OR 641 MIDTERM
Due: 8am, Tuesday, Oct 18, 2011.
1. Attempt all problems. Show all work. Explicitly state any assumptions
you have made.
2. The problem weights are as given in the parentheses.
3. The exam is open book, open notes.
4. No discussion about th
Homework 4
Due: Next week @ 13:40
1. Suppose cfw_Xn , n 0 and cfw_Yn , n 0 are two independent DTMCs with state-space
S = cfw_0, 1, 2, . Prove or give a counterexample to the following statements:
(a) cfw_Xn + Yn , n 0 is a DTMC.
(b) cfw_(Xn , Yn ), n 0 i
Corrections to the Second Edition of
Modeling and Analysis of Stochastic Systems
Vidyadhar Kulkarni
September 3, 2012
Send additional corrections to the author at his email address vkulkarn@email.unc.edu.
Chapter 2
1. Page 11, Equation (2.5): insert , aft
MA2215 Linear Programming
Tutorial 3 Solution
1. Consider the following polyhedron of an LP problem:
2x1 x2 + 5x3
= 1
3x2
+ x4
5
7x1
4x3 + x4
4
x 1 , x2 , x 4 0
(1)
(2)
(3)
Identify all active constraints at each of the following points x = (x1 , x2 ,
Operations Research
Instructor: Mohamed Omar
Lecture 8: Revised Simplex Method
Math 187
Last Time
Revised Simplex Method
In this class, we develop a method for running the Simplex Method, without keeping track
of a tableau. This is helpful because most of
Chapter 14
Solved Problems
14.1
Probability review
Problem 14.1. Let X and Y be two N0 -valued random variables such that X = Y + Z , where
Z is a Bernoulli random variable with parameter p (0, 1), independent of Y . Only one of the
following statements i
Moment Generating Functions
August 29, 2005
1
Generating Functions
1.1
The ordinary generating function
We dene the ordinary generating function of a sequence. This is by far the most
common type of generating function and the adjective ordinary is usuall
IE 561 - Fall 2015
1. Consider a nonhomogeneous Poisson process cfw_N(t), t 0 with rate function (t).
t
Dene the mean value function as m(t) =
(s)ds.
0
a) Derive the probability distribution of N(s) given that N(t) = n, for 0 s t.
b) Prove that, condition
IE 561 - Fall 2015
Assignment 3
Due November 9 - 09:40
Please work alone and show all your work.
Late assignments will not be accepted.
1. There are n organisms in a colony at time 0. The lifetimes of these organisms are
iid exp() random variables. What i
IE 561 - Fall 2015
Assignment 2
Due November 2 - 09:40
Please work alone and show all your work.
Late assignments will not be accepted.
1. Suppose the joint density of X and Y is given by f (x, y) = 4y(x y)e(x+y) , 0 <
x < , 0 y x. Compute E[X|Y = y].
2.
IE 561 - Fall 2015
Assignment 4
Due November 16 - 09:40
Please work alone and show all your work.
Late assignments will not be accepted.
1. A machine works for an exponentially distributed time with rate and then fails. A
repair crew checks the machines a
IE 561 - Fall 2015
Assignment 6
Due December 14 - 09:40
Please work alone and show all your work.
Late assignments will not be accepted.
1. Consider a production system that produces a single item per day. Whether or not a
particular item is defective dep
IE 561 - Fall 2015
Assignment 5
Due November 30 - 09:40
Please work alone and show all your work.
Late assignments will not be accepted.
1. Prove the following proposition:
Let cfw_N(t), t 0 be a Poisson Process with rate . Let A1 , A2 , . . . , An be dis
Mathematical Database
GENERATING FUNCTIONS
1. Introduction
The concept of generating functions is a powerful tool for solving counting problems.
Intuitively put, its general idea is as follows. In counting problems, we are often interested in
counting the
3
Does the Simplex Algorithm Work?
In this section we carefully examine the simplex algorithm introduced in the previous chapter.
Our goal is to either prove that it works, or to determine those circumstances under which it
may fail. If the simplex does n
Chapter 6
The two-phase simplex method
We now deal with the rst question raised at the end of Chapter 3. How do we
nd an initial basic feasible solution with which the simplex algorithm is started?
Phase one of the simplex method deals with the computatio
HOMEWORK 4: SOLUTIONS 1. A Markov chain with state space cfw_1, 2, 3 has transition probability matrix 1 / 3 1/ 3 1/ 3 P = 0 1/2 1/2 0 0 1 Show that state 3 is absorbing and, starting from state 1, nd the expected time until absorption occurs. Solution. L
Operations Research
Instructor: Mohamed Omar
Handout : Revised Simplex Method
Math 187
Revised Simplex Method
1. Begin with a feasible basis B and corresponding basic feasible solution x .
2. Solve AT y = cB for y .
B
3. Find k N such that ck = ck AT y >
MOMENT GENERATING
FUNCTIONS
1
Moments
For each integer k, the k-th moment of X is
E X the k-th moment
*
k
k
k E X the k-th central moment
k
2
MOMENT GENERATING FUNCTION (mgf)
Let X be a rv with cdf FX(x). The moment
generating function (mgf) of X, denot
College of Management, NCTU
Operation Research II
Spring, 2009
Chap16 Markov Chains
We often are faced with making decision based upon phenomena that have
uncertainty associated with them. Their variation can be described by a
probability model.
We presen
Operations Research
Instructor: Mohamed Omar
Lecture 7: Two-Phase Method
Math 187
Last Time
We discussed the detailed iterations of the Simplex Method.
http:/www.math.hmc.edu/ omar/math187/SimplexMethod.pdf
Issues with Simplex Method
1. (Pertinent) How do
Operations Research
Instructor: Mohamed Omar
Handout : Simplex Method
Math 187
Assumptions: The LP problem is of the form
max
z = cT x
s.t.
Ax = b
x0
where A is an m n matrix, n m, rank(A)=m. A feasible basis B is known. We denote
the set cfw_1, 2, . . .
Operations Research
Instructor: Mohamed Omar
Lecture 5: Basic Feasible Solutions/Simplex Method
Math 187
Theorem 1 Let A be an m n matrix, rank(A)=m. Let
F = cfw_x Rn : Ax = b, x 0.
Then x is a basic feasible solution of cfw_x Rn : Ax = b, x 0 if and only
OR641
FINAL
Due: Tuesday, Dec 13, 2011; 8:00am
Attempt all problems. Show all work. All problems carry equal weight.
The test is open books, open notes.
P1. Mr. Dilbert possesses r umbrellas, which he employs in going from
his home to oce and vice versa.