Modified Method for Fixed Charge Transportation Problem

Transcription

1 Iteratioal Joural of Egieerig Ivetios e-issn: , p-issn: Volue, Issue1 (August 201) PP: Modified Method for Fixed Charge Trasportatio Proble Debiprasad Acharya b, Majusri Basu a Ad Atau Das a a Departet of Matheatics; Uiversity of Kalyai; Kalyai ; Idia. b Departet of Matheatics; N. V. College; Nabadwip; Nadia; W.B.; Idia. ABSTRACT: I traditioal ethod for solvig fixed charge trasportatio proble, we itroduce duy colu to ake balace trasportatio proble fro ubalace trasportatio proble. We set zero cost for the duy colu. It is used i both crisp eviroet as well as fuzzy eviroet. But if we use the axiu cost i each row for respective positio i duy colu we get better result. A coparative study betwee the existig ethod ad the odified ethod shows that the latter is uch ore effective. Key words: Fuzzy Nuber, Fuzzy Trasportatio Proble, Fixed charge Trasportatio Proble. Matheatics Subject Classificatio : 90B06, 90C08. I. INTRODUCTION I a trasportatio proble (TP) geerally cost of trasportatio is directly proportioal to aout of coodity which is to be trasported. However, i ay real world probles, i additio to trasportatio cost, a fixed cost, soeties called a set up cost, is also icurred whe a distributio variable assue a positive value. Such proble are called fixed charge trasportatio proble (FCTP). The FCTP differs fro the liear TP oly i the o-liearity of the objective fuctio. While ot beig liear i each of the variables, the objective fuctio has a fixed cost associated with each origi. The fixed charge TP was origially forulated by Hirsch ad Datzig[1]. I 1961, Baliski[4] preseted a techique which provides a approxiate solutio for ay give FCTP. May probles i practice ca be treated as FCTP. FCTP has bee studies by ay researchers [1,2,5,6,8,15,19,21,22,2,24] The otio of fuzzy set has bee itroduced by L. A. Zadeh[25] i order to foralize the cocept of regardless i class ebership, i coectio with the represetatio of hua kowledge [18]. It was developed to defie ad solve the coplex syste with sources of ucertaity or iprecisio which are ostatistical i ature. Fuzzy TP has bee studied by ay authors [,7,9,10,11,12,14,16,17,20,26]. I this paper, we have doe a odificatio. Ubalaced TP coverted ito balaced TP by itroducig duy destiatio with axiu cost i each row. Here we have doe a coparative study betwee existig ethod ad the ew ethod. We see that our odificatio gives better result of the TP. II. PRELIMINARIES Basic Defiitio Defiitio 2.1 Let A be a classical set ad µ A (x) be a fuctio defied over A [0,1]. A fuzzy set A * with ebership fuctio µ A (x) is defied by A * = {(x, µ A (x)) : x A ad µ A (x) [0, 1]} Defiitio 2.2 A real fuzzy uber ã= (a 1, a 2, a, a 4 ), where a 1, a 2, a, a 4 R ad two fuctios f(x); g(x) : R [0; 1], where f(x) is o decreasig ad g(x) is o icreasig, such that we ca defie ebership fuctio µ ã (x) satisfyig the followig coditios f x if a 1 x a 2 1 if a µ ã x = 2 x a g x if a x a 4 0 oterwise Defiitio 2. The ebership fuctio of trapezoidal fuzzy uber ã= (a 1, a 2, a, a 4 ), is defied by µ ã x = x a 1 a 2 a 1 for a 1 x a 2 1 for a 2 x a a 4 x a 4 a for a x a 4 0 oterwise P a g e 67

2 Modified Method for Fixed Charge Trasportatio Proble Defiitio 2.4 The agitude of the trapezoidal fuzzy uber ã = (a 1, a 2, a, a 4 ) is defied by Mag ã = a 1+2a 2, + 2a +a 4 6 Defiitio 2.5 The two fuzzy ubers ã= (a 1, a 2, a, a 4 ) ad b = (b 1, b 2, b, b 4 ) are said to be ã > b if Mag ã > Mag b. Defiitio 2.6 The two fuzzy ubers ã= (a 1, a 2, a, a 4 ) ad b = (b 1, b 2, b, b 4 ) are said to be equal if Mag ã = Mag b Arithetic operatios: Let ã= (a 1, a 2, a, a 4 ) ad b = (b 1, b 2, b, b 4 ) be two trapezoidal fuzzy ubers, where a 1, a 2, a, a 4 ; b 1, b 2, b, b 4 R the the arithetic operatio o ã ad b are: 1. Additio: The additio of ã ad b is ã b = (a 1 + b 1, a 2 + b 2, a + b, a 4 + b 4 ). 2. Subtractio: - b = ( - b 4, - b, - b 2, - b 1 ), the the subtractio of ã ad b is ã b = (a 1 - b 4, a 2 - b, a - b 2, a 4 - b 1 ).. Multiplicatio: The ultiplicatio of ã ad b is ã b = (t 1, t 2, t, t 4 ) where t 1 = i{ a 1 b 1, a 1 b 4, a 4 b 1, a 4 b 4 }; t 2 = i{a 2 b 2, a 2 b, a b 2, a b }. t = ax{ a 2 b 2, a 2 b, a b 2, a b }; t 4 =ax{ a 1 b 1, a 1 b 4, a 4 b 1, a 4 b 4 }. 4. a 1, a 2, a, a 4 for 0. ã = a 4, a, a 2, a 1 for Proble Forulatio I 1994, Basu et. al.[6] cosider a fixed charge trasportatio proble i crisp eviroet as follows: P 1 : Mi Z = j=1 c ij x ij + F i j =1 x ij a i ; for i = 1,2,,.,. x ij = b j ; for j = 1,2,,.,. where x ij = the aout of product trasported fro the i th origi to the j th destiatio, c ij = the cost ivolved i trasportig per uit product fro the i th origi to the j th destiatio, F i = the fixed cost (or fixed charge) associated with origi i, a i = the uber of uits available at the i th origi, b j = the uber of uits required at the j th destiatio. is the uber of origi ad is uber of destiatio. I 2010, A. Kuar et. al.[16] cosider fixed charge trasportatio proble i fuzzy eviroet as follows: P 2 : Mi Z = j=1 c ij x ij f i j =1 x ij a i ; for i = 1,2,,.,. x ij = b j ; for j = 1,2,,.,. where f i = the fixed cost associated with origi i, ad all other otatio are defied i above. I both cases the solutio procedure as follows: First we have to balaced the proble P 1 ad P 2 usig duy destiatios. The we have P : Mi Z = j=1 c ij x ij + F i j =1 x ij = a i ; for i = 1,2,,.,. x ij = b j ; for j = 1,2,,.,,. where c i, = ax { c ij }, 1 i 1 j ad P 4 : Mi Z = j=1 c ij x ij f i j =1 x ij = a i ; for i = 1,2,,.,. x ij = b j ; for j = 1,2,,.,,. where c i, = ax { c ij }, 1 i 1 j P a g e 68

3 Modified Method for Fixed Charge Trasportatio Proble I proble P ad P 4, we cosider the costs associated with the duy cells are all axiu i each correspodig row. Fid a basic feasible solutio of the proble P ad P 4 with respect to the variable costs. Let B be the curret basis. III. ALGORITHM I both the cases algorith are sae while first oe is i crisp eviroet ad secod oe is i fuzzy eviroet. Step 1: Covert ito balaced trasported proble. Step 2: Set k=1, where is the uber of iteratios i the algorith. Step : Fid a basic feasible solutio of the proble P with respect to the variable costs. Let B be the curret basis. Step 4: calculate the fixed cost of the curret basic feasible solutio (without cosiderig duy cells) ad deote this by F 1 (curret), where F 1 (curret) = F i Step 5: Fid (C ij u i v j ); for all (i; j) B ad deote it by (C ij ) 1 ; where u i, v j are the dual variables for i = 1, 2,,.,; j = 1, 2,,..,, + 1. Step 6: Fid A 1 1 ij = (C ij ) 1 (E ij ) 1, where A ij is the chage i cost occurs for itroducig a o-basic (i; j) cell with value (E ij ) 1 (for all i, j B) ito the basis by akig reallocatio. 1 Step 7: Fid F ij (Differece) = F 1 ij (NB) F 1 (curret), where F 1 ij (NB) is the total fixed cost ivolved for itroducig the variable x ij with values (E ij ) 1 (for all i, j B) ito the curret basis to for a ew basis. 1 1 Step 8: Add F ij (Differece) ad A ij ; ad deote it by Δ 1 ij, i.e. Δ 1 1 ij = F ij (Differece) + A 1 ij, for all i, j B. Step 9: If all Δ 1 ij 0, the goto Step 10; otherwise fid i { Δ 1 ij, Δ 1 ij 0, i,j B }. The the variable x ij associated with i (Δ 1 ij ) will eter ito the basis, where I, j B. Cotiue this procedure util all Δ 1 ij 0. Goto Step. Step 10: Let Z 1 be the optiu cost of P 1 ad X 1 be the optiu solutio correspodig to Z 1. Siilar Algorith for the proble P 4. IV. NUMERICAL EXAMPLE Basu et. al. [6] cosider the fixed charge trasportatio proble which is tabulated i Table 1. Table 1 D 1 D 2 D a i O O O b j The fixed cost are F 11 = 100; F 12 = 50; F 1 = 50 F 21 = 150; F 22 = 50; F 2 = 50 F 1 = 200; F 2 = 0; F = 50 Where F i = l=1 δ il F il for i = 1; 2; where δ i1 = 1; if l=1 x ij >0 for i = 1, 2, : where δ i2 = 1; if l=1 x ij >7 for i = 1, 2, : where δ i = 1; if l=1 x ij >10 for i = 1, 2, : Table 2. I [6] the optiu solutio is X * = {x 11 = 5, x 1 = 14, x 2 = 8, x = 1}, with optiu cost Z * = 660. Itroducig duy destiatio D 4 with axiu cost of the correspodig row i Table 1, we get Table 2 D 1 D 2 D D 4 a i O O O P a g e 69

4 Modified Method for Fixed Charge Trasportatio Proble The optiu solutio of this proble are tabulated i Table. Table. D 1 D 2 D D 4 a i O O O The optiu solutio is X 1 ={ x 11 = 5, x 12 = 8, x 1 = 5, x 2 = 10 }, with optiu cost Z 1 = 562. Kuar et. al.[16] cosider the fixed charge trasportatio proble i fuzzy eviroet which is tabulated i Table 4. Table 4 D 1 D 2 D a i O 1 (1,4,5,10) (,6,9,18) (,6,9,18) 19 O 2 (1,,4,8) (2,4,6,12) (0,1,2,5) 10 O (0,1,2,5) (0,0.5,1.5,2) (0,0.5,1.5,2) 11 b j The fixed cost are f 11 = (70, 80, 100, 150); f 12 = (0, 40, 50, 80); f 1 = (0, 40, 50, 80); f 21 = (90, 100, 200, 210); f 22 = (0, 40, 50, 80); f 2 = (0, 40, 50, 80); f 1 = (100; 150; 200; 50); f 2 = (70, 80, 100, 150); f = (0, 40, 50, 80); Where f i = l=1 δ il f il for i = 1; 2; where δ i1 = 1; if l=1 x ij >0 for i = 1, 2, : where δ i2 = 1; if l=1 x ij >7 for i = 1, 2, : where δ i = 1; if l=1 x ij >10 for i = 1, 2, : I [16] the optiu solutio is X = { x 11 = 5, x 1 = 14, x 2 = 8, x = 1}, with optiu cost Z = (47, 498.5, 664.5, 110). Here also f i has cosider three steps as above. Itroducig duy destiatio D 4 with axiu cost of the correspodig row i Table 4, we get Table 5 Table 5 D 1 D 2 D D 4 a i O 1 (1,4,5,10) (,6,9,18) (,6,9,18) (,6,9,18) 19 O 2 (1,,4,8) (2,4,6,12) (0,1,2,5) (2,4,6,12) 10 O (0,1,2,5) (0,0.5,1.5,2) (0,0.5,1.5,2) (0,1,2,5) 11 The optiu solutio of this proble are tabulated i Table 6 Table 6 D 1 D 2 D D 4 a i O 1 (1,4,5,10) (,6,9,18) (,6,9,18) (,6,9,18) O 2 (1,,4,8) (2,4,6,12) (0,1,2,5) (2,4,6,12) O (0,1,2,5) (0,0.5,1.5,2) (0,0.5,1.5,2) (0,1,2,5) P a g e 70

5 Modified Method for Fixed Charge Trasportatio Proble The optiu solutio is X 1 = { x 11 = 5, x 12 = 8, x 1 = 5, x 2 = 10}, with optiu cost Z 1 = (299, 408, 612, 94). Coparative study betwee Basu et. al., Kuar et. al. ad odified ethod, is give i Table 7. Table 7 Optiu solutio Basu et. al X * ={x 11 = 5, x 1 = 14, x 2 = 8, x = 1} Modified X 1 ={ x 11 = 5, x 12 = 8, x 1 = 5, Method x 2 = 10} Kuar et. X ={ x 11 = 5, x 1 = 8, x 2 = 8, al. x = 10} Modified X 1 ={ x 11 = 5, x 12 = 8, x 1 = 5, Method x 2 = 10} Optiu cost Z * =660 Z 1 =562 Z =(47, 498.5, 664.5, 110). Mag.( Z ) = 660 Z 1 =(299, 408, 612, 94). Mag.(Z 1 ) = 562 REFERENCES [1] V. Adlakha, K. Kowalski, A Siple Algorith for the Source-Iduced Fixed-Charge Trasportatio Proble, The Joural of the Operatioal Research Society, Vol. 55, No. 12 (Dec., 2004), pp [2] V. Adlakha, K. Kowalski ad B. Lev, A brachig Method for the fixed charge trasportatio proble, Oega, vol. 8,(2010), p.p [] T. Allahviraloo, F. H. Lot_, M. K. Kiasary, N. A. Kiai ad L. Alizadeh, Solvig fully fuzzy liear prograig proble by the rakig fuctio, Applied Matheatical Scieces, 2008, 2: [4] M. L. Baliski, Fixed cost trasportatio probles, Naval research logistics quarterly, 1961(8), [5] R. S. Barr, F. Glover ad D. Kliga, A ew optiizatio ethod for large scale fixed charge trasportatio probles, Operatios Research, 1981, 29: [6] M. Basu, B. B. Pal ad A. Kudu, A Algorith For The Optiu Tie Cost Trade-off i Fixed Charge Bi-criterio Trasportatio Proble, Optiizatio. 1994, 0: [7] R. E. Bella ad L. A. Zadeh, Decisio akig i a fuzzy eviroet, Maageet Sciece, 1970, 17: [8] W. L. Bruce ad A. W. Chris, Revised-Modified Pealties for Fixed Charge Trasportatio Probles, Maageet Sciece, Vol. 4, No. 10 (Oct., 1997), pp [9] L. Capos ad A. G. Muoz, A subjective approach for rakig fuzzy uber, Fuzzy Sets ad Systes, 1989, 29: [10] D. S. Diagar ad K. Palaivel, The Trasportatio Proble i Fuzzy Eviroet, Iteratioal Joural of Algoriths, Coputig ad Matheatics, 2009, 2: [11] K. Gaesa ad P. Veeraai, Fuzzy liear progras with trapezoidal fuzzy Nubers, Aals of Operatios Research, 2006, 14: [12] P. Gupta ad M. K. Mehlawat, A algorith for a fuzzy trasportatio proble to select a ew type of coal for a steel aufacturig uit, Top, 2007, 15: [1] W. M. Hirsch ad G.B. Datzig, Notes o liear Prograig: Part XIX, The fixed charge proble. Rad Research Meoradu No. 18, Sata Moica; Califoria; [14] A. Kaufa ad M. M. Gupta,. Itroductio to Fuzzy Arithetics: Theory ad Applicatios. Va Nostrad Reihold, New York, [15] K. Kowalski ad B. Lev, O step fixed-charge trasportatio proble, OMEGA: The Iteratioal Joural of Maageet Sciece, 2008, vol. 6, p.p [16] A. Kuar, A. Gupta ad M. K. Shara, Solvig fuzzy bi-criteria fixed charge trasportatio proble usig a ew fuzzy algorith, Iteratioal Joural of Applied Sciece ad Egieerig, 2010; 8: [17] G. S. Mahapatra ad T. K. Roy, Fuzzy ulti-objective atheatical prograig o reliability optiizatio odel, Applied Matheatics ad Coputatio, 2006, 174: [18] A. Ojha, Soe studies o trasportatio probles i differet eviroets, Vidyasagar Uiversity, [19] U. S. Palekar, M. H. Karwa ad S. Ziots, A brach-ad-boud ethod for the fixed charge proble, Maageet Sciece, 1990, 6: [20] P. Padia ad G. Nataraja, A ew algorith for fidig a fuzzy optial solutio for fuzzy trasportatio probles, Applied Matheatical Scieces, 2010, 4: [21] P. Robers ad L. Cooper, A study of the fixed charge trasportatio proble, cop. ad aths. With applicatios, 2, 1976, [22] S. Sadagopa ad A. Ravidra, A vertex rakig algorith for the fixed charge trasportatio Proble, Joural of Optiizatio Theory ad Applicatios, 1982, 7: [2] K. Sadrock, A siple algorith for solvig sall fixed charge trasportatio probles, Joural of the Operatioal Research Society, 1988, vol. 9, p.p [24] D. I. Steiberg, The fixed charge proble, Naval Research Logistics Quarterly, 1970, 17: [25] L. A. Zadeh, Fuzzy sets, Iforatio ad Cotrol, 1965, 8: 8-5. [26] H. J. Ziera, Fuzzy prograig ad liear prograig with several objective fuctios, Fuzzy Sets ad Systes, 1978, 1: P a g e 71