THE LINEAR PROGRAMMING APPROACH TO PRODUCTION PLANNING
| Author | H. C. MACKENZIE |
| Date | 01 February 1957 |
| DOI | http://doi.org/10.1111/j.1467-9485.1957.tb00216.x |
| Published date | 01 February 1957 |
THE
LINEAR PROGRAMMING APPROACH TO
PRODUCTION PLANNING
I
INTRODUCTION
DURING the last few years, considerable attention has been paid by
American economists, both academic and industrial, to the develop-
ment of a new technique of production planning, known as mathe-
matical programming. More recently, interest has been shown in this
country, and a number of firms have already applied it to some
of
their problems. Successful applications-mainly in the United States-
include the allocation of scarce machines to their most profitable uses,
determination of ‘best’ or minimum-cost blends, and the routing
of
transport.’ In this article an attempt is made to illustrate some of the
basic ideas, and to indicate potential applications
of
the technique.
Discussion
is
confined to the special case of
linear
programming, since,
so
far,
no
general method of solving non-linear programming problems
has been developed.
I1
THE
NATURE
OF
LINEAR
PROGRAMMING
In
its analysis
of
the individual firm, the starting-point of mathe-
matical programming is the same as that of traditional economic
theory.2 The aim of production is to transform inputs into outputs
in such a way as to maximise profits. The problem tackled by the
programming approach arises when the supply
of
some inputs is
limited and cannot be increased in the planning period being con-
sidered.
If
there is a choice of uses for these scarce inputs, and
if
profits vary according to the uses chosen, a programming problem
exists
:
to determine the
‘
optimum allocation
’
of
resources
to
produc-
tion,
‘
optimum allocation
’
being defined as that which maximises the
firm’s
profit in the planning period. The linear programming approach
to this problem consists
of
two steps: (i) its formulation as a
mathematical problem, and (ii) the solution of the mathematical
problem. These are discussed in turn in the following paragraphs.
Two simplifying assumptions will be made: that the firm can sell
all that it can make
of
any product at current market prices, and that
Cf. Henderson and Schlaifer,
‘
Mathematical Programming
’,
Harvard
For a full theoretical discussion see
R.
Dorfman,
Application
of
Linear
Business Review,
June
1954.
Programming to
the
Theory
of
the Firm
(1951).
29
30
H.
C.
MACKENZIE
all except the scarce inputs are available in any desired quantities
at current market prices.
A
mathematical formulation of the allocation problem implies the
use of a theory of production.
In
the literature, the appropriate
programming theory is usually discussed under the name of activity
analy~is.~
It
is not proposed here to discuss the theory in detail, but
it may be useful to indicate briefly how it differs from marginal analysis.
Both linear programming and marginal analysis take as given the
techniques by which inputs are transformed into outputs, and both
assume constant returns to scale. They differ radically, however, in
their conceptions of the nature of the techniques. The difference may
be illustrated by considering the case of a two-factor, single-product
firm.
In
marginal analysis, the relation between inputs and output is
expressed in
the
form of a single-valued production function which
shows the maximum output for any given set of inputs. Since this
function is assumed
to
be continuous within limits set by the tech-
nology, it follows that
if
the quantity
of
one input to be used is fixed.
output can still be increased
or
decreased by varying the quantity
of the remaining input. The firm has therefore to choose the optimum
from an infinite set of possible outputs. Linear programming takes the
view that the technology does not allow such a choice, and that, in
fact, if the quantity of one input
to
be used is fixed, then the quantities
of the remaining input and of output are also determined.
In
other
words, the ratios of inputs to other inputs, and of inputs to outputs,
are fixed. Hence a limited supply of one input would provide an
effective upper boundary to output, and the only possible variation
would be in scale. Given sufficient demand for the product, therefore,
output will be completely determined by the available supply of the
limited input. Let us suppose, however, that the technology provides
not one, but two ways of making the same product (or two different
products), the two ways being distinguished by two sets of inflexible
input-output ratios,
i.e.
some possibility of factor substitution exists.
though not of continuous substitution as in marginal analysis.
In
this
case variations in output become possible since the scarce input can
be allocated to the two methods of production in different quantities,
and the methods can be used at different levels
of
production provided
they can be carried on independently.
In
programming terminology,
the two methods are referred
to
as activities or processes,’ and the
Cf.
Activity Analysis
of
Production and Allocation.
edited
by
T.
C.
Koop-
‘Koopmans,
op.
cit.,
uses the term ‘activjty’ in his general theory
of
in the particular context
of
mans
(1951).
production
;
Dorfman,
op.
cit.,
uses ‘process
the
firm.
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