Linear Programming: Algorithms and applications

156 Pages·1981·3.56 MB·English

by  S. Vajda (auth.)

Linear Programming
Algorithms and applications
Linear Programming
Algorithms and applications
S. VAJDA
Visiting Professor
University of Sussex
LONDON NEW YORK
CHAPMAN AND HALL
First published in 1981
by Chapman and Hall Ltd
11 New Fetter Lane, London EC4P 4EE
Published in the USA by
Chapman and Hall
in association with Methuen, Inc.,
733 Third Avenue, New York NY 10017
© 1981 S. Vajda
Softcover reprint of the hardcover 1s t edition 1981
Typeset by Style set Limited, Salisbury, Wiltshire
ISBN-13: 978-0-412-16430-9 e-ISBN-13: 978-94-011-6924-0
DO I: 10.1 007/978-94-011-6924-0
This paperback edition is sold subject to the condition that it shall
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All rights reserved. No part of this book may be reprinted, or
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British Library Cataloguing in Publication Data
Vajda, Steven
Linear programming. - (Science paperbacks; 167).
1. Linear programming
I. Title II. Series
519.7'2 T57.74 80-41232
Contents
Preface vii
1. Linear Programming
2. Algorithms 16
3. Duality 50
4. Theory of Games 65
5. Transportation and Flow in Networks 78
6. Integer Programming 116
7. Linear Programming under Uncertainty 135
Answers to Problems 140
References 148
Index 149
Preface
This text is based on a course of about 16 hours lectures to students
of mathematics, statistics, and/or operational research.
It is intended to introduce readers to the very wide range of
applicability of linear programming, covering problems of manage
ment, administration, transportation and a number of other uses
which are mentioned in their context.
The emphasis is on numerical algorithms, which are illustrated by
examples of such modest size that the solutions can be obtained
using pen and paper. It is clear that these methods, if applied to
larger problems, can also be carried out on automatic (electronic)
computers. Commercially available computer packages are, in fact,
mainly based on algorithms explained in this book.
The author is convinced that the user of these algorithms ought
to be knowledgeable about the underlying theory. Therefore this
volume is not merely addressed to the practitioner, but also to the
mathematician who is interested in relatively new developments in
algebraic theory and in some combinatorial theory as well. The
chapters on duality, and on flow in networks, are particularly directed
towards this aim and they contain theorems which might not be
directly relevant to methods of computation. The application of the
concept of duality to the theory of games is of historical interest.
It is hoped that the figures, which illustrate the results, will be
found illuminating by readers with active geometrical imagination.
The references record the sources which contributed to the
foundation and to the development of the subject. However, the
contents of this book are meant to be self-contained.
A further development of linear programming is the field of non
linear or more generally of mathematical programming, but this is
beyond the scope of the present exposition.
The mathematics used are elementary. Of algebraic concepts we
use those of vectors and matrices and of their multiplication. In one
place we refer to the solution of linear homogeneous equations,
dependent on their rank.
CHAPTER 1
Linear programming
To introduce our topic, we start with two examples.
Example 1.1
A manufacturer produces two types of cloth, TI and T2. For a unit
length of TI he needs 4 units of some raw material Rt. 5 units of
raw material R2, and 1 unit of raw material R3• For a unit length of
T2 the requirements are, in the same order, 1,3 and 2 units, respect
ively.
The amounts of R I, R 2 and R 3 which are available are, respect
ively, 56, 105, and 56.
The manufacturer estimates that from the sale of one unit of TI
he can make a profit of 4 (in some monetary unit), and that from
the sale of one unit of T2 his profit will be 5.
He wants to know how much of Tl and of T2 he should produce
from the available raw material, so as to maximize his total profit.
We assume that he can sell all he produces.
We shall now formulate this problem algebraically. Denote the
amount of T I produced by x I, and that of T2 by X2. He then needs
4x I + X2 of raw material R I, and this must not exceed 56. Thus
4xI +X2 ~ 56
Similarly, regarding R 2 and R 3
5xI + 3X2 ~ 105
and
xI+2x2~56
His profit will be 4xI + 5x2 = B (for 'benefit'), and he should choose
XI and X2 so as to maximize this expression, not forgetting that
neither XI nor X2 must be negative; explicitly
XI ~0,X2 ~O.
2 LINEAR PROGRAMMING
We consider now a second problem, which is called the 'nutrition
problem' in the literature of our subject.
Example 1.2
There are two types of food on the market, F land F 2. They contain
three different nutritional ingredients, N 1 , N 2 and N 3. One unit of
Fl contains, respectively, 3, 4 and 1 units of these, and one unit of
F2 contains 1,3 and 2 units.
A housewife has been told that she needs, for her family, 3 units
of Nl, 6 of N2, and 2 of N3. She wants to obtain these ingredients
by purchasing the available food items at the smallest total cost. One
unit of Fl costs 20, and one unit of F2 costs 10 (in some monetary
unit).
The housewi(e's problem can be formulated as follows, if Yl is
the amount of Fl andY2 that of F2 to be bought.
Minimize
20Yl + lOY2 = C(for 'cost')
subject to
3Yl + Y2 ~3
4Yl+3Y2~6
Yl+2Y2~2
Yl ~0,Y2 ~O
We have written the conditions as inequalities, because it might be
cheaper to obtain more of some ingredient than necessary if it is
contained abundantly in some other food item which supplies other
ingredients as well.
The two examples which we have given are instances of linear
programming problems. In such formulations it is required to
optimize (maximize or minimize) a linear expression, the 'objective
function', subject to linear inequalities, the 'constraints'.
There is no fundamental algorithmic difference between maximiz
ing and minimizing, because max [(x) equals -min [-[(x)].
In the maximizing problem the left hand side of a constraint was
not larger than the right hand side (a constant), while in the minimiz
ing problem the opposite. was true. Again, there is no fundamental
LINEAR PROGRAMMING 3
algebraic difference between the two cases, because if [(x) ~ c, then
-[(x) ~-c.
An inequality a 1X l + ... + anXn :>;; c can be written as an equation
alxl+···+anXn+Xn+l=C, with Xn+l~O. We call such an
additional variable a 'slack' variable. In an inequality alXl + ... +
anXn ~ C we would subtract a non-negative slack variable from the
left hand side to convert it into an equivalent equation.
Conversely, an equation a 1 x 1 + ... + anXn = C can be written as a
pair of two simultaneous inequalities, thus
and
Another way of converting the given equation, in non-negative
variables, into an inequality, consists in choosing one of the aj which
is not zero, sayan, solving for Xn, namely
and writing
If a variable, say x, is not required to be non-negative, then in order
to keep within the usual framework we replace it by two new
variables which are required to be non-negative, thus x = x' - x".
We now define the term linear programming. We mean by it maxi
mizing or minimizing a linear function of non-negative variables,
subject to linear constraints (equations or inequalities).
In algebraic matrix notation we have a linear programming prob
lem. Maximize b'x, subject to Ax:>;; c, where A is a matrix of m rows
and n columns, c is an m-vector, and x as well as bare n-vectors. The
co-ordinates of x are the variables to be determined, while A, b, and
c are known. The vector x is also required to have non-negative
co-ordinates (transposition of a vector or of a matrix will be denoted
by a prime).
If we introduce slack variables explicitly, then we have Ax +
Imx = c, where 1m is the identity matrix of order m, and x is the
4 LINEAR PROGRAMMING
m-vector of slack variables. The number of all variables is then
n +m =N, say.
In order to obtain a diagrammatic representation, we think of the
components of x as ordinates of a point in a space of n dimensions.
=
In the examples above we had n 2, and we can therefore use dia
grams in a plane. The diagram for Example 1.1 is Fig. 1.1. We have
drawn
=
the Xl-axis X2 0
the x2"axis Xl =0
and the straight lines
4XI +X2 = 56
5XI + 3X2 =1 05
Xl + 2x2 =5 6
Introducing slack variables x3, X4, and Xs into the inequality con
straints, the last three equations can be written
X3 =0,X4 =O,xs =0
Every straight line of the diagram is thus described by one of 5
variables being zero.
Xl = 0
\\
\\
\\
'~~'-""
..... ",~~. ..
X2 = 0
Figure 1.1