Optimization Methods: Linear Programming- Revised Simplex Method. Module 3 Lecture Notes 5. Revised Simplex Method, Duality and Sensitivity analysis

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1 Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod Module Lectue Notes Revsed Smple Meod, Dualty and Senstvty analyss Intoducton In e pevous class, e smple meod was dscussed whee e smple tableau at each teaton needs to be computed entely. Howeve, evsed smple meod s an mpovement ove smple meod. Revsed smple meod s computatonally moe effcent and accuate. Dualty of LP poblem s a useful popety at makes e poblem ease n some cases and leads to dual smple meod. Ths s also helpful n senstvty o post optmalty analyss of decson vaables. In s lectue, evsed smple meod, dualty of LP, dual smple meod and senstvty o post optmalty analyss wll be dscussed. Revsed Smple meod Beneft of evsed smple meod s clealy compehended n case of lage LP poblems. In smple meod e ente smple tableau s updated whle a small pat of t s used. The evsed smple meod uses eactly e same steps as ose n smple meod. The only dffeence occus n e detals of computng e enteng vaables and depatng vaable as eplaned below. Let us consde e followng LP poblem, w geneal notatons, afte tansfomng t to ts standad fom and ncopoatng all equed slack, suplus and atfcal vaables. ( ) ( ) Z c + c + c + LLL + c + Z = 0 ( ) ( ) c + c + c + LLL + c = b n n c + c + c + LLL + c = b n n M M M M M M c + c + c + LLL+ c = b l m m m mn n m As e evsed smple meod s mostly benefcal fo lage LP poblems, t wll be dscussed n e contet of mat notaton. Mat notaton of above LP poblem can be epessed as follows: n n ML

2 Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod T Mnmze z = C X subect to : AX = B w : X 0 b c 0 whee = X, c b C =, B =, 0 0 =, Μ Μ M Μ n c n b 0 m c c A = Μ cm c c c Μ m Λ Λ Ο Λ c n c n Μ cmn It can be noted fo subsequent dscusson at column vecto coespondng to a decson vaable k s Let ck ck. Μ cmk X S s e column vecto of basc vaables. Also let C S s e ow vecto of costs coeffcents coespondng to X and S s e bass mat coespondng to X. S S. Selecton of enteng vaable Fo each of e nonbasc vaables, calculate e coeffcent ( WP c), whee, P s e coespondng column vecto assocated w e nonbasc vaable at hand, c s e cost coeffcent assocated w at nonbasc vaable and W = CS S. Fo mamzaton (mnmzaton) poblem, nonbasc vaable, havng e lowest negatve (hghest postve) coeffcent, as calculated above, s e enteng vaable.. Selecton of depatng vaable a. A new column vecto U s calculated as U = S B. b. Coespondng to e enteng vaable, anoe vecto V s calculated as V = S P, whee P s e column vecto coespondng to enteng vaable. c. It may be noted at leng of bo U and V s same ( = m ). Fo =, Λ, m, e atos, U () V(), ae calculated povded () > 0 V. =, fo whch e ato s least, s noted. The basc vaable of e cuent bass s e depatng vaable. If t s found at () 0 V fo all, en fue calculaton s stopped concludng at bounded soluton does not est fo e LP poblem at hand. ML

3 Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod. Update to new bass Old bass S, s updated to new bass S, as = [ ES ] new S new whee E 0 L η L L η L 0 0 M M O M L M M = M M L η L M M M M L M O M M 0 0 L ηm L L ηm L 0 () () V V and η = V () fo fo = column S s eplaced by S new and steps ough ae epeated. If all e coeffcents calculated n step,.e., ( WP c) s postve (negatve) n case of mamzaton (mnmzaton) poblem, en optmum soluton s eached and e optmal soluton s, X S = S B and z = CXS Dualty of LP poblems Each LP poblem (called as Pmal n s contet) s assocated w ts countepat known as Dual LP poblem. Instead of pmal, solvng e dual LP poblem s sometmes ease when a) e dual has fewe constants an pmal (tme equed fo solvng LP poblems s dectly affected by e numbe of constants,.e., numbe of teatons necessay to convege to an optmum soluton whch n Smple meod usually anges fom. to tmes e numbe of stuctual constants n e poblem) and b) e dual nvolves mamzaton of an obectve functon (t may be possble to avod atfcal vaables at oewse would be used n a pmal mnmzaton poblem). The dual LP poblem can be constucted by defnng a new decson vaable fo each constant n e pmal poblem and a new constant fo each vaable n e pmal. The coeffcents of e vaable n e dual s obectve functon s e component of e pmal s equements vecto (ght hand sde values of e constants n e Pmal). The dual s equements vecto conssts of coeffcents of decson vaables n e pmal obectve functon. Coeffcents of each constant n e dual (.e., ow vectos) ae e ML

4 Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod column vectos assocated w each decson vaable n e coeffcents mat of e pmal poblem. In oe wods, e coeffcents mat of e dual s e tanspose of e pmal s coeffcent mat. Fnally, mamzng e pmal poblem s equvalent to mnmzng e dual and e espectve values wll be eactly equal. When a pmal constant s less an equal to n equalty, e coespondng vaable n e dual s non-negatve. And equalty constant n e pmal poblem means at e coespondng dual vaable s unestcted n sgn. Obvously, dual s dual s pmal. In summay e followng elatonshps ests between pmal and dual. Pmal Mamzaton Mnmzaton vaable constant 0 vaable unestcted constant w = sgn RHS of constant Cost coeffcent assocated w vaable n e obectve functon Dual Mnmzaton Mamzaton constant vaable Inequalty sgn of Constant: f dual s mamzaton f dual s mnmzaton constant w = sgn vaable unestcted Cost coeffcent assocated w vaable n e obectve functon RHS of constant See e pctoal epesentaton n e net page fo bette undestandng and quck efeence: ML

5 Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod Mak e coespondng decson vaables n e dual Opposte fo e Dual,.e., Mnmze Cost coeffcents fo e Obectve Functon Mamze Z = c + c + LLL + c n n Subect to Coeffcents of e st constant c + c + LLL + c nn = b y c + c + LLL + cnn b y M M c + c + LLL+ c b y m m mn n m m 0, unestcted, L, 0 Coespondng sgn of e st constant s n Rght hand sde of e st constant Thus e Obectve Functon, Mnmze by + by + L + bmym Thus, e st constant, c y + c y + L + c y c m m Coeffcents of e nd constant Coespondng sgn of e nd constant s = Rght hand sde of e nd constant Thus, e nd constant, c y + c y + L + c y = c m m M M M M Detemne e sgn of y Detemne e sgn of y LL Detemne e sgn of y m Dual Poblem Mnmze Z = b y + b y + LLL + b y Subect to c y + c y + LLL + c y c m m c y + c y + LLL + c y = c m m M M c y + c y + LLL+ c y c n n mn m n y unestcted, y 0, L, y 0 m m m ML

6 Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod It may be noted at, befoe fndng ts dual, all e constants should be tansfomed to lessan-equal-to o equal-to type fo mamzaton poblem and to geate-an-equal-to o equal-to type fo mnmzaton poblem. It can be done by multplyng w bo sdes of e constants, so at nequalty sgn gets evesed. An eample of fndng dual poblem s llustated w e followng eample. Pmal Dual Mamze Z = + Mnmze Z = 000y 000y + 000y Subect to unestcted 0 Subect to y y + y = y + y y 0 y 0 y 0 It may be noted at second constant n e pmal s tansfomed to befoe constuctng e dual. Pmal-Dual elatonshps Followng ponts ae mpotant to be noted egadng pmal-dual elatonshp:. If one poblem (ee pmal o dual) has an optmal feasble soluton, oe poblem also has an optmal feasble soluton. The optmal obectve functon value s same fo bo pmal and dual.. If one poblem has no soluton (nfeasble), e oe poblem s ee nfeasble o unbounded.. If one poblem s unbounded e oe poblem s nfeasble. ML

7 Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod 7 Dual Smple Meod Computatonally, dual smple meod s same as smple meod. Howeve, e appoaches ae dffeent fom each oe. Smple meod stats w a nonoptmal but feasble soluton whee as dual smple meod stats w an optmal but nfeasble soluton. Smple meod mantans e feasblty dung successve teatons whee as dual smple meod mantans e optmalty. Steps nvolved n e dual smple meod ae:. All e constants (ecept ose w equalty (=) sgn) ae modfed to less-anequal-to ( ) sgn. Constants w geate-an-equal-to ( ) sgn ae multpled by ough out so at nequalty sgn gets evesed. Fnally, all ese constants ae tansfomed to equalty (=) sgn by ntoducng equed slack vaables.. Modfed poblem, as n step one, s epessed n e fom of a smple tableau. If all e cost coeffcents ae postve (.e., optmalty condton s satsfed) and one o moe basc vaables have negatve values (.e., non-feasble soluton), en dual smple meod s applcable.. Selecton of etng vaable: The basc vaable w e hghest negatve value s e etng vaable. If ee ae two canddates fo etng vaable, any one s selected. The ow of e selected etng vaable s maked as pvotal ow.. Selecton of enteng vaable: Cost coeffcents, coespondng to all e negatve elements of e pvotal ow, ae dentfed. The atos ae calculated afte changng Cost Coeffcents e sgn of e elements of pvotal ow,.e., ato =. Elements of pvotal ow The column coespondng to mnmum ato s dentfed as e pvotal column and assocated decson vaable s e enteng vaable.. Pvotal opeaton: Pvotal opeaton s eactly same as n e case of smple meod, consdeng e pvotal element as e element at e ntesecton of pvotal ow and pvotal column.. Check fo optmalty: If all e basc vaables have nonnegatve values en e optmum soluton s eached. Oewse, Steps to ae epeated untl e optmum s eached. ML

8 Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod 8 Consde e followng poblem: Mnmze subect to Z = By ntoducng e suplus vaables, e poblem s efomulated w equalty constants as follows: Mnmze subect to Z = = = = = Epessng e poblem n e tableau fom: Iteaton Bass Z Vaables Z Ratos 0. / b Pvotal Row Pvotal Column Pvotal Element ML

9 Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod 9 Tableaus fo successve teatons ae shown below. Pvotal Row, Pvotal Column and Pvotal Element fo each tableau ae maked as usual. Iteaton Bass Z Vaables Z -/ / 0 b / 0 0 / / 0 0 -/ 0 0 / / 7 Ratos / Iteaton Bass Z Vaables Z 0 0 -/ 0 -/ 0 / b / / 0 8/ 0 0 / 0 -/ 0 / / 0 -/ -/ Ratos Iteaton Bass Z Vaables Z b Ratos ML

10 Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod 0 As all e b ae postve, optmum soluton s eached. Thus, e optmal soluton s Z =. w = and =.. Soluton of Dual fom Fnal Smple Tableau of Pmal Pmal Mamze Z = subect to,, Fnal smple tableau of pmal: + Dual Mnmze subect to Z' = y y y, y + y y + y y y, y + 0y + y y y 0 + y y y Z y As llustated above soluton fo e dual can be obtaned coespondng to e coeffcents of slack vaables of espectve constants n e pmal, n e Z ow as, y =, y = and Z =Z=/. y = and ML

11 Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod Senstvty o post optmalty analyss A dual vaable, assocated w a constant, ndcates a change n Z value (optmum) fo a small change n RHS of at constant. Thus, Z = y b whee y s e dual vaable assocated w e constant, b s e small change n e RHS of constant, and Z s e change n obectve functon owng to b. Let, fo a LP poblem, constant be + 0 and e optmum value of e obectve functon be 0. What f e RHS of e constant changes to,.e., constant changes to +? To answe s queston, let, dual vaable assocated w e constant s y, optmum value of whch s. (say). Thus, = 0 = and y =.. b So, Z = y b =. =. and evsed optmum value of e obectve functon s ( 0 +.) =.. It may be noted at bass. b should be so chosen at t wll not cause a change n e optmal ML

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