A New Method for Finding an Optimal Solution. of Fully Interval Integer Transportation Problems

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1 Applied Matheatical Scieces, Vl. 4, 200,. 37, A New Methd fr Fidig a Optial Sluti f Fully Iterval Iteger Trasprtati Prbles P. Padia ad G. Nataraja Departet f Matheatics, Schl f Advaced Scieces, VIT Uiversity, Vellre-4, Tail Nadu, Idia Crrespdig authr. padia6@rediffail.c ( P. Padia ) Abstract A ew ethd aely, separati ethd based zer pit ethd [9] is prpsed fr fidig a ptial sluti fr iteger trasprtati prbles where trasprtati cst, supply ad dead are itervals. The prpsed ethd is a fuzzy ethd ad als, has bee develped withut usig the idpit ad width f the iterval i the bjective fucti. The sluti prcedure is illustrated with a uerical exaple. The separati ethd ca be served as a iprtat tl fr the decisi akers whe they are hadlig varius types f lgistic prbles havig iterval paraeters. Further, the prpsed ethd is exteded t fuzzy trasprtati prbles. Matheatics Subject Classificati: 90C08, 90C70, 90B06, 90C29, 90C90 Keywrds: Iterval iteger trasprtati prble, Optial sluti, Zer pit ethd, Fuzzy trasprtati prble. Itrducti Varius efficiet ethds were develped fr slvig trasprtati prbles with the assupti f precise surce, destiati paraeter, ad the pealty factrs. I real life prbles, these cditis ay t be satisfied always. T deal with iexact cefficiets i trasprtati prbles, ay researchers [-3, 5-8, 0, ] have prpsed fuzzy ad iterval prgraig techiques fr slvig the. Das et al. [3] prpsed a ethd, called fuzzy techique t slve iterval trasprtati prble by csiderig the right bud ad the idpit f the

2 820 P. Padia ad G. Nataraja iterval. Segupta ad Pal [0] prpsed a ew fuzzy rietated ethd t slve iterval trasprtati prbles by csiderig the idpit ad width f the iterval i the bjective fucti. I this paper, we prpse a ew ethd aely, separati ethd t fid a ptial sluti fr iteger trasprtati prbles where trasprtati cst, supply ad dead are itervals. We develp the separati ethd withut usig the idpit ad width f the iterval i the bjective fucti f the fully iterval trasprtati prble which is a -fuzzy ethd. The prpsed ethd is based zer pit ethd [9]. The sluti prcedure is illustrated with a uerical exaple. The ew ethd ca be served as a iprtat tl fr the decisi akers whe they are hadlig varius types f lgistic prbles havig iterval paraeters. Further, the prpsed ethd is exteded t fuzzy trasprtati prbles. 2 Preliiaries Let D dete the set f all clsed buded itervals the real lie R. That is, D = {[ a, b], a b ad a ad b are i R }. We eed the fllwig defiitis f the basic arithetic peratrs ad partial rderig clsed buded itervals which ca be fud i [6, 4]. Defiiti : Let A = [ a, b] ad B = [ c, d] be i D. The, A B = [ a + c, b + d] ; A ΘB = [ a d, b c] ; ka = [ ka, kb] if k is a psitive real uber; ka = [ kb, ka] if k is a egative real uber ad A B = [ p, q] where p = i{ ac, ad, bc, bd} ad q = ax.{ ac, ad, bc, bd} Defiiti 2: Let A = [ a, b] ad B = [ c, d] be i D. The, A B if a c ad b d A B if B A, that is, a c ad b d ad A = B if A B ad B A, that is, a = c ad b = d. 3 Fully Iterval Iteger Trasprtati Prbles Csider the fllwig fully iterval iteger trasprtati prble (FIITP): [ 2 y Miiize z, z ] = [ c, d ] [ x, ] subject t

3 Fully iterval iteger trasprtati prbles 82 [ x, y] = [ ai, pi ], i =,2,..., () [ x, y] = [ bj, qj], j =,2,..., (2) x 0 0 j =,2,..., ad are itegers (3) where c ad d are psitive real ubers fr all i ad j, a i ad pi are psitive real ubers fr all i ad b ad q are psitive real ubers fr all j. j Defiiti 3: The set { [ x, y], fr all i =,2,..., ad j =,2,..., } is said t be a feasible sluti f (FIITP) if they satisfy the equatis (), (2) ad (3). Defiiti 4: A feasible sluti {[ x, y], fr i =,2,..., ad j =,2,..., } f the prble (FIITP) is said t be a ptial sluti f (FIITP) if [ c, d] [ x, y] [ c, d] [ u, v], fr i =,2,..., ad j =,2,..., ad fr all feasible {[ u, v ] fr i =,2,..., ad j =,2,..., }. Nw, we prve the fllwig there which fids a relati betwee ptial slutis f a fully iterval iteger trasprtati prble ad a pair f iduced trasprtati prbles ad als, is used i the prpsed ethd. There : If the set { y fr all i ad j } is a ptial sluti f the upper bud trasprtati prble (UBITP) f (FIITP) where (UBITP) Miiize z2 = dy subject t y = pi y = qj j, i =,2,..., (4), j =,2,..., (5) y 0, i =,2,..., ad j =,2,..., ad are itegers (6) ad the set { x fr all i ad j } is a ptial sluti f the lwer bud trasprtati prble (LBITP) f (FIITP) where (LBITP) Miiize z = cx

4 822 P. Padia ad G. Nataraja subject t x = ai x = bj 0, i =,2,..., (7), j =,2,..., (8) x, i =,2,..., ad j =,2,..., ad are itegers, (9) [ y the the set f itervals { x, y ] fr all i ad j } is a ptial sluti f the prble (FIITP) prvided x, fr all i ad j. Prf: Let {[ x, y], fr all i ad j } be a feasible sluti f the prble (FIITP). Therefre, { x, fr all i ad j } ad { y, fr all i ad j } are feasible slutis f the prbles (UBITP) ad (LBITP). Nw, sice { x, fr all i ad j } ad { y, fr all i ad j } are ptial slutis f (UBITP) ad (LBITP), we have ad dy y dy ; cx x, i =,2,..., ad j =,2,...,. This iplies that, c x, i = cx dy cx, i = That is, [ c, d ] [ x, y ] c, d ] [ x, y ], Nw, sice { x y [ d y, fr all i ad j } ad { y, fr all i ad j } satisfy (4) t (9) ad x, fr all i ad j, we ca cclude that the set { [ x, y ] fr all i ad j } is a feasible sluti f (FIITP). Thus, the set f itervals { x, y ], fr all i ad j } is a ptial sluti f the prble (FIITP). Hece the there. [ 3 Separati ethd We, w itrduce a ew algrith aely, separati ethd fr fidig a ptial sluti fr a fully iterval iteger trasprtati prble. The separati ethd prceeds as fllws.

5 Fully iterval iteger trasprtati prbles 823 Step. Cstruct the UBITP f the give FIITP. Step 2. Slve the UBITP usig the zer pit ethd. Let { a ptial sluti f the UBITP. Step 3. Cstruct the LBITP f the give FIITP. y, fr all i ad j } be Step 4. Slve the LBITP with the upper bud cstraits x y, fr all i ad j usig the zer pit ethd. Let { x, fr all i ad j } be the ptial sluti f LBITP with x y, fr all i ad j. [ Step 5. The ptial sluti f the give FIITP is { x, y ], fr all i ad j } (by the There ). The prpsed algrith is illustrated by the fllwig exaple. Exaple : Csider the fllwig FIITP [,2] [,3] [5,9] [4,8] [7,9] [,2] [7,0] [2,6] [3,5] [7,2] [7,9] [7,] [3,5] [5,7] [6,8] Dead [0,2] [2,4] [3,5] [5,7] [40,48] Nw, the UBITP f the give prble is give belw: Dead Nw, usig the zer pit ethd, the ptial sluti t the UBITP is y = 5, y 4 2 =, y 7 2 =, y 4 24 =, y 5 33 = ad y 3 34 =. Nw, the LBITP f the give prble with the upper buded cstraits is give belw: Dead

6 824 P. Padia ad G. Nataraja ad als, y x, i =,2,..., ad j =,2,..., ad are itegers. Nw, usig the zer pit ethd, usig the zer pit ethd, the ptial sluti t the LBITP with the upper buded cstraits is x = 5, x 2 2 =, x 5 2 =, x 2 24 =, x 3 33 = ad x 3 34 =. = 2 = 2 34, y ] = [3,3 34 Thus, a ptial sluti t the give FIITP is [ x, y ] [5,5], [ x, y ] [2,4], 2 = 2 24 = 24 [ = [ x, y ] [5,7], [ x, y ] [2,4], x, y ] [3,5] ad [ x ] ad als, the iiu trasprtati cst is [02,202]. 4 Iterval Trasprtati Prbles Csider the fllwig iterval iteger trasprtati prble (IITP): 2 ] (IITP) Miiize [ z, z ] = [ c, d w subject t w [ i i a, p ] w [ bj, qj], i =,2,...,, j =,2,..., w 0, i =,2,..., ad j =,2,..., ad itegers where c ad d are psitive real ubers fr all i ad j, a i ad pi are psitive real ubers fr all i ad b j ad q j are psitive real ubers fr all j. Let w = αx + ( α) y, 0 α ad x ad y are itegers with x y fr all i ad j. Csider the fllwig FIITP [ 2 y Miiize z, z ] = [ c, d ] [ x, ] subject t [ i i [, y x, y ] = [ a, p ] [ x, y] = [ bj, qj], i =,2,...,, j =,2,..., x ] 0, i =,2,..., ad j =,2,..., ad are itegers

7 Fully iterval iteger trasprtati prbles 825 The abve prble ca be slved by usig separati ethd. Let { [ x, y ], fr all i ad j } be a ptial sluti f the abve prble. Sice w, fr all i ad j are itegers, chse ay α such that w = α x + ( α ) y, fr all i ad j are itegers. The, ptial slutis t IITP are give belw. w = α x + ( α ) y 0 α. The slvig prcedure f btaiig ptial slutis t IITP usig the separati ethd is illustrated by the fllwig exaple. Exaple 2: Csider the fllwig IITP with iteger real decisi variables [3,5] [2,6] [2,4] [,5] [7,9] [4,6] [7,9] [7,0] [9,] [7,2] [4,8] [,3] [3,6] [,2] [6,8] Dead [0,2] [2,4] [3,5] [5,7] [40,48] Let w = αx + ( α) y, 0 α ad x ad y are itegers with x y fr all i ad j. Nw, we csider the fllwig FIITP with variables [ x, y] fr all i ad j crrespdig t the give IITP. Nw, the UBITP f the FIITP is give belw Dead Nw, usig the zer pit ethd, the ptial sluti t the UBITP is y 3 = 9, y 2, y 3, y 6 23 =, y 32 = ad y 7 34 =. 2 = 22 = Nw, the LBITP f the FIITP with the buded cstraits is give belw Dead

8 826 P. Padia ad G. Nataraja ad als, x, i =,2,..., ad j =,2,..., ad are itegers. y Nw, usig the zer pit ethd, usig the zer pit ethd, the ptial sluti t the LBITP with buded cstraits is x 7, x 0, x, x 6, x 32 = ad x 5 34 =. 3 = 2 = 22 = 2 = 2 34, y ] = 34 Thus, a ptial sluti t the FIITP is [ x, y ] [7,9], [ x, y ] [0,2], 3 = 3 [ x 22, y ] = [,3], [ x, ] [6,6] y =, [ x, ] [, ] y32 = ad [ x [5,7] ad als, the iiu trasprtati cst is [9,232]. Nw, w = αx + ( α) y, 0 α are itegers fr all i ad j. S.N. α Sluti z -value Nuber f uits trasprted 0 3 = w 9, w 2 2 =, w 3 22 =, w 23 = 6, 32 = w ad w 7 34 = [47,232] = w 8, w 2 =, w 2 22 =, w 23 = 6, 32 = w ad w 6 34 = [33,2] = w 7, w 0 2 =, w 22 =, w 23 = 6, 32 = w ad w 5 34 = [9,90] 40 Nte: This type f slutis set is very uch useful fr decisi akers t select a sluti accrdig t their eeds, sice the set f slutis t the ITP is a fucti f the uber f uits trasprted. 23 = 5 Fully fuzzy iteger trasprtati prbles Csider the fllwig fuzzy iteger trasprtati prble (FFITP) where (FFITP) Miiize ~ z c~ ~ x subject t ~ x ~ = ai, i =,2,..., ~ ~ x = bj, j =,2,...,

9 Fully iterval iteger trasprtati prbles 827 ~ x ~ 0, i =,2,..., ad j =,2,..., ad are itegers where = the uber f supply pits ; = the uber f dead pits ; ~ x is the ucertai uber f uits shipped fr supply pit i t dead pit j ; ~c is the ucertai cst f shippig e uit fr supply pit i t the dead pit j ; ~a i is the ucertai supply at supply pit i ad ~ b j is the ucertai dead at dead pit j. A trapezidal fuzzy uber ( a, b, c, d) ca be represeted as a iterval uber fr as fllws. ( a, b, c, d) = [ a + ( b a) α, d ( d c) α] ; 0 α. (0) Usig the relati (0), we ca cvert the give fuzzy trasprtati prble it a iterval trasprtati prble. Usig the separati ethd, we btai a ptial sluti t the iterval trasprtati. The, agai usig the relati (0), we ca btai a ptial sluti t the give fuzzy trasprtati prble. The sluti prcedure f btaiig a ptial sluti t a fuzzy trasprtati prble usig the separati ethd is illustrated by the fllwig exaple. Exaple 3: Csider the fllwig FFITP: (,2,3,4) (,3,4,6) (9,,2,4) (5,7,8,) (,6,7,2) (0,,2,4) (-,0,,2) (5,6,7,8) (0,,2,3) (0,,2,3) (3,5,6,8) (5,8,9,2) (2,5,6,9) (7,9,0,2) (5,0,2,7) Dead (5,7,8,0) (,5,6,0) (-,3,4,8) (,2,3,4) The give fuzzy trasprtati prble is a balaced e. Nw, the FIITP crrespdig t the abve prble is give belw. [+α,4-α ] [+2α,6-2α ] [9+2α,4-2α ] [5+2α,-3α ] [+5α,2-5α ] [0+α, 4-2α ] [-+α,2-α ] [5+α, 8-α ] [0+α,3-α ] [0+α,3-α ] [3+2α,8-2α ] [5+3α,2-3α ] [2+3α,9-3α ) [7+2α,2-2α ] [5+5α,7-5α ] Dead [5+2α,0-2α ] [+4α,0-4α ] [-+4α, 8-2α ] [+α,4-α ] [6+α,32-9α ]

10 828 P. Padia ad G. Nataraja Nw, the UBITP f the FIITP is give belw. 4-α 6-2α 4-2α -3α 2-5α 4-2α 2-α 8-α 3-α 3-α 8-2α 2-3α 9-3α 2-2α 7-5α Dead 0-2α 0-4α 8-2α 4-α 32-9α Nw, usig the zer pit ethd, belw. a ptial sluti t the UBITP is give Dead 0-4α 2-α 3-α 0-2α 3-2α 4-α Nw, the LBITP f the FIITP with the cstraits is give belw. ad als, +α +2α 9+2α 5+2α +5α 0+α -+α 5+α 0+α 0+α 3+2α 5+3α 2+3α 7+2α 5+5α Dead 5+2α +4α -+4α +α x y, i =,2,..., ad,2,..., j = ad are itegers. Nw, usig the zer pit ethd, a ptial sluti t the LBITP is give belw. Dead +4α α α 5+2α -+2α +α Therefre, a ptial sluti t the FIITP is give belw.

11 Fully iterval iteger trasprtati prbles 829 [+4α,0-4α ] [α,2-α ] [+5α,2-5α ] [α, 3-α ] [0+α,3-α ] [5+2α,0-2α ] [-+2α, 3-2α ] [+α, 4-α ] [5+5α,7-5α ] Dead [5+2α,0-2α ] [+4α,0-4α ] [+2α,6-2α ] [+α,4-α ] Thus, the fuzzy ptial sluti fr the give FFITP is ~ x 2 (,5,6,0), ~ x 3 (0,,,2), ~ x 23 (0,,2,3 ), ~ x 3 (5,7,8,0), ~ x 23 (,,,3 ) ad ~ x 34 (,2,3,4 ) with the fuzzy bjective value ~ z = (4,00,44,297 ) ad the crisp value f the ptiu fuzzy trasprtati cst fr the prble, z is Reark : The fuzzy trasprtati prble with crisp decisi variables ca als be slved by usig the separati ethd siilar t the ethd f slvig iterval iteger trasprtati prbles. 6 Cclusi The separati ethd based the zer pit ethd prvides a ptial value f the bjective fucti fr the fully iterval trasprtati prble. This ethd is a systeatic prcedure, bth easy t uderstad ad t apply ad als it is a -fuzzy ethd. The prpsed ethd prvides re ptis ad ca be served a iprtat tl fr the decisi akers whe they are hadlig varius types f lgistic prbles havig iterval paraeters. Refereces [] S. Chaas ad D. Kuchta, A ccept f sluti f the trasprtati prble with fuzzy cst c-efficiet, Fuzzy Sets ad Systes, 82 (996), [2] J.W. Chieck ad K. Raada, Liear prgraig with iterval cefficiets, Jural f the Operatial Research Sciety, 5 (2000), [3] S.K. Das, A. Gswai ad S.S. Ala, Multibjective trasprtati prble with iterval cst, surce ad destiati paraeters, Eurpea Jural f Operatial Research, 7 (999), [4] Gerge J.Klir ad B Yua, Fuzzy sets ad fuzzy lgic: Thery ad Applicatis, Pretice-Hall, 2008.

12 830 P. Padia ad G. Nataraja [5] Ishibuchi, H., Taaka, H., Multibjective prgraig i ptiizati f the iterval bjective fucti. Eurpea Jural f Operatial Research, 48 (990), [6] R.E. Mre, Methd ad applicatis f iterval aalysis, SLAM, Philadelphia, PA, 979. [7] C. Oliveira ad C.H. Atues, Multiple bjective liear prgraig dels with iterval cefficiets a illustrated verview, Eurpea Jural f Operatial Research, 8(2007), [8] P.Padia ad G.Nataraja, A ew algrith fr fidig a fuzzy ptial sluti fr fuzzy trasprtati prbles, Applied Matheatical Scieces, 4 ( 200), [9] P.Padia ad G.Nataraja, A ew ethd fr fidig a ptial sluti fr trasprtati prbles, Iteratial J. f Math. Sci. ad Egg. Appls., 4 ( 200) ( accepted). [0] A. Segupta ad T.K. Pal, Iterval-valued trasprtati prble with ultiple pealty factrs, VU Jural f Physical Scieces, 9(2003), 7 8. [] S.Tg, Iterval uber ad fuzzy uber liear prgraig, Fuzzy sets ad systes, 66 (994), Received: Nveber, 2009