Department of Industrial Engineering & Operations Research
IEOR 160: Operations Research I (Fall 2014)
Homework 5
Due: Friday, Oct 10th
Question 1. Sheldon and Amy are trying to divide a pile of items into two portions. The values of the items are
$4, $14
Department of Industrial Engineering & Operations Research
IEOR 160: Operations Research I (Fall 2014)
Homework 7
Due: Friday, Oct 24th
Question 1. Use a Binary Search to determine (to within an interval of length 0.5) the optimal solution to
maximize 3x
Department of Industrial Engineering & Operations Research
IEOR 160: Operations Research I (Fall 2014)
Homework 3
Due: Friday, Sept 26th
Question 1. A university department wishes to create a daily schedule for its classes using integer
programming. The d
Department of Industrial Engineering & Operations Research
IEOR 160: Operations Research I (Fall 2014)
Manpower planning
Certain types of facilities operate seven days each week and face the problem of allocating person power
during the week as stang requ
Homework 4 Solution
Page 502-problem 3:
Define
Define
,
as the number of product produced.
to denote whether we produce any product ,
.
Then the formulation is
is a very large number.
Page 503-problem 10:
Assume
is a very large number:
where is a binary d
1
SOLUTIONS TO MP CHAPTER 5 PROBLEMS
SECTION 5.1 1. Typical isoprofit line is 3x1+c2x2=z. This has slope -3/c2. If slope of isoprofit line is <-2, then Point C is optimal. Thus if -3/c2<-2 or c2<1.5 the current basis is no longer optimal. Also if th
Homework 1 Solution
Problem 40
Define:
number of units of A produced.
number of units of B produced.
number of units of raw materials purchased.
number of units of raw materials produced.
Note that producing a unit of B requires 2 hrs of assembly 1, 2 hrs
Department of Industrial Engineering & Operations Research
IEOR 160: Operations Research I (Fall 2014)
Homework 1 Solutions
Due: Friday, Sept 12th
Question 1. We are helping Sheldon plan his upcoming meals. To simplify the problem, we assume
that there ar
Department of Industrial Engineering & Operations Research
IEOR160 Operations Research I
Final Exam
Fall 2005
Name:
Grade:
1. (15 points) For an inequality constrained optimization problem with concave objective function f , suppose none of the constraint
IEOR 160 - Practice Final
Fall 2008
Problem 1 Three cities are located at the vertices of an equilateral triangle. (That is, the distance between any two cities is the same as the distance between any two other cities.) An airport is to be built at a loca
Department of Industrial Engineering & Operations Research
IEOR 160: Operations Research I (Fall 2014)
Homework 1
Due: Friday, Sept 10th
Question 1. We are helping Sheldon plan his upcoming meals. To simplify the problem, we assume
that there are three fo
Department of Industrial Engineering & Operations Research
IEOR 160: Linear Programming (Fall 2014)
Homework 0
Due: Friday, Sept 5th
This assignment will be graded but will not count for credit.
In your answers be sure to clearly dene all variables and pa
Non Linear Programming modeling
examples
l Location of facilities with Euclidean distances
l Portfolio optimization
l Regression (line/curve fitting)
1
IEOR160 2010
Linear Programming Model
Maximize c1 x1 + c2 x2 + . + cn xn
ASSUMPTIONS:
subject to
a11x1
Nonlinear Programming Theory
Why is nonlinear programming
different from linear programming?
1
IEOR160 2014
Difficulties of NLP Models
Linear
Program:
Nonlinear
Programs:
2
IEOR160 2014
Graphical Analysis of Non-linear programs
in two dimensions: An examp
Integer Programming leading algorithm
1
IEOR160 2014
Overview of Techniques for Solving
Integer Programs
Enumeration Techniques
Complete Enumeration
list all solutions and choose the best
Branch and Bound- implicit
Implicitly search all solutions, but
Convex functions vs. convex sets
If y = f(x) is convex, then
cfw_(x,y) : f(x) y is a convex set
y
f(x)
x
IEOR160 2014
1
Local Minimum Property
A local min of a convex function on a convex feasible
region is also a global min.
Strict convexity implies that
Introduction to Integer Programming
Integer programming models
1
IEOR160 2014
A 2-Variable Integer program
maximize
3x + 4y
subject to
5x + 8y 24
x, y 0 and integer
What is the optimal solution?
2
IEOR160 2014
The Feasible Region
4
5
Question: What is the
Using Cutting Planes
y
Optimum
(integer)
solution
P
x
Example. Minimize x + 10y
subject to x, y are in P
x, y integer
IEOR160-2014
Optimum
fractional
(i.e. infeasible)
solution
1
Using Cutting Planes
Idea: add
constraints that
eliminate fractional
solutio
Department of Industrial Engineering & Operations Research
IEOR 160: Operations Research I (Fall 2014)
Homework 12
Due: Friday, Dec 5th
Questions 1 and 2. Use Dijkstras algorithm to solve the shortest path problem for the two networks shown on
the next pa
Department of Industrial Engineering & Operations Research
IEOR 160 Operations Research I (Fall 2012)
Class time & location
Lecture: MW 2-3P, 150 GSPP (Goldman School of Public Policy)
Discussion:F 8-9A, 3113 Etcheverry or F 11-12P, 3107 Etcheverry
Instru
Department of Industrial Engineering & Operations Research
IEOR 160: Operations Research I (Fall 2014)
Homework 7 Solutions
Due: Friday, Oct 24th
Question 1. Use a Binary Search to determine (to within an interval of length 0.5) the optimal solution to
ma
Department of Industrial Engineering & Operations Research
IEOR 160: Operations Research I (Fall 2014)
Homework 4 Solutions
Due: Friday, Oct 3th
Question 1. A basketball coach needs to choose ve players to form the starting lineup of his
team. The team co
Department of Industrial Engineering & Operations Research
IEOR 160: Operations Research I (Fall 2014)
Homework 6 Solutions
Due: Friday, Oct 17th
Question 1. Consider the following integer program:
max
s.t.
x1 + 4x2
10x1 + 20x2 22
5x1 + 10x2 49
x1 5
x1 ,
Department of Industrial Engineering & Operations Research
IEOR 160: Operations Research I (Fall 2014)
Homework 8 Solutions
Due: Friday, Oct 31th
Question 1. On the given set S, determine whether each function is convex, concave, or neither.
1. f (x1 , x2
Department of Industrial Engineering & Operations Research
IEOR 160: Operations Research I (Fall 2014)
Homework 3
Due: Friday, Sept 26th
Question 1. A university department wishes to create a daily schedule for its classes using integer
programming. The d
IEOR 160 HW2 solution
Exercises from the book Introduction to mathematical programming by Winston and Venkataramanan
12.2.8
The optimization problem can be formulated as
0.5
0.3
0.6
max 10x0.4
1 + 8x2 + 12x3 + 16x4
s.t. x1 + x2 + x3 + x4 100
(Budget const
IEOR 160 HW1 solution
Prove or disprove the following statements on continuous functions f , g : Rn R
and a nonempty, closed, bounded, convex subset S of Rn .
1. min cfw_f (x) : x S = max cfw_f (x) : x S
Solution. True. Let z = min cfw_f (x) : x S and let
IEOR 160 Midterm solution
Problem 1
(a) F
(b) T
(c) F
(d) F
(e) F
(f) F
(g) T
Problem 2
Let x1 , x2 Rn and [0, 1]. We have that
f (x1 + (1 )x2 ) f (x1 ) + (1 )f (x2 )
= g f (x1 + (1 )x2 ) g f (x1 ) + (1 )f (x2 )
g f (x1 ) + (1 )g f (x2 )
(Concavity of f
IEOR 160 Quiz 1 solution
Find all stationary points, local min/max of f (x, y) = 2x3 + 4xy 2 3x2 y + 4y 3 .
Setting the gradient to 0, we have that
f
= 6x2 + 4y 2 6xy = 0
x
f
= 8xy 3x2 + 12y 2 = 0.
y
Thus we find from the first equation that xy = x2 + 32
IEOR 160 HW5 solution
Exercises from the book Introduction to mathematical programming by Winston and Venkataramanan
9.2.1
Solution taken from the discussion (files uploaded in bcourses).
Define
Sets:
N : Set of players - N = cfw_1, . . . , 7
NG : Set