prelim2sol - ENGRI 1101 Engineering Applications of QR Fall...

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Unformatted text preview: ENGRI 1101 Engineering Applications of QR Fall ’08 Prelim 2 SOLUTIONS Name: 1 Points Possible Ii You have 1 hour and 30 minutes to answer all the questions in the exam. The total number of points is 100. For questions with multiple parts, the point value of each part is given at the start of that part. Be sure to give full explanations for all of your answers. In most cases, the correct reasoning alone is worth more credit than the correct answer by itself. Your explanation need not be wordy, but it should be sufficient to justify your answer. You receive much more credit for an incorrect answer if you recognize that something that you have computed is not correct than if you merely pretend that it is correct. Not all the parts are in the same level of difficulty! Please make sure that you do not spend too much time on one part. Good Luck! l l 1_ (30) Consider the following linear programming problem: . O t 0 max 2: 22:1—x2+x3 s.t. . 3x1+x2+x3$60 1121—9324—2273310 $1+$2'III3S2O $17$27$320 (a) (5) Put the problem into dictionary form suitable for application of the simplex ‘ algorithm. ' - . . \ I . X in’br‘oolocufl shack chflals\€5 XL“ X5, (MA 6 J wa jut one “twig ' clx‘c’o‘onood‘. X1+ :: X5 2 \0 ~— m + X1‘ 2X3 Xe = .10 ~ ><\ - M. + x3 ’2 3 1X3 v- X3+ X3 . k. (b) (5) What is the solution corresponding'to the starting dictionary in (a)? Is it feasible? What is its objective function value? ‘ O Elbe Solu'bbn corresoonobhj *0 this olicbmotj ‘3 2t _ )(b‘: M: O, X3: U, $3-0, H I a . , n \e ‘ ‘ ' \\ (table T la “£233 0‘3 \l)\€. \Oc‘a gflfiw , .__x_, ml _ , if): V i! K. I ‘ML‘§1’J>\J(./lqlfa~ i, g ’l 1.“: w LWMWWWWWMM. ' ‘ r i _ i ;. ~i‘w\‘vf :xidjgxxt’fl w” ‘ xxx, u: awe- \ ~ \- llmg dew; l é \ s ‘2; t: C - i (d) (5) What is the solution conresponding to themew dictionary you found in (C)? What is its objective function value? Is it‘an inlprovement over (b)? i : H“ ‘ X, 3;)(y: X1: X520} .nggl q l l infi " I cubed ‘ve qfimc’ii on value \5 ‘ i : i ‘ t \\ \ SQ m \6 on {mpmuweni o , _ I as exoe (Lieci‘ (e) (5) Can you terminate the algorithm, or should more steps be carried out? Explain . the reasoning behind your conclusion. ‘ * l . r ’ . . . > \ \ glow, ‘ ilxenzi is a variable, w (W 305 ‘ i V ’ ' olo ELK ’ “ I 006 ‘m file? you» comes *0 J ‘ WW (AA WOKACL “Wm I ’Rwiipo» v VOW; ingreoswfl fine/iii)“ biker, more Ave ‘ lobaecliue are I fleeciecl \‘Q X3) '6 ‘cheoseA /. JAM; \Oww \MVOSQA on "\\¢\€ om0004 0%- “We. inc r ease, over VXHZOS: X3 $ 60 1H M \nxesx hood I >/ w (\e XE) \oou c} L $mc€ CCDGQQ\UQJ\+OQ *3 ‘5 <3 XE >/ O .5) {m A XV» w\ Re \ace secmé‘ (00‘) _ ” ll 2 , S X\ I 2 + X311 " Xb ‘ ' X : "" _ x H 3 ‘20 X ‘. a CA\\ Q'\\’\€i’ (0W5 ‘ f $ ‘ Oxna ‘ .‘ \la. ’45 H, 511 X\ ” 3/1 7(9— ’V x j ,. \ X5 H (‘3 x]; N ’V \ll x1 I; XE : \‘ , .. X5 X6 M .— 3/2 PH "3’2, 1* Wm ............................................ ,_ i ‘ A “2 x2 ’ ‘ X5 5 «x- 3/1 7“ Z :2 (c) (10) Carry out one iteration of the simplex algorithm (choose :53 variable!). Show your work! 6 . as the ent‘ermg l I KDerg‘ma 1‘ Cl 0 2. (25) During the next 2 months General Cars must meet (on time) the following demands for trucks and cars: month 1, 400. trucks and 800 cars; month 2, 300 trucks and 300 cars. During each month a total of at most 1000 vehicles can be produced. At the beginning of month 1, 100 trucks and 200 cars are in inventory, available for immediate delivery. ‘ Each truck uses 2 tons of steel, and each car uses 1 ton of steel. During month 1, steel costs $400 per ton; during month 2, steel costs $600 per ton. At most 2500 tons of steel can be purchased each month.‘ (Steel can be used only during the month in which it is purchased.) ‘ There is an option of holding vehicles that remain on hand after satisfying the demand in month 1 in inventory, and using them to satisfy the demand in month 2. A holding cost of $150 per each vehicle held in inventoryis assessed. Each car gets 20 mpg (miles per gallon), and each truck «gets 10 mpg. During each month, the vehicles produced by the company must average at least 16 mpg. Formulate a linear programming model to determine how to meet the demand and mileage requirements at minimum total cost (do not worry about integrality). Begin by clearly defining all the decision variables, and express the objective function and constraints in terms of these variables. (/ (I // If 2 a g )6 V A \G (“Cl ll" ‘ 0’? 5‘ €\ (in purchase : Omgun ’ ‘ $ \ // /; .— 1 r // g m /‘ /i ( \J \V“ 3 N ,,I, ‘ ‘ » “A \QKEQQ J 3 1., {l “ \‘wflc, H V \. Z ’ it 4* C " L153 w (1 k (am-xx, l l 3 R v “A xJLM \\/~\\: 4-,", \X a H T C // ” “€15 I I, , l 0 Additional space fbr problem 2 min 400 3‘ + £0 Ck C; + 1’“ 31 T1 + H L I .y {A T \ é K, C +\@ 1,‘ \EKLx-r 1) k P gwfik \ 39’ M £0 T >/ \e (€139) & Ikrgod“ \Q 3. 6* QOcl f > ’096 T S 8; HQ HT 7/0 Sac“ Cx, CQIT‘) 62’ H ) ~ _\’ . 3‘ l l l 3. (30) Consider the following partial branch-and-bound tree for an input to the integer- constrained diet problem (which is, as you should recall, a minimization problem). You may assume that each objective function coefficient is integer. At each node, you are given the optimal value of the current version’s.LP relaxation, and the lowest indexed variable (i.e. the variable with the smallest subscript) that has a fractional value in \ the LP optimal solution. (If no such variable exists, then it is indicated that this LP optimum is integer.) LP infeasible (a) (8) Suppose that you are given only this information: what can you conclude about the optimal value for this integer program? That is, give the best upper and lower . bounds on the optimal value. ' ’ .13 //\ N in. u p O \lJ \j Q\\\,M<L (b) (6) Suppose that you next decide to branch at the, node labeled (*). What additional constraints should be imposed for each of these two branches. l (c) (8) Suppose that the LP values for these two wrenches are equal to 19.1 and ‘20.1 and the corresponding solutions are not integer. What can you conclude about the optimal value of the integer program now? Justify your answer. Q flkfl m\ \IOIKVLf. :34) “Wt LP ‘ ‘ ' *lrx‘ . \n gm . \ l :1 0 Q3? aeosé Aka boom \ 3 . \gfl / OyJPQ’lWUQ ’{JVOCSBOfl/fi) Uglue» one \ql‘psgf' T _. 3)., Owe . ne‘m LO€ goA w, ‘03“ +~\’:fi\\::)\€ ~\\/\e \omcl/l' wlu" 3‘? UGA‘Ue OM; more” glad. @ gmsége, solulx‘m) NM“ vome 3" (d) (8) Suppose that the coefficients of the objective function were not integer. What would your answer to part (c) be now? Justify your answer. 5 of {QAQROf‘ ~ was b w AW: Lek/6" how we “ ‘ ‘ l 006 (\O'l mleber ) owe we lA 0A3 \‘quej booocl ( Obfiuweol \H r ' g _ , .,, $803"! 5/ C bill“; I ©#A\\v\;\ '\ CK“: D (x x 4‘4” ‘ V <\ \ch fgguz C; l? 1 N46 W0+ eke boomdA eule 3.? V0 \ bound v’ . , ’> ‘ O Wows be com», 6 W00 S)“ be O JA/be ‘ ‘\‘\I\O/\ m W13 brewa ( L10 l a \ 3 ‘ “paler bound on \Acmcl (10> ' 4. (15 points) Derive the dual linear program to the following linear program. (That is, show how to find it, rather than just writing it down.) maximize 43:1 + 8372 .. 6x3 _ 83:4 subjectto $1+4§E2-—x4 S 4 "xl“$2+$3 2 -8 xi 2 0: 2=1,. .,4. A ‘ £an ’1 I x) \ r $er wngn\co\ hair firng rm . \{\’\‘O m ésx ~ 3X4 A, )(\ + 8 >41 3 ~ max ’ xu S H - 4 X9. x + . +X\ 4” X1 I “ Xv. '. X” X2, X3, \ / O >/ . - Q 09$ 31 v \ fl 1 uor‘taHeS 3x 2 n ‘ Adéxw Anomaler duo » 3‘ WW (jdoA Va ‘ ‘XCOHQ 3 , \3o \ MM me \Me om NO ’935 m9 OWN.“ . _ L S A l o 7‘: + H X2 ya 95 \ // A 7,1, ‘31 v; >\.__ ‘*» ~ ‘ W44” M, M ,. - , ,. w. AM wage 4 143+ g3), L 74K ’V X2 ' ¥>\ V v "(,b<\+q"l’x”\ ‘L 33’ 8 K“ - 4.} yr W\A\\C\/i \5 9 x ’¥ X\ ( a :j\ *3). bomox m KW?” Q?) RWY/Rm \ \J V N o $ :(‘QU\C)\€ 0“ m0 A ‘x C * wawm a 2 \mm o s, . ‘ ~ . 5&6: a $8 $ * 3‘ 9mm M deg xiv, k) w °\\r\a 88x (Woke/9% «WU WNW / we 08%;“ ~\\r\~e Qo\\ow;® Au \ gawk/w mm A 3‘ + 3 5‘2, 33‘ xx " '+ :52 5 A W ‘ flaky" :59, ->/' 8 . . f 3?, 2/6 2»% ...
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prelim2sol - ENGRI 1101 Engineering Applications of QR Fall...

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