Review Solutions, Exam 2, Operations Research
1. Prove the weak duality theorem: For any x feasible for the primal and y feasible for the
dual, then.
HINT: Put the primal so that Ax b and the dual so that AT y c
SOLUTION: With the primal and dual in norma
Example 1, Section 4.16
The Leon Burnit Ad Agency is trying to determine a TV schedule for Priceler Auto.
Ad
HIM LIP HIW
Cost
Football
7
10
5
100, 000
3
5
4
60, 000
Soap Opera
Goals
40
60
35 600, 000
This means, for example, that the ad agency wants to re
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
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
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
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 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 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
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 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
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 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
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
Example 2, Section 4.16 (Goal Programming)
The Dewright company is considering three new products to replace current models. Primary consideration should be given to three factors:
Long-run prot of at least 125 million dollars.
Maintain current employme
Homework Solutions to Section 3.9
1. Dene variables rst. In this case, we need to know the hours to run each process per week,
and the barrels of stu being bought/sold. In this case, we might dene:
x1
x2
x3
g2
o1
o2
Hours of Process 1 run per week
Hours o
Scientic Applications of LP
These notes will show how to interpret curve tting problems as linear programs. In order
to start, we need to dene dierent ways of measuring the error between an unknown set of
points in the plane:
cfw_(x1 , y1 ), (x2 , y2 ), (
Matrix Notation and Simplex (6.2)
Prof. D.R. Hundley
Whitman College
Fall 2013
DRHundley (WHI)
Math 339
10/16/2013
1 / 14
Introduction
Consider the LP in standard form,
max z = cT x
st Ax = b
x0
where A is m n with rank m. We will re-partition these matri
Operations Research
Prof. D.R. Hundley
Whitman College
Fall 2013
DRHundley (WHI)
Math 339
10/16/2013
1 / 18
Introduction
We started this last time:
Were buying advertising time for HIW and HIM.
Let x1 , x2 be the number of ads purchased during a comedy sh
Homework Solutions to Section 3.1-2
These are the solutions to the problems not turned in: 3.1- 1, 5 and 3.2-1, 2, 4
3.1 Solutions
1. In this case, x1 , x2 are given to you- Normally, you would want to dene those terms rst.
Let x1 , x2 be the number of ac
HW Set 3 Solutions
3.10, 7 Let st be the pairs of shoes made during quarter t.
Let it be inventory of pairs of shoes at the end of each quarter.
Let xt be the number workers getting quarter t o during each year. So, for example,
x1 would be working all qu
Review Material, Exam 1, Ops Research
The exam will cover material from Chapter 3 (we skipped 3.6, 3.7), up through section 4.8.
Background Material: Linear Algebra
Though these may not be asked explicitly, you should be able to do the following (and may
Review Material, Exam 2, Ops Research
1. Prove the weak duality theorem: For any x feasible for the primal and y feasible for the
dual, then.
HINT: Put the primal so that Ax b and the dual so that AT y c
2. Show that the solution to the dual is y = (cT B
Solutions to Review Questions, Exam 1
1. If x = [1, 1, 0, 2]T and y = [2, 1, 0, 4]T , then compute the distance from x to y using
the (a) 1-norm, and (b) the innity norm.
SOLUTION:
xy
1
= |1 2| + | 1 + 1| + |0 0| + |2 4| = 1 + 0 + 0 + 2 = 3
xy
= max cfw_1
Final Exam, Operations Research (Fall 13)
This is a take home exam. You may use your text, any class notes and anything on our class
website to help, and you may use LINDO, Maple and/or Matlab. You should not look for
solutions on the internet, or get hel