21-257 Models and Methods for Optimization, Fall 2013
Solutions to Homework 2
Section 1.4:
14. Show that the following linear program does not have a unique solution. Describe the
set of optimal solutions.
Minimize : 4y1 + 2y2
Subject to : y1 + 3y2
y1
2y1
21-257 Lectures
1
The Two-Phase Method
Variables to be inserted into a linear program:
A nonnegative slack variable is inserted with a +1 coecient into each () resource
constraint. Recall that a slack variable:
represents unused resources and has the sa
21-257 Models and Methods for Optimization, Fall 2013
Solutions to Homework 11
1. We begin as usual by converting the cost matrix into an opportunity matrix and calculating regrets:
1
2
3
4
1
M
25
33
30
2
25
M
20
15
3
20
23
M
25
4
40
45
30
M
1
2
3
4
1234
21-257 Models and Methods for Optimization, Fall 2013
Solutions to Homework 7
Section 5.3:
2. We begin by constructing a CPM network: notice that I simplied construction by
creating a new initial node for each activity with multiple predecessors.
We next
21-257 Models and Methods for Optimization, Fall 2013
Solutions to Homework 8
1. a. The minimum completion time for this project is 27 days, and the critical path is
(0, 2) , (2, 3) , (3, 5) , (5, 6).
b. Let t01 , t02 , t13 , t23 and t35 represent the dur
21-257 Models and Methods for Optimization, Fall 2013
Solutions to Homework 9
1. This is a knapsack problem in which the cost of each item serves as its weight, and the
purchased items are the knapsack, where its maximum weight is simply the budget (so
W
21-257 Models and Methods for Optimization, Fall 2013
Homework 10
Solution to #4 Updated on 11/19: Comments regarding this given in blue text.
1. Let A, B, C, D, E, F be cfw_0, 1 variables representing whether we build a police station in
each community:
21-257 Models and Methods for Optimization, Fall 2013
Solutions to Homework 6
Section 4.3:
4. a. If we change the RHS of the second constraint by adding a change b2 , then b2 and
s2 have the same coecients, and so the new objective function value would be
31> FMTm/mmqm'FZ/e 0m OLJDXQC/i/Q/ A
WW ox Cammvn WM,
mm 0 jedpe amdpn ' f
0\ CORSWk-f wfoh MLbe Y3 @emwol £7
WA éN m egémkfyg/g/céwiilv Cowgmfwfh 95) MMMZQA SQH 1;, V
A f-
EUEJ ea.+ ~§3 r" ._
r "f~-.m_uti_n_._x«~ 21 25 7 Lectures 2
A scheduling examp
21257 Lectures 4
A project management example:
Example 1.3.12. Consider the following tasks involved in building a frame houSe. Given
the relationships among the tasks and the estimated time in days required for each, determinemm L
the shortest time requi
Test 1 will be Monday, February 8
Each chapter in the text concludes with a list of objectives. Most objectives include
an example that illustrates the objective and suggests one or more exercises that test the
objective.
Here are the objectives that desc
Department of Mathematical Sciences
CARNEGIE MELLON UNIVERSITY
21-257 Models and Methods for Optimization, 9 units, Spring, 2016
Instructor: Dr. Russ Walker, WEH 6219, [email protected], 412-268-9657
Text: Walker, Introduction to Mathematical Programmin
MM c1>c¢+cp>gtw 6mm v\ vmak
gwbjecfé CHINAE OLD/XL?" mm X S bl
(KaliF F ' F g. J9; m Comshrukf_ Q) MAX a921, BOX; WW
[Xxx X; cm: s[ .
5X! FHlL/{z 5:200 37
DC; >(; >/ O.
"2.0 \ i 2 f 00 lo 0
Observe; X: 20 i5 {7k {gal O O [Era
O 0 29¢
go 34.00!
J»
A; Di E
21-257 Models and Methods for Optimization, Fall 2013
Solutions to Homework 5
Section 4.2:
2. Applying the Duality Theorem to solve a minimization LP consists of three steps: rst,
nd the dual maximization LP, then solve it using the simplex algorithm, and
21-257 Models and Methods for Optimization, Fall 2013
Solutions to Homework 1
Section 2.6:
1(a). This question asks a solution to the following system of equations:
2x1 x2 + 2x3 = 2
4x1 x2 + 3x3 = 5 .
x1 + 2 x2 + x3 = 4
We solve this by applying Gauss-Jor
21-257 Models and Methods for Optimization, Fall 2013
Solutions to Homework 3
Section 3.3:
2. The rst step is to insert slack variables - which Ill denote by s1 , s2 , s3 and form the
initial simplex tableau. Notice that Im leaving out the z column entire
M241: Matrices and Linear Transformations
Homework Assignment 6
Answers
Graded Problems: Section 3.1 Problems 2, 12, 22; Section 3.2 Problems 14, 22; Problem 3.
Section 3.1
Problem 2. (1, 1) and (1, 2) are not orthogonal; (1, 1) and (0, 0) are not indepen
M241: Matrices and Linear Transformations
Homework Assignment 5
Answers
Graded Problems: Section 2.6 Problems 8, 14; Problems A (iii), B, C(i), E.
Section 2.6
Problem 8. Let T (p(t) = (2 + 3t)p(t). Apply T to the elements of B1 = cfw_1, t, t2 , t3 , then
M241: Matrices and Linear Transformations
Homework Assignment 4
Answers
Graded Problems: Section 2.3 Problems 14, 22; Problems A (a), A (b), B.
Section 2.2
10|
b1
.
b2
Problem 6. The echelon form of (A|) is 0 1 |
b
0 0 | b3 2b1 3b2
b1
= b2 is in the colu
M241: Matrix Algebra
Homework Assignment 1
Answers
Graded Problems: Section 1.3 Problems 4, 8, 10; Section 1.4 Problems 4, 10, 38.
Section 1.2.
Problem 4. No solution; zero right sides produce 3 lines through origin; right sides 1, 1, 0 give
x = 1 and y =
Matrix Algebra
Homework Assignment 10
Due on Wednesday, December 1, at the start of class.
Your homework should have a cover sheet with the following information: course title,
section number, subsection number if you are in Section A, last and rst name (