ESE 403 Operations Research Fall Examination 1

Transcription

1 Name: Slutin ESE 403 Operatins Research Fall 2010 Examinatin 1 Clsed bk/ntes/hmewrk/cellphne examinatin. Yu may use a calculatr. Please write n ne side f the paper nly. Extra pages will be supplied upn request. Yu will nly receive full credit if yu shw all yur wrk. Questin Pint value Yur Scre Ttal 100 1

2 - -.:.: lwing is nt a necessary assumptin befre we can apply the Simplex 0 assumptin."'"?::;,xlj"ti nality assumptin. V".e 0 licativity assumptin. ibility assumptin. V- f the abve are necessary. ints] Fr maximizatin prblems, if the bjective functin evaluated at a Basic Feasible (BF~ slutin is n smaller than its value at every adjacent BF slutin, then Qthe slutin is ptimal. b. the slutin is unbunded. c. the prblem is infeasible. d. the prblem has multiple ptimal sl ti e. Nne f the abve is true. 3. (3 pints) f a prblem has tw ptimal Basic Feasible (BF) Slutins, then a. it has infinitely many ptimal BF slutins.. b. it has infinitely many ptimal epe slutins. Qit has infinitely many ptimal slutins. d. All f the abve are true. e. Nne f the abve is true. 4. (3 pints) The Simplex methd a~s chse the entering basic variable that leads t a. the best adjacent Basic Feasible (BF) slutin. -&.- the best adjacent bjective functin (largest the directin f the maximum imprvement._ d. All f the abve are true. e. Nne f the abve is true. 5. (3 pints) Simplex methd's minimum rati rule fr chsing the leaving basic variable a. always chse t remain in the feasible regin as the entering variable increases. b. always chses t stp at the first cnstraint intersectin as the entering variable increases. c. ~a~chses the basic variable that will g t zer fir as entering yariable All f the abve are true. e. Nne f the abve is true. 2

3 6. (3 pints) n a particular iteratin f the Simplex methd, if there is a tie fr which variable shuld be the leaving basic variable, then the next BF slutin a. must nt have any basic variable equal t zer. b. must have at mst tw basic variables equal t zer. c. must have exactly ne basic variable equal t must have at least ne basic variable equal t zer. /./ e. must have infinitely many basic variables equal t zer. ~ 7. (3 pints) n the Simplex methd, if there is n leaving basic variable at sme iteratin, a. then the prblem has n feasible slutins. b. then the prblem has multiple ptimal slutins. c. then the prblem has exactly ne ptimal slutin. J / d. then the prblem has multiple unbunded ptimal slutb/" G) then the prblem has unbunded ptimal slutin. 8. (3 pints) When an artificial prblem is created using the Big M methd, if the basic slutin f any iteratin cntains an artificial then the crrespnding crner pint slutin is nt feasible fr th lglnal prblem. 'b, then the crrespnding crner pint slutin is ptimal fr e riginal prblem. ~ then the crrespnding crner pint slutin is unb ed fr the riginal prblem. X then the crrespnding crner pint slutin is the single ptimal slutin. "e... Nne f the abve is true. 9. (3 pints) n Simplex methd, adjacent crner pint slutin f a prblem with n decisin variable shares a. n cnstraints. n-l cnstraints. c. n+l cnstraints. d. n+2 cnstraints. e. Nne f the abve is true. 10. (3 pints) n Simplex methd, if we have n decisi variable and m cnstraints, a. there will be n nnbasic variable in e basic feasible slutin. b. there will be m nnba ic varia in the basic feasible slutin. c. there will be n+m nn ariable in the basic feasible slutin. d. there will be m*n nnbasic variable in the basic feasible slutin. GJ Nne f the abve is true. W\ \aq~k V6..t-\.b<.QS V - VV' V\ On b ((,S i"c, \fa,f"\-a\::f(!> 3

4 11. (15 pints) The prfessr in charge f ESE230 needs t schedule the staffing f the helpdesk. The helpdesk pens frm 8AM until midnight. Frm histrical data f the demand, the fllwing minimum number ftutrs are required t be n duty: 8AM-nn: 4 Nn-4PM: 8 4PM-8PM: 10 8PM-midnight: 6 Tw types f tutrs can be hired: graduate students and undergraduate students. Graduate tutrs wrk fr 8 cnsecutive hurs in any f the fllwing shifts: mrning (8AM-4PM), ~ afternn (nn-8pm), and evening (4PM-midnight). Graduate tutrs are paid $20 an hur Undergraduate tutrs can be hired t wrk any f the fur shifts listed abve. Undergraduate tutrs are paid $10 an hur. T make sure there is adequate supervisin fr every time perid, there must be at least 2 graduate tutrs n duty fr every undergraduate tutr n duty. The prfessr in charge wuld like t determine hw many graduate and hw many undergraduate wrkers shuld wrk each shift t meet the requirement at the minimum pssible cst. Frmulate an NTEGER linear prgramming mdel fr this prblem. b = ji. { ~('"t\a"+c..c-tv.a en..\..l \ f\ 3 AA-li p~ s ~;x.f \2. PM - 6pf\.. \.{'L:. ;\>\ ~A.M.- 12-~M dtit+ l2.-?m - 4 PM ' " 4

5 12. (15 pints) Cnsider the fllwing prblem. Maximize Subject t Xl +2X2+ X3s 10 xl~2x2 +2X3 ~20 and :6 -\ "7 -L.- - \ D D 0? """"'-"";;: C- ')(4- '(L; LO ", - \ 0 t "'l. ~q '",.'.': Z -"'2. ~ 0 z. -"2. \ - D ~V ~t 't. t( y Y-Z f'\~t -:..~ ~ -:k-4 ~'7 \.4,v.~ '1..&> 0 '2.0 -~ v... -~ -r 't- D 7 -'f't - ~... ~~...v~r C.OA.c.-\-~,~a ->;> u~'o~~,~ ~ 1e7' CD 0 7 -~ » J.ce \. ~f' 4 "''e,d OK. -- -"2, ~~..k,.. trm ~(\~ v.r\e.~ (~G..r~+",,\.~ ') t -1c 1 1c~ '1c+- 1c:~ "K4- -:k-., ~\-\- -"2- t:- 5

6 13. (15 pints) Cnsider the fllwing prblem. Minimize Z=Xl-2x2 Subject t Xl::; 4 X2::;3 Xl+2x2 ~11 and Slve using Simplex in tabular frm. Shw all steps. kytr hlwj ~e,.z.= &; -)) -ix~-;l.) \ )(1-2.,x., ~ Lf»:i-).+)(y<:~ ~ &1'-')) + j L '/;;.' -~) t --. x> \ -J} ') - 2 -z... 7\\ ~'J..1-,.. 1 'X'\ 1- X) =- (, )(~ ~)(0j ~5 - )(,\ t-),(; - X'J t Y6 ::. Ṯ ')(' (, CO., x:s ~X~ )(;1 X~ X'11 Xs Yt -). 0 0 M 0.\ (j) V 0 0 ) () 0 ) 0 {) '-1 6

7 . 'X l fz X, - \ XL! ~- )\& ~~ b >(d enws - -M 0 0 ;v\ 0 ~-lra (J b \ 0 \ () X4lw-vc-5 ~." -2- ~Z- 0 ( ~ \ \ t - >{J' eh4v-5 0" -;f+ja M,~O \;1-72;11\ 0 \.,0 0 Q' c -/ ")(3 le--a\'-'0..-. tl ~ 0 B -;A -\ ~\- -:r:. ~_e... ""--.. O--~'~-9~iM-0 '6 -JV\ Jw-~~J~ () 0 0 c,/t JOt-S ,.. Q \ 0 0 '5.-1 -?. -\ --_. ~\._ f~~"'1Jz:.e..

8 14. (25 pints) Cnsider the fllwing prblem. Minimize subject t 2Xl + X2 ~ 10-3Xl + 2X2 ~ 6 Xl + X2 ~ 6 and a. Slve this prblem graphically. b. Using the Big M methd, cnstruct the cmplete first simplex tableau fr the simplex methd and identify the crrespnding initial (artificial)' BF slutin. Als identify the initial entering basic variable and the leaving basic variable. c. Wrk thrugh the Simplex methd step by step t slve the prblem. Make sure yu identify the entering basic variable, leaving basic variable, and the crrespnding BF slutin. d. What is the ptimal slutin value? What are the crrespnding decisin variable values? Q. X2. fi ~ ~ <.. 5 '\ J ) :2x 1- Y ; t '1~lO-2J{ 1.) -3x+2.'1 : ~ L "?'hn..~ a+- po")'r.~ A, S ~,~ 10-2.)(.(g-X \ \ l./ ~ X 'X.t.= 2 2~ 3~, -\-~.lcz =-.3(f ') +- 2{Z') l2;~~ 7

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