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Applications of Linear Programming

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APM 236H1S Course outline, Spring 2003

** Professor: ** Mary Pugh,
mpugh@math.utoronto.ca, (416) 978-5233,
room 3141,
Earth Sciences Centre,
22 Russell Street.

Office hours: Mondays, 2:30-3:30, Tuesdays, 2:30-3:30 (until Tues April 8)

** TA: ** Joon Song,
song@math.utoronto.ca, (416) 978-3201, room 209, 1 Spadina Crescent.

Office hours are by appointment: please call or email.

**
Prerequisites: ** MAT 223H or MAT 240H

**
Text: ** Bernard Kolman and Robert E. Beck, * Elementary Linear
Programming with Applications, 2nd edition * .

** Additional Materials: ** nine problem sets are given below.
Solutions for the first eight problem sets are available for
short-term loan in the Gerstein Science Information Centre, as are
copies of some term tests (including solutions). I
will post solutions for the other problem sets after we cover that
material. The Association of Part-time Undergraduate Students in SS
1089 has old final exams for most courses currently offered at the
St. George campus.

**
Marking Scheme: ** Two term tests, each worth 20% of the course
mark, and a two-hour final exam worth 60% of the course mark. Calculators
are not allowed during the term tests and the final exam. The times and
locations of the term tests will be announced later.

**Course Summary (subject to change): ** review of linear algebra
(Chapter 0, you read it on your own), introduction to linear
programming (Chapter 1), the simplex method (Chapter 2), further
topics in linear programming (Sections 3.1-3.4), matrix games
(supplementary class notes), problems from integer programming
(Chapter 5).

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**
Recommended problems:
Make sure you can do the following problems. Note:
when the time comes, I will add problems concerning matrix games.
**

** problem set 1: ** p 21, #6b. (Additional instructions: (i).
Solve for x, y, and z in terms of w. (ii). Solve for x, z, and w
in terms of y.) p 21, #9a, p 28, #6c, #8b, p 42, #5d, #6b

** problem set 2: ** p 57, #2, #4. p. 28, #6, #8, p 59, #10. Put
these problems in canonical and in standard form.

** problem set 3: ** p 82, #14. p 83, #16. In the preceding
questions, replace instructions (b) and (c) with "draw the line
z = c^T x = k, where k is the optimal value of z." p 91, #4, #8, #12.
p 100, #6, #8.

Supplementary problems:

1. Prove that the set of optimal solutions of any linear programming
problem is convex.

2. Prove that the set of objective values which a linear programming
problem attains over its feasible region is convex.

3. Give an example of a convex set in R^2 which
is not a line segment, and hwihc has (1,0) and (0,2) as its only
extreme points.

4. Find all extreme points (in R^3) of the set

{(x1,x2,x3) such that x1 - x2 + x3 = -1, 3 x1 - 2 x2 + 4 x3 = 2, x1 + x2 + 3 x3 = 9, x1 >=0, x2 >= 0, x3 >= 0}.

** problem set 4: **
p 120, #6 (solve the problem which this tableau represents), #8 (insert
row labels and solve the problem). p.212, #14, #16, #19. p. 122, #22, #23.
p 131, #6.

** problem set 5: **
p. 150 #2, p. 152, #10, #20.

Supplementary problems:

1. Maximize z = -x1 + 5 x2 + 3 x3 subject to the constraints

2 x1 + x2 + x3 = 5, 3 x1 + 2 x2 >= 6, 4 x1 + 3 x2 - x3 <= 7, x1 >= 0,
x2 >= 0, x3 >= 0.

2. Maximize z = -9 x1 - 4 x3 subject to the constraints

9 x1 + x2 + x3 >= 27, 3 x1 + x2 + 2 x3 = 9, x1 >= 0, x2 >= 0, x3 >= 0.

** problem set 6: ** p 165, #2, #4. p. 166 #6, p. 183, #9. Modify
this problem by replacing "x1, x2, x3 unrestricted" with
"x1 >= 0, x2 >= 0, x3 >= 0". p 184, #11.

Supplementary problems pertaining to section 3.3:

Consider the standard problem: Maximize z = -76 x1 + 13 x2 - 11 x3 - 27 x4 + 2 x5, subject to constraints

11 x1 + x2 - x3 - 7 x4 + x5 <= 3,
-15 x1 - x2 + 2 x3 + 23 x4 - 3 x5 <= 2,
-7 x1 + 2 x2 - 2 x3 - 9 x4 + x5 <= 4,
x1, x2, x3, x4, x5 >= 0.

a) insert slack variables x6, x7, x8 for the first, second, and third
constraints respectively, and write the initial simplex tableau
for the problem.

b) The final simplex tableau for the problem has basic variables
x5, x3, x2 (in that order). Without using the simplex method to find it,
write the final simplex tableau for the problem.

c) Let w_i (i=1,2,3) be the dual variable associated with the ith primal
constraint. What is the optimal solution of the dual problem?

** problem set 7: ** p 325, #8, #10, #11, #12.
Here are the
solutions
to problem set 7.

** problem set 8: ** p 338, #1-7.
Here are the
solutions
to problem set 8.