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
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,
Practice
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IEOR 4004
Other methodologies for Integer Programming
1
Frequency assignment revisited
In the following picture we show (in blue) the signal frequencies that can be operated at any of the nodes. Each node
needs to pick one frequency, or it can choose not
IEOR 4004
Integer Programming Models
Integer Programming problems are optimization models where some or all variables are required to take integral
values. Frequently, these variables are in fact binary i.e. they must take value 0 or 1; this choice is use
IEOR 4106, Final Exam, Columbia University
Prof. S. Kou, Dec 21, 2004, from 4:10pm to 7pm.
1. (10 pts). Suppose cars arrive at a gas service station according to a Poisson process with
a rate cars per hour.
(a) Let Y be the number of arrivals within t hou
Midterm Solution. Fall 2005.
E3106. Prof. S. Kou, Columbia University
1. The state space of the Markov chain is cfw_(i, j)i 1, j 1. Suppose
that the Markov chain is currently in the state (i, j). Let T denote the time the
Markov chain stays in state (i,
IEOR 4106, Final Exam, Columbia University
Prof. S. Kou, Dec. 21, 2004
1. (10 pts). (a). Y has a Poisson distribution with parameter t. E[Y ] = t.
(b). X has an exponential distribution with rate . E[X] = 1/.
2. (10 pts). (M/M/1 queue). This is a birthde
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 4004
More integer programming models
1
Multilevel location problems
A company is designing a distribution network. In the following figure,
8
25
4
1
100
9
5
2
760
10
6
3
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7
1000
400
12
In the figure, the square nodes indicate warehouses, the diamo
Practice
Questions
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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 4004
Lecture 3  Intro to Linear Programming Algorithms
1
Motivation
We assume an LP in standard form:
(LP):
Subject to
max wT x
(1a)
Ax = b
x 0.
(1b)
(1c)
Here we are assuming that A has m rows and n columns, and so x 2 Rn and b 2 Rm . The
algorithm
IEOR 4004
Simplex Method: duality
1
Pricing interpretation
Consider a manufacturing problem with two resources, blocks of wood and cans of paint, and two products,
toy soldiers and toy trains.
Manufacturer
Max 3x1 + 2x2
x1 + x2 80 [wood]
2x1 + x2 100 [pai
IEOR 4004
Network Problems, III
1
MinimumCost Flow problem
A delivery company runs a delivery network between major US cities. Selected cities are connected by routes as
shown below. On each route a number of delivery trucks is dispatched daily (indicate
IEOR 4004
Simplex Method: duality
1
Pricing interpretation
Consider a manufacturing problem with two resources, blocks of wood and cans of paint, and two products,
toy soldiers and toy trains.
Manufacturer
Max 3x1 + 2x2
x1 + x2 80 [wood]
2x1 + x2 100 [pai
IEOR E4004: Deterministic Models
Assignment 3: Due October 26
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
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