IE301 Operations Research II
Fall 2014
Instructor: Ali Ekici
The OR World You Know
Linear Programming
minimizeormaximizealinearobjective
subjecttolinearequalitiesandinequalities
maximize 3x+4y
subj
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 t
Probability Review
1
Basic Rules of Probability
Definition: Any situation where the outcome is
uncertain is called an experiment.
Definition: For any experiment, the sample space S
of the experiment
Question-1:
The following data have been estimated for two mutually exclusive investment alternatives, A and B,
associated with a small engineering project for which revenues as well as expenses are i
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 resal
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 conc
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
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Continuous Time
Markov Chains
1
Continuous Time Markov Chain
So far, we looked at the evolution of a process
cfw_Xt=cfw_X0,X1,X2, at discrete times
Frequently, the evolution for a process is
observe
Constrained Optimization of a
Function of Multiple Variables
1
Constrained Optimization with
Equality Constraints
Consider NLPs of the following type:
max (or min) f ( x1 , x2 ,., xn )
s.t.
g1 ( x1 ,
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 combina
Markov Chains
(Classification of States and
Steady State Probabilities)
1
Classification of States of a MC
Definition: Given two states of i and j, state j is
accessible from state i if p(n)ij>0 for
Markov Chains
(Passage Times and
Absorbing Chains)
1
First Passage Times
The number of transitions made by the process
going from state i to state j is the first passage
time from state i to state j.
The Wagner-Whitin Algorithm
Wagner and Whitin have developed a
method that greatly simplifies the
computation of optimal production
schedules when storage and production
capacities are infinite.
1
The
Markov Chains
1
Description
Sometimes we are interested in how a
random variable changes over time.
The study of how a random variable
evolves over time includes stochastic
processes.
We will study
some examples
for the last weeks
course
1
Computer Example
A computer is inspected at the end of every hour. It
is found to be either up or down. If it is found to be
up, the probability of its remai
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 combina
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
Minimiz
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 chai
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
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
Minimiz
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
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 ma
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 conc