Duality in linear programming

Transcription

1 MPRA Munch Personal RePEc Archve Dualty n lnear programmng Mhaela Albc and Dela Teselos and Raluca Prundeanu and Ionela Popa Unversty Constantn Brancoveanu Ramncu Valcea 7 January 00 Onlne at MPRA Paper No 986, posted 9 January 00 :58 UTC

2 DUALITY IN LINEAR PROGRAMMING Albc Mhaela, Unversty Constantn Brancoveanu, Rm Valcea Teselos Dela, Unversty Constantn Brancoveanu, Ptest Prundeanu Raluca, Unversty Constantn Brancoveanu, Rm Valcea Popa Ionela, Unversty Constantn Brancoveanu, Rm Valcea Abstract Any lnear programmng problem marked as P and called prmal can be seen n connecton th another lnear programmng problem marked as D and called dual The economc nterpretaton of the dual model brngs about ne nformaton hen analyzng such phenomena and hen substantatng decson makng JEL classfcaton: C60, C6 Keyords: Lnear programmng problem, dualty The dea of a lnear programme s dualty and the theory of lnear programmng along th the dualty markng manner have played a specal role n economc analyses by the ay n hch they have emphaszed the nature of prces Ever snce margnal analyss onards, no other dea has proven to be that mportant to the fundamental theory of prces Dual Problem Let the eample of lnear programmng n ts general form be : mn n = n = n = n ( ma ) f = a a a b b = b 0, = n = c, = k, = k p, = p m We shall call ths problem «prmal» and mark t as P The prmal problem can be assocated th another lnear programmng problem marked as D and called «dual» The transton from the prmal problem to the dual one s done accordng to the follong rules : before the transton takes place, the prmal problem must be turned nto ts canoncal form ; f the prmal lnear programmng problem s mamum, then the dual lnear programmng one s mnmum and the other ay round ; Lancaster K (97) Mathematcal Economc Analyss, Scentfc Publshng House, Bucharest

3 the number of restrctons n the prmal lnear programmng problem equals the number of varables n the dual lnear programmng problem ; the number of varables n the prmal lnear programmng problem equals the number of restrctons n the dual lnear programmng problem ; 5 vector c n the prmal lnear programmng problem s vector "b n the dual lnear programmng problem ; 6 vector b n the prmal lnear programmng problem s vector «c» n the dual lnear programmng problem ; 7 the factors matr n the dual lnear programmng problem s the transposed matr of the prmal lnear programmng problem Observaton The dualty relaton s symmetrc: the dualty s dualty s the prmal problem Correspondence rules beteen the prmal lnear programmng problem and the dual lnear programmng one []: LPP P mnmum mamum number of varables number of restrctons free terms of restrctons factors of obectve functon columns of restrctons matr varable 0 varable 0 varable R harmonous restrcton non-harmonous restrcton equalty restrcton LPP D mamum mnmum number of restrctons number of varables factors of obectve functon free terms of restrctons ros of restrctons matr harmonous restrcton non-harmonous restrcton equalty restrcton varable 0 varable 0 varable R Let the lnear programmng problem be: mn f ( ) = c A b (P) 0 Its dual nature s to be the lnear programmng problem: ma g( ) = b A c (D) 0 Analogous as to the lnear programmng problem n a mamum canoncal form: ma f ( ) = c A b (P) 0 Its dual nature s to be: mn g( ) = b A c (D) 0 Eample Let us rte the dual nature of the lnear programmng problem: ma f = ( )

4 ,, 0 = The problem s canoncal form s: ( ) ma f =,, 0 = The problem s dual nature s to be: ( ) mn g = 0, 0 Eample Let us rte the dual nature of the lnear programmng problem: ( ) mn f = 0 0 The problem s turned back nto ts canoncal form: ( ) mn f = 0 0 The problem s dual nature s: ( ) ma g =,, 0 = It s mportant to understand that dualty s frst and foremost a formal mathematcal relaton Once a problem has been suggested, one makes up ts double nature accordng to the rules above If the prmal problem s consstent, one naturally epects ts dualty to prove nterestng One s epectaton can be grounded or not, but dualty s presence as a formal feature of the prmal problem s not affected []

5 Dualty Theorems Heren are the dualty theorems that sho the connecton beteen the prmal problem and the dual problem n a canoncal form Let the couple of problems be: mn f ( ) = c ma g( ) = b (P) A b (D) A c 0 0 A M m, n, M n,, M, m Let P = { 0, A b} and P = { 0, A c} be the set of admssble solutons of prmal lnear programmng problem (P), and of dual lnear programmng problem (D), respectvely Proposton Irrespectve of hat P and solutons of the to problems, there s the follong nequalty: g( ) f ( ) dec b c Demonstraton: P A b P are a ( ), couple of The relaton s multpled by on the left and the result s A b P hence A c The relaton s multpled by on the rght and the result s A c Therefore, b A c Proposton If soluton couple ( ~, ~ ) of the to problems has the feature that g ( ~ ) = f ( ~ ), then ~ s the optmal soluton of (P) prmal lnear programmng problem and ~ s the best soluton of (D) dual lnear programmng one Demonstraton: We suppose by reducto ad absurdum that ~ s not the optmal soluton of (P) prmal lnear programmng problem; then, there s soluton * P so that ( * f ) < f ( ~ ) ( (P) prmal lnear programmng problem beng mnmum) But f ( ~ ) = g ( ~ ) ( ) ( ) ( ) * f < f ~ = g ~ *, so there s couple (, ~ ) of admssble solutons that contradcts Proposton Corollary ) If (P) prmal lnear programmng problem does not have fnte optmzaton, then (D) dual lnear programmng problem does not have any admssble solutons (namely P = Φ); ) If (D) dual lnear programmng problem does not have fnte optmzaton, then (P) prmal lnear programmng problem does not have any admssble solutons (namely P = Φ) Theorem 5 If the soluton of the (P) prmal lnear programmng problem s ( ~ P ) and fnte, then the best soluton of the (D) dual lnear programmng problem s stll ( ~ P ) and fnte, and the optmal values of the obectve functons concde: f ( ~ ) = g ( ~ ) It s ntutvely deduced from Propostons, and Corollary by negaton Addtonally, f ~ s the basc optmal soluton of the (P) prmal lnear programmng problem for base B ~ made up th m ndependent lnear column vectors n A = ( a, a,, a,, am ), then ~ ~ B ~ ~ ~ ~ = = B b ; = c ~ B B here c ~ B are the m costs correspondng to the vectors n base B ~ The values of the obectve functons are: f ( ~ ~ ) = c ~ B b B ş g( ~ ~ ) = c ~ B b B hence f ( ~ ) = g ( ~ )

6 Ths theorem leads to the concluson that the fnal smple table correspondng to the prmal problem ncludes the optmal solutons of both problems (prmal and dual) The soluton T B = cb B of the dual problem s obtaned on ro z at the cross th the vectors columns that have formed the orgnal base Analogously, f the dual problem s solved, the result s that the soluton of the (P) prmal lnear programmng problem s to be found n the last smple table of the (D) dual B lnear programmng problem, on ro z c, ust belo the columns that have orgnally formed the base Ths consequence gves the possblty to solve a (P) prmal lnear programmng problem by ts dual one f the latter s easer to solve, and the solutons of the prmal one are read accordng to the above Theorem 6 (The Fundamental Theorem of Dualty) For any couple of dual problems, one and only one of the follong stuatons s possble: ) Both problems have solutons: therefore, they have optmal solutons and the optmal values of the obectve functons concde; ) One of the problems has a soluton, the other does not: therefore, the former problem has fnte optmzaton; ) Nether of the to problems has a soluton Theorem 7 (The Theorem of Complementary Spacng) Takng account of the couple of lnear programmng problems (P), (D) stated above, the maor and suffcent condton for ~ solutons P and P ~ to be optmal s: ~ ( A ~ b) = 0 ( c A ~ ) ~ = 0 Demonstraton: In order to demonstrate emergency, let ~ and ~ be the optmal solutons of the dual A ~ b A ~ c problems, namely ~ and 0 ~ 0 Then ~ ( A ~ b) 0 and ( c A ~ ) ~ 0 But c ~ = ~ b, c ~ A ~ ~ = b ~ A ~ ~, ( c A ~ ) ~ ~ ( A ~ b) = 0 Snce the to addton terms n the left member of the obtaned nequalty are nonnegatve, the result s that ether s nule and therefore pursues the desred condtons For suffcency, f these relatons are added: ~ ( A ~ b) = 0 ( c A ~ ) ~, = 0 the result s: ~ A ~ b ~ c ~ A ~ ~ = 0 c ~ = ~ b and so ~ s the optmal soluton of the prmal problem, and ~ s the best soluton of the dual problem, accordng to Proposton Lemma 8 (The Fundamental Lemma) If and are possble vectors of the prmal, respectvely the dual problem, the follong relatons are true: c A b Demonstraton: It s notced that A b 0 s obtaned from the prmal problem s restrctons Snce 0 f s possble, ( A b) 0 Hence, A b Usng the dual problem s restrctons A c 0 and the non-negatvty restrctons upon, there s, f and are possble: c A 5

7 Theorem 9 (The Equlbrum Theorem of Lnear Programmng) a) If, are possble ponts for the prmal and dual problem, they are optmal f and only f: () = 0 henever a < b ; () = 0 henever a > c, that s the k-th varable of a problem s nule hen the k-th restrcton of the other one s not effectve b) The optmal pont (or optmal ponts f t s about beng non-strctly optmal) shall be so that the number of non-nule varables of ether problem shall not eceed the number of restrctons n that problem The equlbrum theorem s mportant for to reasons Frstly, t allos one to verfy a prmal soluton s ablty to be optmal even f one does not have the optmal dual soluton Then, even more remarkably, t leads one to a number of nterpretatons of economc models condtons to be optmal, models havng the eact form requred by lnear programmng Economc Interpretaton Let us consder a lnear manufacturng model th n outputs and m b nputs beteen hch there s a relaton defned by a constant manufacturng factors The factors sho hat nput amount s necessary to manufacture a output unt In ths case, a s the total nput amount necessary for the manufacturng of compound output A means vector b of the nputs necessary to manufacture ths compound output Vector p of products prces and vector b of all avalable resources are stated The optmal manufacturng s defned as beng ma p and ts features are analyzed Therefore, there s a lnear programmng problem ma p Its dual problem s then A b 0 mn b A p 0 It s knon from the dualty theorem that p = b, p s a value epresson (prces multpled by amounts); therefore, t s epected that b have the same meanng Snce b s the amount vector, s a random vector of prces hch n ths case are nputs prces Due to the ne dmenson that dual varables get by ther connecton th prces n economc matters, dual varables shall be often knon as shado prces Let us no consder the dual problem s restrctons, each havng the follong form: Snce a p a s nput amount necessary to manufacture a output amount, a a s the represents the value of nput necessary to manufacture a output unt, and Lancaster K (97) Mathematcal Economc Analyss, Scentfc Publshng House, Bucharest 6

8 total value of nputs necessary for the manufacturng of a output unt, all nputs beng assessed n shado prces The anser to ths queston s pursued: hch s the loest value that s to be attached to the vector of b avalable resources knong that there s a possblty to turn resources nto products and then to sell them? The restrctons of the dual problem epress the fact that f the value of the nputs ncorporated n a product s loer than the product s prce, t s more advantageous to sell the products nstead of the resources Once the, optmal ponts have been reached, the economy (or the enterprse) does no longer care f t sells the product obtanng p, or f t sells the resources at prces, because the total cashng s the same: p = b Thus, t can be stated that: any resource that cannot be entrely used for the manufacturng of an optmal compound output shall be gven a shado prce equallng zero or t shall be consdered that ts optmal value s nule; once an optmal state has been acheved, no product shall be seen as such f ts unt cost eceeds ts prce (the nputs beng assessed by optmal shado prces) In other ords, the resources that make up the ecess supply are free goods and the manufacturng generatng losses shall be left out n case the shado prces are real ones These relatons correspond to the equlbrum of a compettve economy If only the -th restrcton vares, t s deduced that the -th dual varable (n the optmal pont) can be consdered the margnal value of the problem modfyng the -th restrcton In typcal economc contets, there s gong to be the margnal socal value (or margnal revenue) of the ncrease n a proper resource amount Thus, one can ustfy the common nterpretaton of dual varables as shado prces Eample Let the lnear programmng problem be: ( ) = 5 ma f 0, =,, ) State the optmal soluton by usng prmal smple algorthm; ) Wrte the dual problem assocated th the one above and then rte ts optmal soluton; ) Interpret the dual problem s solutons from the economc pont of ve ) The lnear programmng problem s brought back to ts standard form: ma f ( ) = 5 0( y y ) y = y = 0, =,, y, y 0 0 If the optmal state s not unque, the last statement s vald for at least one optmal pont 7

9 0 B C B B θmn y y y 0-0 y 0 0 / y 5/ / 0 -/ -/ 5/ 5 / / / 0 / 5/ -/ 0 9/ 0 5/ y Snce the problem s mamum, all the problem s optmal soluton s: =, = 0, = 0, y =, y = 0 Products P and P are not manufactured - they are not effcent The mamum proft s b) The dual problem of the orgnal one s: m n g( ) = 5, 0 The dual problem s solutons are: =, =, y = 9, y =, y 0 = c) mn g( ) = Product P s manufactured n the amount and resources y ş y reman unbought References Bădn V, Frcă O, (995) Mathematcs Course for Economsts, The Romanan- Amercan Unversty, Bucharest Dumtru M, Manole S (coord) (006) Economc Mathematcs, Economc Independence Publshng House, Pteşt Lancaster K (97) Mathematcal Economc Analyss, Scentfc Publshng House, Bucharest 0 8

10 Oprescu Gh (coord) (999) Economcs Appled Mathematcs, Independence Publshng House, Brăla Economc 9