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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 , $3 .>_ 0 (b) If (P) is unbounded, then ﬁnd a feasible solution with objective value 2 100.
[4] If (P) has an optimal solution, then
Kb (i) write down a basic optimal solution, and
(ii) explain whether or not (P) has a unique optimal solution. / '3
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/ ’2, ﬁnale 4M any X4. an” kt ¢M$§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“; miniion «F. i\ 2 \1 To “A 'H“ “'iKW fonts, we look gr BPS 5,41:
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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
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€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 + 2735 = 1 122 + 1133 + 25114 — $5 + 3226 = 2 \’__,__,________._..J
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[16 marks] _ . Ar} "0 , (l I
Consider the following tableau: K“ ix J
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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 enlas, 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’sWilma
(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 LagC (“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). )
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