

1 Dump truck 119909119868

2 TotalFuel 4000

119868

L 3000119868

L 1000119868

L 4000119868


119868

hrs 180119868

hrs 360119868

hrs 540119868


119868

60119868 40119868 100119868


2688119868




Min (max element)


= min (00125119868

00187119868

) = 00125119868

(21)


Now 00125119868




119909119868

1 119909119868

2

119888119868

119895

119860

119862

119861

(

(

24119868

00125119868

001119868

00042119868

24119868

00094119868

0015119868

00187119868

1119868

1119868

1119868

)

)

(22)


00125119868

119909119868

1119890 + 00094119868

119909119868

2119890 = 1119868

0011198681199091198681119890 + 0015119868

119909119868

2119890 = 1119868(23)


119909119868

1119890 = 5968119868

119909119868

2119890 = 2688119868

(24)


119909119868

1119890 = 5968119868

119909119868

2119890 = 15times 2688119868

= 4032119868

(25)

where 5968119868

= (2047 5968 15814) (minus3567 596824392) and 4032

119868

= (2767 4032 6063) (1568 403212742)



119868









Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 1


119868

1 Dump truck 119909119868

2 TotalFuel 4000

119868

L 3000119868

L 1000119868

L 4000119868


119868

hrs 180119868

hrs 360119868

hrs 540119868


119868

60119868 40119868 100119868


2688119868




Min (max element)


= min (00125119868

00187119868

) = 00125119868

(21)


Now 00125119868




119909119868

1 119909119868

2

119888119868

119895

119860

119862

119861

(

(

24119868

00125119868

001119868

00042119868

24119868

00094119868

0015119868

00187119868

1119868

1119868

1119868

)

)

(22)


00125119868

119909119868

1119890 + 00094119868

119909119868

2119890 = 1119868

0011198681199091198681119890 + 0015119868

119909119868

2119890 = 1119868(23)


119909119868

1119890 = 5968119868

119909119868

2119890 = 2688119868

(24)


119909119868

1119890 = 5968119868

119909119868

2119890 = 15times 2688119868

= 4032119868

(25)

where 5968119868

= (2047 5968 15814) (minus3567 596824392) and 4032

119868

= (2767 4032 6063) (1568 403212742)



119868









Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article the knowledge of expert opinion in

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