IND E 513 Homework 7 Solutions
Problem 1 The journal articles involving stochastic programming were very interesting.
Problem 2 (Exercise 6.2 from text) Write a formulation of the variation of the diet problem
with two children, n foods, m nutrients, and

IND E 513 Homework 6
Due at the beginning of class, 1:00 pm on Wed., November 19, 2014
Reading Assignment: Start reading Chapter 6 in the textbook.
Problem 1
1. Show that, in the simplex method, if a variable xj leaves the basis, it cannot enter the basis

IND E 513 Homework 6 Solutions
Problem 1
(1a.) Show that, in the simplex method, if a variable xj leaves the basis, it cannot enter the basis in the
next iteration.
Consider a basic variable xj that is going to leave the basis and xi is going to enter the

IND E 513 Homework 4 Solutions
Problem 1 Consider the following two linear programming problems formed from the same problem
data.
(P1 ) max c x Ax b
(P2 ) min c x Ax b
Notice that in both these programs, the variables x are assumed to be free.
(a) The du

IND E 513 Homework 7
Due at the beginning of class, 1:00 pm on Wed., November 26, 2014
Reading Assignment: Finish reading Chapter 6 in the textbook.
Problem 1 Find a recent journal article about stochastic programming with recourse, and provide
the comple

IND E 513 Homework 7
Due at the beginning of class, 1:00 pm on Wed., December 3, 2014
Reading Assignment: Read Chapter 7 in the textbook.
Problem 1 Exercise 7.1 from text
Problem 2 Exercise 7.2 from the text.
Problem 3 Exercise 7.7 from the text.
Problem

IND E 513 Homework 2 Solutions
Problem 1 A manufacturing rm faces demands D1 , D2 , . . . , DN for its product over the next N weeks.
The rm decides to produce x1 , x2 , . . . , xN units of this product in weeks 1, 2, . . . , N respectively to meet
this d

IND E 513 Homework 5 Solutions
Problem 1
The basis matrix consists of columns 2,1,3 of matrix
1
B= 2
0
Note x2 is the rst basic variable. If we have
A, which is
2 1
1 0
1 0
ci (cB + e )B1 Ai 0,
1
i = 2,
then the current basis remains optimal. This is equ

IND E 513 Homework 1 Solutions
Problem 1 Problem 1.2 from the textbook.
m
(1.2a) Let f (x) =
fi (x). Since function fi , i = 1, 2, . . . , m are convex we have
i=1
fi (x + (1 )y) fi (x) + (1 )fi (y), i = 1, 2, . . . , m,
for x, y Rn and [0, 1]. Adding the

IND E 513 Homework 8 Solutions
Problem 1 Exercise 7.1 from the text.
We construct a network with 2N + 1 nodes as follows. There is a supply node s. For i = 1, . . . , N , there
is a node Ci corresponding to clean tablecloths, and a node Di corresponding t

IND E 513 Homework 5
Due at the beginning of class, 1:00 pm on Wed., November 5, 2014
Reading Assignment: Finish reading Chapter 5 (5.1-5.6) from the textbook.
Problem 1 Consider the linear programming problem
min x1 2x2
2x1 + x2
x1 + 2x2
x1
x 1 , x2
2
7

IND E 513 Homework 4
Due at the beginning of class, 1:00 pm on Wed., October 29, 2014
Reading Assignment: Please read Chapter 4 (4.1-4.6, and 4.11), and start reading Chapter 5
(5.1-5.6) from the textbook.
Problem 1 Consider the following two linear progr

IND E 513 Homework 1
Due at the beginning of class, 1:00 pm on Wed., October 8, 2014
Reading Assignment: Please read all of Chapter 1 and start Chapter 2 in the textbook.
Problem 1 Problem 1.2 from the textbook. (Hint: to show the sum is piecewise linear,

IND E 513 Homework 3 Solutions
Problem 1 First, write the problem in standard form by adding slack variables:
min 2x1 x2
subject to x1 x2 + s3 = 2
x1 + x2 + s4 = 6
x1 , x2 , s3 , s4 0
The basic feasible solution that is given, (0, 0, 2, 6), is not optimal

Notes 6: Sensitivity Analysis
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Notes 6
Slide 1
Sensitivity Analysis
We study the dependence of the optimal cost and the optimal
solution on problem data A, b, c. Important in practice because we
often have incomplete information about p

Notes 5: Duality Theory
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Notes 5
Slide 1
Motivating Example for Duality Theory
Familys Diet Problem: minimize cost of six basic foods to meet
minimum daily requirements of vitamins A and C, let xj be the kg
of food i eaten daily, i = 1,

Notes 3: From Geometry to Simplex Method
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Notes 3
Slide 1
Polyhedra in Standard Form
Denition
A polyhedron in standard form is given by cfw_x 2 R n |Ax = b, x
where A is an m n matrix and b is a vector in R m .
I
I
0,
There are m equali

Notes 7: Large Scale Optimization
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Notes 7
Slide 1
Delayed Column Generation
Suppose the problem in standard form has A so large that it
cannot be stored in memory. However A may be very sparse (not
many non-zero entries), and A may hav

Notes 2: Geometry of LPs
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Notes 2
Slide 1
Piecewise Linear Convex Functions
Theorem
Let f1 , . . . , fm : n be convex functions. Then, the function f
dened by f (x) = maxi=1,.,m fi (x) is also convex.
Proof: Let x, y
n
and let [0, 1].

Notes 8: Stochastic Programming
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Notes 8
Slide 1
Decision Making under Uncertainty
Sensitivity analysis on a deterministic LP is not the same thing as
involving the randomness in the model directly:
Is Sensitivity Analysis of Any Use?
S

Notes 1: Introduction and Background
IND E 513
Prof. Zelda Zabinsky
IND E 513
Notes 1
Slide 1
Introduction
1. What is a Linear Program (LP)?
2. Simple examples of LPs
I
I
I
I
The
The
The
The
Diet Problem
Production Problem
Transportation Problem
Alloy Pro

Notes 9: Network Flow Problems
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Notes 9
Slide 1
I
A variety of networks pervade our daily lives transportation
networks, communication networks, electrical networks, social
networks, nancial networks .
I
Several optimization problems in

IND E 513 Homework 2
Due at the beginning of class, 1:00 pm on Wed., October 15, 2014
Reading Assignment: Please nish reading Chapter 2 and read Chapter 3 (3.1-3.5) from the
textbook.
Problem 1 A manufacturing rm faces demands D1 , D2 , . . . , DN for its

IND E 513 Homework 3
Due at the beginning of class, 1:00 pm on Wed., October 22, 2014
Reading Assignment: Finish reading Chapter 3 (read 3.1-3.5, skip 3.6-3.7), and read the beginning of Chapter 4 (4.1-4.4) in the textbook.
Problem 1 Consider the followin

Notes 4: Conceptual Development of the Simplex
Method
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Notes 4
Slide 1
Development of the Simplex Method
In this section, we assume that every basic feasible solution is
nondegenerate.
Suppose we are at a basic feasible solution x and w