lecture07

# lecture07 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 1 SensitivityandComplexityIssue YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 2 LPexampleagain maximize x 1 +2 x 2 subjectto x 1 ≤ 1 x 2 ≤ 1 x 1 + x 2 ≤ 1 . 5 x 1 , x 2 ≥ . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 3 a1 a2 a3 a2 a3 a4 a4 a5 a norm direction cone contained by the norm LP Geometry depicted in two variable space If the direction of c is Objective contour Each corner point has the point is optimal. cone of a corner point, then c Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 4 SolutionData BeginningTableau: B- 1- 2000 3 1 100 1 4 010 1 5 1 001 1 1 2 FinalTableau: B 000 1 1 2 1 2 3 001 1- 1 1 2 2 010 1 1 1 100- 1 1 1 2 Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 5 WhenaBSFisoptimal? GivenaBFSintheLPstandardform A B x B = b , x B ≥ , x N = , anditscompanionshadowpriceandreducedcostvectors: A T B y = c B , ( and r = c- A T y ) . Ifthereducedcostvector r ≥ ),thentheBFSwithbasicvariableset B ,then theBFS optimal .(–Whataboutthe maximization ?) Atoptimalitywealwayshave c T x = b T y . Whatdotheshadowpricesandreducedcostsmean? Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 6 Whatdotheshadowprices y mean? • All inactive constrainthave zeroshadowprice . • Ingeneral,theshadowpriceonagivenactiveconstraintisthe rateofchange intheOVasthe RHS oftheconstraint increases ,ceterisparibus. • IftheOSis degenerate ,theshadowpricemaybevalidfor one-sided changes intheRHS. • TheconstraintRHS ranges givetherangesoftheconstraintRHSoverwhich nochangeinthe optimalbasis willoccur. • Oneofthe allowableincreaseanddecrease foran inactive constraintis infiniteandtheotherequalstotheslackorsurplus. • Ingeneral,whentheRHSofan active constraintchanges,boththeOVand OSwillchange. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 7 Whatdothereducedcosts r mean? • All basic variableshave zeroreducedcost . • Ingeneral,thereducedcostcoefficientofany non-basic variableisthe amounttheobjectivecoefficientofthatvariablewouldhavetochange,withall otherdataheldfixed,inorderforittobecomea basic variableatoptimality. • IftheOSis degenerate ,theobjectivecoefficientofa non-basic variablewould havetochangebyatleast,andpossiblymorethan,thereducedcostinorder tobecomea basic variableintheOS. • Theobjectivecoefficient ranges givetherangesoftheobjectivefunctionover whichnochangeinthe optimalbasis willoccur. • Oneofthe allowableincreaseanddecrease fora non-basic variableisinfinite andtheotheristhereducedcost. • Ifa non-basic variablehas zero reducedcost,thentheremayexistan alternative optimalsolution....
View Full Document

## This note was uploaded on 06/16/2010 for the course MS&E 211 taught by Professor Yinyuye during the Fall '07 term at Stanford.

### Page1 / 17

lecture07 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online