IE301 Operations Research II
Fall 2014
Instructor: Ali Ekici
The OR World You Know
Linear Programming
minimizeormaximizealinearobjective
subjecttolinearequalitiesandinequalities
maximize 3x+4y
subjectto5x+8y 24
x,y 0
A feasible solution satisfies all o
IE 301 - Practice Test II
INSTRUCTIONS
Anything we talked about dynamic programming (both deterministic and probabilistic) in
class might show up on your exam. Just because a topic is not covered in these questions
does not mean that I will not test you o
Sample Problems 2
1. Given f(x,y)=x2-3xy-y2, identify the stationary points of f. Determine whether they are
local/global max, min or saddle points.
Answer: (0,0) is the only stationary point which is a saddle point.
2. Given f(x1,x2)= X1X2+3X2-X12-X22, i
IE301 Operations Research II
Fall 2015
Instructor: Ali Ekici
The OR World You Know
Linear Programming
minimize or maximize a linear objective
subject to linear equalities and inequalities
maximize 3x + 4y
subject to
5x + 8y 24
x, y 0
A feasible solutio
Probabilistic Dynamic
Programming
1
Probabilistic Dynamic Programming
In deterministic dynamic programming, a
specification of the current state and current
decision was enough to tell us with certainty the new
state and the costs/rewards during the curr
IE301 Operations Research II
Fall 2016
Course Description
The aim of this course is to introduce the most widely used nonlinear mathematical programming
and dynamic programming methods and Markov chains. Topics covered in this course include
nonlinear opt
Unconstrained Maximization and
Minimization with Several Variables
Consider this unconstrained NLP
max (or min) f ( x1 , x2 ,.xn )
s.t.
( x1 , x2 ,., xn ) R n
1
Unconstrained Maximization and
Minimization with Several Variables
A necessary condition for
Dynamic Programming
1
Description
Dynamic Programming (DP) is a technique that
can be used to solve many optimization
problems.
It provides a systematic procedure for
determining the optimal combination of
decisions.
In most applications, dynamic progr
ShawIaIfurS=Rixhxmx=x +x+21xix2xg3 xlx1isanumnex
flmctinn.
Slhiiln 'l'heHessaanis-giwnby
2 l 1
HII],I2EXJ=[l 2 l.]
l l it
Bydnhngmsiiandmhlmm] 1 andEafHassiamwpubtainlemt-m'dpmi
pal mier 4 3:- 1]. y deleting rm [and claims] 1 and 3 of Haasian. we obtain H
Quadratic Programming
1
Quadratic Programming
A quadratic programming problem (QPP) is an
NLP in which each term in the objective
function is of degree 2, 1, or 0 and all
constraints are linear
max f ( x1 , x2 ) 15 x1 30 x2 4 x1 x2 2 x12 4 x22
s.t.
x1 2
NLPs with One Variable
One Variable Unconstrained
Optimization
Consider f(x) defined from R to R
Lets assume we are trying to find the
maximum value f(x) can take
max f ( x)
s.t.
x
If f(x) is concave, any local maximum is also
a global maximum
One Var
IE 301 - Assignment 2
Due: Friday, November 11, 2016 at 12.00
(Papers will be collected before the midterm)
Question 1
Suppose that a new car costs $10,000 and that the annual operating cost and resale value of
the car are as shown in the following table.
NAME: f0 L U 7/0/Vf
Midterm Exam #3
IE 301 Fall 2013
January 8, 2014
INSTRUCTIONS
Exam duration: 110 minutes.
This is a closed-book, closednotes exam. You are NOT allowed to use and refer to the book,
class notes, homework assignments, homework solutions,
Sample Problems 1
1. If K units of capital and L units of labor are used, then a company can produce KL units of a
manufactured good. Capital can be purchased at $4/unit and labor can be purchased at $1/unit. A
total of $800 is available to purchase capit
Sample Problems (NLPs with One Variable)
1. Consider the following function:
2
3
f(x) = 48x - 60x + x
(a) Use the first and second order derivatives to find the local maxima and local minima of f(x)
(b) Use the first and second order derivatives to show t
NLPs with One Variable
One Variable Unconstrained
Optimization
Consider f(x) defined from R to R
Lets assume we are trying to find the
maximum value f(x) can take
max f ( x)
s.t.
x
If f(x) is concave, any local maximum is also
a global maximum
One Var
Solutions to Sample Problems (NLPs with One Variable)
1.
2. In the excel file.
3. (0,0) is the only stationary point which is a saddle point.
4. (1,2) is the only stationary point which is a global maximum since the function is a concave function.
1
Solutions to Modeling Questions
3. Decision Variables:
R = gallons of regular gasoline produced daily
U = gallons of unleaded gasoline produced daily
P = gallons of premium gasoline produced daily
A1 = gallons of A1 crude purchased daily
A2 = gallons of A
Unconstrained Maximization and
Minimization with Several Variables
Consider this unconstrained NLP
max (or min) f ( x1 , x2 ,.xn )
s.t.
( x1 , x2 ,., xn ) R n
1
Unconstrained Maximization and
Minimization with Several Variables
A necessary condition for
IE 301 - Assignment 1
Due: Monday, November 4, 2013 at 5pm
1. Given f(x,y,z)=2x2+xy-y2+yz+z2-6x-7y-8z+9, identify the stationary points
of f. Determine whether they are local/global max, min or saddle points.
2. Use the bisection method to solve the follo
NAME: j? Z. U 77 0A,;
Midterm Exam #1
IE 301 Fall 2013
November 6, 2013
INSTRUCTIONS
Exam duration: 100 minutes.
This is a closed-book, closed-notes exam. You are NOT allowed to use and refer to the book,
class notes, homework assignments, homework soluti
IE 301 - Assignment 3
Due: Friday, December 27, 2013 at 11:55pm
1. Consider the Markov chain that has the following (one-step) transition matrix.
(a) Determine the classes of this Markov chain and, for each class, determine
whether it is recurrent or tran
IE 301 - Practice Test I
INSTRUCTIONS
Anything we talked about in class might show up on your exam. Just because a topic is
not covered in these questions does not mean that I will not test you on it.
This exam is longer than the test you will have, but I
NAME: $524 H 770/05
Midterm Exam #2
IE 301 Fall 2013
December 4, 2013
INSTRUCTIONS
0 Exam duration: 100 minutes.
0 This is a closed-book, closed-notes exam. You are NOT allowed to use and refer to the book,
class notes, homework assignments, homework so
Modeling Questions
1. If K units of capital and L units of labor are used, then a company can produce KL units of a
manufactured good. Capital can be purchased at $4/unit and labor can be purchased at
$1/unit. A total of $800 is available to purchase capi