21-257 Models and Methods for Optimization, Fall 2012
Solutions to Homework 1
Section 1.3:
4. The constraint imposed by the blanking department is
x1 + x2 + 2x3 + .5x5 120,
and the constraint imposed by the forming department is
x1 + 3x3 + 2x4 + x5 120.
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21-257 Models and Methods for Optimization, Fall 2012
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
21-257 Models and Methods for Optimization, Fall 2012
Solutions to Homework 2
Section 2.6:
1(a). This question asks us to solve the following system of equations:
2x1 x2 + 2x3 = 2
4x1 x2 + 3x3 = 5 .
x1 + 2x2 + x3 = 4
Ill give two solutions, both of which
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Name
Instruc ' awed
1. TlWmok, closednotes, closed-homework.
2. You may also use a calculator, such as a TI89, to help with computations, but you must show all
theoretical work (numerical computations need not be shown).
3. No collaboration of an
21-257 Models and Methods for Optimization, Fall 2013
The Simplex Algorithm
(Taken from Introduction to Mathematical Programming by R. Walker)
Rule 1. Obtain the initial basic solution by setting each slack variable equal to the
corresponding resource and
21-257 Models and Methods for Optimization, Fall 2013
Solutions to Integer Programming Modeling Examples #1-#4
Example #1. (Example 8.6.5.) At the heart of this problem is determining
whether or not well build a paramedic base at each of the proposed site
21-257 Models and Methods for Optimization, Fall 2012
Solution to Integer Programming Modeling Examples #5
Example #5 Solution. Well need to know when each robot order is completed to
determine how late it is (and thus the penalty cost). Since DEM works n
21-257 Models and Methods for Optimization, Fall 2013
TSP Branch-and-bound Examples
Example #1 (From Lecture on 12/2): Solve the traveling salesman problem with
distances in the table below.
1
2
3
4
1
M
2
9
7
2
9
M
3
8
3
8
5
M
1
4
4
6
9
M
Solution: We beg
In Class Sensitivity Analysis Problems & Solutions
In Class Sensitivity Analysis Problems & Solutions
In Class Sensitivity Analysis Problems & Solutions
V Exam 3
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2. You may also use a. calculator, such as a. (Fl89, to help with C()111pttluti()ns. but you must show all
theoretical work (numerical computati
21-257 Models and Methods for Optimization, Fall 2013
Test 3 Prep - Select Problem Answers
The following are the answers to some even-numbered suggested problems (not previously assigned as homework) which are relevant to studying for Test 3.
Section 5.3,
21-257 Models and Methods for Optimization, Fall 2013
Examples of Applying the Simplex Algorithm
Example 1.3.1: Farmer Brown, via the Simplex Algorithm
Initial Simplex Tableau: Basis = cfw_s1 , s2 , s3
P
W
s1
s2
s3
z
10
20
1
0
0
0 1100
1
4
0
1
0
0
160
1
21-257 Models and Methods for Optimization, Fall 2013
Formulation Examples from Lecture, 9/11/13
Below, set up the linear program which solves each of the problems. Provide all necessary
constraints, and clearly identify each variable.
Example #1: Animal
M241: Matrix Algebra
Homework Assignment 8
Answers
Graded Problems: Section 4.2 Problems 4, 12; Section 4.3 Problems 14, 28; Section 4.4
Problems 2, 3.
Section 4.2
Problem 4. The determinant of the rst matrix is 20; the determinant of the second matrix is
M241: Matrix Algebra
Homework Assignment 7
Answers
Graded Problems: Section 3.3 Problems 12, 14; Section 3.4 Problems 8, 16, 18, 28.
Section 3.3
23/11
Problem 6. p = A(AT A)1 AT = 14/11.
b
65/11
12/11
Since = p + q , we have that q = p = 36/11 .
b
b
12/11
M241: Matrix Algebra
Homework Assignment 6
Answers
Graded Problems: all of them.
Section 3.2.
Problem 14. The vector (1, 2, 1) is on the line of intersection of the planes x + y + t = 0 and
x t = 0. The projection matrix is
1/6 1/3 1/6
1/3 2/3 1/3 .
1/6 1
Matrix Algebra
Homework Assignment 6
Due on Wednesday, October 13, 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 (a