TRAPEZOIDAL FUZZY NUMBERS FOR THE TRANSPORTATION PROBLEM. Abstract

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1 TRAPEZOIDAL FUZZY NUMBERS FOR THE TRANSPORTATION PROBLEM ARINDAM CHAUDHURI* Lecturer (Mathematcs & Computer Scence) Meghnad Saha Insttute of Technology, Kolkata, Inda *correspondng author KAJAL DE Professor n Mathematcs School of Scence Neta Subhas Open Unversty, Kolkata, Inda kaalde@redffmal.com DIPAK CHATTERJEE Dstngushed Professor Department of Mathematcs St. Xaver s College, Kolkata, Inda PABITRA MITRA Assstant Professor Department of Computer Scence Engneerng Indan Insttute of Technology, Kharagpur, Inda Abstract Transportaton Problem s an mportant problem whch has been wdely studed n Operatons Research doman. It has been often used to smulate dfferent real lfe problems. In partcular, applcaton of ths Problem n NP-Hard Problems has a remarkable sgnfcance. In ths Paper, we present the closed, bounded and non empty feasble regon of the transportaton problem usng fuzzy trapezodal numbers whch ensures the exstence of an optmal soluton to the balanced transportaton problem. The mult-valued nature of Fuzzy Sets allows handlng of uncertanty and vagueness nvolved n the cost values of each cells n the transportaton table. For fndng the ntal soluton of the transportaton problem we use the Fuzzy Vogel s Approxmaton Method and for determnng the optmalty of the obtaned soluton Fuzzy Modfed Dstrbuton Method s used. The fuzzfcaton of the cost of the transportaton problem s dscussed wth the help of a numercal example. Fnally, we dscuss the computatonal complexty nvolved n the problem. To the best of our knowledge, ths s the frst work on obtanng the soluton of the transportaton problem usng fuzzy trapezodal numbers. Keywords: Transportaton Problem, Lnear Programmng Problem, Fuzzy trapezodal numbers, Fuzzy Vogel s Approxmaton Method, Fuzzy Modfed Dstrbuton Method 1. Introducton The transportaton problem s a specal type of the lnear programmng problem. It s one of the earlest and most frutful applcatons of lnear programmng technque. It has been wdely studed n Logstcs and Operatons Management where dstrbuton of goods and commodtes from sources to destnatons s an mportant ssue 10. The orgn of the transportaton methods dates back to 1941 when F. L. Htchcock 7 presented a study enttled The Dstrbuton of a Product from Several Sources to Numerous Localtes. Ths presentaton s consdered to be the frst mportant contrbuton to the soluton of the Transportaton Problems. In 1947 T. C. Koopmans 11 presented an ndependent study called Optmum Utlzaton of the Transportaton System. These two contrbutons helped n the development of transportaton methods whch nvolve a number of shppng sources and a number of destnatons. An earler approach was gven by Kantorovch 10. The lnear programmng formulaton and the assocated systematc method for soluton were frst gven by G. B. Dantzg 4. The computatonal procedure s an adaptaton of the smple method appled to the system of equatons of the assocated lnear programmng problem. The task of dstrbutor s decsons can be optmzed by reformulatng the dstrbuton problem as generalzaton of the classcal transportaton problem. The conventonal transportaton problem can be represented as a mathematcal structure whch comprses an obectve functon subect to certan constrants. In classcal approach, transportng costs from m sources or wholesalers to the n destnatons or consumers are to be mnmzed. It s an optmzaton problem whch has been appled to solve varous NP-Hard problems. Wthn a gven tme perod each shppng source has a certan capacty and each destnaton has certan requrements wth a gven cost of shppng from the source to the destnaton. The obectve functon s to mnmze total transportaton costs and satsfy the destnaton requrements wthn the source requrements 8, 13. However, n real lfe stuatons, the nformaton avalable s of mprecse nature and there s an nherent degree of vagueness or 1

2 uncertanty present n the problem under consderaton. In order to tackle ths uncertanty the concept of Fuzzy Sets can be used as an mportant decson makng tool 15. Imprecson here s meant n the sense of vagueness rather than the lack of knowledge about the parameters present n the system. The Fuzzy Set Theory thus provdes a strct mathematcal framework n whch vague conceptual phenomena can be precsely and rgorously studed 3. In ths work, we dscuss the balanced transportaton problem and related theorems whch descrbe mportant mathematcal characterstcs, and then develop n terms of the revsed smplex method computatonal procedure for solvng the problem. Though ths problem can be solved by usng the smplex method, ts specal structure allows us to develop a smplfed algorthm for ts soluton. Ths model s not representatve of a partcular stuaton but may arse n many physcal stuatons. Consderng ths pont of vew we develop the model usng trapezodal fuzzy numbers 1, 2. The nature of the soluton s closed, bounded and non empty feasble whch ensures the exstence of an optmal soluton to the balanced transportaton problem. The cost values of each cell n the transportaton table are represented n terms of the trapezodal fuzzy numbers whch allows handlng of uncertanty and vagueness nvolved. The ntal soluton of the transportaton problem s calculated usng the Fuzzy Vogel s Approxmaton Method and optmalty test of the soluton s performed usng Fuzzy Modfed Dstrbuton Method. Ths paper s organzed as follows. In secton 2, the trapezodal membershp functon s defned. In the next secton, the general transportaton problem wth fuzzy trapezodal numbers s dscussed. Ths s followed by the soluton of transportaton problem usng fuzzy trapezodal numbers n secton 4. Secton 5 llustrates the soluton of transportaton problem through a numercal example. The computatonal complexty of the problem s gven n secton 6, followed by dscussons n secton 7. Fnally, n secton 8 conclusons are gven. 2. Trapezodal Membershp Functon The trapezodal membershp functon 14 follows: s specfed by four parameters {a, b, c, d} as 0, x a ( x a) /( b a), a x b trapezod( x; a, b, c, d) 1, b x c ( d x) /( d c), c x d 0, x d (1) The fgure 1 below llustrates an example of a trapezodal membershp functon defned by trapezod(x; 10, 20, 60, 96). 3. Transportaton Problem wth Fuzzy Trapezodal Numbers Assume a stuaton havng m orgns or supply centers whch contan varous amounts of commodty that has to be allocated to n destnatons or demand centers. Consder that the th (1) (2) (3) (4) orgn must supply the fuzzy quantty A [ a, a, a, a ]( [,0,0, ]), whereas the th destnaton must receve the fuzzy quantty (1) (2) (3) (4) (1) (2) (3) (4) B [ b, b, b, b ]( [,0,0, ]). Let the fuzzy cost C [ c, c, c, c ] of shppng a unt quantty from the orgn to the destnaton be known for all the orgns and destnatons. As t s possble to transport from any one orgn to any one destnaton, and the problem s to determne the number of unts to be transported from orgn to the destnaton such that all requrements are satsfed at a total mnmum transportaton cost. Ths scenaro holds for a balanced transportaton problem. Further n an unbalanced transportaton problem 5, 12 the sum avalabltes or supples of the orgns are not equal to the 2

3 sum of the requrements or demands at the destnatons. In order to solve ths problem we frst convert the unbalanced problem nto a balanced one by artfcally convertng t to a problem of equal demand and supply. For that, we ntroduce a fcttous or dummy orgn or destnaton that wll provde the requred supply or demand respectvely. The costs of transportng a unt from the fcttous orgn as well as the costs of transportng a unt to the fcttous destnaton are taken as zero. Ths s equvalent to not transportng from a dummy source or to a dummy destnaton wth zero transportaton cost Fgure 1: Trapezodal membershp functon defned by trapezod(x; 10, 20, 60, 96) The mathematcal formulaton of the problem s as follows. Let (1) (2) (3) (4) X [ x, x, x, x ] be the fuzzy number of unts suppled from the orgn to the destnaton. Then the problem can be wrtten as: MnmzeZ C m 1 subect to constrants: n 1 m 1 n 1 n 1 X X A, 1,2,......, m (1) (2) (3) (4) (here, Z [ z, z, z, z ] ) (2) (supply constrants) (3) X B, 1,2,......, n (destnaton constrants) (4) X [,0,0, ] for all and (5) where, A [,0,0, ] and B [,0,0, ] for all and. It should be noted that the Transportaton Problem s a lnear program. Suppose that there exsts a feasble soluton to the problem. Then, t follows, from equatons (3) and (4), that m 1 n m X A n B Thus, for the problem to be consstent, we must have the followng consstency equaton: m 1 n A B 1 If the problem s nconsstent, the followng equaton holds: (6) 3

4 m 1 n A B 1 (7) If the consstency condton (6) holds, then the transportaton problem s balanced, such that the total supply s equal to the total demand the problem s unbalanced. It s obvous from the constrants (3), (4) and (5) that every component X of a fuzzy feasble soluton vector X s bounded,.e., [,0,0, ] [ x (1), x (2), x (3), x (4) ] mn([ a. (1), a (2), a (3), a (4) ],[ b Thus, the feasble regon of the problem s closed, bounded and non-empty. Hence, there always exsts an optmal soluton to the balanced transportaton problem. Constrant equatons (3) and (4) can be wrtten n the matrx form as follows: AX B T wth X X, X,..., X, X,..., X,... X ), ( n 21 2n T ( A1, A2,..., Am, B1, B2,..., B n B ) mn (1), b (2), b and A as an (m + n) mn matrx gven by A I.. I.. I.. I.. I.. I... I Here, 1 s the 1 n matrx wth all the components as 1 and I s the n n dentty matrx. Snce the sum of m equatons (3) equals the sum of n equatons (4), the (m + n) rows of A are lnearly dependent. Ths mples that rank (A) m + n - 1. The transportaton model has a specal structure whch enables us to represent t n the form of rectangular array called the transportaton table as gven n fgure 2. In ths table, each of the mn cells corresponds to a varable; each row corresponds to one of the m constrants (3) called row constrants and each column corresponds to one of the n constrants (4) called column constrants. The (, ) th cell at the ntersecton of the th row and th column contans cost C and decson varable X. The cells n the transportaton table can be classfed as occuped cells and unoccuped cells. The allocated cells n the transportaton table are called occuped cells and empty cells are called unoccuped cells. 3.1 Defntons: Fuzzy feasble soluton of the Transportaton Problem Now we gve some mportant defntons relevant for developng the feasble soluton for the transportaton problem usng the trapezodal fuzzy numbers. They are brefly enumerated as follows: a) Fuzzy feasble soluton: Any set of fuzzy non-negatve allocatons (X > [-δ, 0, 0, δ], δ beng a small postve number) whch satsfes the row and column sum s a fuzzy feasble soluton. (3), b (4) ]) 4

5 b) Fuzzy basc feasble soluton: A fuzzy feasble soluton s a fuzzy basc feasble soluton f the number of non-negatve allocatons s at most (m + n -1); where, m s the number of rows, n s the number of columns n the transportaton table. c) Fuzzy non degenerate basc feasble soluton: Any fuzzy feasble soluton to the transportaton problem contanng m orgns and n destnatons s sad to be fuzzy non degenerate, f t contans exactly (m + n - 1) occuped cells. d) Fuzzy degenerate basc feasble soluton: If the fuzzy basc feasble soluton contans less than (m + n - 1) non-negatve allocatons, t s sad to be fuzzy degenerate. D 1 D D n O 1 C 11 C C 1n A 1 O 2 C 21 C C 2n A 2 ORIGINS O m C m1 C m C mn A m B 1 B B n Fgure 2: Transportaton Table 3.2 Theorems Fnally, we present some mportant theorems often found useful whle developng the feasble soluton for the transportaton problem usng the trapezodal fuzzy numbers as well as testng the optmalty of the obtaned soluton. The proofs of these theorems can be found n 5, 6, 12. Theorem 1: The number of basc varables n a transportaton problem s at most (m + n 1). Theorem 2: The transportaton problem always has a feasble soluton. Theorem 3: All bases for the transportaton problem are trangular (upper or lower) n nature. Theorem 4: The values of the basc varables n a basc feasble soluton to the transportaton problem are gven by the expressons of the form X Ap Bq (8) somep where, (n and ) the upper sgns apply to some basc varables and the lower sgns apply to the remanng basc varables. someq Theorem 5: The soluton of the transportaton problem s never unbounded. 5

6 Theorem 6: A subset of the columns of the coeffcent matrx of a transportaton problem s lnearly dependent, f and only f, the correspondng cells or a subset of them can be sequenced to form a loop. Theorem 7: If there be a feasble soluton havng (m + n 1) ndependent postve allocatons and f there be numbers u and v, ( = 1,.,m; = 1,,n) satsfyng c rs = u r + v s for each occuped cell (r, s), then the cell evaluaton correspondng to the unoccuped cell (, ) wll be gven by = c (u + v ). 4. Soluton of the Transportaton Problem usng Fuzzy Trapezodal Numbers The soluton of the fuzzy transportaton problem s generally obtaned n followng two stages: a) Intal basc feasble soluton b) Test of optmalty for the soluton 4.1 Fuzzy Vogel s Approxmaton Method (FVAM) The ntal basc feasble soluton can be easly obtaned usng the methods lke North West Corner Rule, Least Cost Method or Matrx Mnma Method, Vogel s Approxmaton Method (VAM) etc. VAM s preferred over the other methods, snce the ntal basc feasble soluton obtaned by ths method s ether optmal or very close to the optmal soluton. Here, we dscuss only VAM and usng the fuzzy trapezodal numbers. The steps nvolved n FVAM for fndng the fuzzy ntal soluton are brefly enumerated below 12 : Step 1: The penalty cost s found by consderng the dfference the smallest and next smallest costs n each row and column. Step 2: Among the penaltes calculated n step 1, the maxmum penalty s chosen. If the maxmum penalty occurs more than once then any one can be chosen arbtrarly. Step 3: In the selected row or column found n step 2, the cell havng the least cost s consdered. An allocaton s made to ths cell by takng the mnmum of the supply and demand values. Step 4: Fnally, the row or column s deleted whch s fully fuzzy exhausted. Now, consderng the reduced transportaton tables repeat steps 1-3 untl all the requrements are fulflled. 4.2 Fuzzy Modfed Dstrbuton Method (FMODIM) Once the fuzzy ntal basc feasble soluton has been obtaned, the next step s to determne whether the soluton obtaned s fuzzy optmum or not. Optmalty test can be conducted to any ntal basc feasble soluton of the transportaton problem provded such allocatons have exactly (m + n 1) non negatve allocatons, where m s the number of orgns and n s the number of destnatons. Also these allocatons must be n ndependent postons. To perform the optmalty test, we make use of the FMODIM usng the fuzzy trapezodal numbers. The varous steps nvolved n FMODIM for performng the optmalty test are gven below 12 : Step 1: Fnd the fuzzy ntal basc feasble soluton of the fuzzy transportaton problem by usng FVAM. 6

7 (1) (2) (3) (4) Step 2: Fnd a set of numbers U = [ u u, u, u ] and V = [ v, (1) (2) (3) (4), v, v, v each row and column satsfyng U (+) V = C for each occuped cell. We start by assgnng a number fuzzy zero (whch may be [-0.05, 0, 0, 0.05] to any row or column havng the maxmum number of allocatons. If the maxmum number of allocaton s more than one, we choose any one arbtrarly. Step 3: For each empty or unoccuped cell, we fnd the sum of U and V and wrte t n each cell. Step 4: Fnd the net evaluaton value for each empty cell gven as, Δ = C (-) (U (+) V ) and also wrte t n each cell. Ths gves the optmalty concluson whch may be any of the followng: a) If all Δ > [-δ, 0, 0, δ], the soluton s fuzzy optmum and a fuzzy unque soluton exsts. b) If Δ [-δ, 0, 0, δ], then the soluton s fuzzy optmum, but an alternate soluton exsts. c) If at least one Δ < [-δ, 0, 0, δ], the soluton s not fuzzy optmum. In ths case we go to the next step, to mprove the total transportaton cost. Step 5: Select the empty cell havng the most negatve value of Δ. From ths cell we draw a closed path by drawng horzontal and vertcal lnes wth the corner cells occuped. Assgn postve and negatve sgns alternately and fnd the mnmum allocaton from the cell havng the negatve sgn. Ths allocaton s to be added to the allocaton havng postve sgn and subtracted from the allocaton havng negatve sgn. Step 6: The step 5 yelds a better soluton by makng one or more occuped cell as empty and one empty cell as occuped. For ths new set of basc feasble allocatons repeat from steps 2 5 untl an optmum basc feasble soluton s obtaned. 5. Numercal Example In ths secton, we consder the fuzzy transportaton problem and obtan the fuzzy ntal basc feasble soluton of the problem by FVAM and determne the fuzzy membershp functons of costs and allocatons. We then test the optmalty of the soluton obtaned usng FMODIM. 5.1 Intal Basc Feasble Soluton by FVAM The transportaton problem conssts of 3 orgns and 4 destnatons. The cost coeffcents are denoted by trapezodal fuzzy numbers. The correspondng avalablty (supply) and requrement (demand) vectors are also gven n the fgure below. In the above Table, A = [ , 1010, 1010, ] (column sum) and B = [ , 1010, 1010, A and ] for B are fuzzy equal, dfferng by the fuzzy zero ] (row sum). Snce, vz., [-0.35, 0, 0, 0.35] the gven problem s balanced and there exsts a fuzzy feasble soluton to the problem. We frst fnd the row and column penalty as the dfference between the fuzzy least and next fuzzy least cost n the correspondng rows and columns respectvely. In the above problem the maxmum penalty s [6.9, 7, 7.1] correspondng to D 2 column. In ths allocaton the cell havng the fuzzy least cost s (1, 2). To ths cell we allocate the mnmum of Supply A 1 and Demand B 2.e., ([279.95, 280, 280, ], [249.95, 250, 250, ]) = [249.95, 250, 250, ] as gven n Fgure 2. Ths exhausts the second column 7

8 by fuzzy zero, [-0.10, 0, 0, 0.10] and supply s reduced to ([279.95, 280, 280, ] (-) [249.95, 250, 250, ]) = [29.90, 30, 30, 30.10]. The second column s deleted from the Fgure 2 such that we have the followng shrunken matrx gven n Fgure 3. As the second column s deleted the values of penaltes are also changed. ORIGINS D 1 D 2 D 3 D 4 Supply (A ) O 1 [12.95, 13, [14.95, 15, [15.95, 16, [17.95, 18, [279.95, 280, 13, 13.05] 15, 15.05] 16, 16.05] 18, 18.05] 280, ] O 2 [19.95, 20, [21.95, 22, [10.95, 11, [7.95, 8, [329.95, 330, 20, 20.05] 22, 22.05] 11, 11.05] 8, 8.05] 330, ] O 3 [18.95, 19, [24.95, 25, [16.95, 17, [10.95, 11, [399.95, 400, 25, 25.05] 17, 17.05] 11, 11.05] 400, ] Demand [299.95, 300, [249.95, 250, [279.95, 280, [179.95, 180, (B ) 300, ] 250, ] 280, ] 180, ] Fgure 1: Transportaton Problem wth 3 Orgns and 4 Destnatons ORIGINS D 1 D 2 D 3 D 4 Supply (A ) Row Penalty O 1 [12.95, 13, [14.95, 15, [15.95, 16, [17.95, 18, [279.95, 280, [1.9, 2, 13, 13.05] 15, 15.05] 16, 16.05] 18, 18.05] 280, ] 2, 2.1] ([249.95, 250, 250, ]) O 2 [19.95, 20, [21.95, 22, [10.95, 11, [7.95, 8, [329.95, 330, [2.9, 3, 20, 20.05] 22, 22.05] 11, 11.05] 8, 8.05] 330, ] 3, 3.1] O 3 [18.95, 19, Demand [299.95, 300, (B ) 300, ] Column Penalty [5.9, 6, 6, 6.1] [24.95, 25, 25, 25.05] [249.95, 250, 250, ] [-0.10, 0, 0, 0.10] [6.9, 7, 7, 7.1] [16.95, 17, 17, 17.05] [279.95, 280, 280, ] [5.9, 6, 6, 6.1] [10.95, 11, 11, 11.05] [179.95, 180, 180, ] [2.9, 3, 3, 3.1] [399.95, 400, 400, ] [5.9, 6, 6, 6.1] Fgure 2: 1 st Allocaton to the Transportaton Problem In Fgure 3, the maxmum value of Penalty s [5.9, 6, 6, 6.1] correspondng to D 1 column and O 3 row. Takng ether of the row or column vz., f we consder row O 3 ; the cell havng the least fuzzy cost s (3, 4). To ths cell we allocate the mnmum of Supply A 3 and Demand B 4.e., ([399.95, 400, 400, ], [179.95, 180, 180, ]) = [179.95, 180, 180, ] as gven n Fgure 3. Ths exhausts the fourth column by fuzzy zero, [-0.10, 0, 0, 0.10] and supply s reduced to ([399.95, 400, 400, ] (-) [179.95, 180, 180, ]) = [219.90, 220, 220, ]. The fourth column s deleted from the Fgure 3 such that we have the followng shrunken matrx gven n Fgure 4. The values of penaltes are also changed. In Fgure 4, the maxmum value of Penalty s [7.9, 8, 8, 8.1] correspondng to O 2 row. In ths allocaton the cell havng the fuzzy least cost s (2, 3). To ths cell we allocate the mnmum of Supply A 2 and Demand B 3.e., ([329.95, 330, 330, ], [279.95, 280, 280, ]) = [279.95, 280, 280, ] as gven n Fgure 4. Ths exhausts the thrd column by fuzzy zero, [-0.10, 0, 0, 0.10] and supply s reduced to ([329.95, 330, 330, ] (-) [279.95, 280, 280, ]) = [49.90, 50, 50, 50.10]. The thrd column s deleted from the Fgure 4 such that we have the followng shrunken matrx gven n Fgure 5. As the thrd column s deleted the values of penaltes are also changed. ORI GIN D 1 D 3 D 4 Supply (A ) Row Penalty O 1 [12.95, 13, [15.95, 16, [17.95, 18, [29.90, 30, [2.9, 3, 13, 13.05] 16, 16.05] 18, 18.05] 30, 30.10] 3, 3.1] 8

9 O 2 [19.95, 20, 20, 20.05] O 3 [18.95, 19, Demand (B ) Column Penalty [299.95, 300, 300, ] [5.9, 6, 6, 6.1] [10.95, 11, 11, 11.05] [16.95, 17, 17, 17.05] [279.95, 280, 280, ] [4.9, 5, 5, 5.1] [7.95, 8, 8, 8.05] [10.95, 11, 11, 11.05] ([179.95, 180, 180, ]) [179.95, 180, 180, ] [-0.10, 0, 0, 0.10] [2.9, 3, 3, 3.1] [329.95, 330, 330, ] [399.95, 400, 400, ] [2.9, 3, 3, 3.1] [5.9, 6, 6, 6.1] Fgure 3: 2 nd Allocaton to the Transportaton Problem ORIGINS D 1 D 3 Supply (A ) Row Penalty O 1 [12.95, 13, [15.95, 16, [29.90, 30, [2.9, 3, 13, 13.05] 16, 16.05] 30, 30.10] 3, 3.1] O 2 [19.95, 20, [10.95, 11, [329.95, 330, [7.9, 8, 20, 20.05] 11, 11.05] 330, ] 8, 8.1] [279.95, 280, O 3 [18.95, 19, Demand [299.95, 300, (B ) 300, ] Column Penalty [5.9, 6, 6, 6.1] 280, ] [16.95, 17, 17, 17.05] [279.95, 280, 280, ] [-0.10, 0, 0, 0.10] [4.9, 5, 5, 5.1] [219.90, 220, 220, ] [1.9, 2, 2, 2.1] Fgure 4: 3 rd Allocaton to the Transportaton Problem In Fgure 5, the maxmum value of Penalty s [19.95, 20, 20, 20.05] correspondng to O 2 row. As there s only one element n ths row, ths cell.e., (2, 1) s the fuzzy least cost cell and s consdered for allocaton. We allocate the mnmum of Supply A 2 and Demand B 1.e., ([49.95, 50, 50, 50.05], [299.95, 300, 300, ]) = [49.95, 50, 50, 50.05] as gven n Fgure 5. Ths exhausts the second row by fuzzy zero, [-0.10, 0, 0, 0.10] and demand s reduced to ([299.95, 300, 300, ] (-) [49.95, 50, 50, 50.05]) = [249.90, 250, 250, ]. The second row s deleted from the Fgure 5 such that we have the followng shrunken matrx gven n Fgure 6. As the second row s deleted the values of penaltes are also changed. ORIGINS D 1 Supply (A ) Row Penalty O 1 [12.95, 13, [29.90, 30, [12.95, 13, 13, 13.05] 30, 30.10] 13, 13.05] O 2 [19.95, 20, [49.95, 50, [19.95, 20, 20, 20.05] 50, 50.05] 20, 20.05] [49.95, 50, [-0.10, 0, 50, 50.05] 0, 0.10] O 3 [18.95, 19, Demand [299.95, 300, (B ) 300, ] Column [5.9, 6, Penalty 6, 6.1] [219.90, 220, 220, ] [18.95, 19, Fgure 5: 4 th Allocaton to the Transportaton Problem In Fgure 6, the maxmum value of Penalty s [18.95, 19, correspondng to O 3 row. As there s only one element n ths row, ths cell.e., (3, 1) s the fuzzy least cost cell and s consdered for allocaton. We allocate the mnmum of Supply A 3 and Demand B 1.e., 9

10 ([219.90, 220, 220, ], [249.90, 250, 250, ]) = [219.90, 220, 220, ] as gven n Fgure 6. Ths exhausts the thrd row by fuzzy zero, [-0.10, 0, 0, 0.10] and demand s reduced to ([249.90, 250, 250, ] (-) [219.90, 220, 220, ]) = [29.85, 30, 30, 30.15]. The thrd row s deleted from the Fgure 6 such that we have the followng shrunken matrx gven n Fgure 7. As the thrd row s deleted the values of penaltes are also changed. D 1 Supply (A ) Row Penalty O 1 [12.95, 13, [29.90, 30, [12.95, 13, 13, 13.05] 30, 30.10] 13, 13.05] ORIGINS O 3 [18.95, 19, [219.90, 220, 220, ] [219.90, 220, 220, ] [-0.10, 0, 0, 0.10] [18.95, 19, Demand (B ) Column Penalty [249.90, 250, 250, ] [5.9, 6, 6, 6.1] Fgure 6: 5 th Allocaton to the Transportaton Problem In Fgure 7, only one cell value correspondng to D 1 column and O 1 row remans. There s only one Penalty value.e., [12.95, 13, 13, 13.05]. In ths allocaton the cell havng the fuzzy least cost s (1, 1). To ths cell we allocate the mnmum of Supply A 1 and Demand B 1.e., ([29.90, 30, 30, 30.10], [29.85, 30, 30, 30.15]) = [29.90, 30, 30, 30.10] as gven n Fgure 7. Ths exhausts the frst row and frst column by fuzzy zero, [-0.20, 0, 0, 0.20] and all the requrements are fulflled n the 6 th allocaton. The demand and supply are dffered by fuzzy zero ([-0.35, 0, 0, 0.35] (-) [-0.20, 0, 0, 0.20]) = [-0.55, 0, 0, 0.55] due to fuzzness. Fnally, the ntal basc feasble soluton s shown n fgure 8. There are 6 postve ndependent allocatons gven by m + n -1 = Ths ensures that the soluton s a fuzzy non degenerate basc feasble soluton. The total transportaton cost = {C 11 () X 11 } (+) {C 12 () X 12 } (+) {C 21 () X 21 } (+) {C 23 () X 23 } (+){C 31 () X 31 } (+) {C 34 () X 34 } = {[12.95, 13, 13, 13.05] () [29.90, 30, 30, 30.10]} (+) {[14.95, 15, 15, 15.05] () [249.95, 250, 250, ]} (+) {[19.95, 20, 20, 20.05] () [49.95, 50, 50, 50.05]} (+){[10.95, 11, 11, 11.05] () [279.95, 280, 280, ]} (+) {[18.95, 19, () [219.90, 220, 220, ]} (+) {[10.95, 11, 11, 11.05] () [179.95, 180, 180, ]} = [ , 14380, 14380, ] (9) Here, C are cost coeffcents and X are allocatons ( = 1, 2, 3; = 1, 2, 3, 4). ORIGINS D 1 Supply (A ) Row Penalty O 1 [12.95, 13, [29.90, 30, [12.95, 13, 13, 13.05] 30, 30.10] 13, 13.05] [29.90, 30, [-0.60, 0, 30, 30.10] 0, 0.60] Demand [29.85, 30, (B ) 30, 30.15] Column Penalty [12.95, 13, 13, 13.05] Fgure 7: 6 th Allocaton to the Transportaton Problem ORIGI NS D 1 D 2 D 3 D 4 Supply (A ) O 1 [12.95, 13, [14.95, 15, [15.95, 16, [17.95, 18, [279.95, 280, 13, 13.05] 15, 15.05] 16, 16.05] 18, 18.05] 280, ] ([29.90, 30, ([249.95, 250, 10

11 30, 30.10]) 250, ]) O 2 [19.95, 20, 20, 20.05] ([49.95, 50, 50, 50.05]) O 3 [18.95, 19, ([219.90, 220, 220, ]) [21.95, 22, 22, 22.05] [24.95, 25, 25, 25.05] [10.95, 11, 11, 11.05] ([279.95, 280, 280, ]) [16.95, 17, 17, 17.05] [7.95, 8, 8, 8.05] [10.95, 11, 11, 11.05] ([179.95, 180, 180, ]) [329.95, 330, 330, ] [399.95, 400, 400, ] Demand (B ) [299.95, 300, 300, ] [249.95, 250, 250, ] [279.95, 280, 280, ] [179.95, 180, 180, ] Fgure 8: The fnal allocated matrx wth the correspondng allocaton values 5.2 Fuzzy Membershp Functons of the costs and allocatons Now, we present the fuzzy membershp functon of C and X and then the transportaton costs. ( x 12.95) / 0.05,12.95 x 13 1,13 x 13 c 11 (13.05 x) / 0.05,13 x , Let C 11 be the nterval of confdence for the level of presumpton, [0, 1]. ( ) ( ) Thus, C 11 = [ c c ] = [ , ] (10) 1, 2 X 11 ( x 29.90) / 0.10,29.90 x 30 1,30 x 30 (30.10 x) / 0.10,30 x , Let X 11 be the nterval of confdence for the level of presumpton, [0, 1]. ( ) ( ) Thus, X 11 = [ x x ] = [ , ] (11) 1, 2 C 11 () X 11 = [( ) ( ), ( ) ( )] = [ , ] (12) Smlarly, we can wrte the other fuzzy membershp functons as follows: c12 ( x 14.95) 1,15 x 15 (15.05 x) 0, / 0.05,14.95 / 0.05,15 x 15 x C 12 = [ , ] (13) 11

12 X 12 ( x ) / 0.05, x 250 1,250 x 250 ( x) / 0.05,250 x , X 12 = [ , ] (14) C 12 () X 12 = [( ) ( ), ( ) ( )] = [ , ] (15) c21 ( x 19.95) / 0.05,19.95 x 20 1,20 x 20 (20.05 x) / 0.05,20 x , C 21 = [ , ] (16) X 21 ( x 49.95) / 0.05,49.95 x 50 1,50 x 50 (50.05 x) / 0.05,50 x , X 21 = [ , ] (17) C 21 () X 21 = [( ) ( ), ( ) ( )] = [ , ] (18) c23 ( x 10.95) / 0.05,10.95 x 11 1,11 x 11 (11.05 x) / 0.05,11 x , C 23 = [ , ] (19) X 23 ( x ) / 0.05, x 280 1,280 x 280 ( x) / 0.05,280 x , X 23 = [ , ] (20) C 23 () X 23 = [( ) ( ), ( ) ( )] = [ , ] (21) 12

13 X 31 c31 ( x 18.95) 1,19 x 19 (19.05 x) 0, / 0.05,18.95 / 0.05,19 x 19 x C 31 = [ , ] (22) ( x ) / 0.10, x 220 1,220 x 220 ( x) / 0.10,220 x , X 31 = [ , ] (23) C 31 () X 31 = [( ) ( ), ( ) ( )] = [ , ] (24) c34 ( x 10.95) / 0.05,10.95 x 11 1,11 x 11 (11.05 x) / 0.05,11 x , C 34 = [ , ] (25) X 34 ( x ) / 0.05, x 180 1,180 x 180 ( x) / 0.05,180 x , X 34 = [ , ] (26) C 34 () X 34 = [( ) ( ), ( ) ( )] = [ , ] (27) Now, the total transportaton cost s gven by: Cost = {C 11 () X 11 } (+) {C 12 () X 12 } (+) {C 21 () X 21 } (+) {C 23 () X 23 } (+) {C 31 () X 31 } (+) {C 34 () X 34 } = [ , ] (28) The expresson gven n equaton (19) s obtaned usng the equatons (12), (15), (18), (21), (24) and (27). From equaton (28) we solve the followng equatons whose roots [0, 1]: From equaton (29) we have, x 1 = 0 (29) x 2 = 0 (30) 2 { ((56.51) ( x1 ))}/(2 0.02) 13

14 From equaton (30) we have, cos t 2 { ((53.78) ( x2 ))}/(2 0.02) 2 { ((56.51) ( x1 ))}/(2 0.02), x ,14380 x {53.78 ((53.78) ( x2 ))}/(2 0.02),14380 x , whch s the requred fuzzy membershp functon of the transportaton cost (usng equaton (9)). 5.3 Optmalty Test by FMODIM To determne the fuzzy optmal soluton for the above problem, we make use of the FMODIM. We determne the set of numbers U and V for each row and column of the matrx gven n fgure 8 wth U (+) V = C for each occuped cell. We assgn the value of fuzzy zero to U 1 = [-0.05, 0, 0, 0.05] arbtrarly, as all the rows have the dentcal number of allocatons. From the occuped cells we have, V 1 = C 11 (-) U 1 = [12.95, 13, 13, 13.05] (-) [-0.05, 0, 0, 0.05] = [12.90, 13, 13, 13.10] V 2 = C 12 (-) U 1 = [14.95, 15, 15, 15.05] (-) [-0.05, 0, 0, 0.05] = [14.90, 15, 15, 15.10] U 2 = C 21 (-) V 1 = [19.95, 20, 20, 20.05] (-) [12.90, 13, 13, 13.10] = [6.85, 7, 7, 7.15] V 3 = C 23 (-) U 2 = [10.95, 11, 11, 11.05] (-) [6.85, 7, 7, 7.15] = [3.80, 4, 4, 4.20] U 3 = C 31 (-) V 1 = [18.95, 19, (-) [12.90, 13, 13, 13.10] = [5.85, 6, 6, 6.15] V 4 = C 34 (-) U 3 = [10.95, 11, 11, 11.05] (-) [5.85, 6, 6, 6.15] = [4.80, 5, 5, 5.20] Now we calculate the sum of U and V for each unoccuped cell. The values of U (+) V are gven below the C value of the cells whch are as follows: U 1 (+) V 3 = [-0.05, 0, 0, 0.05] (+) [3.80, 4, 4, 4.20] = [3.75, 4, 4, 4.25] U 1 (+) V 4 = [-0.05, 0, 0, 0.05] (+) [4.80, 5, 5, 5.20] = [4.75, 5, 5, 5.25] U 2 (+) V 2 = [6.85, 7, 7, 7.15] (+) [14.90, 15, 15, 15.10] = [21.75, 22, 22, 22.25] U 2 (+) V 4 = [6.85, 7, 7, 7.15] (+) [4.80, 5, 5, 5.20] = [11.65, 12, 12, 12.35] U 3 (+) V 2 = [5.85, 6, 6, 6.15] (+) [14.90, 15, 15, 15.10] = [20.75, 21, 21, 21.25] U 3 (+) V 3 = [5.85, 6, 6, 6.15] (+) [3.80, 4, 4, 4.20] = [9.65, 10, 10, 10.35] Next we fnd the net evaluatons = C (-) (U (+) V ) for each unoccuped cell. The values of are gven below the values of U (+) V whch are as follows: 13 = C 13 (-) (U 1 (+) V 3 ) = [15.95, 16, 16, 16.05] (-) ([-0.05, 0, 0, 0.05] (+) [3.80, 4, 4, 4.20]) = [11.80, 12, 12, 12.20] 14 = C 14 (-) (U 1 (+) V 4 ) = [17.95, 18, 18, 18.05] (-) ([-0.05, 0, 0, 0.05] (+) [4.75, 5, 5, 5.25]) = [12.75, 13, 13, 13.25] 22 = C 22 (-) (U 2 (+) V 2 ) = [21.95, 22, 22, 22.05] (-) ([6.85, 7, 7, 7.15] (+) [14.90, 15, 15, 15.10]) = [-0.20, 0, 0, 0.20] 24 = C 24 (-) (U 2 (+) V 4 ) = [7.95, 8, 8, 8.05] (-) ([6.85, 7, 7, 7.15] (+) [4.80, 5, 5, 5.20]) = [3.70, 4, 4, 4.30] 32 = C 32 (-) (U 3 (+) V 2 ) = [24.95, 25, 25, 25.05] (-) ([9.65, 10, 10, 10.35] (+) [14.90, 15, 15, 15.10]) = [-0.40, 0, 0, 0.40] 33 = C 33 (-) (U 3 (+) V 3 ) = [16.95, 17, 17, 17.05] (-) ([9.65, 10, 10, 10.35] (+) [3.80, 4, 4, 4.20]) = [2.50, 3, 3, 3.50] 14

15 The above calculated values are gven n fgure 9. Here, as all [-0.40, 0, 0, 0.40] the soluton obtaned s optmal n nature and an alternatve soluton exsts gven by 32 = [-0.40, 0, 0, 0.40]. Therefore, the optmal allocatons are gven by the followng: X 11 = [29.90, 30, 30, 30.10]; X 12 = [249.95, 250, 250, ] X 21 = [49.95, 50, 50, 50.05]; X 23 = [279.95, 280, 280, ] X 31 = [219.90, 220, 220, ]; X 34 = [179.95, 180, 180, ] The total optmum transportaton cost = [ , 14380, 14380, ] The fuzzy membershp functons of the optmalty test can be obtaned usng smlar technque as n the case for the fuzzy ntal basc feasble soluton. 6. Computatonal Complexty Here, we nvestgate the computatonal complexty of the transportaton problem usng Vogel s Approxmaton Method wth m orgns and n destnatons. Let the total computatonal tme be gven by T (m, n). The tme to calculate the penalty values for m rows = (n + 1) m. The tme to calculate the penalty values for n columns = (m + 1) n. The tme to search for the maxmum value of the penalty for the correspondng least value of the cost = {m / (m + n)} n, f maxmum penalty s found n a row and the least value of the cost = {n / (m + n)} m, f maxmum penalty s found n a column. Now, the tme requred to obtan the feasble soluton correspondng to (m + n -1) cell allocatons m n( m n 1) = ( m n). n m( m n 1) ( m n) Hence, the total computatonal tme requred s gven by, m ( n 1) m n( m n 1) T (m, n) = ( m n) n ( m 1) n m( m n 1) ( m n) Thus, the total tme complexty s O (mn). The computatonal tme grows as the values of m and n ncreases, as a result of whch the problem becomes ntractable n nature. ORIGINS O 1 [12.95, 13, 13, 13.05] ([29.90, 30, 30, 30.10]) D 1 D 2 D 3 D 4 U [14.95, 15, 15, 15.05] ([249.95, 250, 250, ]) [15.95, 16, 16, 16.05] [3.75, 4, 4, 4.25] [11.80, 12, 12, 12.20] [17.95, 18, 18, 18.05] [4.75, 5, 5, 5.25] [12.75, 13, 13, 13.25] [-0.05, 0, 0, 0.05] O 2 [19.95, 20, 20, 20.05] ([49.95, 50, 50, 50.05]) [21.95, 22, 22, 22.05] [21.75, 22, 22, 22.25] [10.95, 11, 11, 11.05] ([279.95, 280, 280, ]) [7.95, 8, 8, 8.05] [11.65, 12, 12, 12.35] [6.85, 7, 7, 7.15] 15

16 [-0.20, 0, 0, 0.20] [3.70, 4, 4, 4.30] O 3 [18.95, 19, ([219.90, 220, 220, ]) [24.95, 25, 25, 25.05] [20.75, 21, 21, 21.25] [-0.40, 0, 0, 0.40] [16.95, 17, 17, 17.05] [9.65, 10, 10, 10.35] [2.50, 3, 3, 3.50] [10.95, 11, 11, 11.05] ([179.95, 180, 180, ]) [5.85, 6, 6, 6.15] V [12.90, 13, 13, 13.10] [14.90, 15, 15, 15.10] [3.80, 4, 4, 4.20] [4.80, 5, 5, 5.20] Fgure 9: The fnal allocated matrx wth the sum of the U and V values and the cell evaluatons for the unoccuped cells 7. Dscussons The degeneracy n transportaton problem and unbalanced transportaton problem can be smlarly represented usng fuzzy trapezodal numbers. Ths model of fuzzy trapezodal numbers for the transportaton problem can be easly extended to the assgnment problem whch s a specal class of the transportaton problem where the number of orgns s equal to the number of destnatons. However, f the FVAM algorthm for obtanng the soluton of transportaton problem s appled to solve the assgnment problem a large number of teratons have to be performed for the resoluton of degeneracy tll the optmal soluton s obtaned. In assgnment problem t s observed that a basc feasble soluton for the constrant equatons wll consst of (2m - 1) varables. But t s observed that every basc soluton wll consst of m basc varables equal to 1 and (m 1) basc varables equal to 0 and as such the basc feasble soluton wll have a hgh level of degeneracy [5], [12]. Ths wll ncrease the overall computatonal complexty of the soluton. The effectveness of the solutons obtaned for the transportaton problem can greatly be enhanced by ncorporatng the genetc algorthms alongwth the fuzzy trapezodal numbers such that the computatonal complexty s greatly reduced. 8. Conclusons Ths Paper presents the closed, bounded and non empty feasble regon of the transportaton problem usng fuzzy trapezodal numbers whch ensures the exstence of an optmal soluton to the balanced transportaton problem. The mult-valued nature of Fuzzy Sets allows handng of uncertanty and vagueness nvolved n the cost values of each cells n the transportaton table. For fndng the ntal soluton of the transportaton problem we use the FVAM and for determnng the optmalty of the obtaned soluton FMODIM s used. The fuzzfcaton of the cost of the transportaton problem s dscussed wth the help of a numercal example. We also dscuss the computatonal complexty nvolved n the problem. The effectveness of the solutons obtaned for the problem can greatly be enhanced by ncorporatng the genetc algorthms alongwth the fuzzy trapezodal numbers such that the computatonal complexty s greatly reduced. References 1. S. Chanas, M. Delgado, J. L. Verdegay and M. A. Vla, Interval and fuzzy extensons of classcal transportaton problems, Transportaton Plannng Technology 17 (1993). 2. S. Chanas and D. Kuchta, Fuzzy nteger transportaton problem, Fuzzy Sets and Systems 98 (1998). 16

17 3. A. Chaudhur and K. De, A Comparatve study of the Transportaton Problem under Probablstc and Fuzzy Uncertantes, Proceedngs of 15 th Mathematcs Conference, Department of Mathematcs, Unversty of Dhaka, Dhaka, Bangladesh, 2007, pp G. B. Dantzg, Applcaton of the Smplex Method to a Transportaton Problem, Chapter 23 n Koopman s Actvty Analyss of Producton and Allocaton, Cowles Commsson Monograph 13, John Wley and Sons, Inc., New York, P. K. Gupta and Man Mohan, Problems n Operatons Research, Sultan Chand and Sons, S. I. Gauss, Lnear Programmng: Methods and Applcatons, ffth edton, McGraw Hll Book Company, New York, F. L. Htchcock, Dstrbuton of a product from several Sources to numerous Localtes, Journal of Mathematcal Physcs 12:3 (1978). 8. H. Isermann, The enumeraton of all effcent soluton for a lnear multple obectve transportaton problem, Naval Research Logstcs Quarterly 26 (1979). 9. N. S. Kambo, Mathematcal Programmng Technques, revsed edton, EWP, New Delh, L. V. Kantorovch, On the Translocaton of Masses, Doklady Akad. Nauk SSR 37 (1942). 11. T. C. Koopmans, Optmum Utlzaton of the Transportaton System, Econometrca 17 (1949). 12. V. K. Kapoor and S. Kapoor, Operatons Research: Technques for Management, Sultan Chand and Sons, J. L. Rnguest and D. B. Rnnks, Interactve solutons for the lnear mult obectve transportaton problem, European Journal of Operatonal Research 32 (1987). 14. J. Yen and R. Langar, Fuzzy Logc: Intellgence, Control and Informaton, Pearson Educaton, H. J. Zmmermann, Usng Fuzzy Sets n Operatonal Research, European Journal of Operatonal Research 13 (1983). 17