1
Introduction
1.1
THE ORIGINS OF OPERATIONS RESEARCH
Since the advent of the industrial revolution, the world has seen a remarkable growth in
the size and complexity of organizations. The artisans small shops of an earlier era have
evolved into the billi
2
Overview of the
Operations Research
Modeling Approach
The bulk of this book is devoted to the mathematical methods of operations research (OR).
This is quite appropriate because these quantitative techniques form the main part of what
is known about OR.
7
Other Algorithms for
Linear Programming
The key to the extremely widespread use of linear programming is the availability of an
exceptionally efficient algorithmthe simplex methodthat will routinely solve the largesize problems that typically arise in p
20
Forecasting
How much will the economy grow over the next year? Where is the stock market headed?
What about interest rates? How will consumer tastes be changing? What will be the hot
new products?
Forecasters have answers to all these questions. Unfort
14
Game Theory
Life is full of conflict and competition. Numerous examples involving adversaries in conflict include parlor games, military battles, political campaigns, advertising and marketing campaigns by competing business firms, and so forth. A basi
4
Solving Linear
Programming Problems:
The Simplex Method
We now are ready to begin studying the simplex method, a general procedure for solving
linear programming problems. Developed by George Dantzig in 1947, it has proved to be
a remarkably efficient m
16
Markov Chains
The preceding chapter focused on decision making in the face of uncertainty about one
future event (learning the true state of nature). However, some decisions need to take into
account uncertainty about many future events. We now begin l
8
The Transportation and
Assignment Problems
Chapter 3 emphasized the wide applicability of linear programming. We continue to
broaden our horizons in this chapter by discussing two particularly important (and related)
types of linear programming problems
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5
The Theory of the
Simplex Method
Chapter 4 introduced the basic mechanics of the simplex method. Now we shall delve a
little more deeply into this algorithm by examining some of its underlying theory. The
first section further develops the general geome
17
Queueing Theory
Queues (waiting lines) are a part of everyday life. We all wait in queues to buy a movie
ticket, make a bank deposit, pay for groceries, mail a package, obtain food in a cafeteria,
start a ride in an amusement park, etc. We have become
6
Duality Theory and
Sensitivity Analysis
One of the most important discoveries in the early development of linear programming
was the concept of duality and its many important ramifications. This discovery revealed
that every linear programming problem h
13
Nonlinear Programming
The fundamental role of linear programming in OR is accurately reflected by the fact that
it is the focus of a third of this book. A key assumption of linear programming is that all
its functions (objective function and constraint
21
Markov Decision
Processes
Chapter 16 introduced Markov chains and their analysis. Most of the chapter was devoted
to discrete time Markov chains, i.e., Markov chains that are observed only at discrete
points in time (e.g., the end of each day) rather t
25W
Reliability
The many deﬁnitions of reliability that exist depend upon the viewpoint of the user.
However, they all have a common core that contains the statement that reliability,
[2(1), is the probability that a device performs adequately over the in
Appendix 6
Simultaneous Linear
Equations
Consider the system of simultaneous linear equations
allxl + “u": + "'+ alnxn = biv
auxl + 422x: + . . + aux" = b2,
amxl + amzxz + - - t + amx" = bm.
It is commonly assumed that this system has a solution, and a un
Supplement to Chapter 18
The Evaluation of Travel Time
As discussed in Sec. 18.4, one of the important considerations for deciding how many
service facilities to provide is the amount of time that customers must spend traveling
to and from a facility. The
Supplement to Appendix 3.1
More About LINGO
Appendix 3.1 describes and illustrates how LINGO can be used to formulate and solve relatively
small models. We now will show how LINGO can formulate a huge model like the one for the production
planning problem
Supplement to Chapter 8
An Algorithm for the Assignment Problem
In Sec. 8.3, we pointed out that the transportation simplex method can be used to
solve assignment problems but that a specialized algorithm designed for such
problems should be more efficien
11
Dynamic Programming
Dynamic programming is a useful mathematical technique for making a sequence of interrelated decisions. It provides a systematic procedure for determining the optimal combination of decisions.
In contrast to linear programming, ther
in 1797 Lagrange published in Paris his treatise ‘ _
on the Theory of Analytic Functions. In this ﬁindamental
book, he clearly describes his method for the
solution of constrained optimization problems:
“.when a function of several Variables has to be
unm
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Use the LINDO output in Figure 6 to answer the following
questions:
a If only 40 acres of land were available, what would
Leary’s proﬁt be?
II If the price of wheat dropped to $26, what would be
the new optim
Elements of Linear Algebra
A matrix is any rectangular array of nubmer
12 123 1 column [3 4] row
3 4 ’ 4 5 6 ’ 2 vector ’ 7 vector
if A has In rows and n columns then
col j
the order of the matrix A is mxn.
A=B<=>aij=bij [0, 0, ., 0] zero row vector.
Scal