An Algorithm to Solve Fuzzy Trapezoidal Transshipment Problem

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1 Iteratioal Joural of Systems Sciece ad Applied Mathematics 206; (4): doi: 0.648/j.ssam A Algorithm to Solve Fuzzy Trapezoidal Trasshipmet Problem P. Gayathri, K. R. Subramaia 2 Departmet of Mathematics, A.V.C.College (Autoomous), Maampadal, Mayiladuthurai 2 Departmet of Computer Applicatios, Srimati Idira Gadhi College, Trichy, Tamiladu, Idia address: pgayathrisudar@gmail.com (P. Gayathri) To cite this article: P. Gayathri, K. R. Subramaia. A Algorithm to Solve Fuzzy Trapezoidal Trasshipmet Problem. Iteratioal Joural of Systems Sciece ad Applied Mathematics. Vol., No. 4, 206, pp doi: 0.648/j.ssam Received: September 5, 206; Accepted: October 0, 206; Published: November 9, 206 Abstract: The fuzzy trasportatio problem i which available commodity frequetly moves from oe source to aother source or destiatio before reachig its actual destiatio is called a fuzzy trasshipmet problem. I this paper, a ew method is proposed to fid the fuzzy optimal solutio of fuzzy trasportatio problems with the followig trasshipmet: From a source to ay aother source, from a destiatio to aother destiatio, ad from a destiatio to ay source. I the proposed method all the parameters are represeted by trapezoidal fuzzy umbers. To illustrate the proposed method a fuzzy trasportatio problem with trasshipmet is solved. The proposed method is easy to uderstad ad to apply for fidig the fuzzy optimal solutio of fuzzy trasportatio problems with trasshipmet occurrig i real life situatios. Keywords: Trasportatio Problem, Fuzzy Trasshipmet Problem, Trapezoidal Fuzzy Numbers. Itroductio I a fuzzy trasportatio problem shipmet of article of trade takes place betwee sources ad destiatios. But as a alterative of direct shipmets to destiatios, the commodity ca be trasported to a meticulous destiatio through oe or more itermediated or fuzzy trasshipmet poits. Each of these poits i tur supply to other poits. Thus, whe the shipmets pass from destiatio to destiatio ad from source to source, fuzzy trasshipmets exists here. Such a problem caot be solved as such by the usual fuzzy trasportatio algorithm, but slight alteratio is required before applyig to the fuzzy trasshipmet problem. Trasshipmet usually takes place i trasport hubs. Much iteratioal trasshipmet also takes place i selected customs areas, thus avoidig the eed for customs checks or duties, otherwise a major difficulty for efficiet trasport. For example, stockiest reserves the goods viz. medicies, food grais ad other items at warehouses for emergecies. O the other had, whe there is a extra demad i the market due to high storage cost at some sources or durig festive ad marriage seasos or durig fire ad military services, the total flow eeds to be improved compellig some of the factories to icrease their productios i order to meet this extra demad. I literature, Trasshipmet Problem was first itroduced by Orde []. He gave a extesio of the origial trasportatio problem to iclude the possibility of trasshipmet i.e., ay shippig or receivig poit is also permitted to act as a itermediate poit. The Trasshipmet techique is used to fid the shortest route from oe poit i a etwork to aother. The problem of determiig simultaeously the flows of primary products through processors to the market as fial products has bee formulated alteratively as a trasshipmet model by Kig ad Loga [2]. A extesio of this problem to a multiregioal, multiproduct ad multi-plat problem formulated i the geeral liear programmig model has bee proposed by Judge et al. [3]. Afterwards, various alteratives formulatios of the trasshipmet problem withi the framework of the trasportatio model that permits solutio of problems of the type discussed by Kig ad Loga without the eed for subtractio of artificial variables were discussed by Hurt ad Tramel [4]. Brigde [5] cosidered the trasportatio problem (TP) with mixed costraits. He solved this problem by cosiderig a related stadard trasportatio problem havig two additioal supply poits ad two additioal destiatios. Kligma ad Russel [6] itroduced a specialized method for solvig a trasportatio problem with several additioal liear

2 59 P. Gayathri ad K. R. Subramaia: A Algorithm to Solve Fuzzy Trapezoidal Trasshipmet Problem costraits. I 985, Garg ad Prakash [7] studied time miimizig trasshipmet problem. After that Gupta et al. [8, 9] have also worked o liear ad o-liear trasportatio problems. I the recet years, idefiite Quadratic Trasportatio Problem has also bee studied [0, 2]. Dahiya ad Verma [] studied capacitated trasportatio problem with bouds o the rim coditios. Khuraa et al. [3, 4] studied three dimesioal time miimizig trasshipmet problem. I literature, much effort has bee cocetrated o trasportatio problems as well as trasshipmet problems with equality costraits. We have solved the problems for balaced as well ubalaced cases ad have discussed the various situatios emergig out of ubalaced trasshipmet problems. The algorithms ad trasformatio are easy to uderstad ad to apply. The solutio method ca serve as a effective tool to the maagers havig productio allocatio problems. 2. Prelimiaries I this sectio some basic defiitios ad arithmetic operatios are reviewed. 2.. Basic Defiitios I this sectio, some basic defiitios are reviewed. [9] ) A fuzzy set A is defied by A = x, µ ( x) : x A, µ ( x) [0,]. {( A ) A } I the pair ( x, ( x) ) µ, the first elemet x belog to the A classical set A, the secod elemet µ A( x ) belog to the iterval [0, ], called Membership fuctio. 2) A trapezoidal fuzzy umber A = ( a, b, c, d ) is said to be o-egative trapezoidal fuzzy umber if ad oly if a 0. 3) A trapezoidal fuzzy umber A = ( a, b, c, d ) is said to be zero, if ad oly if a = 0, b = 0, c = 0, d = 0. 4) Two trapezoidal fuzzy umbers A = ( a, b, c, d) ad A 2 = ( a2, b2, c2, d2 ) are said to be equal i.e., A = A if 2 ad oly if a = a 2, b = b 2, c = c 2, d = d 2. 5) A rakig fuctio is a fuctio R:F(R) R, where F(R) is a set of fuzzy umbers defied o set of real umbers, which maps each fuzzy umber ito the real lie. Let A = ( a, b, c, d ) be a trapezoidal fuzzy umber the R (A) = ( a b) ( d c) 2.2. Arithmetic Operatios Let A = ( a, b, c, d) ad A = ( a, b, c, d ) be two trapezoidal fuzzy umbers the (i) A A 2 = ( a + a2, b + b2, c + c2, d + d2 ) (ii) A A 2 = ( a, b, c, d ) a = mi( a a, a d, a d, d d ), where b = mi( bb 2, bc 2, cb 2, cc2 ), c = bb 2 bc 2 cb 2 cc2 d = max( a a, a d, a d, d d ) Formulatio of the Fuzzy Trasshipmet Problem max(,,, ), The fuzzy trasportatio problem assumes that direct routes exist from each source to each destiatio. However, there are situatios i which uits may be shipped from oe source to aother or to other destiatios before reachig their destiatios. This is called a fuzzy trasshipmet problem. The purpose of trasshipmet, the distictio betwee a source ad destiatio is dropped so that a trasportatio problem with m sources ad destiatios gives rise to a trasshipmet problem with m + sources ad m + destiatios. The basic feasible solutio to such a problem will ivolve [(m + ) + (m + ) -] or 2m basic variables ad if we omit the variables appearig i the (m + ) diagoal cells, we are left with m + basic variables. Thus the fuzzy trasshipmet problem may be writte as: Maximize ji i j=, j i j=, j i = Z c x Subject to i= j=, j i x x = a, i =, 2, 3,.., m x x = b, j= m +, m + 2, m + 3,, m + ji j i=, i j i=, i j where x 0, I, j =, 2, 3,., m +, j i m m where a = b j the the problem is balaced otherwise i i= j = ubalaced. The above formulatio is a fuzzy trasshipmet model, the trasshipmet model is reduced to trasportatio model: Miimize Z 2, 3,.., m = i= j=, j i x = T, i = m + to m +, x j= i= j i= x = b + T, j= m + to m + where x 0, i, j = to m +, j i c x Subject to x = a i + T, i =, j= = T, j = to m The above mathematical model represets a stadard balaced trasportatio problem with (m + ) origis ad (m + ) destiatios. T ca be iterpreted as a buffer stock at each origi ad destiatio. Sice we assume that ay amout o goods ca be trasshipped at each poit, T should be large eough to take care of all trasshipmets. It is clear that the volume of goods trasshipped at ay poit caot

3 Iteratioal Joural of Systems Sciece ad Applied Mathematics 206; (4): exceed the amout produced or received ad hece we take m m = i j i= j= T a or b 4. Proposed Algorithm Step : Balace the give trasshipmet problem if either (total supply > total demad) or (total supply < total demad). Now the trasshipmet table i fuzzy eviromet looks like fuzzy trasportatio table. Step 2: I this fuzzy trasportatio table all the cost, supply ad demad are i fuzzy. Defuzzify all the cost elemets i the balaced trasshipmet table. Step 3: To apply the proposed algorithm, subtract the differece of largest ad smallest cost elemets of the above ad below mai diagoal elemets from all the elemets of the cost matrix i the trasportatio table. Step 4: Arrage all the costs of trasportatio table i ascedig order. (If the costs are repeated, cosider it oce). Assig umbers, 2, 3, to these arraged costs ad form a ew Trasportatio table with their correspodig umbers for the costs. Step 5: Sice the supply & demad are i fuzzy, for rakig of trapezoidal fuzzy umbers use the rakig formula metioed i prelimiaries. Fid the total values assiged i the place of cost elemets row-wise as well as colum-wise. Step 6: Idetify a row (or colum) for this max total ad allocate the miimum trapezoidal Fuzzy umber correspodig to supply (or demad). Allocate the mi. fuzzy umber of supply (or demad) correspodig to the mi. cost of the idetified row (or colum) Step 7: Subtract the allocated fuzzy umber from the correspodig supply ad demad values. Delete a row (or colum) with zero supply (or demad). Cotiue this process util all rows ad colums are satisfied. Step 8: Compute the total fuzzy trasportatio cost for the feasible cost for the feasible allocatio usig the origial balaced fuzzy trasshipmet cost matrix. 5. Numerical Example Cosider the followig trasshipmet problem cotaiig two sources ad three destiatios havig demad ad availability represeted as tables values give below. Table. Fuzzy Trasshipmet problem (Cotais 4 tables as give below). S (0,0,0,0) (0.5,,,2) S2 (0.5,,,2) (0,0,0,0) S S2 D D2 AVAILABILITY S (,4,,3) (2,5,,3) (3,6,2,4) S2 (0.2,2.8,5,2.5) (2,5,,3) (3,28,8,26) DEMAND (2.3,2.2,6.5,9.5) (3.7,2.8,3.5,0.5) D D (0,0,0,0) (0.5,,,2) D2 (0.5,,,2) (0,0,0,0) D2 S D (,4,,3) (0.2,2.8,5,2.5) D2 (2,5,,3) (2,5,,3) (6,34,0,30) (6,34,0,30) Step: The Ubalaced Trasportatio Table is give by Table 2. Fuzzy Trasportatio problem. S S2 D D2 AVAILABILITY S (0,0,0,0) (0.5,,,2) (,4,,3) (2,5,,3) (3,6,2,4) S2 (0.5,,,2) (0,0,0,0) (0.2,2.8,5,2.5) (2,5,,3) (3,28,8,26) D (,4,,3) (0.2,2.8,5,2.5) (0,0,0,0) (0.5,,,2) - D2 (2,5,,3) (2,5,,3) (0.5,,,2) (0,0,0,0) - DEMAND - - (2.3,2.2,6.5,9.5) (3.7,2.8,3.5,0.5) Step2: The Balaced Trasportatio Table is formed S2

4 6 P. Gayathri ad K. R. Subramaia: A Algorithm to Solve Fuzzy Trapezoidal Trasshipmet Problem Table 3. Fuzzy Balaced Trasportatio problem. S S2 D D2 AVAILABILITY S (0,0,0,0) (0.5,,,2) (,4,,3) (2,5,,3) (9,40,2,34) S2 (0.5,,,2) (0,0,0,0) (0.2,2.8,5,2.5) (2,5,,3) (29,62,8,56) D (,4,,3) (0.2,2.8,5,2.5) (0,0,0,0) (0.5,,,2) (6,34,0,30) D2 (2,5,,3) (2,5,,3) (0.5,,,2) (0,0,0,0) (6,34,0,30) DEMAND (6,34,0,30) (6,34,0,30) (28.3,55.2,6.5,49.5) (9.7,46.8,3.5,40.5) Step 3: The Defuzzified Balaced Trasportatio Table is calculated by subtractig all the cost values from the differece of 4.5 ad 0.25 Table 4. Defuzzified Trasportatio problem. S S2 D D2 AVAILABILITY S (9,40,2,34) S (29,62,8,56) D (6,34,0,30) D (6,34,0,30) DEMAND (6,34,0,30) (6,34,0,30) (28.3,55.2,6.5,49.5) (9.7,46.8,3.5,40.5) Step 4: The revised Balaced Trasportatio Table is calculated by subtractig all the cost from the differece of 4.5 ad 0.25 ad assigig values,2,3 accordig to the values Table 5. Revised Trasportatio problem. S S2 D D2 AVAILABILITY S (9,40,2,34) S (29,62,8,56) D (6,34,0,30) D (6,34,0,30) DEMAND (6,34,0,30) (6,34,0,30) (28.3,55.2,6.5,49.5) (9.7,46.8,3.5,40.5) S Step 5: The solved Trasportatio table is give by S2 3 Table 6. Solved Trasportatio problem. S S2 D D2 AVAILABILITY 3 (9,40,2,34) 4 5 (6,34,0,30) (3,6,2,4) (3,6,2,4) D 4 2 D DEMAND (6,34,0,30) 2 (29,62,8,56) 5 (3,28,8,26) (6,34,0,30) (3,28,8,26) 3 (6,34,0,30) (2.3,2.2,6.5,9.5) (3.7,2.8,3.5,0.5) (2.3,2.2,6.5,9.5) (6,34,0,30) (6,34,0,30) (28.3,55.2,6.5,49.5) (9.7,46.8,3.5,40.5) (3,28,8,4) (6,34,0,30) (3.7,2.8,3.5,0.5) (6,34,0,30) Number of fuzzy uits trasported from origi to destiatios S to S is (6,34,0,30), S to S2 is (3,6,2,4), S2 to S2 is (3,28,8,26), S2 to D is (6,34,0,30), D to D is (2.3,2.2,6.5,9.5), D to D2 is (3.7,2.8,3.5,0.5), D2 to D2 is (6,34,0,30). The optimum fuzzy trasshipmet cost is (0.5,,,2)*(3,6,2,4)+(0.2,2.8,5,2.5)*(6,34,0,30)+(0.5,,,2 )* (3.7,2.8,3.5,0.5) = For the above trasshipmet problem, the fuzzy trasshipmet cost by Liear Programmig Problem method is Hece the proposed algorithm gives more optimum result tha the result give by LPP method. 6. Coclusio I this paper, a algorithm is developed to solve the fuzzy trasshipmet problem by usig trapezoidal fuzzy umber to get a optimum solutio. The algorithm is easy to uderstad ad less time cosumable. Numerical example is solved to illustrate the theory. I today s highly

5 Iteratioal Joural of Systems Sciece ad Applied Mathematics 206; (4): competitive market, the pressure o orgaizatios to fid better ways to create ad deliver value to customers becomes stroger. How ad whe to sed the products to the customers i the quatities, they wat i a cost-effective maer, become more challegig. The optimum solutio of Trasshipmet models provides a powerful framework to meet this challege. Refereces [] Orde A (956) Trasshipmet problem. Maag Sci 2 (3): [2] Kig GA, Loga SH (964) Optimum locatio, umber, ad size of processig plats with raw product ad fial product shipmets. J Farm Eco 46: [3] Judge GG, Havlicek J, Rizek RL (965) A iterregioal model: its formulatio ad applicatio to thelivestock idustry. Agric Eco Rev 7: 9. [4] Hurt VG, Tramel TE (965) Alterative formulatios of the trasshipmet problem. J Farm Eco 47 (3): [5] Brigde MEV (974) A variat of trasportatio problem i which the costraits are of mixed type. Oper Res Quaterly 25 (3): [6] Kligmaa D, Russel R (975) Solvig costraied trasportatio problems. Oper Res 23 (): [7] Garg R, Prakash S (985) Time miimizig trasshipmet problem. Idia J Pure Appl Math 6 (5): [8] Gupta A, Khaa S, Puri MC (992) Paradoxical situatios i trasportatio problems. Cah CetEtudesde Rech Operatioell 34: [9] Gupta A, Khaa S, Puri MC (993) A paradox i liear fractioal trasportatio problems with mixedcostraits. Optimizatio 27: [0] Arora SR, Khuraa A (2004) Three dimesioal fixed charge bicriterio idefiite quadratictrasportatio problem. Yugoslavia J Oper Res 4 (): [] Dahiya K, Verma V (2007) Capacitated trasportatio problem with bouds o the rim coditios. Eur JOper Res 78: [2] Khuraa A, Thirwai D, Arora SR (2009) A algorithm for solvig fixed charge bi-criterio idefiitequadratic trasportatio problem with restricted flow. It J Optim Theory Methods Appl (4): [3] Khuraa A, Arora SR (20) Fixed charge bi-criterio idefiite quadratic trasportatio problem withehaced flow. Rev Ivestig Operacioal 32: [4] Khuraa A, Verma T, Arora SR (202) A algorithmfor solvig time miimizig trasshipmetproblem. It J Maag Sci Eg Maag 7 (3): [5] Dubois, D., Prade, H.: Fuzzy Sets ad Systems: Theory ad Applicatios. Academic, New York (980).