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CO-350-1051-Midterm_exam

CO-350-1051-Midterm_exam - Surname(Print R.— First Name H...

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Unformatted text preview: Surname:(Print) R.— First Name: H Signature'L _ _ Id.#: a“— University of Waterloo C0350 -— Linear Optimization Midterm Test February 16, 2005 7:00 — 9:00 p.m. C] J.Cheriyan LEC 001 MWF 2:30—3:20 931 X.Zhou LEC 002 MWF10:30—11:20 INSTRUCTIONS: 1. Write your name, Id.#, and signature in the blanks above. Put a check mark in the box next to your instructor’s name and lecture time. 2. There are 9 pages in this exam including the cover page. Make sure that you have all the pages. 3. Answer each of the questions in the space provided; use the back of the previous. page for additional space. 4. You may NOT use calculators. #1. [18 marks] [14] (a) Solve the following linear programming problem (P) by using the simplex method. 3 Stop after 3 iterations, even if your solution is incomplete. Show all of your \ work. maximize z 2 53:1 + 6$2 —- 4:03 (P) subject to $1 + 3$2 - 133 S 2 .731 S 3 . 21131 + 4272 S 6 $1 7 $2 , $3 .>_ 0 (b) If (P) is unbounded, then find a feasible solution with objective value 2 100. [4] If (P) has an optimal solution, then Kb (i) write down a basic optimal solution, and (ii) explain whether or not (P) has a unique optimal solution. / '3 LJ ‘ ,w‘ a) T0) 2 - Sx‘ M4,,“ _ - X\ «his, +=m3fl (37):") %) Manse Tl) X. a 3x1 — x345“ /,="i z *qx? 4x: +5,“ 1 to X\ [/‘XP/ = X\ *35‘2 "Xi *‘xt : *1 \ “i" ix”. 4x». 4X; =6 ‘SXz .. x; —7‘4 4‘": ._ \, —l ‘ x.,..,,x¢ Bo —J1x1'+1,3_u Aged: -2 344,5,“ B=i\.s.al ' /l \1 A 19")” ( x3 w‘l'cts'l=Mih(_)T)%) Ami“ Ms: lexi'cosrafln'g rulg, way (_ , ammo) , «2,-2.0.0) | '2'. = min U L G, (\,—l,1,o).(\,-\,o,\)). x‘ was T2) 2 +8X1 +4X4 Jix ’ z 6 "l xl*2Y1 J -x 3 =3 1 4' X3 \ X+ ‘ 43X; =l '2’“ ”(S -31 X5 =0 )m LAW ‘. ‘3 D PB={"3'5} l b) opix'mal mluc,V=H ‘ basic oy‘limcl go‘ujmh I (x: ( 7) 08/. )T N( ca ‘ H O o o g“ H: 'Hunl: “mi 411i: L? Muffler: "How 9h,¢._,.,.,9.!li‘l“e OP' "“5“ka Par: £2: 6’03); one L; I {g B “ml is 0 (£5 0) 7L1:d i on. D r .r i ‘ l 1', ( bash The“ ‘lliéfe Can log move ill” ong Wfi‘malm / ’2, finale 4M any X4. an” kt ¢M$§clieel in Le #2. [13 marks] Let F be the set of a: E R4 satisfying ‘— .231 + 1'2 — 2333 + $4 = -2 2x1 + $2 - 31173 + 114 = 6 $1 + $2 + $3 + $4 = 7 $1 7 $2 9 $133 7 $4 2 0 7 (i) Find two extreme points of F. Justify your anSWer. (ii) Find a point of F that is not an extreme point. Justify your answer. 1) One they,” meni'iuheoi {in He (purge moi—vs Siaic 'ii‘e‘i: 5“} ‘3 a 9"“ xv \s 01 ENG fem“; mini-ion «F. i\ 2 \1 To “A 'H“ “'iK-W fonts, we look gr BPS 5,41: 1 2 \51 V ”I k -‘2 \ —2 A: 1 \ a =[61 \ K '4 3&8,‘i\.; :(‘(\‘%3\O)T Xff3‘50\3v ‘p /\ , /\ ii) \vi a new; is no‘ an 6&wa fem“, “Mn “(are Exis‘Vs X\, We “:2 W'A XGR, 014m such 4M X ’ M‘ 4 040x" 6‘: QuivA in F (tam; DAY {C #3. [16 marks] (a) Consider the LP problem (P) max 2 = x1 + 1:2 subject to —— 2x1 —— .132 S —4 $1 — at; S ~—l — 2531 + 5132 S 2 $1 a $2 2 0 [7] (i) Write down an auxiliary problem (Phase 1 problem) for (P) Clearly in- dicate the artificial variables and the slack variables. How many artificial variables are there? . (Do NOT solve the auxiliary problem.) [3] (ii) Write down an initial feasible basis B for the auxiliary problem, and write down an initial basic feasible solution :1:*. see next page for part (b) Ci ) max 2. - X] + X: = 0 $9193ch in -x* w Y1. +Y3 : 4 a X\ "‘ ‘Xz +xf __\ ‘ZX‘ ~4- Xz +X, =2 W: "‘(e 'Y-i x‘ + \(1 ~X3 ‘ Y in): 'Xl a y - I” :4 X\ ‘- XQ «ex? ___ A -ZX. -~xI +5“ ~54 vyz :2 x" “"1~¥‘ #2344 2’“ Y a ' Y“ *1 He +\ "(11, *1;an 4f The“ ave ivuo mii‘b‘cia] Vaviablas is begin, N'riL {WCQ Xs ‘5 5“ ll” L05“: Xe does mi 9’“; We comic] yhcwevw CoMSicié’r “WEN in be one m'bb‘qal WP] 54‘ ilk-e Lam's, 5669/ {is 5| , lbw] ’Hmi WY bf cmgidqgéi a W New ivu‘lial.x¥=(o' O O 0 Z web]: ivC we easily X] 4-, La 0, we coulzi do an «ii a FWD] is brn'rg X] M anal Xe ovi'iX—a Mi €0]\Jiu5 ii, So I] i3 l 05 i5_ (#3, continued) (b) (NOTE: The LP problem in this part is different from the LP problem in part (a .) Consider an LP problem in standard equality form with four variables x1 , $2, x3, x4 and objective function maximize :cl — $2 + 3x3. Suppose that the auxiliary problem (Phase 1 problem) has two artificial variables, $5 and 2:6. The final tableau for Phase 1, with basis {1,2}, is given below. Write down an initial tableau for Phase 2. (Do NOT solve the Phase 2 problem.) w + 23:4 + x6 = 0 $1 + 2133 + 1'4 + 2-735 = 1 122 + 1133 + 25114 — $5 + 3226 = 2 \’__,__,________._..J a (A hi3" ‘ \ ables Ye?\flc€ LA-YDN Nil/k 2 .YDN‘“ Véyo$éfdvfgfllfic \b&‘:¢\ Z — Y\ 4’5"). " 3X3 : O y‘ 4 7Y3 4’ X4 9' \ Y1 "Y3 *2». =2 eh‘min/A’lfi \oasfi VQ‘MU“ in)” Z JON gage}. H) / K2. it) w t) i 5 [16 marks] _ -. Ar}- "0 , (l I Consider the following tableau: K“ ix J t Z " (8 — fl)1'2 — (fl — 3).?)3 3 10/8 + 30 in) $1 + _ $2 ’ (5—5)$3 2 [3+3 3'5 b + (2mm — 10x3 + x4 = —fl+6 i! \i (a) Find the range of values of B E R over which the basis B = { 1, 4} is (i) feasible, ’ XV: (P3 0 o' 3pc) 1322. 4%-?) sin.) (ii)optimal. * X ‘($*3 0 0 -FN) t“ (b) Find the range of values of B 6 1R over which there exists a pivot from B = {1, 4} with x2 as the entering variable (the tableau need not be feasible). in»? (c) Suppose x3 is the choice of entering variable. Find the range of values of ,3 E R over which this tableau (without further pivots) shows that the LP is unbounded (the tableau need not be feasible). we ,3 (d) Suppose fl = 6. List all the pairs (x,,‘:ck) such that as, could be the leaving variable and :ck could be the entering variable on a simplex iteration beginning from this tableau. (xv , x“) XL enters , «i=mm(% ’f) )(3 en-las, Luau/‘0? ) 7% sely o—fl fllY‘El (X4’x2), “(4.3%) / (e) Suppose ,3 = 0. Assume that the initial basis was B = {3,4} with A3 = I. Identify the entering variable and use the lexicographical simplex method to identify the leaving mriable. Explain your reasoning in brief. X1. would mlvr 51V“? 2450‘ . 5 . . +=mm (:71) 3) ‘— ((19% '3 0‘ it? ”'5 iCXI‘coe’Ya,{)iflllc Ulla; 7 #5. [22 marks, each part has 4—6 marks] \ 2/1 For each of the following statements, answer whether it is 31119 false If tr , give a complete explanation, and if false, give a counter—example or a compl e explanation. o / (b) An LP problem may have exactly two optimal solutions. Fab: (c) An LP problem in standard inequality form T max 2 = c a: subject to Am 3 b a: 2 0. has a feasible solution if and only if b 2 0. -ge for parts (d,e) (#5, continued) (d) Consider an LP problem such that every tableau has at most one basic variable of value zero (that is, for every basis B at most one i E B has I),- = 0). Cycling does not occur while solving this problem with the simplex method. WM: : As [0% as we WOYL W‘ L‘ ”W ' ’(V‘Dfimpbk. ”l! l“ '3!le fi‘mifilflk V’s-Wilma (Yfls‘wo Jars 1401 (2ch 3 wt” /‘ A (e) If the simplex method executes a non—degenerate iteration starting from a tableau (T), then (T) does not occur again in the execution. Tvuei Fww (T) in (1"), W6 lwc reduced (0?; vaolouég‘kh . Vast» ~olcl ~W€Ld "‘ V CM never l“ #6. [15 marks] (a) Define what is meant by [2] (i) a convex set; F]; a amyex Se} 4’ “we Exirl X‘1l‘2 6“" ml AG'R ,05As‘ when: X: XX‘ ‘04)? GF. [3] (ii) an extreme point of a convex set. 4’ 1 x 14+ “1625; Ms 5H3 «W 319:1“ $51 and Milk. 0<>~<l ulnar: Xi KX' +04“ xl e F. [4] (b) State the Fundamental Theorem of Linear Programming for problems in standard inequality form. Guam m, L? in slander} (wiequmuiéwm wli‘b «1 rnsiminl ma'lnx ”e L“ YML‘ HAM flc lime (g a loan: Soliibdh,ll1enllemx,'gls a Lag-C (“gut sch—how 2/” a ‘ ~9i¥ 15M“; 6 a» 113‘“er Soluhw, limp “104 (Kids a bag; bd—[ma‘ ga‘u,hmq.w - ( LP is eilber {Miamible'mbuunbA hr 11;: 5.3,. Whit/J ‘ialmi’i’flr‘. Z/- [6] (c) Let A be an m X 71 matrix. let 1) E R’" be a nonnegative vector, and let F be the set of x E lR" satisfying Ax S b and x Z 0. Prove that F has at least one extreme point. You may use results from the course (other than this one) Without proof, pro— vided that you quote them in full. (NI/l 7 ) (Hint: Try using part (b). ) O Qwehil’bai ‘o \s HOMWC‘sQAVW; W m“ CMJWA “V‘ UP 5011ch [Ms 0‘ Lenglak’ Sowbm, W M 1f 41 FTLP 4130, SM Q2 0 111 6:15 Ly Mam? T111 FTLO’ It will lime we OW'Ml éolchlion. > a o ...
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