EM 222
Deterministic Operations
Research
2016-2017 Spring
Weeks 1&2
Integer Programming
LP relaxation of IPs
Uncapacitated Facility Location Problem
There are n potential depots to serve m cities. fj is the fixed cost of
operating depot j. cij is the cost
1) Consider the following tabulated data composed of xi and yi .
xi
yi
1.6
2
2
8
2.5
14
3.2
15
4
8
4.5
2
If it is known that, y is a function of x, that is to say y=f(x), then calculate f(2.8) using Newtons interpolating
polynomials of order 1 to 3. In ot
EM 221
Introduction to Optimization
and Modeling
2016-2017 Fall
Week 9
Review of Linear Algebra
Matrices and Vectors
Matrix a rectangular array of numbers
1 2
3 4
1 2 3
4 5 6
Typical m x n matrix
having m rows and n
columns. We refer to
m x n as the o
EM 221
Introduction to Optimization and
Modeling
Ceren Tuncer akar
2016-2017 Fall
Week 1
OUTLINE
Introduction to Operations Research (OR)
Successful Applications
OR Characteristics
Methodology of OR
Introduction to Linear Programming
2
Operations Resear
EM 221
Introduction to Optimization and
Modeling
2016-2017 Fall
Week 3
Sensitivity Analysis
A Graphical Approach to Sensitivity Analysis
Sensitivity analysis is concerned with how changes
in an LPs parameters affect the optimal solution.
Reconsider the
Gi
EM 221
Introduction to Optimization
and Modeling
2016-2017 Fall
Week 13
Sensitivity Analysis
Sensitivity Analysis and Duality
Part I
Sensitivity Analysis
Suppose we solved an LP to optimality. How do the changes in the
values of parameters effect the opti
EM 221
Introduction to Optimization
and Modeling
2016-2017 Fall
Week 14
Duality
Sensitivity Analysis and Duality
Part II
Economic Interpretation of the Dual Problem
Economic Interpretation of the Dual Problem
Economic Interpretation of the Dual Problem
Ec
EM 221
Introduction to Optimization and
Modeling
2016-2017 Fall
Week 4
Shadow Prices
The shadow price (SP) of a constraint of an LP is the
amount by which the optimal z-value changes per unit
increase in the RHS of the constraint. This definition gives
a
R SESSION
Question 1. (Linear Programming)
Coalco produces coal at three mines and ships it to four customers. The cost per ton of producing coal, the
ash and sulfur content (per ton) of the coal, and the production capacity (in tons) for each mine are gi
Operations Research I
IE 416
California State Polytechnic University, Pomona
Linear programming Homework #4
on Page 97
TEAM 5
Serina Alkejek
Harmeet Hora
Kaveh Kevin Shamuilian
Outline
Problem
Statement
Summary of problem
Formulation of problem
WinQSB Inp
A Capital Budgeting Problem
Star Oil Company is considering five different investment opportunities. The cash outflows and
net present values (in millions of dollars) are given in the following table.
Star Oil has $40 million available for investment at t
Sheet1
Enkelt diet problem
(Urprungligt exempel: http:/www.mcs.vuw.ac.nz/courses/OPRE251/2006T1/Labs/lab09.pdf , men den sidan r on
My diet requires that all the food I eat come from one of the four .basic
food groups. (chocolate cake, ice cream, soft dri
Exam 1 solutions (M126C)
1. Determine if the sequence converges or diverges. If it converges, nd its limit: an =
(ln(n)2
n
SOLUTION: Use lHospitals rule (twice):
(ln(n)2
2 ln(n) (1/n)
ln(n)
1/n
= lim
= lim 2
= lim 2
=0
n
n
n
n
n
1
n
1
lim
Therefore, the s
Exam 2 Review Solutions
1. State the Fundamental Theorem of Calculus: Let f be continuous on [a, b].
x
If g(x) =
f (t) dt, then g (x) = f (x).
a
b
f (x) dx = F (b) F (a), where F is any antiderivative of f .
a
n
b
2. Give the denition of the denite integ
Selected Solutions, Section 5,1
2. The purpose of this exercise is to be sure we have a little experience with estimating
area using rectangles. The midpoint rule is to evaluate the heights of the rectangles
at the midpoint of each rectangle.
Recall that
Final Exam Review
Calculus II
Sheet 3
1. Determine if the series converges (absolute or conditional) or diverges:
(a)
(1)n ln(n)
n
n=1
SOLUTION: We might rst check to see if it converges absolutely. It will not, by
the integral test. To check that the fun
Review Sheet 1 Solutions
n
i2 =
1. Prove by induction:
i=1
n(n + 1)(2n + 1)
6
SOLUTION:
Prove the rst case: If n = 1, then does 12 =
123
?
6
Yes.
k
k(k + 1)(2k + 1)
.
6
i=1
Use the assumption to prove that the statement is true if n = k + 1. In that case
Final Exam Review
Calculus II
Sheet 2
1. True or False, and give a short reason:
2n+3
(a) The Ratio Test will not give a conclusive result for 3n4 +2n3 +3n+5
TRUE. The ratio test fails for plike series (the limit will be 1). To show convergence, use a dir
Solutions to the Review Questions, Exam 3
1. A cross section of a tank of water is the bottom half of a circle of radius 10 ft, and is 50
ft long. Find the work done in pumping the water over the rim of the tank if it lled to
a depth of 7 feet (set up the
Selected Solutions, Section 5.4
10. Hint: Multiply it out rst,
1
v 5 + 4v 3 + 4v dv = v 4 + x4 + 2v 2 + C
6
v(v 2 + 2)2 dv =
12.
1 3
x
3
+ x + tan1 (x) + C.
18. The hint is that sin(2x) = 2 sin(x) cos(x). Using that,
sin(2x)
dx =
sin(x)
2 sin(x) cos(x)
dx
Selected Solutions, Section 5.2
1. This is good practice in taking left endpoints.
In this case, f (x) = 3 x/2, and the interval is [2, 14]. The Riemann sum using 6
rectangles will use:
Width of each rectangle: (14 2)/6 = 12/6 = 2.
The height of the rec
Review Problems: Chapter 11
1. What does it mean to say that a series converges (Im looking for the denition; be
sure you dene any notation you use).
2. Does the given sequence or series converge or diverge?
1
(a)
n
n=2 n
(b)
(e)
(6)
n1 1n
5
(j)
n=1
(f)
Review Solutions: Chapter 11
1. What does it mean to say that a series converges?
SOLUTION: We dene the nth partial sum sn as follows:
n
S 1 = a1
S 2 = a1 + a2
S3 = a1 + a2 + a3
Sn =
an
k=1
The partial sums Sn form a sequence (of numbers). The (innite) se
Quiz 2 Solutions
Part of this quiz was to see if you could follow the directions!
Solutions are written neatly, clearly and completely using your own paper (up to 10 pts)
Quiz is stapled (up to 10 pts).
Quiz turned in on time (up to 10 pts).
Here are t
Selected Solutions, Section 4.9
10. Note that e2 is a constant, so the antiderivative is e2 C
17. The antiderivative is 2 cos() tan() + C, but notice that the C can change because
sec() has a lot of vertical asymptotes, breaking up the real line. Therefor