IEOR E4004: Introduction to Operations Research: Deterministic Models
Jay Sethuraman
Final exam
3 hours; open book/notes; no calculators
1. (20 points) Consider the linear integer programming problem
Max 2x1 4x2
subject to:
2x1 + x2 5,
4x1 + 4x2 5,
x1 , x
IEOR E4004: Introduction to Operations Research: Deterministic Models
Solutions by Stergios Athanassoglou
Sample Final Solutions
Problem 1.
Please refer to HW 9.
Problem 2.
(a) x13 = 2, x24 = 2, x45 = 0, x43 = 1 is a basic feasible solution. This solution
IEOR 4404
Intro OR: Deterministic Models
Prof. Jay Sethuraman
Recitation #1
January 25, 2011
Page 1 of 2
Recitation #1
1. The Temporary Help Company must provide secretaries to its clients over the next year on
the following estimates schedule: spring, 60
IEOR 4404
Intro OR: Deterministic Models
Prof. Jay Sethuraman
Homework #4
February 28, 2011
Page 1 of 5
Homework #4
1. a. What is the optimal production mix? What contribution can the rm anticipate by producing this mix?
The optimal production mix is:
xch
6. (Problem 7.22 from text)
a) The objective function is :
max
cij xik xjk .
ijk
b) Constraints:
Every person is assigned to a car pool:
xik = 1, i
k
The number of people in a car pool is between L and U :
L
xik U, k.
i
c) Dene new variables:
yikl = cfw
4. Let fj (A) be the optimal prot the vendor can make in days j, j + 1, . . . , N , assuming he
starts day j in location A. Similarly, let fj (B) be the optimal prot in days j through N
assuming he starts day j in location B. Then,
fj1 (A) = Aj1 + maxcfw_
IEOR E4004: Introduction to Operations Research: Deterministic Models
Jay Sethuraman
HW 9 Solutions
1.
(a) Lets introduce for each node i V the binary variables: ri , bi , yi . These indicate if node i
is colored red, blue, or yellow, respectively. We now
IEOR E4004: Introduction to Operations Research: Deterministic Models
Jay Sethuraman; email: jay@ieor.columbia.edu
338 Mudd; tel: 212-854-4931
Description. This class is (intended to be) an introduction to the fundamental methods used in deterministic ope
Solving Linear Programs
2
In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the
simplex method, proceeds by moving from one feasible solution to another, at each step improving the value
of the objectiv
Converting a Linear Program to Standard Form
Hi, welcome to a
tutorial on converting
an LP to Standard
Form.
Amit, an MIT Beaver
We hope that you
enjoy it and find it
useful.
Mita, an MIT
Beaver
2
Linear Programs in Standard Form
We say that a linear prog
IEOR E4004: Introduction to Operations Research:
Deterministic Models
Jay Sethuraman
HW 4
Problems not written out explicitly are from the text: Applied Mathematical Programming, by Bradley, Hax and Magnanti.
1. Problem 3.1
2. Problem 3.10
3. Problem 4.2
IEOR 4404
Intro OR: Deterministic Models
Prof. Jay Sethuraman
Homework #3
March 5, 2011
Page 1 of 4
Homework #3
Problem 1. (a) Let xk be the number of ocers assigned to highway k, k = 1, 2, . . . , K. The following
linear program
K
max
k=1 xk rk
s.t. lk x
IEOR E4004: Introduction to Operations Research: Deterministic Models
Jay Sethuraman
HW 3 (due 02/16)
1. You are given the task of assigning D patrol ocers to K highway segments. Each ocer
assigned to segment k reduces speeding violations on that segment
IEOR 4404
Intro OR: Deterministic Models
Prof. Jay Sethuraman
Recitation #2
February 1, 2011
Page 1 of 3
Recitation #2
1. Consider the problem of minimizing a cost function of the form cT x + f (dT x), subject to
the linear constraints A x b. Here, d is a
IEOR 4404
Intro OR: Deterministic Models
Prof. Jay Sethuraman
Recitation #3
February 8, 2011
Page 1 of 3
Recitation #3
2.3 a. Reduce the following system to canonical form. Identify slack, surplus, and articial
variables.
2 x1 + x2 4
(1)
3 x1 + 4 x2 2
(2)
IEOR 4404
Intro OR: Deterministic Models
Prof. Jay Sethuraman
Recitation #4
February 15, 2011
Page 1 of 2
Recitation #4
2.18 Frequently linear programs are formulated with intervals constraints of the form:
5 6 x1 x2 + 3 x3 8
a. Show that this constraint
IEOR 4404
Intro OR: Deterministic Models
Prof. Jay Sethuraman
Recitation #5
February 22, 2011
Page 1 of 2
Recitation #5
1. problem 3.7 b. from the book
The optimal solution is :
2
6
0
0
4
The shadow prices are the negative of the reduced cost of the sla
If we set b = c = 0, A = AT , then (14) and (15) will be the same problem. Thus we can construct
another example as follows:
max 0
s.t.
x1 x2 + 2x3 = 0
(16)
x1 + 2x2 + 3x3 = 0
2x1 + 3x2 + 3x3 = 0
x1 , x2 , x3 unrestricted.
Problem 4.
If cj 0 for all j N,
IEOR 4404
Intro OR: Deterministic Models
Prof. Jay Sethuraman
Recitation #9
April 6, 2011
Page 1 of 2
Recitation #9
1
The simplex method for uncapacitated network ow problems
1. A typical iteration starts with a basic feasible solution f associated with a
IEOR 4404
Intro OR: Deterministic Models
Prof. Jay Sethuraman
Recitation #11
April 25, 2011
Page 1 of 1
Recitation #11
A vendor can set up his truck in one of two locations A, B each day. His prots on the ith day for location
A and B are Ai and Bi respect
Sensitivity Analysis
3
We have already been introduced to sensitivity analysis in Chapter 1 via the geometry of a simple example.
We saw that the values of the decision variables and those of the slack and surplus variables remain unchanged
even though so