Applied Soft Computing 19 (2014) 171176

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Applied Soft Computing

j ourna l h o mepage: www.elsev ier .com/ locate /asoc

A simp anwith ge

Ali EbrahDepartment of

a r t i c l

Article history:Received 6 JulReceived in reAccepted 30 JaAvailable onlin

Keywords:Fuzzy transpoRanking functGeneralized tr

12) py assubershich

uctios shoproan pro

1. Introdu

Transpoear programdeservedly received a great deal of attention in the literature. Thecentral concept in the problem is to nd the least total transporta-tion cost of a commodity in order to satisfy demands at destinationsusing available supplies at origins. Transportation problem can beused for a wide variety of situations such as scheduling, produc-tion, investscheduling,are solved wvalues of sucrisp environo crisp infportation por elementsfuzzy sets, anatural way

Since theming probleing fuzzy linto the FTP.solutions wdata.

Tel.: +98 0E-mail add

eral 2,13ns fuds fo

[25] proposed an algorithm for solving transportation problemswhere the supplies and demands are fuzzy sets with linear ortriangular membership functions. Chanas et al. [4] investigatedthe transportation problem with fuzzy supplies and demands andsolved them via the parametric programming technique. Their

http://dx.doi.o1568-4946/ ment, plant location, inventory control, employment and many others. In general, transportation problemsith the assumptions that the transportation costs and

pplies and demands are specied in a precise way i.e., innment. However, in many cases the decision maker hasormation about the coefcients belonging to the trans-roblem. In these cases, the corresponding coefcients

dening the problem can be formulated by means ofnd the fuzzy transportation problem (FTP) appears in a.

transportation problem is essentially a linear program-m, one straightforward approach is to apply the exist-ear programming techniques [2,11,14,27,31,32,34,35]

But, some of these techniques [2,32] only give crisphich represent a compromise solution in terms of fuzzy

9112129663.resses: aemarzoun@gmail.com, a.ebrahimnejad@srbiau.ac.ir

method provided solution which simultaneously satises the con-straints and the goal to a maximal degree. In addition, Chanas andKuchta [5] discussed the type of transportation problems with fuzzycost coefcients and converted the problem into a bicriterial trans-portation problem with crisp objective function. Their method onlygives crisp solutions based on efcient solutions of the convertedproblems. Jimenez and Verdegay [15,16] investigated the fuzzysolid transportation problem in which supplies, demands and con-veyance capacities are represented by trapezoidal fuzzy numbersand applied a parametric approach for nding the fuzzy solution.Liu and Kao [24] developed a procedure, based on extension prin-ciple to derive the fuzzy objective value of FTP in that the costcoefcients and the supply and demand quantities are fuzzy num-bers. Gani and Razak [12] presented a two stage cost minimizingFTP in which supplies and demands are as trapezoidal fuzzy num-bers and used a parametric approach for nding a fuzzy solutionwith the aim of minimizing the sum of the transportation costsin the two stages. Yang and Liu [33] investigated the xed chargesolid transportation problem under fuzzy environment, in whichthe direct costs, the xed charges, the supplies, the demands andthe conveyance capacities are supposed to be fuzzy variables. Li

rg/10.1016/j.asoc.2014.01.0412014 Elsevier B.V. All rights reserved.lied new approach for solving fuzzy trneralized trapezoidal fuzzy numbers

imnejad

Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

e i n f o

y 2013vised form 29 January 2014nuary 2014e 22 February 2014

rtation problemionapezoidal fuzzy number

a b s t r a c t

In a recent paper, Kaur and Kumar (20fuzzy transportation problem (FTP) bby generalized trapezoidal fuzzy numthe FTP is converted into crisp one, wThe main contribution here is the redsolving two application examples, it isolving any FTP. Since the proposed apand to apply on real life transportatio

ction

rtation problem is an important network structured lin-ming problem that arises in several contexts and has

Sev[1,4,5,1ermanmethosportation problems

roposed a new method based on ranking function for solvingming that the values of transportation costs are represented. Here it is shown that once the ranking function is chosen,

is easily solved by the standard transportation algorithms.n of the computational complexity of the existing method. Bywn that it is possible to nd a same optimal solution withoutch is based on classical approach it is very easy to understandblems for the decision makers.

2014 Elsevier B.V. All rights reserved.

researchers have carried out investigations on FTP,1526,28,29,33]. Zimmermann [35] developed Zimm-zzy linear programming into several fuzzy optimizationr solving the transportation problems. Oheigeartaigh

172 A. Ebrahimnejad / Applied Soft Computing 19 (2014) 171176

et al. [22] proposed a new method based on goal programming forsolving FTP with fuzzy costs. Lin [23] used genetic algorithm forsolving transportation problems with fuzzy coefcients. Dinagarand Palanivel [10] investigated FTP, with the help of trapezoidalfuzzy numbobtain the and Natarajpoint methwhich the tby trapezoinew methoing the optproposed aas well as ttransportat[1] proposewhether maestablishedsolid transpnew methotion problefuzzy numthe fuzzy vfuzzy initiarespectively

Chen [6for situatiothe normalhave devotreal life prized fuzzy proposed aFTP by assuare represerecent papecial type ofabout the pno uncertathat study, trapezoidalfor nding mal solutioare represepaper, we sis convertedtransportatin this studyproposed mnique is basand to applmakers.

This papspecial typenumbers an3, a simpliof FTP. In Sshown usinare discussesuggestion

2. Fuzzy tr

The FTP,cise values

destination, but sure about the supply and demand of the product,can be formulated as follows [18]:

m1

nj=1

xij =

= bj

i =

ai is temanr tran

to thouldision

ui an coluen in

m

1

aiu

i v

r andix Aest cd (Gd (GFTP gsicaled din of on duropoeral

ing s the Fm an

of g

in re

ordinenws tapez

w =en inthis, umbnumlent ndarre dos dec

word [18

. Fin or ers and applied fuzzy modied distribution method tooptimal solution in terms of fuzzy numbers. Pandianan [26] introduced a new algorithm namely, fuzzy zerood for nding fuzzy optimal solution for such FTP inransportation cost, supply and demand are representeddal fuzzy numbers. Kumar and Kaur [19] proposed ad based on fuzzy linear programming problem for nd-imal solution of FTP. Chakraborty and Chakraborty [3]

method for the minimization of transportation costime of transportation when the demand, supply andion cost per unit of the quantities are fuzzy. Basirzadehd a systematic procedure for solving all types of FTPximize or minimize objective function. Tao and Xu [30]

a rough multiple objective programming model for aortation problem. Gupat and Kumar [13] proposed ad to nd solution of a linear multi-objective transporta-m by representing all parameters as interval-valuedbers. Shanmugasundari and Ganesan [28] developedersion of Vogels and MODI methods for obtaining thel basic feasible solution and fuzzy optimal solution,.

] proposed the concept of generalized fuzzy numbersns that the membership function is not restricted to

form. Since then, a high number of researchers [79]ed their efforts to use generalized fuzzy numbers inoblems, but there are few papers in which general-numbers are used to solve FTP. Kaur and Kumar [17]

new method based on ranking function for solvingming that the parameters of transportation problemnted by generalized trapezoidal fuzzy numbers. In ar in this journal, Kaur and Kumar [18] studied a spe-

FTP by assuming that a decision maker is uncertainrecise values of transportation cost only but there isinty about the supply and demand of the product. Intransportation costs were represented by generalized

fuzzy numbers. They modied some existing methodsthe initial basic feasible solution (IBFS) and fuzzy opti-n using ranking function, in which transportation costsnted as generalized trapezoidal fuzzy numbers. In thishow that once the ranking function is chosen, the FTP

into crisp one, which is easily solved by the standardion algorithms. It is demonstrated that the method used

is simpler and computationally more efcient than theethod by Kaur and Kumar [18]. Since the proposed tech-ed on classical approach it is very easy to understandy on real life transportation problems for the decision

er is organized as follows: In Section 2, formulation of of the FTP in terms of generalized trapezoidal fuzzyd summary of the existing method are given. In Sectioned new method is proposed to nd optimal solutionection 4, the application of the proposed method isg two application examples and the obtained resultsd. Section 5 ends this paper with a brief conclusion andfor future directions.

ansportation problem

in which a decision maker is uncertain about the pre- of transportation cost from the ith source to the jth

min

i=

s.t.

nj=1

mi=1

xij

xij0,

wheretotal dcost fosourcethat shor deccost.

Letand jthFTP giv

maxi=

s.t.u

KauAppennorthwmethomethoof the of clasmodisolutiobased their pthe genfollowtion ofprobleinstead

3. Ma

Accble to dIt folloized trwhereFTP givTo do fuzzy nfuzzy equivathe stations aeffort i

It ismetho

Step 1GFLCMcijxij

ai, i = 1, 2, . . ., m,

, j = 1, 2, . . ., n,

1, 2, . . ., m, j = 1, 2, . . ., n.

(1)

he total availability of the product at ith source; bj is thed of the product at jth destination; cij is the approximatesporting one unit quantity of the product from the ithe jth destination; xij is the number of units of the product

be transported from the ith source to the jth destination variables;

mi=1n

j=1cijxij is total fuzzy transportation

d vi be the fuzzy dual variables associated with ith rowmn constraints, respectively, then the fuzzy dual of the

Eq. (1) will be as follows [18]:

i n

j=1bj vj

jcij i = 1, 2, . . ., m, j = 1, 2, . . ., n.(2)

d Kumar [18] based on the ranking function given in, introduced three methods namely generalized fuzzyorner method (GFNCWM), generalized fuzzy least-costFLCM) and generalized fuzzy Vogels approximationFVAM) to nd the initial basic feasible solution (IBFS)iven in Eq. (1). These methods are the direct extension

approaches. Then, they applied the generalized fuzzystribution method (GFMDM) to nd the fuzzy optimalFTP given in Eq. (1) with the help of IBFS. The FGMDM isal feasibility and complementary slackness theorem. Insed method all arithmetic operations are performed onized trapezoidal fuzzy numbers, i.e., ui, vj and cij . In theection, we show that it is possible to nd the same solu-TP given in Eq. (1) with the help of crisp transportationd so all arithmetic operations are done on real numberseneralized trapezoidal fuzzy numbers.

sults

g to ranking function given in Appendix A, it is possi-e a rank for each generalized trapezoidal fuzzy number.hat if Ai = (ai, bi, ci, di; wi), (i = 1, 2, .., k) be k general-oidal fuzzy numbers, then (Ai) = w(ai + bi + ci + di)/4,

min{

wi, i = 1, 2, . . ., k}. This helps us to convert the

Eq. (1) into an equivalent crisp transportation problem.we substitute the rank of each generalized trapezoidaler instead of the corresponding generalized trapezoidalber in the FTP under consideration. This leads to ancrisp transportation problem which can be solved byd transportation algorithms. Then, all arithmetic opera-ne on the crisp numbers. As a result, the computationalreased signicantly in our proposed approach.th noting that the general steps of the Kaur and Kumars] to the FTP given in Eq. (1) are as follows:

d an IBFS of the FTP given in Eq. (1) using GFNCWM orGFVAM.

A. Ebrahimnejad / Applied Soft Computing 19 (2014) 171176 173

Table 1Tabular representation of the chosen FTP.

D1 D2 D3 Availability (ai)

S1 (1,4,9,19;0.5) (1,2,5,9;0.4) (2,5,8,18;0.5) 10S2 (7,9,13,28;0.4) 14S3 (4,5,8,11;0.6) 15Demand (bj) 10

Step 2. CoStop or selStep 3. DetStep 4. Obt

Our maitional comp[18]. In parless numbecations, andmethod [18

Accordinthe Step 1 fuzzy transorder m nthis, it is reqranking funFTT of ordeparison of fGFVAM it isthe FTT of onext smallea fuzzy penm n in all mined baseare repeatewe see to dmetic operaon generalizposed metharithmetic o

In additiby Kaur anui vj=cijvariables. Anon-basic vthe enteringrank of dij . Ttions on genproposed many fuzzy sy

These recomputatioand Kumar

4. Applicat

In this sto show theresults are d

4.1. Examp

Example 4uct availablat three de

orting one unit quantity of product from each source to eachtion is represented by generalized trapezoidal fuzzy num-termine the fuzzy optimal transportation of products suche total fuzzy transportation cost is minimum.

rst in T

the e 2.

crisp stans med andse thel solnd be us

follon G

10,

5, x

0, x

apn Eq

the Fmarlized(3). B

on oposrs.dditi

the fon G

0, x

10,

5, x

his ced IBree

10,

5, x

0, x

repres(8,9,12,26; 0.5) (3,5,8,12;0.2) (11,12,20,27;0.5) (0,5,10,15;0.8)

15 14

mpute dij = cij (ui vj) for each non-basic variable.ect an entering column.ermine an existing column.ain the new BFS and repeat Step 2.

n contribution here is the reduction of the computa-lexity of the proposed method by Kaur and Kumar

ticular, it is shown that our proposed method need tor of elementary operations such as additions, multipli-

comparisons as compared to the Kaur and Kumars].g to the Kaur and Kumars method [18], to carry outusing GFCLM it is required to determine the smallestportation cost in fuzzy transportation tableau (FTT) of

until the FTT is reduced into a FTT of order 1 1. To douired to compare the fuzzy transportation costs usingction given in Appendix A until the FTT is reduced into ar 1 1. While based on our proposed method the com-uzzy costs is done once. Moreover, to nd IBFS using

required to compute a fuzzy penalty for each row ofrder m n by subtracting the smallest entry from thest entry in each step of this method. In a similar way,alty is computed for each column of the FTT of orderiterations. After that, the highest fuzzy penalty is deter-d on ranking function given in Appendix A. These stepsd until the FTT is reduced into a FTT of order 1 1. Aso Step 1 using GFVAM, it is required a lot of fuzzy arith-tions such as additions, subtractions and comparisoned trapezoidal fuzzy numbers. While based on our pro-od the comparison of fuzzy costs is done once and allperations are done on real numbers.on, to carry out the Step 2 using the GFMDM proposedd Kumar [18] it is required to solve the fuzzy systemwith m + n 1 fuzzy equations corresponding to basicfter solving this fuzzy system, the fuzzy value dij for eachariable is obtained based on dij = cij (ui vj). Finally,

column is determined according to the most negativehis step requires to a lot of fuzzy additions and subtrac-eralized trapezoidal fuzzy numbers. While based on ourethods, the entering column is found without solvingstem and without any fuzzy arithmetic operations.sults conrm that the proposed method is simpler andnally more efcient than the proposed method by Kaur[18].

ion

ection, two FTP (adopted from [18]) are used in order applicability of the proposed method and the obtainediscussed.

les

transpdestinaber. Dethat th

We(givenobtainin Tabl

Theby thefamoumethothen uoptimaone fou

If wget thebased

x11 = x21 = x31 =

Nowgiven ition ofand Kugenerain Eq. formedthe prnumbe

In awe getbased

x11 = x21 = x31 =

In tobtainafter th

x11 = x21 = x31 =

Table 2Tabular .1. Table 1 gives the availability (ai) of the prod-e at three sources Si, i = 1, 2, 3 and their demand (bj)stinations Dj, j = 1, 2, 3, and the approximate cost for

S1S2S3Demand (bj) put the rank order of each fuzzy transportation costable 1) instead of corresponding fuzzy number toclassical transportation problem. The results are given

transportation problem given in Table 2 can be solveddard transportation algorithms. We rst apply threethods namely, the north-west corner method, least cost

Vogels approximation method for nding the IBFS and modied distribution method on the IBFS to obtain theution. Finally, we compare the obtained result with they Kaur and Kumar [18].e the north-west corner method to obtain the IBFS, weowing solution which is matched with IBFS obtainedFNWCM proposed by Kaur and Kumar [18]:

x12 = 0, x13 = 0,

22 = 9, x23 = 0,

32 = 5, x33 = 10.(3)

plying the modied distribution method on the IBFS. (3) shows that this initial solution is the optimal solu-TP given in Table 1, too. It needs to point out that Kaur

[18] obtained the same optimal solution by using the fuzzy modied distribution method on the IBFS givenut, in their method all arithmetic operations are per-the generalized trapezoidal fuzzy numbers, while ined method all arithmetic operations are done on real

on, if we apply the least cost method to obtain the IBFS,ollowing solution which is matched with IBFS obtainedFLCM which proposed by Kaur and Kumar [18]:

12 = 10, x13 = 0,x22 = 4, x23 = 0,

32 = 0, x33 = 10.(4)

ase, using the modied distribution method on theFS given in Eq. (4) gives the following optimal solutioniterations:

x12 = 0, x13 = 0,

22 = 9, x23 = 0,

32 = 5, x33 = 10.(5)

entation of classical transportation problem.D1 D2 D3 Availability (ai)

1.65 0.85 1.65 102.75 1.4 2.85 143.5 1.5 1.4 15

15 14 10

174 A. Ebrahimnejad / Applied Soft Computing 19 (2014) 171176

Table 3Summary of the fuzzy transportation problem.

Colliery Washery ai

D1 D2 D3 D4 D D

S1 (20,27,35,41;0.7) (9,11,12,14;0.6) (10,15,18,20;0.7) (15,20,22,24S2 (20,25,35,41;0.7) (9,11,12,16;0.8) (9,11,12,14,0.6) (10,15,21,23S3 (9,10,12,16;0.7) (65,70,74,76;0.8) (20,25,35,41;0.8) (12,15,22,24S4 (9,11,12,14;.6) (10,15,21,24;0.7) (20,25,35,41;0.6) (10,15,18,20bj 112 90 84 92

We note that Kaur and Kumar [18] obtained the same opti-mal solution by using the generalized fuzzy modied distributionmethod on the IBFS given in Eq. (4).

Finally, if we use the Vogels approximation method to nd theIBFS, we reaobtained ba

x11 = 10, x21 = 0, xx31 = 5, x

In this caIBFS given two iteratio

x11 = 10, x21 = 5, xx31 = 0, x

Kaur andthe generalgiven in Eq.

The min

3i=1

3j=1

cijx (0, 5

= (10, 40, 9 5, 50= (117, 205

As showmethod proderived fromproposed isthe fuzzy m

Example 4.company (tof the compIndia) is sho

The ownand demanrespectivelyin Table 3.

Table 4Summary of th

Colliery

S1S2S3S4bj

and severatain about to differentthe transpo

umbptimortat

rstiven o thehis c

soled ope as

0, x

0, x

112,

0, x

cijxi

woron usesuln, thg me

sults

his se ex

worher liay bn. Incisions o

exam use, d; wch to the following solution which is matched with IBFSsed on GFVAM proposed by Kaur and Kumar [18]:

x12 = 0, x13 = 0,

22 = 14, x23 = 0,

32 = 0, x33 = 10.(6)

se, if we apply the modied distribution method on thein Eq. (6), we nd the following optimal solution afterns:

x12 = 0, x13 = 0,

22 = 9, x23 = 0,

32 = 5, x33 = 10.(7)

Kumar [18] obtained the same optimal solution usingized fuzzy modied distribution method on the IBFS

(6).imum fuzzy transportation cost is achieved as follows:

ij = 10(1, 4, 9, 19; 0.5) 5(8, 9, 12, 26; 0.5) 9(3, 5, 8, 12; 0.2) 5

0, 190; 0.5) (40, 45, 60, 130; 0.5) (27, 45, 72, 108; 0.2) (0, 2, 352, 613; 0.2)

n here, the fuzzy optimal solution of the classicalposed in this study is equivalent to the optimal solution

Kaur and Kumars method [18]. However, the method by far simpler and computationally more efcient thanethod proposed by Kaur and Kumar [18].

2. The data, collected from an owner of a regional coalhe data is provided with a legal agreement that the nameany will not be disclosed) situated in Jharia (Dhanbad,wn in Table 3.er of the company is certain about the availabilitiesds of the coal at different collieries and washeries,. These parameters are represented by real numbersBecause of frequently variation in the rates of diesel

e classical transportation problem.

fuzzy nfuzzy otransp

Wecost (gleads t

In tcan beobtaincost ar

x11 = x21 = x31 = x41 =

3i=1

3j=1

It isbased These rfunctioexistin

4.2. Re

In tover th

It isany ottion msolutiothe desolutio

Fortion is(a, b, cWashery ai

D1 D2 D3 D4 D5 D6

18.45 6.9 9.45 12.15 9.45 8.1 12418.15 7.2 6.9 10.35 9.9 8.1 1207.05 42.75 18.15 10.95 8.1 13.35 1506.9 10.5 16.35 9.45 6.6 13.2 170

112 90 84 92 106 80

(A) = w[

where [degree of a decision mmaker is co5 6

;0.7) (10,15,18,20;0.8) (9,12,15,18;0.7) 124;0.8) (10,15,18,23;0.6) (9,12,15,18;0.8) 120;0.7) (10,12,14,18;0.6) (15,20,26,28;0.8) 150;0.6) (8,10,12,14;0.8) (15,20,25,28;07) 170

106 80

, 10, 15; 0.8) 10(4, 5, 8, 11; 0.6)

, 75; 0.8) (40, 50, 80, 110; 0.6)

l other reasons the owner of the company is not cer-the transportation cost (in Rs) from different collieries

washeries. According to past experiences of the ownerrtation cost is represented by generalized trapezoidalers. The owner of the company wants to determine theal transportation of products such that the total fuzzyion cost is minimum.

substitute the rank order of each fuzzy transportationin Table 3) instead of corresponding fuzzy number. This

classical transportation problem given in Table 4.ase, the crisp transportation problem shown in Table 4ved by the standard transportation algorithms. Thetimal solution and minimum total fuzzy transportationfollows:

12 = 80, x13 = 0, x14 = 0, x15 = 0, x16 = 44,

22 = 0, x23 = 84, x24 = 0, x25 = 0, x26 = 0,x32 = 0, x33 = 0, x34 = 38, x35 = 0, x36 = 0,

42 = 10, x43 = 0, x44 = 54, x45 = 106, x46 = 0.

(8)

j = (5148, 6474, 7802, 9244; 0.6).

th to note that the same optimal solution can be founding all the proposed methods by Kaur and Kumar [18].ts conrm that if we want to solve a FTP based on rankingen the proposed approach is more effective than thethod [18].

and discussions

ection, the main advantages of the proposed methodisting methods are explored.th noting that, as mentioned in Remark 2, one can usenear ranking function, and although the obtained solu-e different but the results are still valid for the new

fact, depending on the ranking index to be used byn-maker, the algorithm can return a set of the optimalr a unique solution.ple, assume that the following linear ranking func-

d for the generalized trapezoidal fuzzy number A =):[(1 )a + d] + 12

[(1 )(b a) + (c d)]]

(9)

0, 1] is the optimism index reecting the optimismdecision maker. The larger is, the more optimistic theaker is. The two extreme cases are = 0, the decisionmpletely pessimistic; and = 1, the decision maker is

A. Ebrahimnejad / Applied Soft Computing 19 (2014) 171176 175

completely optimistic. The case = 1/2 reects a linear decisionattitude.

It can be veried that the linear ranking function given inAppendix A is a particular case of linear ranking function given inEq. (9) whe

Solving tin Eq. (9) fthe Kaur anresult:

x11 = 10, x21 = 5, xx31 = 0, x

This solupessimisticbased on thgiven in Eqother wordunder consand Kuamr

Let us ex

(1) The proapproac

(2) By applmal soluprogramear progmaker.

(3) The protranspoable sofmethodgrammi

(4) To solveneed tobers. Wthere isproves compar

(5) Moreovcomparsuch a wto solve(or prop

5. Conclus

A large nels of sophisome of thethe convendata for theContrary togated imprdeveloped athese shortconsidered resented byof supply anHere we conis convertedtransportatconclude thsolving any

Here, we shall point out that the FTP studied in this paper isnot in the form of a problem whose model involves interval-valuedtrapezoidal fuzzy numbers. Therefore, further research on extend-ing the proposed method to overcome these shortcomings is an

ting of th

wled

authous

whicthorh ofuppo

dix Aions

tion 1ralizen by:

tion , c2,

R. T

0, A 0,

A2{w1

A2

k 1. R, wrs, fo1 = (g fune rea

, i2+c2+4

, i2+c2+4

2, 2+c2+4

k 2. f the

usen ob

new and mpan man = 1/2.he FTP of Example 4.1 based on ranking function givenor = 0 by use of both the proposed algorithm andd Kumars algorithm [18] produce the following same

x12 = 0, x13 = 0,

22 = 0, x23 = 9,

32 = 14, x33 = 1.(10)

tion is obtained based on point of view of a completely decision maker. Although the obtained optimal solutionis ranking function is different with the optimal solution. (7) but the main result is valid for this solution yet. Ins, if a unique ranking function is used for solving FTPideration, then both of our proposed method and Kaurs approach produce a same optimal solution.plore the main advantages of the proposed method.

posed technique does not use the goal and parametriches which are difcult to apply in real life situations.ying the proposed approach for nding the fuzzy opti-tion, there is no need of much knowledge of fuzzy linearming technique, Zimmerman approach and crisp lin-ramming which are difcult to learn for a new decision

posed method to solve FTP is based on traditionalrtation algorithms. Thus, the existing and easily avail-tware can be used for the same. However, the existing

[18] to solve FTP should be implemented into a pro-ng language.

the FTP by using the existing method [18], there is use arithmetic operations of generalized fuzzy num-hile, if the proposed technique is used for the same then

need to use arithmetic operations of real numbers. Thisthat it is much easy to apply the proposed method ased to the existing method [18].er, it is possible to assume a generic ranking index foring the fuzzy numbers involved in the FTP problem, inay that each time in which the decision maker wants

the FTP problem under consideration (s)he can chooseose) the ranking index that best suits the FTP problem.

ions and future work

umber of transportation problems with different lev-stication have been studied in the literature. However,se problems have limited real-life applications becausetional transportation problems generally assume crisp

transportation cost, the values of supplies and demands. the conventional transportation problems, we investi-ecise data in the real-life transportation problems andn alternative method that is simple and yet addressesfalls in the existing models in the literature. In the FTPin this study, the values of transportation costs are rep-

generalized trapezoidal fuzzy numbers and the valuesd demand of products are represented by real numbers.cluded that once the ranking function is chosen, the FTP

into crisp one, which is easily solved by the standardion algorithms. By solving two numerical examples, weat it is possible to nd a same optimal solution without

FTP.

interesresults

Ackno

Theanonymmentsthe auresearccially s

Appenoperat

Denia geneis give

A(x) =

Deni(a2 , b2and by:

i) ii)