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Unformatted text preview: . ALL RIGHTS RESERVED Roam Glam; COPYRIGHT © 1981 BY exceﬁﬂ’s Ptorw an Article frOm JUNE. I981 VOL. 244. SCIENTIFIC
AMERICAN NO. 6 The Allocation of Resources
by Linear Programming Abstract, crystallike Structures in many geometrical dimensions can help to solve problems in planning and management. A new algorithm has set upper limits on the complexity ofsucli problems ( :onsider the situation of a small
brewery whOse ale and beer are
always in demand but whose pro duction is limited by certain raw materi als that are in short supply. Suppose the
scarce ingredients are com, hops and
barley malt. The recipe for a barrel of
ale calls for the ingredients in propor—
tions different from those in the recipe
for a barrel of beer. For instance, ale
requires more malt per barrel than beer
does. Furthermore, the brewer sells ale at a proﬁt of $ 13 per barrel and beer at a proﬁt of $23 per barrel. Subject to these conditions, how can the brewer maxi—
mize his proﬁt? It may seem that the brewer‘s best
plan would be to devote all his resources
to the production of beer, his more prof
itable product. This choice may not be
well advised, however, because making
beer may consume some of the avail~
able resources much faster than making
ale does. If ﬁve pounds of corn are re
quired for brewing a barrel of ale and 15
pounds are needed for brewing a barrel
of beer, it may be possible to make three
times as much ale as beer. Moreover, in
brewing only beer the brewer may ﬁnd
that all his corn is used up long before
his supplies of hops and malt are ex
hausted. It may turn out that by produc
ing some beer and some ale he can take
better advantage of his resources and
thereby increase his proﬁt. Determining
such an optimum production program is
not a trivial problem. It is the kind of
problem that can be solved by the tech
nique of linear programming. Linear programming is a mathemati~
cal ﬁeld of study concerned with the ex
plicit formulation and analysis of such
questions. It is a part of the broader ﬁeld
of inquiry called operations research, in
which various methods of mathematical
modeling and quantitative analysis are
applied to large organizations and un
dertakings. Linear programming was
developed shortly after World War H
in response to logistical problems that 126 by Robert G. Bland arose during the war and immediately
after it. One of the earliest publications
on the uses of linear programming dis
cussed a model of the 1948 Berlin airlift. Although the computer is an indis~
pensable tool for solving problems in
linear programming, the term "pro
gramming” is employed in the sense of
planning, not computer programming.
“Linear” refers to a mathematical prop
erty of certain problems that simpliﬁes
their analysis. in the brewery problem
the amount of any one resource needed
to make either ale or beer is assumed
to be proportional to the amount of
the beverage produced. Doubling the
amount of beer doubles the amount of
each ingredient required for the brewing
of beer, and it also doubles the prof
it attributable to the sale of beer. If
the amount of corn consumed in mak»
ing beer is plotted as a function of the
amount of beer produced, the graph is a
straight line. In order to apply the tech
niques of linear programming one must
also assume that products and resources
are divisible, or at least approximately
so. For example, half a barrel of beer
can be produced, and it has half the val
ue of a full barrel. Problems in linear programming are
generally concerned with the al
location ol scarce resources among a
number of products or activities, under
the proportionality and divisibility con—
ditions I have described. The scarce re
sources may be raw materials, partly
ﬁnished products, labor, investment
capital or processing time on large, ex
pensive machines. An optimum alloca—
tion may be one that maximizes some
measure of beneﬁt or utility, such as
proﬁt, or minimizes some measure of
cost. In this era of declining productivi
ty and dwindling resources a technique
that can aid in allocating resources with
the greatest possible efﬁciency may be
well worth examining. The properties of linearprogramming problems derive from elementary prin
ciples of algebra and geometry. Solving
such problems eﬂiciently depends on
algorithms. or stepbystep procedures,
that cleverly exploit these algebraic and
geometrical principles. The algorithms
are also simple in conception, although
the details of their Operation can be
quite intricate. It is the efﬁciency and
versatility of a single algorithm called
the simplex method that is largely re
sponsible for the economic importance
of linear programming. The simplex method was introduced
in 194'}r by George B. Dantzig, who is
now at Staaford University. It has sub
stantial value because it is fast, it is rich
in applications and it can answer impor
tant questions about the sensitivity of
solutions to variations in the input data.
Such questions might take the follow
ing form: How should the production
plan of the brewer change if the avail
ability of hops or the proﬁtability of
beer is altered? How much should he be
willing to pay for additional supplies of
the scarce resources? What price should
he ask from another entrepreneur who
wants to buy some of his supplies? The
simplex method can help in determin
ing whether to buy a machine or sell
one, whether to borrow money or lend
it and whether to pay overtime wages
or discontinue overtime labor. With
the simplex method one can also im
pose additional constraints and solve the
problem again to examine their effect.
For example, the method can quickly
tell an entrepreneur the cost of provid
ing an unproﬁtable service in order to
maintain the goodwill of a customer. The simplex method has proved ex~
tremer eﬁicient in solving complex lin—
earprogramming problems with thou
sands of constraints. For theoreticians,
however, its Speed in solving such prob
lems has been somewhat puzzling.
There is no deﬁnitive explanation of
why the method does so well. Indeed.
there are problems devised by mathe‘ maticians for which the simplex method
is intolerably slow For reasons that are
not entirely clear. such problems do not
seem to arise in practice.
Mathematicians in the U.S.S.R. have
recently developed a new algorithm for
linear programming that in a certain
sense avoids some of the theoretical dif
ficulties that have been attributed to the
simplex method. The development was
reported in frontpage articles in news papers throughout the world, which sug—
gests the economic signiﬁcance of linear
programming today. Unfortunately the
new algorithm. which is called the ellip
soid method. has so far shown little
prospect of outperforming the simplex
method in practice. For now there is
a curious divergence of practical and
theoretical measures of computational
performance. Even when an efﬁcient algorithm is employed. the setup costs of solving a
large linear—programming problem can
be considerable. Expressing a practical
set of circumstances in terms of linear
programming is not a trivial enterprise.
and neither is the collection and organi
zation of the data describing the circum
stances. Moreover, solving the problem
is feasible only with the aid of a high
speed computer. Nevertheless. the economic beneﬁts SIMPLEX METHOD of solving problems in linear programming
ﬁnds an optimum allocation of resources by moving from one vertex
to another along the edges of a polytope, a threedimensional solid
whose faces are polygons. Each point in the region vrbere the polytope
is constructed corresponds to a particular program, or plan, for allo
cating labor, capital or other resources. Associated with each such
program is a net beneﬁt or a net cost. The object of linear program
ming is to ﬁnd the program with the maximum beneﬁt or the mini
mum coat. The polytope deﬁne a region of feasibility: all programs
of allocation repreSented by a point within the polytope or on its sur face are feasible, whereas those represented by a point outside the
polytope are infeasible because of a scarcity of resources. “hen the
relation between resources and beneﬁts or costs is linear, the maxi
mum and minimum values must lie at one of the vertexes of the poly
tope. The simplex algorithm examines the vertexw selectively, trac
ing a path (ABC. . . M) along the edges of the polytope. At each step
along the path the measure of beneﬁts or costs is improved until at
the point M the maximum or minimum is reached. Often there are
many paths from A, the starting vertex, to M. The polytope need not
be threedimensional and it commonly has thousands of dimensions. 12'.lr LABORATORY
6 7 B \I \ ““““‘ BUILDING A BUILDING A BUILDING B BUILDING C BUILDING A
BUILDING B BUILDING C ASSIGNIHEZVT PROBLEM seeks to minimize the cost of matching
buildings available for renovation with functions that must each be
conﬁned to one building. With three buildings and three functions
there are 32, or nine. renovation costs that must be considered. The
costs [which are given here in millions of dollars) are conveniently
represented in a matrix of numbers: a feasible assignment picim one 123 LIBRARY TENNIS COURTS 1 6 2 2 5 4 BUILDING C 2+5+6=14 number from each row and from each column of the matrix. There
are 3 X 2 X I. or six, ways of accomplishing this. In general, for as
signments of size nbyn. the number of assignments is n factorial
(written III}. which is equal to n multiplied by all positive integers less
than it. Because the value of it! increases rapidly. determining the min
imum cost by enumeration at all possible assignments is impractical. of linear programming are often sub—
stantial. In the midl950‘s. when linear
programming methods were developed
to guide the blending of gasoline, the
Exxon Corporation began saving from
2 to 3 percent of the cost of its blend—
ing operations. The application soon
spread within the petroleum industry to
the control of additional reﬁnery opera—
tions, including catalytic cracking, dis—
tillation and polymerization. At about
the same time other industries, notably
paper products, food distribution, agri
culture, steel and metalworking, began
to adopt linear programming. Charles
Boudrye of Linear Programming, Inc,
of Silver Spring, Md, estimates that a
paper manufacturer increased its proﬁts
by $15 million in a single year by em
ploying linear programming to deter
mine its assortment of products. Today “packages” of computer pro
grams based on the simplex algorithm
are offered commercially by some 10
companies. Roughly 1,000 customers
make use of the packages under licenses
from the developers. Each customer
pays a sizable monthly fee, and so it is
likely that each makes use of the meth
od regularly. Many more organizations
have access to the packages through
consultants. In addition specialpurpose
programs for solving problems of ﬂows
in networks have been developed, and
they may be in even wider service than
the generalpurpose linearprogram
ming algorithms. The broad applicability of linear pro
gramming can be illustrated even with
in a single organization. Exxon current
ly applies linear programming to the
scheduling of drilling operations, to the
allocation of crude oil among reﬁneries,
to the setting of reﬁnery operating con
ditions, to the distribution of products
and to the planning of business strategy.
David Smith of the Cemmunication and
Computer Sciences Department of Ex
xon reports that linear programming
and its extensions account for from 5 to
10 percent of the company's total com
puting load. This share has kept pace for
the past 20 years with rapidly expand
ing general applications of information
processing. Athough the simplex method is a pow
erful tool, it is founded on elemen
tary ideas. in order to understand some
of these ideas it is useful to examine a
specially structured allocation problem
called the assignment problem. Consid
er the situation of a university planning
committee with three buildings avail
able for renovation and three functions
the buildings are to serve. Suppose the
functions are those of a laboratory, a.
library and indoor tennis courts, so that
there can be only one function to a
building. The tables in the illustration
on the opposite page indicate the cost
of renovation for each of the nine possi ble matches of a building with a func~
tion. How can the cummittee minimize
the renovation cost? The problem can be solved by mark
ing three of the squares in the 3by3
table. Precisely one Square must be
marked in each row and one must be
marked in each column, so that each
building has a function and all the func
tions are accommodated. The solution
being sought is the one in which the sum
of the costs in the marked squares is as SUB‘i'HACT 4
SUBTHACT 6 small as possible. Finding an optimum
solution is not dilﬁcult because there are
only a few possible ways of marking the
squares. Once one of the three squares in
the ﬁrst row is marked, only two squares
in the second row remain available for
marking. In the third row the choice is
forced, because only one square appears
in a heretofore unmarked column.
Hence there are 3 X 2 X l, or six, ways
of making the assignment. It is an easy
matter to enumerate all six possibilities, SUBTFIACT 7
SUBTFlACT 1
SUBTRACT 6 0’ SUBTFtACTS
4: SUBTHACTZ SUBTFIACT 2 SUBTRACT 5 ‘0 SUBTRACT 5 SU BTFIACT 6
SUBYHACT 4 SUBTRACT 8
SUBTFIACT 5
SUBTRACT 3
SUBTHACT 1
SUBTRACT 0 SUBTRACT 7 CONVINCING A SKEP‘I'IC that an assignment is optimum (colored arms in upper illustra
tion) does not entail evaluating the cost of all the feasible assignments. Here the cost matrix is n
lObylﬂ one, and so there are 10!, or 3,628,800, possible assignments. If a number is subtract
ed from each entry In a row or column, the number is also subtracted from the total cost of any
possible assignment null the relative costs remain unchanged. The reason is that every assign
ment picks exactly one number from the transformed row or column of the matrix. The set ot
numbers to be subtracted can be chosen so that the original matrix is transformed into n matrix
that has no negative entries Ind has at least one entry with a cost of zero in each row and col
umn (loner illustration). Becattse no assignment can have a total cost less than zero the opti
mum nssignment for the transformed matrix must be one that has 1 total cost of zero. It lol
lows that the entries in the corresponding positions in the original matrix are also optimum.
Efﬁcient algorithms have been devised for generating the set of numbers to be subtracted. I29 CORN
480 POUNDS HOPS
160 OUNCES BARLEY MALT
1,190 POUNDS 5 POUNDS CORN 15 POUNDS CORN 4 OUNCES HOPS
35 POUNDS MALT 4 OUNCES HOPS
20 POUNDS MALT $13 PROFIT $23 PROFIT MOST
PROF ITABLE PRODUCT MIX BREWER’S DILEMMA illustrates an application of linear programming to problems of op
timizing the apportionment of resources among various products. The brewer’s production of
beer and ale is limited by the scarcity of three essential ingredients: corn, hops and malt. Feasi
ble production levels are determined not only by the total amount of each ingredient on hand
but also by the proportions of the ingredients required to make the two products. The objective
function, or the quantity to be optimized, is the brewer’s proﬁt. In linear programming all the
resources, all the products and all the beneﬁts are assumed to be divisible quantities: the brewer
can use half a pound of corn, sell a fourth of a barrel of ale and realize proportional proﬁts. \ 130 evaluate the total cost of each one and
select the least expensive assignment.
This enumerative approach solves the
problem of a 3by3 matrix, but it be
comes impractical for larger problems.
Suppose there are four buildings and
four functions; the number of possible
assignments is then 4 X 3 X 2 X 1, or
24. In the general statement of the prob
lem there are 71 buildings and n func
tions, and the number of assignments is
n factorial (written n!), which signiﬁes n
.multiplied by all the integers from 1 to n — 1. For n = 10 there are 10!, or more
than 3.6 million, distinct assignments. The rapid growth of n! dispels any en
thusiasm one might have for the enu
merative method. Suppose one had to
solve a 35by35 assignment problem by
enumeration and one had a computer
that could sort through the possible as
signments, evaluate the cost of each one
and compare it with the lowestcost as
signment encountered up to that point at
a rate of a billion assignments per sec
ond. (A computer capable of this speed
would be much faster than any availa
ble now.) Even if the task of enumerat
ing the 3 5! assignments were to be shared
by a billion such computers, only an in
signiﬁcant fraction of the required com
putations would be completed after a
billion years. The.3Sby~35 assignment problem is
not large. If the problem were one of
assigning personnel to jobs in order to
minimize the total cost of job training, It
might well be equal to 35. There are nu
merous other assignment problems in
which n is equal to 1,000 or more. Clear
ly such problems require a procedure
cleverer than enumeration. he burden of enumeration might be
greatly reduced if one could avoid
examining assignments that are costlier than the ones already checked. This ef fect would be achieved if there were
a stopping rule or optimality criterion
that would allow an optimum assign
ment to be recognized quickly once it
was encountered. Any algorithm that in
corporates such a criterion offers impor
tant collateral beneﬁts. The beneﬁts are
summarized by what Jack Edmonds of
the University of Waterloo in Ontario
calls “the principle of the absolute su
pervisor,” or what might also be called
“the problem of the skeptical boss.”
Suppose after tedious enumeration
you have solved the 10by10 assign
ment problem indicated in the upper il
lustration on the preceding page by ex
amining all 3,628,800 possible assign
ments. The optimum assignment, you
maintain, is the one shaded in color in
the illustration. You present the solu
tion to your boss. who looks you in the
eye, puﬂs on hafcigar and demands,
“How do I know there is no less costly
solution?” You might swallow hard at this question, because it seems the only
way to demonstrate the merits of your
soiution is to repeat the examination
of all 3,628,800 possibilities under the
scrutiny of your boss. A stepping rule oﬁ’ers a concise proof
of optimality. Suppose you go to your
boss not only with the optimum assign
ment but also with a set of numbers
to be subtracted from the entries in
each row and column. Setting forth how
these numbers are obtained would re
quire a detailed discussion of the assign
ment problem; it will sufﬁce to point out
that the set of numbers can be speciﬁed
by an efﬁcient computer algorithm. The
utility of the numbers, once they have
been found, is readily appreciated. Note
that subtracting the same number from
every entry in a given row or column is
equivalent to subtracting this amount
from the total cost of every pessible as
signment. The reason is that every feasi— ble assignment must choose one and
only one entry from each row and each
column. For example, if 5 is subtracted
from every entry in the sixth row, every
possible assignment will include exact
ty one entry that is 5 less than the cor—
responding assignment made with the
original array of costs. The relative costs
of all the assignments will therefore re
main unchanged. Such subtraction can
be done repeatedly, prov'tded it is always
applied uniformly to every entry in a
given row or column. By means of repeated subtraction you
can transform the original cost matrix
into the matrix shown in the lower illus
tration on page 129. The latter array of
costs has a remarkable property. You
can now point out to your boss that the
costs corresponding to the squares se
lected by your assignment are all zero...
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