IEOR 4004
Name_
Exam 1
Fall 2014
UNI_
No materials other than a pen/pencil are allowed.
1. (30 points) Consider the following dictionary for a maximization problem:
x3 =
x4 =
x6 =
z=
b 4x1 + A1x2 + A2x5
2 + x1 + 5x2 + x5
3 + A3x1 + 3x2 + 4x5
10 + C1x1 + C
BHM 1.2
Consider the following linear program:
Maximize z = 2x1 + x2
subject to: 12x1 + 3x2 6
3x1 + x2 7
x2 10
x1 0, x2 0.
a) Draw a graph of the constraints and shade in the feasible region. Label the vertices of this region with
their coordinates.
If we
IEOR E4004: Deterministic Models
Assignment 4: Due November 3
1. Problem 6 (Page 322).
2. Problem 12 (Page 349).
3. Problem 1a (Page 471). Check your answer by writing the shortest path problem as an LP
and having Gurobi solve it.
4. Consider the followin
1. (30 points) Consider the following dictionary for a maximization problem:
x3 =
x4 =
x6 =
z=
b 4x1 + A1x2 + A2x5
2 + x1 + 5x2 + x5
3 + A3x1 + 3x2 + 4x5
10 + C1x1 + C2x2
a. Give all conditions on A1, A2, A3, b, C1, and C2 necessary to make the following
Midterm exam (10/25/2013)
IEOR E4004: Introduction to Operations Research: Deterministic Models
Time: 3 hours (from 10am to 1pm)
Instructions:
write answer to each question on a separate sheet of paper
clearly mark your name and question number on each
IEOR 4004
Name_
Exam 2
UNI_
Fall 2014
No materials other than a pen/pencil are allowed.
1. (25 points) Consider the following linear program and its optimal dictionary.
max = 61 + 2
. .
1 + 2 5
21 + 2 6
1 , 2 0
1 = 2 0.52 + 0.52
1 = 3 0.52 0.52
= 18 22 3
IEOR 4004
Getting started with the Simplex method
Consider the following example:
max
Subject to
3x1
+2x2
x1
2x1
x1
+x2
+x2
80
100
40
x0
Convert to standard form:
max
Subject to
3x1
+2x2
x1
2x1
x1
+x2
+x2
+x3
+x4
+x5
=
=
=
80
100
40
x0
One further elabora
IEOR 4004
Lecture 3 - Intro to Linear Programming Algorithms
We assume an LP in standard form:
max wT x
(1a)
Ax = b
x 0.
(LP):
Subject to
(1b)
(1c)
Here we are assuming that A has m rows and n columns, and so x Rn and b Rm . The
algorithmic ideas we will
IEOR 4004
Lecture 2 - Basic Linear Programming Formulations
Basic concepts related to linear programming:
Feasible region, infeasible problems
Convexity
Bounded, unbounded feasible regions
Extreme points = corner solution
Unique optimum vs multiple o
"Files & Resources" section of the Courseworks site has a couple of files. The file entitled 'Lec1.zip' has
materials related to our first lecture:
1. lecture1.pdf: a writeup with introductory material
2. cashexample.lp: an LP file with the cash example w
IEOR 4004
Lecture 1 - Introduction to Optimization
Example 1. Cash ows
We have $100 to invest. There are three investment vehicles.
a. for every $1 invested now, we get 0.1 one year from now, and $1.3 three years from
now.
b. for every $1 invested now, we
IEOR E4004: Deterministic Models
Assignment 1: Due Lecture 6 (September 28)
Notes
Each assignment must be submitted at the beginning of the class it is due. Late submissions
will NOT be accepted.
You are allowed to discuss the assignment with others but
0.1. GRAPHICAL METHOD
0.1
1
Graphical method
Max 3x1 + 2x2
x1 + x2
2x1 + x2
x1
x1 , x2
80
100
40
0
1. Find the feasible region.
Plot each constraint as an equation line in the plane
Feasible points on one side of the line plug in (0,0) to nd out which
x
IEOR 4004: Optimization Models and Methods
01/23/17
Lecture 2
Instructor: Shipra Agrawal
Summary of last lecture and matrix notation for LP
Linear program (LP) :
Decision variables: x1 , x2 , x3 , . . . ,
maximize or minimize an objective
Objective fun
IEOR 4004
Formal Simplex Method
Here we outline key steps in the Simplex Method applied to an LP in standard form:
max z = cT x
Subject to:
Ax = b
x 0.
Here A is an m n matrix.
Step I. Find an initial basic feasible solution. This is not a trivial step. A
Xiaopei Zhang E4004 HW1 Soln.
Solution for Homework 1
Xiaopei Zhang xz2363
October 2, 2015
Winston and Venkataraman Chapter 3: Problems 4, 9, 28, 44, 45
Problem 4, 9, 44 and 45 could be formulated in different ways. This sulution
only provides with one po
IEOR 4004: Optimization Models and Methods
01/25/17
Lecture 3: Intro to Linear Programming Algorithms
Instructor: Shipra Agrawal
1
Standard form for LPs
There are many kinds of forms for linear programs. We could have a constraint like
2x1 5x2 + x4 = 10
o
IEOR 4004: Optimization Models and Methods
1/18/17
Lecture 1
Instructor: Shipra Agrawal
1
Mathematical modeling by example
Example 1.(Transportation.)
A paper company wants to ship truckloads of paper from warehouses to stores.
There are two warehouses w
IEOR 4004: Optimization Models and Methods
Homework 1
Instructor: Shipra Agrawal
Due on: Wednesday, February 1, 2016
Each assignment must be submitted either on paper at the beginning of the class it is due, or online on
courseworks before the class. NO
Introduction to OR-Deterministic Model (IEOR
E4004) - Sample Midterm #1
Cun Mu - [email protected]
October 7, 2015
Modied by Michael Hamilton.
Problem 1
The problem can be formulated as
m
maxcfw_|xi x| + |yi y|, |xi x| + |yi y |
min
x,y,
xy
i=1
s.t. xL
IEOR E4004: Introduction to OR: Deterministic Models
Midterm 1 Examination Sample Questions
Professor D. Goldfarb
1. (50 points) 1. Formulate the following problem as an LP.
Suppose we wish to locate two facilities to serve m customers located at
position
Mathematical Programming:
An Overview
1
Management science is characterized by a scientic approach to managerial decision making. It attempts
to apply mathematical methods and the capabilities of modern computers to the difcult and unstructured
problems c
0.1. GRAPHICAL METHOD
0.1
1
Graphical method
Max 3x1 + 2x2
x1 + x2
2x1 + x2
x1
x1 , x2
80
100
40
0
1. Find the feasible region.
Plot each constraint as an equation line in the plane
Feasible points on one side of the line plug in (0,0) to nd out which
x
IEOR 4004
Network Problems
A graph or network consists of:
A set V of vertices (nodes, points) and
A set E of edges (arcs, lines) involving two vertices, each.
We may write V ( G ) to refer to the vertices or E( G ) to refer to the edges. In the example
IEOR 4004
Network Problems
A graph or network consists of:
A set V of vertices (nodes, points) and
A set E of edges (arcs, lines) involving two vertices, each.
We may write V ( G ) to refer to the vertices or E( G ) to refer to the edges. In the example
IEOR E4004: Deterministic Models
Assignment 5: Due November 20
1. Problem 9-10 (Page 430).
2. Problem 11 (Page 430).
3. Problem 13 (Page 431).
4. Problem 14 (Page 431).
5. Problem 15 (Page 431). (a) Solve the problem for the specic case shown in Figure 25
IEOR E4004: Deterministic Models
Assignment 3: Due October 27
Homework Exercises (from Chapter 6 in the textbook by Winston and Venkataraman
1. Problem 3 (Page 346). Do not use Gurobi; just use the optimal tableau.
2. Problem 4 (Page 346). Use Gurobi (not