Duality Theory for Interval Linear Programming Problems

Transcription

1 IOSR Joural of Matheatcs (IOSRJM) ISSN: Volue 4, Issue 4 (Nov-Dec, ), Dualty Theory for Iterval Lear Prograg Probles G. Raesh ad K. Gaesa, Departet of Matheatcs, Faculty of Egeerg ad Techology, SRM Uversty, Kattakulathur, Chea - 6, Ida. Abstract : We defe the pral ad dual lear prograg probles volvg terval ubers as the way of tradtoal lear prograg probles. We dscuss the soluto cocepts of pral ad dual lear prograg probles volvg terval ubers wthout covertg the to classcal lear prograg probles. By troducg ew arthetc operatos betwee terval ubers, we prove the weak ad strog dualty theores. Copleetary slackess theore s also proved. A uercal exaple s provded to llustrate the theory developed ths paper. Keywords: Iterval Nubers, Iterval Arthetc, Lear Prograg, Weak Dualty, Strog Dualty, Copleetary Slackess. I. INTRODUCTION Lear prograg s a ost wdely ad successfully used decso tool the quattatve aalyss of practcal probles where ratoal decsos have to be ade. I order to solve a Lear Prograg Proble, the decso paraeters of the odel ust be fxed at crsp values. But to odel real-lfe probles ad perfor coputatos we ust deal wth ucertaty ad exactess. These ucertaty ad exactess are due to easureet accuracy, splfcato of physcal odels, varatos of the paraeters of the syste, coputatoal errors etc. Iterval aalyss s a effcet ad relable tool that allows us to hadle such probles effectvely. Lear prograg probles wth terval coeffcets have bee studed by several authors, such as Atau Segupta et al. [, ], Btra [5], Chaas ad Kuchta [6], Nakahara et al. [], Steuer [6] ad Tog Shaocheg []. Nuerous ethods for coparso of terval ubers ca be foud as Atau Segupta ad Tapa Kuar Pal [, ], Gaesa ad Veeraa [8, 9] etc. By takg axu value rage ad u value rage equaltes as costrat codtos, Tog Shaocheg [] reduced the terval lear prograg proble to two classcal lear prograg probles ad obtaed a optal terval soluto to t. Raesh ad Gaesa [4] proposed a ethod for solvg terval uber lear prograg probles wthout covertg the to classcal lear prograg probles. The dualty theory for exact lear prograg probles was proposed by Soyster [7 9] ad Thuete []. Falk [7] provded soe propertes o ths proble. However, Poerol [] poted out soe drawbacks of Soyster s results ad provded soe ld codtos to prove the. Masahro Iugucha [4] et al has studed the dualty of terval uber lear prograg probles through fuzzy lear prograg probles. Bector ad Chadra [4] troduced a par of lear pral-dual probles uder fuzzy evroet ad establshed the dualty relatoshp betwee the. Hse-Chug Wu [,] troduced the cocept of scalar product for closed tervals the obectve ad equalty costrats of the pral ad dual lear prograg probles wth terval ubers. He troduced a soluto cocept that s essetally slar to the oto of odoated soluto ultobectve prograg probles by posg a partal orderg o the set of all closed tervals. He the proved the weak ad strog dualty theores for lear prograg probles wth terval ubers. Roh [5] also dscussed the dualty a terval lear prograg proble wth real-valued obectve fucto. I ths paper, we attept to develop the dualty theory for terval lear prograg probles wthout covertg the to classcal lear prograg probles. The rest of ths paper s orgased as follows: I secto, we recall the deftos of terval uber lear prograg, terval ubers ad soe related results of terval arthetc o the. I secto, we defe the terval uber pral ad dual lear prograg probles as the way of tradtoal lear prograg probles. We the prove the weak ad strog dualty theores. Copleetary Slackess theore s also proved. I secto 4, a uercal exaple s provded to llustrate the theory developed ths paper. II. PRELIMINARIES The a of ths secto s to preset soe otatos, otos ad results whch are of useful our further cosderato. 9 Page

2 Dualty theory for terval lear prograg probles IR = {a = [a, a ]: a a ad a, a R} be the set of all proper tervals ad Let be the set of all proper tervals o the real le R. If IR = {a = [a, a ]: a > a ad a, a R} a = a = a = a, the a = [a, a] = a s a real uber (or a degeerate terval). We shall use the ters terval ad terval uber terchageably. The d-pot ad wdth (or half-wdth) of a terval uber are defed as a The d-pot ad wdth (or half-wdth) of a terval uber a + a [a, a ] are defed as (a) = ad a- a w(a) =. The terval uber a ca also be expressed ters of ts dpot ad wdth as a [a,a ] (a),w(a)... A New Iterval Arthetc Mg Ma et al.[8] have proposed a ew fuzzy arthetc based upo both locato dex ad fuzzess dex fucto. The locato dex uber s take the ordary arthetc, whereas the fuzzess dex fuctos are cosdered to follow the lattce rule whch are the least upper boud ad greatest lower boud the lattce L. That s for a,bl we defe a b = ax{a,b} ad a b = {a,b}. a = [a, a ], b = [b, b ] IR ad for * +, -,,, the arthetc operatos o a ad b For ay two tervals are defed as: a * b = [a, a ]*[b, b ] = (a),w(a) (b),w(b) (a) (b), ax w(a),w(b). I partcular (). Addto : a + b = (a), w(a) (b), w(b) (a) (b), ax w(a), w(b). (). Subtracto : a - b = (a), w(a) (b), w(b) (a) (b), ax w(a), w(b). (v). Multplcato : a b = (a), w(a) (b), w(b) (a) (b), ax w(a), w(b). (v). Dvso : a b (a), w(a) (b), w(b) (a) (b), ax{w(a), w(b)}, provded (b). III. Ma Results Now we are a posto to prove terval aalogue of soe portat relatoshps betwee the pral ad dual lear prograg probles. We cosder the pral ad dual lear prograg probles volvg terval ubers as follows: Cosder the followg lear prograg proble volvg terval ubers (P) axz c x = = subect to a x b, =,, ad x for all,,,.,, where a, c, x, b IR, =,,,. ad =,,,.,. We call the above proble (P) as the pral terval lear prograg proble, ad t ca be rewrtte as (P) ax z cx subect to Ax b ad x,. where A, b, Let c, x are ( ), (), ( ),( ) atrces volvg terval ubers. X ={ x = (x x x x ) : A x b, x } be the feasble rego of proble (.). We say that,,,, (.) x s a feasble soluto to the pral proble (.), f x X. A feasble soluto x X s sad to be * a optu soluto to the pral proble (.), cx cx for all x X, where = cx = (cx + cx + c x ++ cx ) ad c,,,, cx = (c c c. c ). Now we cosder the followg lear prograg proble volvg terval ubers: 4 Page

3 Dualty theory for terval lear prograg probles (D) w b y = subect to ay c, =,, = ad y for all,,,.,, where a, c, x, b IR, =,,,. ad =,,,.,. We call the above proble (D) or (.) as the dual terval lear prograg proble of the pral proble (P), ad t ca be rewrtte as (D) w by subect to Ay c ad y,.4 where A, b, c, y are (), ( ), ( ), ( ) atrces volvg terval ubers. Let Y y y, y, y,, y : A y c, y be the feasble rego of proble (.). We say that (.) y s a feasble soluto to the dual proble (.), f y Y. A feasble soluto y Y s sad to be a optu soluto to the dual (.) f by by for all y Y, where by = b y = b y b y + b y + + b y ad b = (b, b, b,., b ). Stadard For: For the geeral study, we covert the gve terval uber lear prograg proble to ts stadard for as ax z cx subect to Ax b ad x, where A, b, c, x are ( ), (), (), () atrces cosstg of terval ubers. Defto.. Let x = (x, x,x,,x ) solves A x b. If all x [-α,α ] for soe α, the x s sad to be a basc soluto. If x [-α,α ] for soe α, the x has soe o-zero copoets, say x, x, x,, x k,k. The A x b ca be wrtte as: a x + a x + a x ++ a x + a [-β,β ]+ a [-β,β ]++ a [-β,β ] k k k+ k+ k+ k+ k+ k+ b If the colus a,a,a,,a k correspodg to these o-zero copoets x, x, x,..,x k are learly depedet, the x s sad to be a basc soluto. Reark.. Gve a syste of sultaeous lear equatos volvg terval ubers ukows ( ) A x b, bir, where A s a ( ) terval atrx ad rak of A s. Let B be ay ( ) terval atrx fored by learly depedet colus of A. - T Let xb = B b - = (x,x,x,x ) or sply x B = B b = (x,x,x,x ) ad x = (x,x,x,,x,,,,) s a basc soluto. I ths case, we also say that x B s a basc soluto. Theore.. Cosder A x b. where A - = (a ), a IR. The x B =B b s a soluto of A x b. Theore. (Weak dualty theore) If x = (x,x,x,,x ) s ay feasble soluto to the pral terval lear prograg proble (.) ad y, y, y,, y lear prograg proble (.), the y s ay feasble soluto to the dual terval cx by or c x b y. Proof. Sce x s a feasble soluto to the pral terval lear prograg proble (.), we have a x b, x,,,,,. a, w a x, w x b, w b, x,,,,,. Multplyg the th (,,,, ) pral costrat by y y,wy ad addg we have a, w a x, w x y, w y b, w b y, w y a, w a x, w x y, w y b, w b y, w y,,,,, a,w a x,w x y,w y b y by. (.5) 4 Page

4 Multplyg the th ( =,,,,) dual costrat by x x,wx a y x c x, = =,,,, Dualty theory for terval lear prograg probles ad addg we have a, w a y, w y x, w x c, w c x, w x a, w a y, w y x, w x c, w c x, w x a, w a y, w y x, w x c, w c x, w x, = = = = = = = =,,,, a, w a y, w y x, w x c x cx. (.6) Fro equatos (.5) ad (.6), we have b, w b y, w y c, w c x, w x a, w a y, w y x, w x = = c, w c x, w x b, w b y, w y = cx c x b y by cx by. = Proposto.. Suppose that = (x,x,x,,x ) y are feasble solutos to the pral (.) ad the dual (.) respectvely, such that cx by, the x ad y are optal solutos to the pral ad dual probles respectvely. Proof. Let cx by. Fro the weak dualty theore, we have cx by cx for all x X cx cx for all x X. Slarly, by cx y y, y, y,, y Y by by for all y Y. x ) ad y, y, y,, y for all Proposto.. If cx by, the there exst x X ad y Y such that (.7) cx cx for all x X ad by by for all y Y. That s x s a optu soluto to the pral proble (.) ad y a optu soluto to the dual proble (. ). Proof. Sce cx by, there exst x X ad y Y such that cx by. The the results follow edately fro proposto (.). Theore. (Strog dualty theore) If x ( x, x, x, x ) s a optal soluto to the pral proble (.), the there ext a feasble soluto y ( y, y, y, y ) to the dual proble (.) such that cx by. Proof: We covert the pral proble (.) to ts stadard for by addg slack varables as follows: ax z c x subect to a x x b,,,, ad x (.8) for all,,,,,,, where x are slack varables. That s ax z cx subect to Ax b ad x, where A, b, c, x are, (), ( ( )), (( ) ) atrces cosstg of terval ubers. Let xb B b s a optal basc feasble soluto to (.8), where B s the correspodg bass atrx, c B s the cost vector correspodg to x. We kow that B z c By so that y, B B a. Also (z - c ) = (c y - c ) = B - B - cbb a, for =,,,,. (c B e - ), for = +, +, +,, + (.9) Sce x b s a optal basc feasble soluto to (.8), we have (z c ) for all. So that B B 4 Page

5 Suppose that Dualty theory for terval lear prograg probles (z c ) ( c B a c ) ad ( c B e ). ( c B a - c ) ad c B ( c B A - c ) ad c B. B B B B B B - - c BB A c ad c BB. y c the y ad B, B c c y c y c y s a feasble soluto to the dual. B A A A B Also by yb c BB b c BB cx, whe ever x s a optal soluto to the pral (.8). Hece cx by. Theore.4. (Copleetary Slackess theore) = (x,x,x,,x ) y y, y, y,, y s a If x s a feasble soluto to the pral (.) ad feasble soluto to the dual (.), the they ust satsfy the so-called copleetary slackess codtos: Proof. If = = = = (). If y, the a x b. (). a x b, the y. (). If x, the a y c. (v). If a y c, the x. x s a feasble soluto to the pral (.) ad the strog dualty theore x s a optal soluto to the pral (.) ad (.) such that fro equato (.7), we have y s a feasble soluto to the dual (.), the fro y s a optal soluto to the dual c x ax y b y.. e. (c ), w(c ) (x ), w(x ) (a ), w(a ) (x ), w(x ) (y ), w(y ) fro equato (.), we have (b ), w(b ) (y ), w(y ). (.). (a ), w(a ) (x ), w(x ) (b ), w(b ) (y ), w(y ). (a ), w(a ) (x ), w(x ) (b ), w(b ) (y ), w(y ). (a ), w(a ) (x ), w(x ) (b ), w(b ) (or) (y ), w(y ). (a ), w(a ) (x ), w(x ) (b ), w(b ) (or) (y ), w(y ). a x b (or) y (a ), w(a ) (x ), w(x ) (y ), w(y ) (b ), w(b ) (y ), w(y ). (a ), w(a ) (x ), w(x ) (y ), w(y ) (b ), w(b ) (y ), w(y ). So, () f y, the a x b ad () f Slarly fro equato (.), we have a x b, the y. a x y c x That s (a ), w(a ) (x ), w(x ) (y ), w(y ) (c ), w(c ) (x ), w(x ) (a ), w(a ) (x ), w(x ) (y ), w(y ) (c ), w(c ) (x ), w(x ) a y c x a y c x * * * * = = a y - c (or) x a y c (or) x So (). If x, the a y c = ad = (v). If a y c, the x. Now 4 Page

6 Dualty theory for terval lear prograg probles IV. Nuercal Exaples Exaple 4.. Cosder the followg terval uber lear prograg proble: (P) Max z = [9, ]x +[, 4]x +[8, ]x subect to costrats 6x (4.) + 5x + x [5, 7] 4x + x + 5x [6,8] ad x We call, x, x. the above proble as the pral proble. The the correspodg dual proble s gve by (D) M w [5, 7]y +[6,8]y subect to costrats 6y + 4y [9, ] 5y + y [, 4] y + 5y [8, ] ad y, y. (). Optal soluto to the pral terval uber lear prograg proble: Let us apply the terval verso of splex algorth ad the ew terval arthetc to solve the pral proble. The stadard for of the gve pral terval uber lear prograg proble based upo both locato dex (d pot) ad fuzzess dex fucto (wdth) as: (4.) Ital terato: Ital basc feasble soluto s gve by s = 6,, s = 7,. c,, 9, c B y B x B x x x s s s 6, , s 7, 4 5 z (z - c ) -, -, -9,.75, Sce (z c ), for soe, the curret basc feasble soluto s ot Optal. Frst terato: Here s leaves the bass ad x eters to the bass c,, 9, c B y B x B x x x s s, s x 5.5,.75, ,.5, z, 5, 7.5,, 7.5, (z - c ), -8, 8.5,, 7.5, Max z, x +, x + 9, x + s + s subect tocostrats 6x + 5x + x + s + s 6, 4x + x + 5x 4 + s + s 7, ad x, x, x,s,s. Sce (z c ), for soe, the curret basc feasble soluto s ot Optal. The proved basc feasble soluto s gve by s = 5.5,, x =.75,. Secod terato: Here x leaves the bass ad x eters to the bass c,, 9, c B y B x B x x x s s, s x 8.5,.5, z 46,, 57,,.5, (z - c ) 6,, 8.5,,.5, 44 Page

7 Dualty theory for terval lear prograg probles Sce (z - c ) for all, the curret basc feasble soluto s optal. The optal soluto ss = 8.5,, x =.5, ad ax z = 8.5,. Hece the optal soluto for the gve pral terval uber lear prograg proble s x = [,], x = [.5,4.5] ad axz = [79.5,8.5]. (). Optal soluto to the dual terval uber lear prograg proble: The stadard for of the gve dual terval uber lear prograg proble based upo both locato dex (d pot) ad fuzzess dex fucto (wdth) as: (D) w 6, y + 7, y + s + s + s + MR + MR + MR subect to 6y + 4y - s + s + s + R + R + R, 5y + y + s - s + s + R + R + R, y + 5y + s + s - s + R + R + R 9, ad y, y,s,s s Ital terato: Ital basc feasble soluto s gve by R =,, R =, ad R = 9,. b 6, 7, M M M C B Y B X B y y s s s R R R M R, 6 4-5, M R, (5) - 4.6, M R 9, 5-9.6, w 4M M -M -M -M M M M (w b ) 4M - 6, M - 7, -M -M -M Sce (w b ), for soe, the curret basc feasble soluto s ot Optal. Frst terato: Here R leaves the bass ad y eters to the bass b 6, 7, M M c B y B x B y y s s s R R M R.4,. -..5, 6, y 4.6,.4.5, M R 5.,.8-4, w 6, 5.4M +.4, -M.8M - 5., -M M M (w b ), , -M.8M - 5., -M Sce (w b ), for soe, the curret basc feasble soluto s ot Optal. The proved basc feasble soluto s gve by R.4,, y 4.6,, R 5.,. Secod terato: Here R leaves the bass ad y eters to the bass b 6, 7, M c B y B x B y y s s s R 7, 6, M y y R.5, 4, 9.5, , 4, w 6, 7,.7M +., -.5M , -M M (w b ),,.7M +., -.5M , - M Sce (w b ), for soe, the curret basc feasble soluto s ot Optal Page

8 The proved basc feasble soluto s gve by y =.5,, y = 4,, R = 9.5,. Dualty theory for terval lear prograg probles Thrd terato: Here R leaves the bass ad s eters to the bass b 6, 7, c B y B x B y y s s s 7, 6,, y y s 4,, 4, , w 6, 7,, -5.9,.88, (w - b ),,, -5.9,.88, Sce (w b ), for soe, the curret basc feasble soluto s ot Optal. The proved basc feasble soluto s gve by y 4,,y,,s 4,. Fourth terato: Here y leaves the bass ad s eters to the bass b 6, 7, c B y B x B y y s s s 7,,, y s s.5, 8.57, 6., w 7.5, 7,, -.5,, (w b ) -8.5,,, -.5,, Sce (w b ), for all, the curret basc feasble soluto s Optal. The optal soluto s y =.5,, s = 8.57,, s = 6., ad w = 8.5,. Hece the optal soluto for the dual terval uber lear prograg proble s y =[,], y =[.5,.5], y =[,] ad w =[79.5, 8.5]. Fro the optal solutos for the pral ad dual terval uber lear prograg proble, we see that Pral : axz = [79.5, 8.5] ad Dual : w = [79.5, 8.5]. Hece Pral : axz = [79.5, 8.5] = Dual: w We see that both pral ad dual probles have optal solutos ad the two optal values are equal. Also both optal solutos obey the strog dualty theore. III. CONCLUSION We troduced the otato of pral ad dual lear prograg probles volvg terval ubers as the way of tradtoal lear prograg probles. We dscuss the soluto cocepts of pral ad dual lear prograg probles volvg terval ubers wthout covertg the to classcal lear prograg probles. Uder ew arthetc operatos betwee terval ubers, we have proved the weak ad strog dualty theores. Copleetary slackess theore s also proved. These results wll be useful for post optalty aalyss. A uercal exaple s provded to show that both pral ad dual probles have optal solutos ad the two optal values are equal. Ackowledgeets The authors are grateful to the aoyous referees ad the edtors for ther costructve coets ad suggestos. REFERENCES []. G. Alefeld ad J. Herzberger, Itroducto to Iterval Coputatos, Acadec Press, New York 98. []. Atau Segupta, Tapa Kuar Pal, Theory ad Methodology: O coparg terval ubers, Europea Joural of Operatoal Research, 7 (), 8-4. []. Atau Segupta, Tapa Kuar Pal ad Deba Chakraborty, Iterpretato of equalty costrats volvg terval coeffcets ad a soluto to terval lear prograg, Fuzzy Sets ad Systes, 9 () 9-8. [4]. Bector, C. R. ad S. Chadra, O Dualty Lear Prograg uder Fuzzy Evroet, 46 Page

9 Dualty theory for terval lear prograg probles Fuzzy Sets ad Systes 5 (), 7 5. [5]. G. R. Btra, Lear ultple obectve probles wth terval coeffcets, Maageet Scece, 6 (98) [6]. S. Chaas ad D. Kuchta, Multobectve prograg optzato of terval obectve fuctos - a geeralzed approach, Europea Joural of Operatoal Research, 94 (996), [7]. Falk, J.E.: Exact solutos of exact lear progras. Oper. Res. 4 (976), [8]. K. Gaesa ad P. Veeraa, O Arthetc Operatos of Iterval Nubers, Iteratoal Joural of Ucertaty, Fuzzess ad Kowledge - Based Systes, (6) (5), [9]. K. Gaesa, O Soe Propertes of Iterval Matrces, Iteratoal Joural of Coputatoal ad Matheatcal Sceces, () (7), []. E. Hase, Global Optzato Usg Iterval Aalyss, New York: Marcel Dekker, 99 []. Hladk M., Optal value rage terval lear prograg, Faculty of Matheatcs ad Physcs, Charles Uversty, Prague, 7. []. Hse-ChugWu, Dualty theory for optzato probles wth terval-valued obectve fuctos. J. Opt.Theory Appl. 44 (), []. Hse-Chug Wu, Dualty Theory Iterval-Valued Lear Prograg Probles, J Opt Theory Appl, 5 (), [4]. Iuguch,M., J. Rak, T. Tao ad M. Vlach, Satsfcg Solutos ad Dualty Iterval ad Fuzzy Lear Prograg, Fuzzy Sets ad Systes 5 (), [5]. E. Kaucher, Iterval aalyss exteded terval space IR, Coput. Suppl. (98), 49. [6]. W. A. Lodwck ad K. D. Jaso, Iterval ethods ad fuzzy optzato, It. J. Uccertaty, Fuzzess Kowledge-Based Systes, 5 (997), [7]. R. E. Moore, Method ad Applcato of Iterval Aalyss, SIAM Phladelpha, 979. [8]. Mg Ma, Meahe Freda, Abraha kadel, A ew fuzzy arthetc, Fuzzy sets ad systes, 8 (999), 8-9 [9]. Mraz F., Calculatg the exact bouds of optal values LP wth terval coeffcets, Aals of Operatos Research, 8 (998), 5-6. []. Y. Nakahara, M. Sasak ad M. Ge, O the lear prograg wth terval coeffcets, Iteratoal Joural of Coputers ad Egeerg, (99) -4. []. T. Nrala, D. Datta, H. S. Kushwaha ad K. Gaesa, Iverse Iterval Matrx: A New Approach, Appled Matheatcal Sceces, 5 () (), []. Olvera D. ad Atues C.H., Multple obectve lear prograg odels wth terval coeffcets - a llustrated overvew, Europea Joural of Operatoal Research, 8(7), []. Poerol, J.C.: Costrat qualfcato for exact lear progras. Oper. Res. 7 (979), [4]. G. Raesh ad K. Gaesa, Iterval Lear Prograg wth geeralzed terval arthetc, teratoal Joural of Scetfc& Egeerg Research, () () ISSN [5]. Roh, J, Dualty terval lear prograg. I: Nckel, K.L.E. (ed.) Proceedgs of a Iteratoal Syposu o Iterval Matheatcs, pp Acadec Press, New York (98). [6]. R.E. Steuer, Algorth for lear prograg probles wth terval obectve fucto coeffcets, Matheatcs of Operatoal Research, 6 (98), [7]. Soyster, A.L.: Covex prograg wth set-clusve costrats ad applcatos to exact lear prograg. Oper. Res. (97), [8]. Soyster, A.L.: A dualty theory for covex prograg wth set-clusve costrats. Oper. Res. (974), Erratu, 79 8 [9]. Soyster, A.L.: Iexact lear prograg wth geeralzed resource sets. Eur. J. Oper. Res. (979), 6. []. Thuete, D.J, Dualty theory for geeralzed lear progras wth coputatoal ethods. Oper. Res. 8 (98), 5 []. Tog Shaocheg, Iterval uber ad fuzzy uber lear prograg, Fuzzy Sets ad Systes, 66 (994) - 6. []. Thuete, D.J.: Dualty theory for geeralzed lear progras wth coputatoal ethods. Oper. Res. 8 (98), 5. []. M. Walk, Theory of Duahty Matheatcal Prograzg, Sprger-Verlag, Wre, (989) Page