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Unformatted text preview: Ma‘l'k 501. UNIVERSITY OF WATERLOO m {(HQYM diamL EXAMINATION
SPRING TERM 2002 Surname: ——————————————‘—‘——_—' First Name: __————————————‘————_— Name: (Signature) Id.#:
. [:1 Section 1 8:30 TR
LeCture seetlom El Section 2 10:30 MWF
Course Number C850 350
Course Title Linear Optimization
Sections 1 and 2
Instructor (1) K.Cheung (2) J .Cheriyan Date of Exam June 18, 2002
Time Period 5:00 pm. — 7:00 p.m.
Number of Exam Pages 8 pages (including this cover sheet)
Exam Type Closed Book
INSTRUCTIONS: I. Write your name and Student Id.# in the blanks above. Put a check mark in the box
next to your assigned lecture time. 2. Answer each of the questions in the space provided; use the back of the previous page
for additional space. 3. You may NOT use a calculator. 1. [16 marks]
Solve the following linear programming problem (P) by using the simplex method. [15]
' Use the smallest subscript rule for choosing entering variables, and show all of your
work. _ .
max1m1ze ~x1 + :52
(P) subject to
2'1 + :122 — 2233 S 5
—2x1+x2+ x3S2
x1 + 5173 S 3
1‘1 7 $2 , $3 2 0 2. [17 marks] (a) Deﬁne what is meant by (i) a convex set; (ii) an extreme point of a convex set. (b) Let F be the set of m satisfying — 2:81 + 2x2 + 12345671173 — 2.234 = 2
$1 + x2 — 9876543$3 + .734 = 3
1:1 7 1'2 7 $3 7 $4 2 0 (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. [7] 3. [9 marks]
Consider the LP problem
(P) max 2 : 8x1 + 9232 + 10:33
subject to — 3271 + x2 + 5.23 = ——10
2x1 — 3:63 2 —20
— 71131 + 5313 S —30
$1 , 172 , $3 2 0 (a) Write down an auxiliary problem (Phase 1 problem) for Clearly indicate
the artiﬁcial variables and the slack variables. How many artiﬁcial variables are
there? (Do NOT solve the auxiliary problem.) (b) Write down an initial feasible basis B for the auxiliary problem, and write down
an initial basic feasible solution :c*. 4. [8 marks]
[8] Let A be a 3 X 4 matrix, and let 6 2 [10, 20, 30, OF. Consider the LP problem max 2 = cTr
subject to ACE = b
a: Z 0. Suppose that the auxiliary problem (Phase 1 problem) has two artiﬁcial variables, $5
and $6. The ﬁnal tableau for Phase 1, with basis {1, 2, 5}, is given below. Write down an initial tableau for Phase 2. (Do NOT solve the Phase 2 problem.) w + 334 + 25% = 0
$1 + 41173 + 5134 = 1 .232 i  21’3 — .734 + 35% = 1 — $3 + 3554 + $5 + %$5 = 0 5. [20 marks]
(a) Deﬁne what is meant by (i) a basis of an m X n matrix A, where A has rank m; (ii) a basic solution of a system A3: = b (where A is as above); (P) max 2 2 0T1:
(iii) atableau for the linear programming problem subject to At = b
a: 2 0. (b) Consider the following tableau (T) with basis B = {2,3,6}, where [3 is a (real—
valued) parameter. Suppose that the initial basis consists of the slack variables $4a$53$6
Z + {131 ‘— $4 —' {1:5 3 0
371 + $2 + 2134 + $5 = 0
— 3:1 + $3 + 2334 — 411:5 = 0
$1 +ﬂm4—m5+xe=0 (i) Suppose that B =1. List all the pairs (away) such that 33;, could be the
entering variable and 33,. could be the leaving variable on a simplex iteration
beginning from (T). (ii) Suppose that ﬂ = l, and that .134 is the entering variable. Find the leaving
variable using the lexicographic rule. Justify your answer. (iii) Suppose that 274 is the entering variable. Find all values of 3 such that the
lexicographic rule chooses $6 as the leaving variable. Justify your answer. 6. [20 marks]
For each of the following statements, answer whether it is true or false. If true, give a
complete explanation, and if false, give a counter—example. (a) If the feasible region of a linear programming problem is unbounded, then the
linear programming problem is also unbounded. (b) Every subset of a convex set is convex. (c) The set of optimal solutions of a linear programming problem is a convex set. ((1) If we have a tableau (not necessarily feasible) such that E, S 0 for all j E N, then every'feasible solution (of the LP problem) has objective value 3 17 (recall that 1—)
denotes the number on the right hand side of the z—row). (e) In a degenerate iteration of the simplex method, both the old tableau and the
new tableau are degenerate. 7. [10 marks] (a) State the Fundamental Theorem of Linear Programming for LP problems in
standard equality form. (b) Prove that if the set F = {:L' : A1: = b,
at least one extreme point.
(Note that A may not have full row rank.)
(Hint: Try using part (21).) a: _>_ 0} is non—empty, then it has ...
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