Research ArticleThe Intelligence of Dual Simplex Method to Solve LinearFractional Fuzzy Transportation Problem

S Narayanamoorthy and S Kalyani

Department of Applied Mathematics Bharathiar University Coimbatore Tamil Nadu 641 046 India

Correspondence should be addressed to S Narayanamoorthy snm phdyahoocoin

Received 13 October 2014 Revised 30 January 2015 Accepted 4 February 2015

Academic Editor Thomas DeMarse

Copyright copy 2015 S Narayanamoorthy and S Kalyani 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

An approach is presented to solve a fuzzy transportation problem with linear fractional fuzzy objective function In this proposedapproach the fractional fuzzy transportation problem is decomposed into two linear fuzzy transportation problems The optimalsolution of the two linear fuzzy transportations is solved by dual simplex method and the optimal solution of the fractional fuzzytransportation problem is obtained The proposed method is explained in detail with an example

1 Introduction

The transportation problem refers to a special case of linearprogramming problem The basic transportation problemwas developed by Hitchcock in 1941 [1] In mathematicsand economics transportation theory is a name given to thestudy of optimal transportation and allocation of resourcesThe problem was formalized by the French MathematicianGaspard Monge Tolstoi was one of the first person to studythe transportation problem mathematically Transportationproblem deals with the distribution of single commodityfrom various sources of supply to various destinations ofdemand in such a manner that the total transportation costis minimized In order to solve a transportation problem thedecision parameters such as availability requirements and theunit transportation cost of the model must be fixed at crispvalues

When applying OR-methods to engineering problemsfor instance the problems to be modelled and solved arenormally quite clear cut well described and crisp Theycan generally be modelled and solved by using classicalmathematics which is dichotomous in character If vaguenessenters it is normally of the stochastic kindwhich can properlybemodelled by using probability theoryThis is true formanyareas such as inventory theory traffic control and schedulingin which probability theory is applied via queuing theory

One of the most important areas of the kind in which thevagueness is of a different kind compared to randomness isprobably that of decision making In decision making thehuman factor enters with all its vagueness of perceptionsubjectivity and attitudes of goals and of conceptions Intransportation problem supply demand and unit transporta-tion cost may be uncertain due to several factors theseimprecise data may be represented by fuzzy numbers Whenthe demand supply or the unit cost of the transportationproblem takes a fuzzy value then the optimum value of thatproblem is also fuzzy so it becomes a fuzzy transportationproblem

Linear fractional programming deals with that class ofmathematical programming problem in which the relationsamong the variables are linear the constraint relation mustbe in linear form and the objective function to be optimizedmust be a ratio of two linear functions The field of linearfractional programming problem was developed by Hungar-ian mathematician B Matros in 1960 The linear fractionalprogramming problem is an important planning tool for thepast decadeswhich is applied to different disciplines like engi-neering business finance and economics Linear fractionalproblem is generally used for modelling real life problemswith one or more objective(s) such as profit-cost inventory-sales actual cost-standard cost and output-employee multi-ple level programming problems are frequently encountered

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2015 Article ID 103618 7 pageshttpdxdoiorg1011552015103618

2 Computational Intelligence and Neuroscience

in a hierarchical organization manufacturing plant logisticcompanies and so forth From practical point of view itbecomes necessary to consider the large structured program-ming problem Even it is theoretically possible to solve thisproblem in practical it is not easy to solve There are certainlimitations which restrict the endeavours of the analyst Chiefamong these limitations is the problem of dimensionalityThis suggests the idea of developing methods of solution thatshould not use simultaneously all the data of the problem onesuch approach is the decomposition principle due to Dantzig[2] for linear programs The principle requires the solutionof a series of linear programming problems of smaller sizethan the original problem In this paper we proposed a newmethod to find the optimal solution of the fractional fuzzytransportation problem based on dual simplex approach

This paper is organised as follows In Section 2 relatedwork is presented Section 3 provides preliminary back-ground on fuzzy set theory The linear fractional fuzzyprogramming problem is explained in Section 4 In Section 5problem formulation of fractional fuzzy transportation prob-lem appeared The procedure for the proposed method isgiven in Section 6 An illustrative example is provided toexplain this proposed method and the conclusions are givenin Sections 7 and 8 respectively

2 Related Work

Fractional fuzzy transportation problem is a special typeof linear programming problem and it is an active area ofresearch There are a lot of articles in this area which cannotbe reviewed completely and only a few of them are reviewedhere Charnes and Cooper [3] developed a transformationtechnique 119911 = 1(119902

1015840119909+120573) to convert fractional programming

method to linear programming method Swarup [4] intro-duced simplex type algorithm for fractional program andBitran and Novaes [5] converted linear fractional probleminto a sequence of linear programs and the method is widelyaccepted Tantawy [6] developed a dual solution techniquefor the fractional program Hasan and Acharjee [7] usedthe conversion of linear fractional problem into single linearprogramming and computed inMATHEMATICA Joshi andGupta [8] solved the linear fractional problem by primal-dualapproach Sharma and Bansal [9] proposed a method basedon simplex method in which the variables are extended Jainet al [10] proposed ABC algorithm for solving linear frac-tional programming problem using C-language Sulaimanand Basiya [11] used transformation technique

Verma et al [12] developed an algorithm which ranksthe feasible solutions of an integer fractional programmingproblem in decreasing order of the objective function valuesMetev and Yordanova-Markova [13] applied reference pointmethod Pandian and Jayalakshmi [14] used decompositionmethod and the denominator restriction method to convertfractional problem to linear problem Chandra [15] proposeddecomposition principle for solving linear fractional prob-lem Sharma and Bansal [9] gave an integer solution for

linear fractional problem Dheyah [16] gave optimal solutionfor linear fractional problem with fuzzy numbers Guzelet al [17] proposed an algorithm using first order Taylorseries to convert interval linear fractional transportationproblem into linear multiobjective problem Madhuri [18]gave a solution procedure to minimum time for linearfractional transportation problem with impurities Jain andArya [19] solved an inverse transportation problem withlinear fractional objective function

Motivated by thework ofGuzel et al [17] in this paper theauthor proposes a new method to find the optimal solutionof the fractional fuzzy transportation problem Some of thelimitations in the existing methods are that in Charnesand Coopers methods to obtain the solution one needs tosolve the linear problem by two-phase method in each stepwhich becomes lengthy and clumsy In Bitran and Novaersquosmethod one needs to solve a sequence of problems whichsometimes may need many iterations In this method if119901(119909) lt 0 for all 119909 then this method fails In Swaruprsquos methodwhen the constraints are not in canonical form then thismethod becomes lengthy and its computational process iscomplicated in each iteration InHarveyrsquosmethodhe convertsthe linear fractional problem max 119911 = (119888119909 + 120572)(119889119909 + 120573) tolinear problem under the assumption that 120573 = 0 If 120573 = 0 thismethod fails

To overcome these limitations in this paper a newmethod is proposed such that the linear fractional trans-portation problem is decomposed into two linear fuzzytransportation problems under the assumption that 120572 120573

values are zero These linear transportation problems aresolved by dual simplex method

3 Preliminaries

In this section we present the most basic notations anddefinitions which are used throughout this work We startwith defining a fuzzy set

31 Fuzzy Set [20] Let 119883 be a set A fuzzy set 119860 on 119883 isdefined to be a function 119860 119883 rarr [0 1] or 120583

119860 119883 rarr [0 1]

Equivalently a fuzzy set is defined to be the class of objectshaving the following representation 119860 = (119909 120583

119860(119909)) 119909 isin 119883

where 120583119860

119883 rarr [0 1] is a function called membershipfunction of 119860

32 Fuzzy Number [20] The fuzzy number 119860 is a fuzzy setwhose membership function satisfies the following condi-tions

(i) 120583119860(119909)

is piecewise and continuous

(ii) A fuzzy set119860 of the universe of discourse119883 is convex

(iii) A fuzzy set of the universe of discourse 119883 is called anormal fuzzy set if there exists 119909

119894isin 119883

Computational Intelligence and Neuroscience 3

33 Trapezoidal Fuzzy Number [20] A fuzzy number 119886 =

(1198861 1198862 1198863 1198864) is said to be a trapezoidal fuzzy number if its

membership function is given by

120583119886

=

0 119909 le 1198861

119909 minus 119886

119887 minus 119886

1198861le 119909 le 119886

2

1 1198862

le 119909 le 1198863

119888 minus 119909

119889 minus 119888

1198863le 119909 le 119886

4

0 119909 ge 1198864

(1)

34 Arithmetic Operations Let 119886 = (1198861 1198862 1198863 1198864) and

119887 = (119887

1 1198872 1198873 1198874) be two trapezoidal fuzzy numbers the

arithmetic operators on these numbers are as followsAddition

119886 +119887 = (119886

1+ 1198871 1198862+ 1198872 1198863+ 1198873 1198864+ 1198874) (2)

Subtraction

119886 minus119887 = (119886

1minus 1198871 1198862minus 1198872 1198863minus 1198873 1198864minus 1198874) (3)

Scalar multiplication

119896119886 = (1198961198861 1198961198862 1198961198863 1198961198864) 119896 gt 0

119896119886 = (1198961198864 1198961198863 1198961198862 1198961198861) 119896 lt 0

(4)

35 Ranking Function The following ranking functiondefined here is used to compare the fuzzy values when weperform the dual simplex method A ranking function 119877

119865(119877) rarr 119877 whichmaps each fuzzy number into the real line119865(119877) represents the set of all trapezoidal fuzzy numbers If 119877is any ranking function then 119877(119886) = (119886

1+ 1198862+ 1198863+ 1198864)4

36 Efficient Point A feasible point 1199090

isin 119875 is said to be anefficient solution for 119875 if there exists no other feasible point 119909of the problem 119875 such that 119911

119894(119909) ge 119911

119894(1199090) and 119894 = 1 2

37 Ideal Solution Let 1199090 be the optimal solution to the

problem 119875119905 119905 = 1 2 then the value of the objective function

119911119905(1199090) 119905 = 1 2 is called the ideal solution to the problem 119875

38 Best Compromise Solution An efficient solution 1199090

isin 119875

is said to be the best compromise solution to the problem 119875if the distance between the ideal solution and the objectivevalue at 119909

0 119911119905(1199090) 119905 = 1 2 is minimum among the efficient

solution to the problem 119875

Lemma 1 The feasible solution for the problem119875 is consideredto be efficient if and only if there exist no other feasible solutionsfor which we obtain a better value at least for one criterion forwhich the value of the rest of criteria remains unmodified

Theorem 2 If 119883119905is the set of efficient points for problem 119875

119905

119905 = 1 2 then 119883119905is a subset of the set of all efficient points 119883 of

119875

Proof Let1198831be the set of efficient points of 119875

1 Let119883

2be the

set of efficient points of 1198752 As 119883

119905sub 119883 119905 = 1 2 The set of all

efficient points of 1198751 1198752is contained in 119875

Theorem 3 Let 1199090

= 119909119894119895 119909119894119895

isin 119875 be an optimal solution of1198751and let 1199090 = 119909

119894119895 119909119894119895

isin 1198752 be an optimal solution of 119875

2 then

119909 = 11990901199090 is an optimal solution of the given problem 119875 where

all of 11990911

11990912

11990921

11990922

are elements of 1198751 1198752

Proof Let 1199090 be an optimal solution of 119875

1 This solution is

an efficient solution if there is no feasible solution these setsof efficient solutions are contained in 119909 Similarly the set ofefficient solutions of 119875

2are contained in 119909 So byTheorem 2

these solutions are the set of efficient solutions of 119875

4 Introduction to Linear Fractional FuzzyProgramming Problem (LFFPP)

One of the main aims of this paper is to extend the linearfractional problem into linear fractional fuzzy programmingproblem

The LFFPP can be presented in the following way

max 119891 (119909) =

119901 (119909)

119902 (119909)

subject to 119909 isin 119878 sub 119877119899

(5)

where 119901(119909) and 119902(119909) are linear functions and the set 119878 isdefined as 119878 = 119909119860119909 = 119887 119909 gt 0 Here 119860 is a fuzzy 119898 times 119899

matrix we will suppose that 119878 is a bounded polyhedron and119902(119909) gt 0 for all119909 isin 119878 It is well known that the goal function inmax 119891(119909) has a global maximum function 119878 and has not anyother local maximaThismaximum is obtained at an extremepoint of 119878

LFFPP (5) can now be written as

max 119901 (119909)

subject to 119909 isin 119878 sub 119877119899

max 119902 (119909)

subject to 119909 isin 119878 sub 119877119899

(6)

Then by Theorem 3 if 119901(119909)119902(119909) gt 0 forall119894 and forall119909 isin 119878 then theset of efficient points for problem 119875

119905 119905 = 1 2 is a subset of the

set of all efficient points of 119875

5 Problem Formulation

The proposed fuzzy transportation model is based on thefollowing assumption

Index Set Consider the following

119894 is source index for all 119894 = 1 2 3 119898

119895 is destination index for all 119895 = 1 2 3 119899

4 Computational Intelligence and Neuroscience

Parameters Consider the following

119909119894119895is the number of units transported from source 119894

to destination 119895119888119894119895is per unit profit cost of transportation from source

119894 to destination 119895119889119894119895is per unit cost of transportation from source 119894 to

destination 119895

Objective Function Consider the following

(i) Fractional transportation problem

min 119911 =

sum119898

119894=1sum119899

119895=1119888119894119895119909119894119895

sum119898

119894=1sum119899

119895=1119889119894119895119909119894119895

(7)

The values of 119888119894119895and 119889

119894119895are considered to be trape-

zoidal fuzzy number

Constraints Consider the following(i) Constraints on supply available for all sources 119894

119898

sum

119894=1

119909119894119895

= 119886119894

forall119894 (8)

(ii) Constraints on demand for each destination 119895

119899

sum

119895=1

119909119894119895

= 119887119895

forall119895 (9)

Nonnegative Constraints Consider the following(i) Nonnegative constraints on decision variables

119909119894119895

ge 0 (10)

The above LFFTPwill have a basic feasible solution onlywhensum119898

119894=1119886119894= sum119899

119895=1119887119895

6 Procedure for Proposed Method

Here we present a new method for solving linear fractionalfuzzy transportation problem

Consider the linear fractional fuzzy transportation prob-lem as

min 119911 =

sum119898

119894=1sum119899

119895=1119888119894119895119909119894119895

sum119898

119894=1sum119899

119895=1119889119894119895119909119894119895

119875 subject to119898

sum

119894=1

119909119894119895

= 119886119894

119899

sum

119895=1

119909119894119895

= 119887119895

119909119894119895

ge 0

(11)

Since the objective function is a fractional objective functionwe take the problem 119875 into two linear fuzzy transportationproblems by (6) as follows

min 119911119873

=

119898

sum

119894=1

119899

sum

119895=1

119888119894119895119909119873

119894119895

1198751 subject to

119898

sum

119894=1

119909119873

119894119895= 119886119894

119899

sum

119895=1

119909119873

119894119895= 119887119895

119909119873

119894119895ge 0

min 119911119863

=

119898

sum

119894=1

119899

sum

119895=1

119888119894119895119909119863

119894119895

1198751 subject to

119898

sum

119894=1

119909119863

119894119895= 119886119894

119899

sum

119895=1

119909119863

119894119895= 119887119895

119909119863

119894119895ge 0

(12)

We solve these two linear fuzzy transportation problems bydual simplex methods Thus the solution is obtained formin 119911

119873 and min 119911119863 separately By Theorem 3 the optimum

solution of 119875 is obtained when the optimal solution of 1198751and

1198752is obtained

7 Numerical Example

To illustrate the proposed method consider the followingexample for linear fractional fuzzy transportation problem

119875 min 119911 = ((0 2 4 6) 11990911

+ (1 2 6 7) 11990912

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

)

sdot ((0 1 3 4) 11990911

+ (2 3 5 6) 11990912

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

)minus1

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

(13)

The problem 119875 is a linear fractional fuzzy transportationproblem inwhich the objective function has trapezoidal fuzzynumber supply and demand values are crisp values Now by

Computational Intelligence and Neuroscience 5

Table 1

119862119895

(minus6 minus4 minus2 0) (minus7 minus6 minus2 minus1) (minus6 minus5 minus4 minus1) (minus8 minus5 minus4 minus3) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

0 1198781

60 1 1 0 0 1 0 0 00 119878

245 0 0 1 1 0 1 0 0

0 1198783

minus50 minus1 0 minus1 0 0 0 1 00 119878

4minus55 0 minus1 0 minus1 0 0 0 1

119911 = 0 (0 2 4 6) (1 2 6 7) (1 4 5 6) (3 4 5 8) 0 0 0 0119877(sdot sdot sdot ) 3 4 4 5 0 0 0 0

uarr darr

Table 2

119862119895

(minus6 minus4 minus2 0) (minus7 minus6 minus2 minus1) (minus6 minus5 minus4 minus1) (minus8 minus5 minus4 minus3) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

(minus8 minus5 minus4 minus3) 11988321

45 0 0 1 1 minus1 0 minus1 minus10 119878

20 0 0 0 0 1 1 1 1

(minus6 minus5 minus4 minus1) 11988311

5 1 0 0 minus1 1 0 0 1(minus7 minus6 minus2 minus1) 119883

1255 0 1 0 1 0 0 0 minus1

119911 = (minus995 minus755 minus390 minus100) 0 0 0 (minus5 8 6 9) (minus5 0 3 6) 0 (1 4 5 6) (minus4 2 9 13)

our computation technique the above problem can be writtenas

1198751 min 119911

119873= (0 2 4 6) 119909

11+ (1 2 6 7) 119909

12

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

1198752 min 119911

119863= (0 1 3 4) 119909

11+ (2 3 5 6) 119909

12

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

(14)

Consider1198751The linear problem can be expressed in standard

form as

1198751 min 119911

119873= (0 2 4 6) 119909

11+ (1 2 6 7) 119909

12

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

subject to 11990911

+ 11990912

+ 1199041= 60

11990921

+ 11990922

+ 1199042= 45

11990911

+ 11990921

minus 1199043= 50

11990912

+ 11990922

minus 1199044= 55

11990911

11990912

11990921

11990922

1199041 1199042 1199043 1199044ge 0

(15)

Now 1198751is solved by the dual simplex method

The initial iteration of the problem is given in Table 1From the initial iteration we can see that the nonbasic

variable 11988312

enters the basis and the basic variable 1198784leaves

the basisProceeding the dual simplex method and after few itera-

tions we get Table 2 In Table 2 all the values of119883119861are positive

and the optimum solution is obtained as follows

min 119911119873

= (100 390 755 995)

11990911

= 5 11990912

= 55 11990921

= 45

(16)

Consider1198752The linear problem can be expressed in standard

form as1198752 min 119911

119863= (0 1 3 4) 119909

11+ (2 3 5 6) 119909

12

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

subject to 11990911

+ 11990912

+ 1198781= 60

11990921

+ 11990922

+ 1198782= 45

11990911

+ 11990921

minus 1198783= 50

11990912

+ 11990922

minus 1198784= 55

11990911

11990912

11990921

11990922

1198781 1198782 1198783 1198784ge 0

(17)Now 119875

2is solved by the dual simplex method

The initial iteration of the problem is given in Table 3From the initial iteration we can see that the nonbasic

variable 11988312

enters the basis and the basic variable 1198784leaves

the basisProceeding the dual simplex method and after few itera-

tionswe get Table 4 In Table 4 all the values of119883119861are positive

and the optimum solution is obtained as follows

min 119911119863

= (155 305 515 665)

11990911

= 5 11990912

= 55 11990921

= 45

(18)

6 Computational Intelligence and Neuroscience

Table 3

119862119895

(minus4 minus3 minus1 0) (minus6 minus5 minus3 minus2) (minus7 minus5 minus3 minus1) (minus9 minus7 minus6 minus2) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

0 1198781

60 1 1 0 0 1 0 0 00 119878

245 0 0 1 1 0 1 0 0

0 1198783

minus50 minus1 0 minus1 0 0 0 1 00 119878

4minus55 0 minus1 0 minus1 0 0 0 1

119911 = 0 (0 1 3 4) (2 3 5 6) (1 3 5 7) (2 6 7 9) 0 0 0 0119877(sdot sdot sdot ) 2 4 4 6 0 0 0 0

uarr darr

Table 4

119862119895

(minus4 minus3 minus1 0) (minus6 minus5 minus3 minus2) (minus7 minus5 minus3 minus1) (minus9 minus7 minus6 minus2) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

(minus7 minus5 minus3 minus1) 11988321

45 0 0 1 1 minus1 0 minus1 minus10 119878

20 0 0 0 0 1 1 1 1

(minus4 minus3 minus1 0) 11988311

5 1 0 0 minus1 1 0 0 1(minus6 minus5 minus3 minus2) 119883

1255 0 1 0 1 0 0 0 minus1

119911 = (minus665 minus515 minus305 minus155) 0 0 0 (8 minus2 3 3) (minus3 0 4 7) 0 (1 3 5 7) (minus1 39 13)

By Theorem 3 the optimum solution of the fractional fuzzytransportation problem is obtained

Thus

min 119911 =

(100 390 755 995)

(155 305 515 665)

(19)

8 Conclusion

The present paper proposes a method for solving linearfractional fuzzy transportation problem where the trape-zoidal fuzzy numbers are used in the objective functionThe limitations of the existing methods are discussed Theadvantage of the computational technique is that the methodworks even when 120572 120573 values are zero The dual simplexmethod is one of the best methods to get the optimalsolution The method can be easily implemented to solveany type of transportation problem The solution procedureshave been illustrated by an example The result shows thehigh efficiency of the solution and this can be extended tofractional quadratic problems The simplex method can beused instead of dual simplex if all the constraints are of le typeconstraints

Conflict of Interests

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

Acknowledgment

The authors would like to thank the anonymous referees forvarious suggestions which have led to an improvement inboth the quality and the clarity of the paper

References

[1] F L Hitchcock ldquoThe distribution of a product from severalsources to numerous localitiesrdquo Journal of Mathematics andPhysics vol 20 pp 224ndash230 1941

[2] G B Dantzig Linear Programming amp Extension PrincetonUniversity Press Princeton NJ USA 1962

[3] A Charnes and W W Cooper ldquoAn explicit general solution inlinear fractional programmingrdquo Naval Research Logistics vol20 no 3 pp 449ndash467 1973

[4] K Swarup ldquoSome aspects of linear fractional functionalsprogrammingrdquo The Australian Journal of Statistics vol 7 pp90ndash104 1965

[5] G R Bitran and A G Novaes ldquoLinear programming with afractional objective functionrdquo Operations Research vol 21 no1 pp 22ndash29 1972

[6] S F Tantawy ldquoA new method for solving linear fractionalprogramming problemsrdquo Australian Journal of Basic amp AppliedSciences vol 1 no 2 pp 105ndash108 2011

[7] M B Hasan and S Acharjee ldquoSolving LFP by converting it intoa single LPrdquo International Journal of Operations Research vol 8no 3 pp 1ndash14 2011

[8] V D Joshi and N Gupta ldquoLinear fractional transportationproblemwith varying demand and supplyrdquo LeMatematiche vol66 pp 2ndash12 2011

[9] S C Sharma and A Bansal ldquoA integer solution of fractionalprogramming problemrdquo General Mathematics Notes vol 4 no2 pp 1ndash9 2011

[10] S Jain A Mangal and S Sharma ldquoC-approach of ABSalgorithm for fractional programming problemrdquo Journal ofComputer and Mathematical Sciences vol 4 no 2 pp 126ndash1342013

[11] A Sulaiman and K Basiya ldquoUsing transformation technique tosolve multi-objective linear fractional problemrdquo InternationalJournal of Research and Reviews in Applied Sciences vol 14 no3 pp 559ndash567 2013

Computational Intelligence and Neuroscience 7

[12] V Verma H C Bakhshi and M C Puri ldquoRanking in integerlinear fractional programming problemsrdquo ZOR Methods ampModels of Operational Research vol 34 no 5 pp 325ndash334 1990

[13] B SMetev and I T Yordanova-Markova ldquoMulti-objective opti-mization over convex disjunctive feasible sets using referencepointsrdquo European Journal of Operational Research vol 98 no 1pp 124ndash137 1997

[14] P Pandian and M Jayalakshmi ldquoOn solving linear fractionalprogramming problemsrdquo Modern Applied Science vol 7 no 6pp 90ndash100 2013

[15] S Chandra ldquoDecomposition principle for linear fractionalfunctional programsrdquo Revue Francaise drsquoInformatique et deRecherche Operationnelle no 10 pp 65ndash71 1968

[16] A N Dheyab ldquoFinding the optimal solution for fractionallinear programming problems with fuzzy numbersrdquo Journal ofKerbala University vol 10 no 3 pp 105ndash110 2012

[17] N Guzel Y Emiroglu F Tapci C Guler and M SyvryldquoA solution proposal to the interval fractional transportationproblemrdquo Applied Mathematics amp Information Sciences vol 6no 3 pp 567ndash571 2012

[18] D Madhuri ldquoLinear fractional time minimizing transportationproblemwith impuritiesrdquo Information Sciences Letters vol 1 no1 pp 7ndash19 2012

[19] S Jain and N Arya ldquoAn inverse transportation problem withthe linear fractional objective functionrdquo Advanced Modelling ampOptimization vol 15 no 3 pp 677ndash687 2013

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

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httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 Computational Intelligence and Neuroscience

in a hierarchical organization manufacturing plant logisticcompanies and so forth From practical point of view itbecomes necessary to consider the large structured program-ming problem Even it is theoretically possible to solve thisproblem in practical it is not easy to solve There are certainlimitations which restrict the endeavours of the analyst Chiefamong these limitations is the problem of dimensionalityThis suggests the idea of developing methods of solution thatshould not use simultaneously all the data of the problem onesuch approach is the decomposition principle due to Dantzig[2] for linear programs The principle requires the solutionof a series of linear programming problems of smaller sizethan the original problem In this paper we proposed a newmethod to find the optimal solution of the fractional fuzzytransportation problem based on dual simplex approach

This paper is organised as follows In Section 2 relatedwork is presented Section 3 provides preliminary back-ground on fuzzy set theory The linear fractional fuzzyprogramming problem is explained in Section 4 In Section 5problem formulation of fractional fuzzy transportation prob-lem appeared The procedure for the proposed method isgiven in Section 6 An illustrative example is provided toexplain this proposed method and the conclusions are givenin Sections 7 and 8 respectively

2 Related Work

Fractional fuzzy transportation problem is a special typeof linear programming problem and it is an active area ofresearch There are a lot of articles in this area which cannotbe reviewed completely and only a few of them are reviewedhere Charnes and Cooper [3] developed a transformationtechnique 119911 = 1(119902

1015840119909+120573) to convert fractional programming

method to linear programming method Swarup [4] intro-duced simplex type algorithm for fractional program andBitran and Novaes [5] converted linear fractional probleminto a sequence of linear programs and the method is widelyaccepted Tantawy [6] developed a dual solution techniquefor the fractional program Hasan and Acharjee [7] usedthe conversion of linear fractional problem into single linearprogramming and computed inMATHEMATICA Joshi andGupta [8] solved the linear fractional problem by primal-dualapproach Sharma and Bansal [9] proposed a method basedon simplex method in which the variables are extended Jainet al [10] proposed ABC algorithm for solving linear frac-tional programming problem using C-language Sulaimanand Basiya [11] used transformation technique

Verma et al [12] developed an algorithm which ranksthe feasible solutions of an integer fractional programmingproblem in decreasing order of the objective function valuesMetev and Yordanova-Markova [13] applied reference pointmethod Pandian and Jayalakshmi [14] used decompositionmethod and the denominator restriction method to convertfractional problem to linear problem Chandra [15] proposeddecomposition principle for solving linear fractional prob-lem Sharma and Bansal [9] gave an integer solution for

linear fractional problem Dheyah [16] gave optimal solutionfor linear fractional problem with fuzzy numbers Guzelet al [17] proposed an algorithm using first order Taylorseries to convert interval linear fractional transportationproblem into linear multiobjective problem Madhuri [18]gave a solution procedure to minimum time for linearfractional transportation problem with impurities Jain andArya [19] solved an inverse transportation problem withlinear fractional objective function

Motivated by thework ofGuzel et al [17] in this paper theauthor proposes a new method to find the optimal solutionof the fractional fuzzy transportation problem Some of thelimitations in the existing methods are that in Charnesand Coopers methods to obtain the solution one needs tosolve the linear problem by two-phase method in each stepwhich becomes lengthy and clumsy In Bitran and Novaersquosmethod one needs to solve a sequence of problems whichsometimes may need many iterations In this method if119901(119909) lt 0 for all 119909 then this method fails In Swaruprsquos methodwhen the constraints are not in canonical form then thismethod becomes lengthy and its computational process iscomplicated in each iteration InHarveyrsquosmethodhe convertsthe linear fractional problem max 119911 = (119888119909 + 120572)(119889119909 + 120573) tolinear problem under the assumption that 120573 = 0 If 120573 = 0 thismethod fails

To overcome these limitations in this paper a newmethod is proposed such that the linear fractional trans-portation problem is decomposed into two linear fuzzytransportation problems under the assumption that 120572 120573

values are zero These linear transportation problems aresolved by dual simplex method

3 Preliminaries

In this section we present the most basic notations anddefinitions which are used throughout this work We startwith defining a fuzzy set

31 Fuzzy Set [20] Let 119883 be a set A fuzzy set 119860 on 119883 isdefined to be a function 119860 119883 rarr [0 1] or 120583

119860 119883 rarr [0 1]

Equivalently a fuzzy set is defined to be the class of objectshaving the following representation 119860 = (119909 120583

119860(119909)) 119909 isin 119883

where 120583119860

119883 rarr [0 1] is a function called membershipfunction of 119860

32 Fuzzy Number [20] The fuzzy number 119860 is a fuzzy setwhose membership function satisfies the following condi-tions

(i) 120583119860(119909)

is piecewise and continuous

(ii) A fuzzy set119860 of the universe of discourse119883 is convex

(iii) A fuzzy set of the universe of discourse 119883 is called anormal fuzzy set if there exists 119909

119894isin 119883

Computational Intelligence and Neuroscience 3

33 Trapezoidal Fuzzy Number [20] A fuzzy number 119886 =

(1198861 1198862 1198863 1198864) is said to be a trapezoidal fuzzy number if its

membership function is given by

120583119886

=

0 119909 le 1198861

119909 minus 119886

119887 minus 119886

1198861le 119909 le 119886

2

1 1198862

le 119909 le 1198863

119888 minus 119909

119889 minus 119888

1198863le 119909 le 119886

4

0 119909 ge 1198864

(1)

34 Arithmetic Operations Let 119886 = (1198861 1198862 1198863 1198864) and

119887 = (119887

1 1198872 1198873 1198874) be two trapezoidal fuzzy numbers the

arithmetic operators on these numbers are as followsAddition

119886 +119887 = (119886

1+ 1198871 1198862+ 1198872 1198863+ 1198873 1198864+ 1198874) (2)

Subtraction

119886 minus119887 = (119886

1minus 1198871 1198862minus 1198872 1198863minus 1198873 1198864minus 1198874) (3)

Scalar multiplication

119896119886 = (1198961198861 1198961198862 1198961198863 1198961198864) 119896 gt 0

119896119886 = (1198961198864 1198961198863 1198961198862 1198961198861) 119896 lt 0

(4)

35 Ranking Function The following ranking functiondefined here is used to compare the fuzzy values when weperform the dual simplex method A ranking function 119877

119865(119877) rarr 119877 whichmaps each fuzzy number into the real line119865(119877) represents the set of all trapezoidal fuzzy numbers If 119877is any ranking function then 119877(119886) = (119886

1+ 1198862+ 1198863+ 1198864)4

36 Efficient Point A feasible point 1199090

isin 119875 is said to be anefficient solution for 119875 if there exists no other feasible point 119909of the problem 119875 such that 119911

119894(119909) ge 119911

119894(1199090) and 119894 = 1 2

37 Ideal Solution Let 1199090 be the optimal solution to the

problem 119875119905 119905 = 1 2 then the value of the objective function

119911119905(1199090) 119905 = 1 2 is called the ideal solution to the problem 119875

38 Best Compromise Solution An efficient solution 1199090

isin 119875

is said to be the best compromise solution to the problem 119875if the distance between the ideal solution and the objectivevalue at 119909

0 119911119905(1199090) 119905 = 1 2 is minimum among the efficient

solution to the problem 119875

Lemma 1 The feasible solution for the problem119875 is consideredto be efficient if and only if there exist no other feasible solutionsfor which we obtain a better value at least for one criterion forwhich the value of the rest of criteria remains unmodified

Theorem 2 If 119883119905is the set of efficient points for problem 119875

119905

119905 = 1 2 then 119883119905is a subset of the set of all efficient points 119883 of

119875

Proof Let1198831be the set of efficient points of 119875

1 Let119883

2be the

set of efficient points of 1198752 As 119883

119905sub 119883 119905 = 1 2 The set of all

efficient points of 1198751 1198752is contained in 119875

Theorem 3 Let 1199090

= 119909119894119895 119909119894119895

isin 119875 be an optimal solution of1198751and let 1199090 = 119909

119894119895 119909119894119895

isin 1198752 be an optimal solution of 119875

2 then

119909 = 11990901199090 is an optimal solution of the given problem 119875 where

all of 11990911

11990912

11990921

11990922

are elements of 1198751 1198752

Proof Let 1199090 be an optimal solution of 119875

1 This solution is

an efficient solution if there is no feasible solution these setsof efficient solutions are contained in 119909 Similarly the set ofefficient solutions of 119875

2are contained in 119909 So byTheorem 2

these solutions are the set of efficient solutions of 119875

4 Introduction to Linear Fractional FuzzyProgramming Problem (LFFPP)

One of the main aims of this paper is to extend the linearfractional problem into linear fractional fuzzy programmingproblem

The LFFPP can be presented in the following way

max 119891 (119909) =

119901 (119909)

119902 (119909)

subject to 119909 isin 119878 sub 119877119899

(5)

where 119901(119909) and 119902(119909) are linear functions and the set 119878 isdefined as 119878 = 119909119860119909 = 119887 119909 gt 0 Here 119860 is a fuzzy 119898 times 119899

matrix we will suppose that 119878 is a bounded polyhedron and119902(119909) gt 0 for all119909 isin 119878 It is well known that the goal function inmax 119891(119909) has a global maximum function 119878 and has not anyother local maximaThismaximum is obtained at an extremepoint of 119878

LFFPP (5) can now be written as

max 119901 (119909)

subject to 119909 isin 119878 sub 119877119899

max 119902 (119909)

subject to 119909 isin 119878 sub 119877119899

(6)

Then by Theorem 3 if 119901(119909)119902(119909) gt 0 forall119894 and forall119909 isin 119878 then theset of efficient points for problem 119875

119905 119905 = 1 2 is a subset of the

set of all efficient points of 119875

5 Problem Formulation

The proposed fuzzy transportation model is based on thefollowing assumption

Index Set Consider the following

119894 is source index for all 119894 = 1 2 3 119898

119895 is destination index for all 119895 = 1 2 3 119899

4 Computational Intelligence and Neuroscience

Parameters Consider the following

119909119894119895is the number of units transported from source 119894

to destination 119895119888119894119895is per unit profit cost of transportation from source

119894 to destination 119895119889119894119895is per unit cost of transportation from source 119894 to

destination 119895

Objective Function Consider the following

(i) Fractional transportation problem

min 119911 =

sum119898

119894=1sum119899

119895=1119888119894119895119909119894119895

sum119898

119894=1sum119899

119895=1119889119894119895119909119894119895

(7)

The values of 119888119894119895and 119889

119894119895are considered to be trape-

zoidal fuzzy number

Constraints Consider the following(i) Constraints on supply available for all sources 119894

119898

sum

119894=1

119909119894119895

= 119886119894

forall119894 (8)

(ii) Constraints on demand for each destination 119895

119899

sum

119895=1

119909119894119895

= 119887119895

forall119895 (9)

Nonnegative Constraints Consider the following(i) Nonnegative constraints on decision variables

119909119894119895

ge 0 (10)

The above LFFTPwill have a basic feasible solution onlywhensum119898

119894=1119886119894= sum119899

119895=1119887119895

6 Procedure for Proposed Method

Here we present a new method for solving linear fractionalfuzzy transportation problem

Consider the linear fractional fuzzy transportation prob-lem as

min 119911 =

sum119898

119894=1sum119899

119895=1119888119894119895119909119894119895

sum119898

119894=1sum119899

119895=1119889119894119895119909119894119895

119875 subject to119898

sum

119894=1

119909119894119895

= 119886119894

119899

sum

119895=1

119909119894119895

= 119887119895

119909119894119895

ge 0

(11)

Since the objective function is a fractional objective functionwe take the problem 119875 into two linear fuzzy transportationproblems by (6) as follows

min 119911119873

=

119898

sum

119894=1

119899

sum

119895=1

119888119894119895119909119873

119894119895

1198751 subject to

119898

sum

119894=1

119909119873

119894119895= 119886119894

119899

sum

119895=1

119909119873

119894119895= 119887119895

119909119873

119894119895ge 0

min 119911119863

=

119898

sum

119894=1

119899

sum

119895=1

119888119894119895119909119863

119894119895

1198751 subject to

119898

sum

119894=1

119909119863

119894119895= 119886119894

119899

sum

119895=1

119909119863

119894119895= 119887119895

119909119863

119894119895ge 0

(12)

We solve these two linear fuzzy transportation problems bydual simplex methods Thus the solution is obtained formin 119911

119873 and min 119911119863 separately By Theorem 3 the optimum

solution of 119875 is obtained when the optimal solution of 1198751and

1198752is obtained

7 Numerical Example

To illustrate the proposed method consider the followingexample for linear fractional fuzzy transportation problem

119875 min 119911 = ((0 2 4 6) 11990911

+ (1 2 6 7) 11990912

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

)

sdot ((0 1 3 4) 11990911

+ (2 3 5 6) 11990912

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

)minus1

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

(13)

The problem 119875 is a linear fractional fuzzy transportationproblem inwhich the objective function has trapezoidal fuzzynumber supply and demand values are crisp values Now by

Computational Intelligence and Neuroscience 5

Table 1

119862119895

(minus6 minus4 minus2 0) (minus7 minus6 minus2 minus1) (minus6 minus5 minus4 minus1) (minus8 minus5 minus4 minus3) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

0 1198781

60 1 1 0 0 1 0 0 00 119878

245 0 0 1 1 0 1 0 0

0 1198783

minus50 minus1 0 minus1 0 0 0 1 00 119878

4minus55 0 minus1 0 minus1 0 0 0 1

119911 = 0 (0 2 4 6) (1 2 6 7) (1 4 5 6) (3 4 5 8) 0 0 0 0119877(sdot sdot sdot ) 3 4 4 5 0 0 0 0

uarr darr

Table 2

119862119895

(minus6 minus4 minus2 0) (minus7 minus6 minus2 minus1) (minus6 minus5 minus4 minus1) (minus8 minus5 minus4 minus3) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

(minus8 minus5 minus4 minus3) 11988321

45 0 0 1 1 minus1 0 minus1 minus10 119878

20 0 0 0 0 1 1 1 1

(minus6 minus5 minus4 minus1) 11988311

5 1 0 0 minus1 1 0 0 1(minus7 minus6 minus2 minus1) 119883

1255 0 1 0 1 0 0 0 minus1

119911 = (minus995 minus755 minus390 minus100) 0 0 0 (minus5 8 6 9) (minus5 0 3 6) 0 (1 4 5 6) (minus4 2 9 13)

our computation technique the above problem can be writtenas

1198751 min 119911

119873= (0 2 4 6) 119909

11+ (1 2 6 7) 119909

12

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

1198752 min 119911

119863= (0 1 3 4) 119909

11+ (2 3 5 6) 119909

12

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

(14)

Consider1198751The linear problem can be expressed in standard

form as

1198751 min 119911

119873= (0 2 4 6) 119909

11+ (1 2 6 7) 119909

12

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

subject to 11990911

+ 11990912

+ 1199041= 60

11990921

+ 11990922

+ 1199042= 45

11990911

+ 11990921

minus 1199043= 50

11990912

+ 11990922

minus 1199044= 55

11990911

11990912

11990921

11990922

1199041 1199042 1199043 1199044ge 0

(15)

Now 1198751is solved by the dual simplex method

The initial iteration of the problem is given in Table 1From the initial iteration we can see that the nonbasic

variable 11988312

enters the basis and the basic variable 1198784leaves

the basisProceeding the dual simplex method and after few itera-

tions we get Table 2 In Table 2 all the values of119883119861are positive

and the optimum solution is obtained as follows

min 119911119873

= (100 390 755 995)

11990911

= 5 11990912

= 55 11990921

= 45

(16)

Consider1198752The linear problem can be expressed in standard

form as1198752 min 119911

119863= (0 1 3 4) 119909

11+ (2 3 5 6) 119909

12

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

subject to 11990911

+ 11990912

+ 1198781= 60

11990921

+ 11990922

+ 1198782= 45

11990911

+ 11990921

minus 1198783= 50

11990912

+ 11990922

minus 1198784= 55

11990911

11990912

11990921

11990922

1198781 1198782 1198783 1198784ge 0

(17)Now 119875

2is solved by the dual simplex method

The initial iteration of the problem is given in Table 3From the initial iteration we can see that the nonbasic

variable 11988312

enters the basis and the basic variable 1198784leaves

the basisProceeding the dual simplex method and after few itera-

tionswe get Table 4 In Table 4 all the values of119883119861are positive

and the optimum solution is obtained as follows

min 119911119863

= (155 305 515 665)

11990911

= 5 11990912

= 55 11990921

= 45

(18)

6 Computational Intelligence and Neuroscience

Table 3

119862119895

(minus4 minus3 minus1 0) (minus6 minus5 minus3 minus2) (minus7 minus5 minus3 minus1) (minus9 minus7 minus6 minus2) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

0 1198781

60 1 1 0 0 1 0 0 00 119878

245 0 0 1 1 0 1 0 0

0 1198783

minus50 minus1 0 minus1 0 0 0 1 00 119878

4minus55 0 minus1 0 minus1 0 0 0 1

119911 = 0 (0 1 3 4) (2 3 5 6) (1 3 5 7) (2 6 7 9) 0 0 0 0119877(sdot sdot sdot ) 2 4 4 6 0 0 0 0

uarr darr

Table 4

119862119895

(minus4 minus3 minus1 0) (minus6 minus5 minus3 minus2) (minus7 minus5 minus3 minus1) (minus9 minus7 minus6 minus2) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

(minus7 minus5 minus3 minus1) 11988321

45 0 0 1 1 minus1 0 minus1 minus10 119878

20 0 0 0 0 1 1 1 1

(minus4 minus3 minus1 0) 11988311

5 1 0 0 minus1 1 0 0 1(minus6 minus5 minus3 minus2) 119883

1255 0 1 0 1 0 0 0 minus1

119911 = (minus665 minus515 minus305 minus155) 0 0 0 (8 minus2 3 3) (minus3 0 4 7) 0 (1 3 5 7) (minus1 39 13)

By Theorem 3 the optimum solution of the fractional fuzzytransportation problem is obtained

Thus

min 119911 =

(100 390 755 995)

(155 305 515 665)

(19)

8 Conclusion

The present paper proposes a method for solving linearfractional fuzzy transportation problem where the trape-zoidal fuzzy numbers are used in the objective functionThe limitations of the existing methods are discussed Theadvantage of the computational technique is that the methodworks even when 120572 120573 values are zero The dual simplexmethod is one of the best methods to get the optimalsolution The method can be easily implemented to solveany type of transportation problem The solution procedureshave been illustrated by an example The result shows thehigh efficiency of the solution and this can be extended tofractional quadratic problems The simplex method can beused instead of dual simplex if all the constraints are of le typeconstraints

Conflict of Interests

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

Acknowledgment

The authors would like to thank the anonymous referees forvarious suggestions which have led to an improvement inboth the quality and the clarity of the paper

References

[1] F L Hitchcock ldquoThe distribution of a product from severalsources to numerous localitiesrdquo Journal of Mathematics andPhysics vol 20 pp 224ndash230 1941

[2] G B Dantzig Linear Programming amp Extension PrincetonUniversity Press Princeton NJ USA 1962

[3] A Charnes and W W Cooper ldquoAn explicit general solution inlinear fractional programmingrdquo Naval Research Logistics vol20 no 3 pp 449ndash467 1973

[4] K Swarup ldquoSome aspects of linear fractional functionalsprogrammingrdquo The Australian Journal of Statistics vol 7 pp90ndash104 1965

[5] G R Bitran and A G Novaes ldquoLinear programming with afractional objective functionrdquo Operations Research vol 21 no1 pp 22ndash29 1972

[6] S F Tantawy ldquoA new method for solving linear fractionalprogramming problemsrdquo Australian Journal of Basic amp AppliedSciences vol 1 no 2 pp 105ndash108 2011

[7] M B Hasan and S Acharjee ldquoSolving LFP by converting it intoa single LPrdquo International Journal of Operations Research vol 8no 3 pp 1ndash14 2011

[8] V D Joshi and N Gupta ldquoLinear fractional transportationproblemwith varying demand and supplyrdquo LeMatematiche vol66 pp 2ndash12 2011

[9] S C Sharma and A Bansal ldquoA integer solution of fractionalprogramming problemrdquo General Mathematics Notes vol 4 no2 pp 1ndash9 2011

[10] S Jain A Mangal and S Sharma ldquoC-approach of ABSalgorithm for fractional programming problemrdquo Journal ofComputer and Mathematical Sciences vol 4 no 2 pp 126ndash1342013

[11] A Sulaiman and K Basiya ldquoUsing transformation technique tosolve multi-objective linear fractional problemrdquo InternationalJournal of Research and Reviews in Applied Sciences vol 14 no3 pp 559ndash567 2013

Computational Intelligence and Neuroscience 7

[12] V Verma H C Bakhshi and M C Puri ldquoRanking in integerlinear fractional programming problemsrdquo ZOR Methods ampModels of Operational Research vol 34 no 5 pp 325ndash334 1990

[13] B SMetev and I T Yordanova-Markova ldquoMulti-objective opti-mization over convex disjunctive feasible sets using referencepointsrdquo European Journal of Operational Research vol 98 no 1pp 124ndash137 1997

[14] P Pandian and M Jayalakshmi ldquoOn solving linear fractionalprogramming problemsrdquo Modern Applied Science vol 7 no 6pp 90ndash100 2013

[15] S Chandra ldquoDecomposition principle for linear fractionalfunctional programsrdquo Revue Francaise drsquoInformatique et deRecherche Operationnelle no 10 pp 65ndash71 1968

[16] A N Dheyab ldquoFinding the optimal solution for fractionallinear programming problems with fuzzy numbersrdquo Journal ofKerbala University vol 10 no 3 pp 105ndash110 2012

[17] N Guzel Y Emiroglu F Tapci C Guler and M SyvryldquoA solution proposal to the interval fractional transportationproblemrdquo Applied Mathematics amp Information Sciences vol 6no 3 pp 567ndash571 2012

[18] D Madhuri ldquoLinear fractional time minimizing transportationproblemwith impuritiesrdquo Information Sciences Letters vol 1 no1 pp 7ndash19 2012

[19] S Jain and N Arya ldquoAn inverse transportation problem withthe linear fractional objective functionrdquo Advanced Modelling ampOptimization vol 15 no 3 pp 677ndash687 2013

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 3

33 Trapezoidal Fuzzy Number [20] A fuzzy number 119886 =

(1198861 1198862 1198863 1198864) is said to be a trapezoidal fuzzy number if its

membership function is given by

120583119886

=

0 119909 le 1198861

119909 minus 119886

119887 minus 119886

1198861le 119909 le 119886

2

1 1198862

le 119909 le 1198863

119888 minus 119909

119889 minus 119888

1198863le 119909 le 119886

4

0 119909 ge 1198864

(1)

34 Arithmetic Operations Let 119886 = (1198861 1198862 1198863 1198864) and

119887 = (119887

1 1198872 1198873 1198874) be two trapezoidal fuzzy numbers the

arithmetic operators on these numbers are as followsAddition

119886 +119887 = (119886

1+ 1198871 1198862+ 1198872 1198863+ 1198873 1198864+ 1198874) (2)

Subtraction

119886 minus119887 = (119886

1minus 1198871 1198862minus 1198872 1198863minus 1198873 1198864minus 1198874) (3)

Scalar multiplication

119896119886 = (1198961198861 1198961198862 1198961198863 1198961198864) 119896 gt 0

119896119886 = (1198961198864 1198961198863 1198961198862 1198961198861) 119896 lt 0

(4)

35 Ranking Function The following ranking functiondefined here is used to compare the fuzzy values when weperform the dual simplex method A ranking function 119877

119865(119877) rarr 119877 whichmaps each fuzzy number into the real line119865(119877) represents the set of all trapezoidal fuzzy numbers If 119877is any ranking function then 119877(119886) = (119886

1+ 1198862+ 1198863+ 1198864)4

36 Efficient Point A feasible point 1199090

isin 119875 is said to be anefficient solution for 119875 if there exists no other feasible point 119909of the problem 119875 such that 119911

119894(119909) ge 119911

119894(1199090) and 119894 = 1 2

37 Ideal Solution Let 1199090 be the optimal solution to the

problem 119875119905 119905 = 1 2 then the value of the objective function

119911119905(1199090) 119905 = 1 2 is called the ideal solution to the problem 119875

38 Best Compromise Solution An efficient solution 1199090

isin 119875

is said to be the best compromise solution to the problem 119875if the distance between the ideal solution and the objectivevalue at 119909

0 119911119905(1199090) 119905 = 1 2 is minimum among the efficient

solution to the problem 119875

Lemma 1 The feasible solution for the problem119875 is consideredto be efficient if and only if there exist no other feasible solutionsfor which we obtain a better value at least for one criterion forwhich the value of the rest of criteria remains unmodified

Theorem 2 If 119883119905is the set of efficient points for problem 119875

119905

119905 = 1 2 then 119883119905is a subset of the set of all efficient points 119883 of

119875

Proof Let1198831be the set of efficient points of 119875

1 Let119883

2be the

set of efficient points of 1198752 As 119883

119905sub 119883 119905 = 1 2 The set of all

efficient points of 1198751 1198752is contained in 119875

Theorem 3 Let 1199090

= 119909119894119895 119909119894119895

isin 119875 be an optimal solution of1198751and let 1199090 = 119909

119894119895 119909119894119895

isin 1198752 be an optimal solution of 119875

2 then

119909 = 11990901199090 is an optimal solution of the given problem 119875 where

all of 11990911

11990912

11990921

11990922

are elements of 1198751 1198752

Proof Let 1199090 be an optimal solution of 119875

1 This solution is

an efficient solution if there is no feasible solution these setsof efficient solutions are contained in 119909 Similarly the set ofefficient solutions of 119875

2are contained in 119909 So byTheorem 2

these solutions are the set of efficient solutions of 119875

4 Introduction to Linear Fractional FuzzyProgramming Problem (LFFPP)

One of the main aims of this paper is to extend the linearfractional problem into linear fractional fuzzy programmingproblem

The LFFPP can be presented in the following way

max 119891 (119909) =

119901 (119909)

119902 (119909)

subject to 119909 isin 119878 sub 119877119899

(5)

where 119901(119909) and 119902(119909) are linear functions and the set 119878 isdefined as 119878 = 119909119860119909 = 119887 119909 gt 0 Here 119860 is a fuzzy 119898 times 119899

matrix we will suppose that 119878 is a bounded polyhedron and119902(119909) gt 0 for all119909 isin 119878 It is well known that the goal function inmax 119891(119909) has a global maximum function 119878 and has not anyother local maximaThismaximum is obtained at an extremepoint of 119878

LFFPP (5) can now be written as

max 119901 (119909)

subject to 119909 isin 119878 sub 119877119899

max 119902 (119909)

subject to 119909 isin 119878 sub 119877119899

(6)

Then by Theorem 3 if 119901(119909)119902(119909) gt 0 forall119894 and forall119909 isin 119878 then theset of efficient points for problem 119875

119905 119905 = 1 2 is a subset of the

set of all efficient points of 119875

5 Problem Formulation

The proposed fuzzy transportation model is based on thefollowing assumption

Index Set Consider the following

119894 is source index for all 119894 = 1 2 3 119898

119895 is destination index for all 119895 = 1 2 3 119899

4 Computational Intelligence and Neuroscience

Parameters Consider the following

119909119894119895is the number of units transported from source 119894

to destination 119895119888119894119895is per unit profit cost of transportation from source

119894 to destination 119895119889119894119895is per unit cost of transportation from source 119894 to

destination 119895

Objective Function Consider the following

(i) Fractional transportation problem

min 119911 =

sum119898

119894=1sum119899

119895=1119888119894119895119909119894119895

sum119898

119894=1sum119899

119895=1119889119894119895119909119894119895

(7)

The values of 119888119894119895and 119889

119894119895are considered to be trape-

zoidal fuzzy number

Constraints Consider the following(i) Constraints on supply available for all sources 119894

119898

sum

119894=1

119909119894119895

= 119886119894

forall119894 (8)

(ii) Constraints on demand for each destination 119895

119899

sum

119895=1

119909119894119895

= 119887119895

forall119895 (9)

Nonnegative Constraints Consider the following(i) Nonnegative constraints on decision variables

119909119894119895

ge 0 (10)

The above LFFTPwill have a basic feasible solution onlywhensum119898

119894=1119886119894= sum119899

119895=1119887119895

6 Procedure for Proposed Method

Here we present a new method for solving linear fractionalfuzzy transportation problem

Consider the linear fractional fuzzy transportation prob-lem as

min 119911 =

sum119898

119894=1sum119899

119895=1119888119894119895119909119894119895

sum119898

119894=1sum119899

119895=1119889119894119895119909119894119895

119875 subject to119898

sum

119894=1

119909119894119895

= 119886119894

119899

sum

119895=1

119909119894119895

= 119887119895

119909119894119895

ge 0

(11)

Since the objective function is a fractional objective functionwe take the problem 119875 into two linear fuzzy transportationproblems by (6) as follows

min 119911119873

=

119898

sum

119894=1

119899

sum

119895=1

119888119894119895119909119873

119894119895

1198751 subject to

119898

sum

119894=1

119909119873

119894119895= 119886119894

119899

sum

119895=1

119909119873

119894119895= 119887119895

119909119873

119894119895ge 0

min 119911119863

=

119898

sum

119894=1

119899

sum

119895=1

119888119894119895119909119863

119894119895

1198751 subject to

119898

sum

119894=1

119909119863

119894119895= 119886119894

119899

sum

119895=1

119909119863

119894119895= 119887119895

119909119863

119894119895ge 0

(12)

We solve these two linear fuzzy transportation problems bydual simplex methods Thus the solution is obtained formin 119911

119873 and min 119911119863 separately By Theorem 3 the optimum

solution of 119875 is obtained when the optimal solution of 1198751and

1198752is obtained

7 Numerical Example

To illustrate the proposed method consider the followingexample for linear fractional fuzzy transportation problem

119875 min 119911 = ((0 2 4 6) 11990911

+ (1 2 6 7) 11990912

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

)

sdot ((0 1 3 4) 11990911

+ (2 3 5 6) 11990912

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

)minus1

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

(13)

The problem 119875 is a linear fractional fuzzy transportationproblem inwhich the objective function has trapezoidal fuzzynumber supply and demand values are crisp values Now by

Computational Intelligence and Neuroscience 5

Table 1

119862119895

(minus6 minus4 minus2 0) (minus7 minus6 minus2 minus1) (minus6 minus5 minus4 minus1) (minus8 minus5 minus4 minus3) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

0 1198781

60 1 1 0 0 1 0 0 00 119878

245 0 0 1 1 0 1 0 0

0 1198783

minus50 minus1 0 minus1 0 0 0 1 00 119878

4minus55 0 minus1 0 minus1 0 0 0 1

119911 = 0 (0 2 4 6) (1 2 6 7) (1 4 5 6) (3 4 5 8) 0 0 0 0119877(sdot sdot sdot ) 3 4 4 5 0 0 0 0

uarr darr

Table 2

119862119895

(minus6 minus4 minus2 0) (minus7 minus6 minus2 minus1) (minus6 minus5 minus4 minus1) (minus8 minus5 minus4 minus3) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

(minus8 minus5 minus4 minus3) 11988321

45 0 0 1 1 minus1 0 minus1 minus10 119878

20 0 0 0 0 1 1 1 1

(minus6 minus5 minus4 minus1) 11988311

5 1 0 0 minus1 1 0 0 1(minus7 minus6 minus2 minus1) 119883

1255 0 1 0 1 0 0 0 minus1

119911 = (minus995 minus755 minus390 minus100) 0 0 0 (minus5 8 6 9) (minus5 0 3 6) 0 (1 4 5 6) (minus4 2 9 13)

our computation technique the above problem can be writtenas

1198751 min 119911

119873= (0 2 4 6) 119909

11+ (1 2 6 7) 119909

12

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

1198752 min 119911

119863= (0 1 3 4) 119909

11+ (2 3 5 6) 119909

12

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

(14)

Consider1198751The linear problem can be expressed in standard

form as

1198751 min 119911

119873= (0 2 4 6) 119909

11+ (1 2 6 7) 119909

12

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

subject to 11990911

+ 11990912

+ 1199041= 60

11990921

+ 11990922

+ 1199042= 45

11990911

+ 11990921

minus 1199043= 50

11990912

+ 11990922

minus 1199044= 55

11990911

11990912

11990921

11990922

1199041 1199042 1199043 1199044ge 0

(15)

Now 1198751is solved by the dual simplex method

The initial iteration of the problem is given in Table 1From the initial iteration we can see that the nonbasic

variable 11988312

enters the basis and the basic variable 1198784leaves

the basisProceeding the dual simplex method and after few itera-

tions we get Table 2 In Table 2 all the values of119883119861are positive

and the optimum solution is obtained as follows

min 119911119873

= (100 390 755 995)

11990911

= 5 11990912

= 55 11990921

= 45

(16)

Consider1198752The linear problem can be expressed in standard

form as1198752 min 119911

119863= (0 1 3 4) 119909

11+ (2 3 5 6) 119909

12

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

subject to 11990911

+ 11990912

+ 1198781= 60

11990921

+ 11990922

+ 1198782= 45

11990911

+ 11990921

minus 1198783= 50

11990912

+ 11990922

minus 1198784= 55

11990911

11990912

11990921

11990922

1198781 1198782 1198783 1198784ge 0

(17)Now 119875

2is solved by the dual simplex method

The initial iteration of the problem is given in Table 3From the initial iteration we can see that the nonbasic

variable 11988312

enters the basis and the basic variable 1198784leaves

the basisProceeding the dual simplex method and after few itera-

tionswe get Table 4 In Table 4 all the values of119883119861are positive

and the optimum solution is obtained as follows

min 119911119863

= (155 305 515 665)

11990911

= 5 11990912

= 55 11990921

= 45

(18)

6 Computational Intelligence and Neuroscience

Table 3

119862119895

(minus4 minus3 minus1 0) (minus6 minus5 minus3 minus2) (minus7 minus5 minus3 minus1) (minus9 minus7 minus6 minus2) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

0 1198781

60 1 1 0 0 1 0 0 00 119878

245 0 0 1 1 0 1 0 0

0 1198783

minus50 minus1 0 minus1 0 0 0 1 00 119878

4minus55 0 minus1 0 minus1 0 0 0 1

119911 = 0 (0 1 3 4) (2 3 5 6) (1 3 5 7) (2 6 7 9) 0 0 0 0119877(sdot sdot sdot ) 2 4 4 6 0 0 0 0

uarr darr

Table 4

119862119895

(minus4 minus3 minus1 0) (minus6 minus5 minus3 minus2) (minus7 minus5 minus3 minus1) (minus9 minus7 minus6 minus2) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

(minus7 minus5 minus3 minus1) 11988321

45 0 0 1 1 minus1 0 minus1 minus10 119878

20 0 0 0 0 1 1 1 1

(minus4 minus3 minus1 0) 11988311

5 1 0 0 minus1 1 0 0 1(minus6 minus5 minus3 minus2) 119883

1255 0 1 0 1 0 0 0 minus1

119911 = (minus665 minus515 minus305 minus155) 0 0 0 (8 minus2 3 3) (minus3 0 4 7) 0 (1 3 5 7) (minus1 39 13)

By Theorem 3 the optimum solution of the fractional fuzzytransportation problem is obtained

Thus

min 119911 =

(100 390 755 995)

(155 305 515 665)

(19)

8 Conclusion

The present paper proposes a method for solving linearfractional fuzzy transportation problem where the trape-zoidal fuzzy numbers are used in the objective functionThe limitations of the existing methods are discussed Theadvantage of the computational technique is that the methodworks even when 120572 120573 values are zero The dual simplexmethod is one of the best methods to get the optimalsolution The method can be easily implemented to solveany type of transportation problem The solution procedureshave been illustrated by an example The result shows thehigh efficiency of the solution and this can be extended tofractional quadratic problems The simplex method can beused instead of dual simplex if all the constraints are of le typeconstraints

Conflict of Interests

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

Acknowledgment

The authors would like to thank the anonymous referees forvarious suggestions which have led to an improvement inboth the quality and the clarity of the paper

References

[1] F L Hitchcock ldquoThe distribution of a product from severalsources to numerous localitiesrdquo Journal of Mathematics andPhysics vol 20 pp 224ndash230 1941

[2] G B Dantzig Linear Programming amp Extension PrincetonUniversity Press Princeton NJ USA 1962

[3] A Charnes and W W Cooper ldquoAn explicit general solution inlinear fractional programmingrdquo Naval Research Logistics vol20 no 3 pp 449ndash467 1973

[4] K Swarup ldquoSome aspects of linear fractional functionalsprogrammingrdquo The Australian Journal of Statistics vol 7 pp90ndash104 1965

[5] G R Bitran and A G Novaes ldquoLinear programming with afractional objective functionrdquo Operations Research vol 21 no1 pp 22ndash29 1972

[6] S F Tantawy ldquoA new method for solving linear fractionalprogramming problemsrdquo Australian Journal of Basic amp AppliedSciences vol 1 no 2 pp 105ndash108 2011

[7] M B Hasan and S Acharjee ldquoSolving LFP by converting it intoa single LPrdquo International Journal of Operations Research vol 8no 3 pp 1ndash14 2011

[8] V D Joshi and N Gupta ldquoLinear fractional transportationproblemwith varying demand and supplyrdquo LeMatematiche vol66 pp 2ndash12 2011

[9] S C Sharma and A Bansal ldquoA integer solution of fractionalprogramming problemrdquo General Mathematics Notes vol 4 no2 pp 1ndash9 2011

[10] S Jain A Mangal and S Sharma ldquoC-approach of ABSalgorithm for fractional programming problemrdquo Journal ofComputer and Mathematical Sciences vol 4 no 2 pp 126ndash1342013

[11] A Sulaiman and K Basiya ldquoUsing transformation technique tosolve multi-objective linear fractional problemrdquo InternationalJournal of Research and Reviews in Applied Sciences vol 14 no3 pp 559ndash567 2013

Computational Intelligence and Neuroscience 7

[12] V Verma H C Bakhshi and M C Puri ldquoRanking in integerlinear fractional programming problemsrdquo ZOR Methods ampModels of Operational Research vol 34 no 5 pp 325ndash334 1990

[13] B SMetev and I T Yordanova-Markova ldquoMulti-objective opti-mization over convex disjunctive feasible sets using referencepointsrdquo European Journal of Operational Research vol 98 no 1pp 124ndash137 1997

[14] P Pandian and M Jayalakshmi ldquoOn solving linear fractionalprogramming problemsrdquo Modern Applied Science vol 7 no 6pp 90ndash100 2013

[15] S Chandra ldquoDecomposition principle for linear fractionalfunctional programsrdquo Revue Francaise drsquoInformatique et deRecherche Operationnelle no 10 pp 65ndash71 1968

[16] A N Dheyab ldquoFinding the optimal solution for fractionallinear programming problems with fuzzy numbersrdquo Journal ofKerbala University vol 10 no 3 pp 105ndash110 2012

[17] N Guzel Y Emiroglu F Tapci C Guler and M SyvryldquoA solution proposal to the interval fractional transportationproblemrdquo Applied Mathematics amp Information Sciences vol 6no 3 pp 567ndash571 2012

[18] D Madhuri ldquoLinear fractional time minimizing transportationproblemwith impuritiesrdquo Information Sciences Letters vol 1 no1 pp 7ndash19 2012

[19] S Jain and N Arya ldquoAn inverse transportation problem withthe linear fractional objective functionrdquo Advanced Modelling ampOptimization vol 15 no 3 pp 677ndash687 2013

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 Computational Intelligence and Neuroscience

Parameters Consider the following

119909119894119895is the number of units transported from source 119894

to destination 119895119888119894119895is per unit profit cost of transportation from source

119894 to destination 119895119889119894119895is per unit cost of transportation from source 119894 to

destination 119895

Objective Function Consider the following

(i) Fractional transportation problem

min 119911 =

sum119898

119894=1sum119899

119895=1119888119894119895119909119894119895

sum119898

119894=1sum119899

119895=1119889119894119895119909119894119895

(7)

The values of 119888119894119895and 119889

119894119895are considered to be trape-

zoidal fuzzy number

Constraints Consider the following(i) Constraints on supply available for all sources 119894

119898

sum

119894=1

119909119894119895

= 119886119894

forall119894 (8)

(ii) Constraints on demand for each destination 119895

119899

sum

119895=1

119909119894119895

= 119887119895

forall119895 (9)

Nonnegative Constraints Consider the following(i) Nonnegative constraints on decision variables

119909119894119895

ge 0 (10)

The above LFFTPwill have a basic feasible solution onlywhensum119898

119894=1119886119894= sum119899

119895=1119887119895

6 Procedure for Proposed Method

Here we present a new method for solving linear fractionalfuzzy transportation problem

Consider the linear fractional fuzzy transportation prob-lem as

min 119911 =

sum119898

119894=1sum119899

119895=1119888119894119895119909119894119895

sum119898

119894=1sum119899

119895=1119889119894119895119909119894119895

119875 subject to119898

sum

119894=1

119909119894119895

= 119886119894

119899

sum

119895=1

119909119894119895

= 119887119895

119909119894119895

ge 0

(11)

Since the objective function is a fractional objective functionwe take the problem 119875 into two linear fuzzy transportationproblems by (6) as follows

min 119911119873

=

119898

sum

119894=1

119899

sum

119895=1

119888119894119895119909119873

119894119895

1198751 subject to

119898

sum

119894=1

119909119873

119894119895= 119886119894

119899

sum

119895=1

119909119873

119894119895= 119887119895

119909119873

119894119895ge 0

min 119911119863

=

119898

sum

119894=1

119899

sum

119895=1

119888119894119895119909119863

119894119895

1198751 subject to

119898

sum

119894=1

119909119863

119894119895= 119886119894

119899

sum

119895=1

119909119863

119894119895= 119887119895

119909119863

119894119895ge 0

(12)

We solve these two linear fuzzy transportation problems bydual simplex methods Thus the solution is obtained formin 119911

119873 and min 119911119863 separately By Theorem 3 the optimum

solution of 119875 is obtained when the optimal solution of 1198751and

1198752is obtained

7 Numerical Example

To illustrate the proposed method consider the followingexample for linear fractional fuzzy transportation problem

119875 min 119911 = ((0 2 4 6) 11990911

+ (1 2 6 7) 11990912

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

)

sdot ((0 1 3 4) 11990911

+ (2 3 5 6) 11990912

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

)minus1

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

(13)

The problem 119875 is a linear fractional fuzzy transportationproblem inwhich the objective function has trapezoidal fuzzynumber supply and demand values are crisp values Now by

Computational Intelligence and Neuroscience 5

Table 1

119862119895

(minus6 minus4 minus2 0) (minus7 minus6 minus2 minus1) (minus6 minus5 minus4 minus1) (minus8 minus5 minus4 minus3) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

0 1198781

60 1 1 0 0 1 0 0 00 119878

245 0 0 1 1 0 1 0 0

0 1198783

minus50 minus1 0 minus1 0 0 0 1 00 119878

4minus55 0 minus1 0 minus1 0 0 0 1

119911 = 0 (0 2 4 6) (1 2 6 7) (1 4 5 6) (3 4 5 8) 0 0 0 0119877(sdot sdot sdot ) 3 4 4 5 0 0 0 0

uarr darr

Table 2

119862119895

(minus6 minus4 minus2 0) (minus7 minus6 minus2 minus1) (minus6 minus5 minus4 minus1) (minus8 minus5 minus4 minus3) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

(minus8 minus5 minus4 minus3) 11988321

45 0 0 1 1 minus1 0 minus1 minus10 119878

20 0 0 0 0 1 1 1 1

(minus6 minus5 minus4 minus1) 11988311

5 1 0 0 minus1 1 0 0 1(minus7 minus6 minus2 minus1) 119883

1255 0 1 0 1 0 0 0 minus1

119911 = (minus995 minus755 minus390 minus100) 0 0 0 (minus5 8 6 9) (minus5 0 3 6) 0 (1 4 5 6) (minus4 2 9 13)

our computation technique the above problem can be writtenas

1198751 min 119911

119873= (0 2 4 6) 119909

11+ (1 2 6 7) 119909

12

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

1198752 min 119911

119863= (0 1 3 4) 119909

11+ (2 3 5 6) 119909

12

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

(14)

Consider1198751The linear problem can be expressed in standard

form as

1198751 min 119911

119873= (0 2 4 6) 119909

11+ (1 2 6 7) 119909

12

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

subject to 11990911

+ 11990912

+ 1199041= 60

11990921

+ 11990922

+ 1199042= 45

11990911

+ 11990921

minus 1199043= 50

11990912

+ 11990922

minus 1199044= 55

11990911

11990912

11990921

11990922

1199041 1199042 1199043 1199044ge 0

(15)

Now 1198751is solved by the dual simplex method

The initial iteration of the problem is given in Table 1From the initial iteration we can see that the nonbasic

variable 11988312

enters the basis and the basic variable 1198784leaves

the basisProceeding the dual simplex method and after few itera-

tions we get Table 2 In Table 2 all the values of119883119861are positive

and the optimum solution is obtained as follows

min 119911119873

= (100 390 755 995)

11990911

= 5 11990912

= 55 11990921

= 45

(16)

Consider1198752The linear problem can be expressed in standard

form as1198752 min 119911

119863= (0 1 3 4) 119909

11+ (2 3 5 6) 119909

12

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

subject to 11990911

+ 11990912

+ 1198781= 60

11990921

+ 11990922

+ 1198782= 45

11990911

+ 11990921

minus 1198783= 50

11990912

+ 11990922

minus 1198784= 55

11990911

11990912

11990921

11990922

1198781 1198782 1198783 1198784ge 0

(17)Now 119875

2is solved by the dual simplex method

The initial iteration of the problem is given in Table 3From the initial iteration we can see that the nonbasic

variable 11988312

enters the basis and the basic variable 1198784leaves

the basisProceeding the dual simplex method and after few itera-

tionswe get Table 4 In Table 4 all the values of119883119861are positive

and the optimum solution is obtained as follows

min 119911119863

= (155 305 515 665)

11990911

= 5 11990912

= 55 11990921

= 45

(18)

6 Computational Intelligence and Neuroscience

Table 3

119862119895

(minus4 minus3 minus1 0) (minus6 minus5 minus3 minus2) (minus7 minus5 minus3 minus1) (minus9 minus7 minus6 minus2) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

0 1198781

60 1 1 0 0 1 0 0 00 119878

245 0 0 1 1 0 1 0 0

0 1198783

minus50 minus1 0 minus1 0 0 0 1 00 119878

4minus55 0 minus1 0 minus1 0 0 0 1

119911 = 0 (0 1 3 4) (2 3 5 6) (1 3 5 7) (2 6 7 9) 0 0 0 0119877(sdot sdot sdot ) 2 4 4 6 0 0 0 0

uarr darr

Table 4

119862119895

(minus4 minus3 minus1 0) (minus6 minus5 minus3 minus2) (minus7 minus5 minus3 minus1) (minus9 minus7 minus6 minus2) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

(minus7 minus5 minus3 minus1) 11988321

45 0 0 1 1 minus1 0 minus1 minus10 119878

20 0 0 0 0 1 1 1 1

(minus4 minus3 minus1 0) 11988311

5 1 0 0 minus1 1 0 0 1(minus6 minus5 minus3 minus2) 119883

1255 0 1 0 1 0 0 0 minus1

119911 = (minus665 minus515 minus305 minus155) 0 0 0 (8 minus2 3 3) (minus3 0 4 7) 0 (1 3 5 7) (minus1 39 13)

By Theorem 3 the optimum solution of the fractional fuzzytransportation problem is obtained

Thus

min 119911 =

(100 390 755 995)

(155 305 515 665)

(19)

8 Conclusion

The present paper proposes a method for solving linearfractional fuzzy transportation problem where the trape-zoidal fuzzy numbers are used in the objective functionThe limitations of the existing methods are discussed Theadvantage of the computational technique is that the methodworks even when 120572 120573 values are zero The dual simplexmethod is one of the best methods to get the optimalsolution The method can be easily implemented to solveany type of transportation problem The solution procedureshave been illustrated by an example The result shows thehigh efficiency of the solution and this can be extended tofractional quadratic problems The simplex method can beused instead of dual simplex if all the constraints are of le typeconstraints

Conflict of Interests

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

Acknowledgment

The authors would like to thank the anonymous referees forvarious suggestions which have led to an improvement inboth the quality and the clarity of the paper

References

[1] F L Hitchcock ldquoThe distribution of a product from severalsources to numerous localitiesrdquo Journal of Mathematics andPhysics vol 20 pp 224ndash230 1941

[2] G B Dantzig Linear Programming amp Extension PrincetonUniversity Press Princeton NJ USA 1962

[3] A Charnes and W W Cooper ldquoAn explicit general solution inlinear fractional programmingrdquo Naval Research Logistics vol20 no 3 pp 449ndash467 1973

[4] K Swarup ldquoSome aspects of linear fractional functionalsprogrammingrdquo The Australian Journal of Statistics vol 7 pp90ndash104 1965

[5] G R Bitran and A G Novaes ldquoLinear programming with afractional objective functionrdquo Operations Research vol 21 no1 pp 22ndash29 1972

[6] S F Tantawy ldquoA new method for solving linear fractionalprogramming problemsrdquo Australian Journal of Basic amp AppliedSciences vol 1 no 2 pp 105ndash108 2011

[7] M B Hasan and S Acharjee ldquoSolving LFP by converting it intoa single LPrdquo International Journal of Operations Research vol 8no 3 pp 1ndash14 2011

[8] V D Joshi and N Gupta ldquoLinear fractional transportationproblemwith varying demand and supplyrdquo LeMatematiche vol66 pp 2ndash12 2011

[9] S C Sharma and A Bansal ldquoA integer solution of fractionalprogramming problemrdquo General Mathematics Notes vol 4 no2 pp 1ndash9 2011

[10] S Jain A Mangal and S Sharma ldquoC-approach of ABSalgorithm for fractional programming problemrdquo Journal ofComputer and Mathematical Sciences vol 4 no 2 pp 126ndash1342013

[11] A Sulaiman and K Basiya ldquoUsing transformation technique tosolve multi-objective linear fractional problemrdquo InternationalJournal of Research and Reviews in Applied Sciences vol 14 no3 pp 559ndash567 2013

Computational Intelligence and Neuroscience 7

[12] V Verma H C Bakhshi and M C Puri ldquoRanking in integerlinear fractional programming problemsrdquo ZOR Methods ampModels of Operational Research vol 34 no 5 pp 325ndash334 1990

[13] B SMetev and I T Yordanova-Markova ldquoMulti-objective opti-mization over convex disjunctive feasible sets using referencepointsrdquo European Journal of Operational Research vol 98 no 1pp 124ndash137 1997

[14] P Pandian and M Jayalakshmi ldquoOn solving linear fractionalprogramming problemsrdquo Modern Applied Science vol 7 no 6pp 90ndash100 2013

[15] S Chandra ldquoDecomposition principle for linear fractionalfunctional programsrdquo Revue Francaise drsquoInformatique et deRecherche Operationnelle no 10 pp 65ndash71 1968

[16] A N Dheyab ldquoFinding the optimal solution for fractionallinear programming problems with fuzzy numbersrdquo Journal ofKerbala University vol 10 no 3 pp 105ndash110 2012

[17] N Guzel Y Emiroglu F Tapci C Guler and M SyvryldquoA solution proposal to the interval fractional transportationproblemrdquo Applied Mathematics amp Information Sciences vol 6no 3 pp 567ndash571 2012

[18] D Madhuri ldquoLinear fractional time minimizing transportationproblemwith impuritiesrdquo Information Sciences Letters vol 1 no1 pp 7ndash19 2012

[19] S Jain and N Arya ldquoAn inverse transportation problem withthe linear fractional objective functionrdquo Advanced Modelling ampOptimization vol 15 no 3 pp 677ndash687 2013

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 5

Table 1

119862119895

(minus6 minus4 minus2 0) (minus7 minus6 minus2 minus1) (minus6 minus5 minus4 minus1) (minus8 minus5 minus4 minus3) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

0 1198781

60 1 1 0 0 1 0 0 00 119878

245 0 0 1 1 0 1 0 0

0 1198783

minus50 minus1 0 minus1 0 0 0 1 00 119878

4minus55 0 minus1 0 minus1 0 0 0 1

119911 = 0 (0 2 4 6) (1 2 6 7) (1 4 5 6) (3 4 5 8) 0 0 0 0119877(sdot sdot sdot ) 3 4 4 5 0 0 0 0

uarr darr

Table 2

119862119895

(minus6 minus4 minus2 0) (minus7 minus6 minus2 minus1) (minus6 minus5 minus4 minus1) (minus8 minus5 minus4 minus3) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

(minus8 minus5 minus4 minus3) 11988321

45 0 0 1 1 minus1 0 minus1 minus10 119878

20 0 0 0 0 1 1 1 1

(minus6 minus5 minus4 minus1) 11988311

5 1 0 0 minus1 1 0 0 1(minus7 minus6 minus2 minus1) 119883

1255 0 1 0 1 0 0 0 minus1

119911 = (minus995 minus755 minus390 minus100) 0 0 0 (minus5 8 6 9) (minus5 0 3 6) 0 (1 4 5 6) (minus4 2 9 13)

our computation technique the above problem can be writtenas

1198751 min 119911

119873= (0 2 4 6) 119909

11+ (1 2 6 7) 119909

12

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

1198752 min 119911

119863= (0 1 3 4) 119909

11+ (2 3 5 6) 119909

12

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

subject to 11990911

+ 11990912

le 60

11990921

+ 11990922

le 45

11990911

+ 11990921

ge 50

11990912

+ 11990922

ge 55

11990911

11990912

11990921

11990922

ge 0

(14)

Consider1198751The linear problem can be expressed in standard

form as

1198751 min 119911

119873= (0 2 4 6) 119909

11+ (1 2 6 7) 119909

12

+ (1 4 5 6) 11990921

+ (3 4 5 8) 11990922

subject to 11990911

+ 11990912

+ 1199041= 60

11990921

+ 11990922

+ 1199042= 45

11990911

+ 11990921

minus 1199043= 50

11990912

+ 11990922

minus 1199044= 55

11990911

11990912

11990921

11990922

1199041 1199042 1199043 1199044ge 0

(15)

Now 1198751is solved by the dual simplex method

The initial iteration of the problem is given in Table 1From the initial iteration we can see that the nonbasic

variable 11988312

enters the basis and the basic variable 1198784leaves

the basisProceeding the dual simplex method and after few itera-

tions we get Table 2 In Table 2 all the values of119883119861are positive

and the optimum solution is obtained as follows

min 119911119873

= (100 390 755 995)

11990911

= 5 11990912

= 55 11990921

= 45

(16)

Consider1198752The linear problem can be expressed in standard

form as1198752 min 119911

119863= (0 1 3 4) 119909

11+ (2 3 5 6) 119909

12

+ (1 3 5 7) 11990921

+ (2 6 7 9) 11990922

subject to 11990911

+ 11990912

+ 1198781= 60

11990921

+ 11990922

+ 1198782= 45

11990911

+ 11990921

minus 1198783= 50

11990912

+ 11990922

minus 1198784= 55

11990911

11990912

11990921

11990922

1198781 1198782 1198783 1198784ge 0

(17)Now 119875

2is solved by the dual simplex method

The initial iteration of the problem is given in Table 3From the initial iteration we can see that the nonbasic

variable 11988312

enters the basis and the basic variable 1198784leaves

the basisProceeding the dual simplex method and after few itera-

tionswe get Table 4 In Table 4 all the values of119883119861are positive

and the optimum solution is obtained as follows

min 119911119863

= (155 305 515 665)

11990911

= 5 11990912

= 55 11990921

= 45

(18)

6 Computational Intelligence and Neuroscience

Table 3

119862119895

(minus4 minus3 minus1 0) (minus6 minus5 minus3 minus2) (minus7 minus5 minus3 minus1) (minus9 minus7 minus6 minus2) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

0 1198781

60 1 1 0 0 1 0 0 00 119878

245 0 0 1 1 0 1 0 0

0 1198783

minus50 minus1 0 minus1 0 0 0 1 00 119878

4minus55 0 minus1 0 minus1 0 0 0 1

119911 = 0 (0 1 3 4) (2 3 5 6) (1 3 5 7) (2 6 7 9) 0 0 0 0119877(sdot sdot sdot ) 2 4 4 6 0 0 0 0

uarr darr

Table 4

119862119895

(minus4 minus3 minus1 0) (minus6 minus5 minus3 minus2) (minus7 minus5 minus3 minus1) (minus9 minus7 minus6 minus2) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

(minus7 minus5 minus3 minus1) 11988321

45 0 0 1 1 minus1 0 minus1 minus10 119878

20 0 0 0 0 1 1 1 1

(minus4 minus3 minus1 0) 11988311

5 1 0 0 minus1 1 0 0 1(minus6 minus5 minus3 minus2) 119883

1255 0 1 0 1 0 0 0 minus1

119911 = (minus665 minus515 minus305 minus155) 0 0 0 (8 minus2 3 3) (minus3 0 4 7) 0 (1 3 5 7) (minus1 39 13)

By Theorem 3 the optimum solution of the fractional fuzzytransportation problem is obtained

Thus

min 119911 =

(100 390 755 995)

(155 305 515 665)

(19)

8 Conclusion

The present paper proposes a method for solving linearfractional fuzzy transportation problem where the trape-zoidal fuzzy numbers are used in the objective functionThe limitations of the existing methods are discussed Theadvantage of the computational technique is that the methodworks even when 120572 120573 values are zero The dual simplexmethod is one of the best methods to get the optimalsolution The method can be easily implemented to solveany type of transportation problem The solution procedureshave been illustrated by an example The result shows thehigh efficiency of the solution and this can be extended tofractional quadratic problems The simplex method can beused instead of dual simplex if all the constraints are of le typeconstraints

Conflict of Interests

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

Acknowledgment

The authors would like to thank the anonymous referees forvarious suggestions which have led to an improvement inboth the quality and the clarity of the paper

References

[1] F L Hitchcock ldquoThe distribution of a product from severalsources to numerous localitiesrdquo Journal of Mathematics andPhysics vol 20 pp 224ndash230 1941

[2] G B Dantzig Linear Programming amp Extension PrincetonUniversity Press Princeton NJ USA 1962

[3] A Charnes and W W Cooper ldquoAn explicit general solution inlinear fractional programmingrdquo Naval Research Logistics vol20 no 3 pp 449ndash467 1973

[4] K Swarup ldquoSome aspects of linear fractional functionalsprogrammingrdquo The Australian Journal of Statistics vol 7 pp90ndash104 1965

[5] G R Bitran and A G Novaes ldquoLinear programming with afractional objective functionrdquo Operations Research vol 21 no1 pp 22ndash29 1972

[6] S F Tantawy ldquoA new method for solving linear fractionalprogramming problemsrdquo Australian Journal of Basic amp AppliedSciences vol 1 no 2 pp 105ndash108 2011

[7] M B Hasan and S Acharjee ldquoSolving LFP by converting it intoa single LPrdquo International Journal of Operations Research vol 8no 3 pp 1ndash14 2011

[8] V D Joshi and N Gupta ldquoLinear fractional transportationproblemwith varying demand and supplyrdquo LeMatematiche vol66 pp 2ndash12 2011

[9] S C Sharma and A Bansal ldquoA integer solution of fractionalprogramming problemrdquo General Mathematics Notes vol 4 no2 pp 1ndash9 2011

[10] S Jain A Mangal and S Sharma ldquoC-approach of ABSalgorithm for fractional programming problemrdquo Journal ofComputer and Mathematical Sciences vol 4 no 2 pp 126ndash1342013

[11] A Sulaiman and K Basiya ldquoUsing transformation technique tosolve multi-objective linear fractional problemrdquo InternationalJournal of Research and Reviews in Applied Sciences vol 14 no3 pp 559ndash567 2013

Computational Intelligence and Neuroscience 7

[12] V Verma H C Bakhshi and M C Puri ldquoRanking in integerlinear fractional programming problemsrdquo ZOR Methods ampModels of Operational Research vol 34 no 5 pp 325ndash334 1990

[13] B SMetev and I T Yordanova-Markova ldquoMulti-objective opti-mization over convex disjunctive feasible sets using referencepointsrdquo European Journal of Operational Research vol 98 no 1pp 124ndash137 1997

[14] P Pandian and M Jayalakshmi ldquoOn solving linear fractionalprogramming problemsrdquo Modern Applied Science vol 7 no 6pp 90ndash100 2013

[15] S Chandra ldquoDecomposition principle for linear fractionalfunctional programsrdquo Revue Francaise drsquoInformatique et deRecherche Operationnelle no 10 pp 65ndash71 1968

[16] A N Dheyab ldquoFinding the optimal solution for fractionallinear programming problems with fuzzy numbersrdquo Journal ofKerbala University vol 10 no 3 pp 105ndash110 2012

[17] N Guzel Y Emiroglu F Tapci C Guler and M SyvryldquoA solution proposal to the interval fractional transportationproblemrdquo Applied Mathematics amp Information Sciences vol 6no 3 pp 567ndash571 2012

[18] D Madhuri ldquoLinear fractional time minimizing transportationproblemwith impuritiesrdquo Information Sciences Letters vol 1 no1 pp 7ndash19 2012

[19] S Jain and N Arya ldquoAn inverse transportation problem withthe linear fractional objective functionrdquo Advanced Modelling ampOptimization vol 15 no 3 pp 677ndash687 2013

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Computational Intelligence and Neuroscience

Table 3

119862119895

(minus4 minus3 minus1 0) (minus6 minus5 minus3 minus2) (minus7 minus5 minus3 minus1) (minus9 minus7 minus6 minus2) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

0 1198781

60 1 1 0 0 1 0 0 00 119878

245 0 0 1 1 0 1 0 0

0 1198783

minus50 minus1 0 minus1 0 0 0 1 00 119878

4minus55 0 minus1 0 minus1 0 0 0 1

119911 = 0 (0 1 3 4) (2 3 5 6) (1 3 5 7) (2 6 7 9) 0 0 0 0119877(sdot sdot sdot ) 2 4 4 6 0 0 0 0

uarr darr

Table 4

119862119895

(minus4 minus3 minus1 0) (minus6 minus5 minus3 minus2) (minus7 minus5 minus3 minus1) (minus9 minus7 minus6 minus2) 0 0 0 0119884119861

119883119861

11988311

11988312

11988321

11988322

1198781

1198782

1198783

1198784

(minus7 minus5 minus3 minus1) 11988321

45 0 0 1 1 minus1 0 minus1 minus10 119878

20 0 0 0 0 1 1 1 1

(minus4 minus3 minus1 0) 11988311

5 1 0 0 minus1 1 0 0 1(minus6 minus5 minus3 minus2) 119883

1255 0 1 0 1 0 0 0 minus1

119911 = (minus665 minus515 minus305 minus155) 0 0 0 (8 minus2 3 3) (minus3 0 4 7) 0 (1 3 5 7) (minus1 39 13)

By Theorem 3 the optimum solution of the fractional fuzzytransportation problem is obtained

Thus

min 119911 =

(100 390 755 995)

(155 305 515 665)

(19)

8 Conclusion

The present paper proposes a method for solving linearfractional fuzzy transportation problem where the trape-zoidal fuzzy numbers are used in the objective functionThe limitations of the existing methods are discussed Theadvantage of the computational technique is that the methodworks even when 120572 120573 values are zero The dual simplexmethod is one of the best methods to get the optimalsolution The method can be easily implemented to solveany type of transportation problem The solution procedureshave been illustrated by an example The result shows thehigh efficiency of the solution and this can be extended tofractional quadratic problems The simplex method can beused instead of dual simplex if all the constraints are of le typeconstraints

Conflict of Interests

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

Acknowledgment

The authors would like to thank the anonymous referees forvarious suggestions which have led to an improvement inboth the quality and the clarity of the paper

References

[1] F L Hitchcock ldquoThe distribution of a product from severalsources to numerous localitiesrdquo Journal of Mathematics andPhysics vol 20 pp 224ndash230 1941

[2] G B Dantzig Linear Programming amp Extension PrincetonUniversity Press Princeton NJ USA 1962

[3] A Charnes and W W Cooper ldquoAn explicit general solution inlinear fractional programmingrdquo Naval Research Logistics vol20 no 3 pp 449ndash467 1973

[4] K Swarup ldquoSome aspects of linear fractional functionalsprogrammingrdquo The Australian Journal of Statistics vol 7 pp90ndash104 1965

[5] G R Bitran and A G Novaes ldquoLinear programming with afractional objective functionrdquo Operations Research vol 21 no1 pp 22ndash29 1972

[6] S F Tantawy ldquoA new method for solving linear fractionalprogramming problemsrdquo Australian Journal of Basic amp AppliedSciences vol 1 no 2 pp 105ndash108 2011

[7] M B Hasan and S Acharjee ldquoSolving LFP by converting it intoa single LPrdquo International Journal of Operations Research vol 8no 3 pp 1ndash14 2011

[8] V D Joshi and N Gupta ldquoLinear fractional transportationproblemwith varying demand and supplyrdquo LeMatematiche vol66 pp 2ndash12 2011

[9] S C Sharma and A Bansal ldquoA integer solution of fractionalprogramming problemrdquo General Mathematics Notes vol 4 no2 pp 1ndash9 2011

[10] S Jain A Mangal and S Sharma ldquoC-approach of ABSalgorithm for fractional programming problemrdquo Journal ofComputer and Mathematical Sciences vol 4 no 2 pp 126ndash1342013

[11] A Sulaiman and K Basiya ldquoUsing transformation technique tosolve multi-objective linear fractional problemrdquo InternationalJournal of Research and Reviews in Applied Sciences vol 14 no3 pp 559ndash567 2013

Computational Intelligence and Neuroscience 7

[12] V Verma H C Bakhshi and M C Puri ldquoRanking in integerlinear fractional programming problemsrdquo ZOR Methods ampModels of Operational Research vol 34 no 5 pp 325ndash334 1990

[13] B SMetev and I T Yordanova-Markova ldquoMulti-objective opti-mization over convex disjunctive feasible sets using referencepointsrdquo European Journal of Operational Research vol 98 no 1pp 124ndash137 1997

[14] P Pandian and M Jayalakshmi ldquoOn solving linear fractionalprogramming problemsrdquo Modern Applied Science vol 7 no 6pp 90ndash100 2013

[15] S Chandra ldquoDecomposition principle for linear fractionalfunctional programsrdquo Revue Francaise drsquoInformatique et deRecherche Operationnelle no 10 pp 65ndash71 1968

[16] A N Dheyab ldquoFinding the optimal solution for fractionallinear programming problems with fuzzy numbersrdquo Journal ofKerbala University vol 10 no 3 pp 105ndash110 2012

[17] N Guzel Y Emiroglu F Tapci C Guler and M SyvryldquoA solution proposal to the interval fractional transportationproblemrdquo Applied Mathematics amp Information Sciences vol 6no 3 pp 567ndash571 2012

[18] D Madhuri ldquoLinear fractional time minimizing transportationproblemwith impuritiesrdquo Information Sciences Letters vol 1 no1 pp 7ndash19 2012

[19] S Jain and N Arya ldquoAn inverse transportation problem withthe linear fractional objective functionrdquo Advanced Modelling ampOptimization vol 15 no 3 pp 677ndash687 2013

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 7

[12] V Verma H C Bakhshi and M C Puri ldquoRanking in integerlinear fractional programming problemsrdquo ZOR Methods ampModels of Operational Research vol 34 no 5 pp 325ndash334 1990

[13] B SMetev and I T Yordanova-Markova ldquoMulti-objective opti-mization over convex disjunctive feasible sets using referencepointsrdquo European Journal of Operational Research vol 98 no 1pp 124ndash137 1997

[14] P Pandian and M Jayalakshmi ldquoOn solving linear fractionalprogramming problemsrdquo Modern Applied Science vol 7 no 6pp 90ndash100 2013

[15] S Chandra ldquoDecomposition principle for linear fractionalfunctional programsrdquo Revue Francaise drsquoInformatique et deRecherche Operationnelle no 10 pp 65ndash71 1968

[16] A N Dheyab ldquoFinding the optimal solution for fractionallinear programming problems with fuzzy numbersrdquo Journal ofKerbala University vol 10 no 3 pp 105ndash110 2012

[17] N Guzel Y Emiroglu F Tapci C Guler and M SyvryldquoA solution proposal to the interval fractional transportationproblemrdquo Applied Mathematics amp Information Sciences vol 6no 3 pp 567ndash571 2012

[18] D Madhuri ldquoLinear fractional time minimizing transportationproblemwith impuritiesrdquo Information Sciences Letters vol 1 no1 pp 7ndash19 2012

[19] S Jain and N Arya ldquoAn inverse transportation problem withthe linear fractional objective functionrdquo Advanced Modelling ampOptimization vol 15 no 3 pp 677ndash687 2013

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014