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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
and that no entries in the matrix are less
than zero. Since the sum of the costs FEASIBILITY REGION in the brewer‘s problem is made up of the intersection of ﬁve half
planes. Any point (A,B) tn the plane corresponds to a production program that calls {or mak
lllg A barrels of ale and B barrels of beer. The ﬁrst three half planes graphically represent all
the production programs that are achievable, given the available quantities of each ingredient.
For example, the amount of corn required is SA + 153 (the weight in pounds of the corn need—
ed to make a bane! of ale times A plus the Weight o! the corn needed to make a barrel of beer
times H). Thls quantity must not exceed the 480 pounds of corn available to the brewer. Hence any point in the half plane to the lower left of the line 5A + 158 = 480 represents a feasible pro duction program that requires no more than the 480 pounds of com available. The half plan associated with hops and malt can be constructed in a slmllar way. The remaining two half planes express the fact that only those programs having nonnegatlve production are feasible. 13?. marked by your assignment is zero and
there are no negative costs, no other pos
sible assignment can have a lower cost.
In short, you have shown your boss, with
a few hundred calculations rather than
tens of millions, that no assignment can
be less expensive than the one you have
chosen. lthough l have not demonstrated how to solve an assignment problem or
how to ﬁnd the row and column num‘
bers to subtract, the assignment prob
lem does illustrate the necessity of
avoiding enumeration and the possibili
ty of doing so by means of a stopping
rule that recognizes an optimum solu
tion. These are design characteristics of
algorithms that can be applied not mere—
ly to the assignment problem but to lin—
earprogramming problems in general. Consider again the situation of a
brewer whOSe two products, ale and
beer, are made from diﬁ'ercnt propor
tions of corn, hops and malt. Suppose
480 pounds of corn. 160 ounces of hOps
and 1,190 pounds of malt are immedi
ately available and the output is limited
by the scarcity of these raw materials.
Other resources, such as water, yeast.
labor and energy, may be consumed in
the manufacturing process, but they are
considered to be plentiful. Although
they may inﬂuence the brewer‘s willing
ness to produce beer and ale because of
their casts, they do not directly limit
the ability to produce. ASsume that the
brewing of each barrel of ale consumes
ﬁve pounds of corn. four ounces of hops
and 35 pounds of malt, whereas each
barrel of beer requires 15 pounds of
corn, four ounces of hops and 20 pounds
of malt. Assume further that all the ale
and beer that can be produced can be
sold at current prices, which yield a
proﬁt of $13 per barrel of ale and $23
per barrel of beer. The scarcity of corn, hops and malt
limits the feasible levels of production.
For example, although there are enough
hops and malt to brew more than 32
barrels of beer, production of this much
beer would exhaust the supply of corn.
allowing no greater output of beer and
no output of ate at all. Another feasible
production program calls for no beer
and 34 barrels of ale, depleting all 1,190
pounds of malt. The ﬁrst alternative
seems preferable to the second. The
proﬁt realized by the ﬁrst program is
32 X $23, or $736, whereas the second
program yields only 34 X $13, or $442. There are other production programs
that are better than either of these. Six
barrels of ale and 30 barrels of beer use
all 480 pounds of corn, 154 of the 160
ounces of hops and 810 of the 1,190
pounds of malt, yielding a proﬁt of
(6 X $13} + (30 X $23), or $768. Many
programs earn even greater proﬁt. In
this case it is not merely impractical to I; VALUE OF THE OBJECTIVE FUNCTION for any point P in the feasible region can be
graphed as the distance ZU’) above or below P measured on the z axis. (Distances along the z
axis have not been drawn to the same scale as distances In the plane of the feasible region.) One
can think of the distance Z(P) as a point in threedimensional space. If the function is linear,
the graph of the values the function takes along any straight line in the feasible region is also
a straight line [upper graph). For any point P in the interior of the region, there is some line
through P that intersects the boundary of the region at two points, say the points R and S (loner
graph). If the line segment in space connecting 2”?) and 2(3) is not parallel to the plane of the
feasible region, the objective function must assume its maximum value along the line segment
at one of its endpoints, say 2(5), which corresponds to the point 5 on the edge of lthe feasible
region. The graph of the objective function along an edge is also a straight line and it too as
sumes its maximum at one of its endpoints, say 2(0), which corresponds to a vertex of the tea
sible region. Thus there is always an edge point that dominates a given interior point, and there
is always a vertex that dominates any point along an edge. To ﬁnd the maximum value Z(M) of
the objective function one need examine only the vertexes. When the feasible region is two
dimensioual, the objective function forms a plane whose maximum height is attained at a vertex. 134 enumerate all the possibilities. as it was
in the assignment problem; here it is not
even possible. There are inﬁnitely many
production programs that meet the con
ditions of the brewer‘s problem. Each
such program is called a feasible solu
tion. Fortunately there is a small set of
feasible solutions called extremal solu
tions to which attention can be conﬁned. he importance of the extremal solu tions becomes apparent when the set
of all feasible solutions is represented
graphically as a set of points in a plane;
the set of points constitutes the feasi
ble region. Let A designatethc number
of barrels of ale brewed according to
any possible production strategy and lcl
B designate the number of barrels of
beer. A and B are known in linear pro
gramming as decision variables. They
can be associated with the coordinate
axes of the plane. Any pointon the plane
can be speciﬁed by a pair of coordinates
(AB), which also correspond to a partic—
ular set of production levels. Because negative levels of production
are not possible, the feasible region can
immediately be conﬁned to the upper
righthand quadrant of the plane. where
A and Bare both nonnegaiive. How does
the scarcity of malt affect production?
Since 35 pounds of malt are needed for
each barrel of ale and 20 pounds are
needed for each barrel of beer. the to
tal amount of malt needed to make A
barrels of ale and B barrels of beer is
35/! + 203. If all 1,190 pounds of malt
are used. 35A + 203 = 1,190. The setol'
points (AB) that satisfy this equation
form a straight line. All points (AB)
corresponding to production plans that
call for more than 1,190 pounds of malt
lie on one side of the line and the points
that require lees malt lie on the other
side. Only the latter set of points and the
points actually on the line are feasible
because of the limited supply of malt. In a similar way the scarcity of hops
conﬁnes the feasible region to one side
of the line 4.4 + 43 = 160 and the scar—
city of corn conﬁnes the region to one
side of the line 5A + 153 = 480. The
points that satisfy all these requirements
make up the feasible region [388 illustra
lion on page 132]. Note that the feasible
region is convex: any line segment that
connects two points in the region (in~
cluding the points on the perimeter
lines) lies entirely within the region. Since the brewer’s proﬁt is $13 per
barrel of ale and $23 per barrel of beer.
his problem is to maximize his total
proﬁt: 13A + 233. in order to do so he
must ﬁnd a point (AB) in the convex
feasible region where 13.4 + 233 has its
maximum value. In linear programming
such a measure of beneﬁts to be maxi
mized (or sometimes of cests to be mini
mized} is called an objective function. The objective function can be incor~
porated into the graph of the feasible region by adding a third dimension. For
any point (A,B) representing a produc~
tion plan the expected proﬁt is given by
the height of the function 13A + 233
above the plane at that point. Clearly
the task confronting the brewer is to
ﬁnd a point in the feasible region where
the objective function is at its greatest
height. If the function had to be evaluat
ed at~all the points, the task would be
impossible, but two distinctive proper
ties of the problem act to narrow the
search. The properties are the convexity
of the feasible region and the linearity of
the objective function. Since the region is convex, any point
in the interior of the region can be in
cluded in a line segment whose end—
points lie on the boundary of the region.
(Indeed, an inﬁnite number of such line
segments can be drawn through any giv
en point; which line segment is chosen
is immaterial.) In the space above each
such line segment it is possible to con
struct a graph of the objective func—
tion giving the proﬁt for each point on
the line segment. Since the objective
function is linear, the graph is a straight
line [see illustration on page 134]. The
straightline graph may be parallel to
the plane, in which case all the brewing
strategies along the line segment have
the same proﬁt. If the graph of the objec
tive function over the line segment is not
parallel to the plane, it must assume its
‘ maximum value at one of the two end
points, which lie on the boundary of the
feasible region. Hence the maximum of
the objective function over such a line segment must always be attained at one
of the points where the line segment in
tersects the boundary. Because the same
analysis can be applied to any line seg
ment in the feasible region, it follows
that the overall maximum of the objec
tive function is invariably found some
where on the boundary of the region.
The brewer in search of maximum proﬁt
can ignore the entire interior of the fea
sible region and consider only those
brewing strategies that correspond to
points on the boundary. The analysis can be taken a step fur
ther by the same argument. If the fea
sible region is a polygon, every point
on the boundary lies on a line segment
whose endpoints are two‘of the vertexes
of the polygon. A graph of the objective
function for such a boundary segment
can be constructed in the same way it is
for a line segment that crosses the interi
or. Again the maximum must be found
at one of the endpoints (or at both end
points if the objective function is con—
stant and parallel to the plane). Thus a
maximum value of the objective func
tion throughout the feasible region can
be found among the vertexes. The brew
er needs only to check, at most, the val
ue of the function at all the vertexes of
the feasible region and select the one
yielding the best proﬁt. He can then be
certain that no other brewing strategy
would bring a higher proﬁt. In the exam
ple considered here there are ﬁve vertex
es. The one at point (12,28), which rep
resents the production of 12 barrels of
ale and 28 barrels of beer, yields a prof it of (12 X $13) + (28 X $23), or $800.
That is the maximum proﬁt the brewer
can realize. he inclusion of an additional con straint (such as a shortage of yeast)
would not signiﬁcantly alter the geomet
rical interpretation of the brewer’s prob
lem. The polygon might then have six
sides instead of ﬁve. The introduction
of a third product, on the other hand,
would have a more profound effect: the
geometrical model would then be three
dimensional. Inequalities in three varia
bles correspond to half spaces deﬁned
by planes in threedimensional space in
stead of half planes deﬁned by lines in
two—dimensional space. The feasible re
gion is no longer a polygon but might
instead look like a cut gemstone, a three—
dimensional polytope whose faces are
all polygons. As the number n of de
cision variables increases further, the
geometrical interpretation remains valid
but it becomes more difﬁcult to visualize
the ndimensional polytope formed by
the intersections of (n — l)dimensional
hyperplanes. Vertexes retain their spe
cial status, however, and their positions
can be determined by algebraic meth
ods that replace geometrical intuition. It may appear that by conﬁning the
evaluation of the objective function to
the vertexes of the feasible region, the
solution of linearprogramming prob
lems by enumeration becomes practical.
As in the assignment problem,_however,
the number of possibilities to be enu
merated grows explosively. A problem
with 35 decision variables and 35 con
straints would be impossible. Dantzig’s simplex method examines
vertexes, but it does so selectively. In the
brewery example the method might be
gin with the vertex at the origin, (0,0).
Here nothing is produced and the prof
it is zero. Following either of the inci
dent edges away from the origin leads
to points with larger objectivefunction
values. The simplex method selects such
an edge, say the B axis, and follows it to
its other end, the vertex (0,32). The pro
gram here calls for making 32 barrels of
beer but no ale, for a proﬁt of $736.
From this vertex an incident edge leads
to still greater objectivefunction values.
The simplex algorithm therefore pro
ceeds to the vertex (12,28) at the other
end of the edge. Here the proﬁt from
brewing 12 barrels of ale and 28 barrels
of beer is $800. At (12,28) all incident
:dges lead in unfavorable directions; the
algorithm therefore halts with a decla
ation that the vertex (12,28) is optimal. In general the simplex method moves
110ng edges of a polytope from vertex to
idjacent vertex, alWays improving the
Ialue of the objective function. The pro—
:edure can begin at any vertex, and it
ialts when no adjacent vertex has a bet
er objectivefunction value than the
:urrent one. This stopping rule is val id only because the feasible region is
convex: convexity guarantees that any
locally maximum vertex is a globally
maximum vertex. One need look only in
the vicinity of each point to tell whether
improvement is possible. How does one tell that an edge lead 8 A—> ing away from a given vertex will im
prove the value of the objective func
tion? The key is the concept of a “mar—
ginal value“ attributed to each resource
at each vertex of the feasible polytope.
For manydimensional problems in lin
ear programming the algebraic method 4—9 CONVEXITY PROPERTY of linear programming states that for any two points to the feasi
ble region a line segment connecting the points must lie entirely within the region. Only the tea
sible region at the left is convex. The region in the diagram at the right is a set of discrete points.
Convexity ensures that any local maximum of a linear objective function is a global maximum. 138 that is commonly employed to choose a
path from vertex to vertex can be under
stood when one understands the signiﬁ
cance of the marginal values. At the maximum vertex in the brew—
cry problem there are 210 pounds of
excess malt that are not utilized in the
production program represented by that
vertex. The addition or subtraction of a
pound of malt from the initial supply of
1,190 pounds Would not change the ob
taiuable proﬁt. On the other hand, an
other ounce of hops would make possi—
ble an increase of $2 in total proﬁt.
This increase is the marginal value of
hops at the maximum vertex. it can be
interpreted as the effect of "pushing"
the constraint line for hops farther
from the origin to reﬂect the extra avail
able ounce of hops [see illustration on
opposite page]. Similarly, the marginal
value of corn at this vertex is $1. Marginal prices have a natural eco
nomic interpretation. If the brewer
could buy an additional ounce of hops.
he could increase his proﬁt by $2. If
hops were available for less than $2 per
ounce, it would be worthwhile to buy
hops. 0n the other hand, if a price high
er than $2 per ounce were offered for
hops. it would be worthwhile for the
brewer to divert hops from the produc
tion of beer and ale and sell them on the
market. This does not mean that the
buying or selling of hops should contin
ue indeﬁniter at $2 per ounce, but for
increases of about 19 ounces or decreas
es of up to 32 ounces, $2 remains the
breakeven price in this example. Marginal values are sometimes called
shadow prices or imputed prices. They
indicate the relative amount that each
scarce resource contributes to the proﬁt—
ability of each item in production. For
example, a barrel of ale requires ﬁve
pounds of corn with an imputed price of
$1 per pound, four ounces of hops with
an imputed price of $2 per ounce and
35 pounds of malt with an imputed
price of zero. The total imputed price
of ale is equal to the proﬁt of $13 per
barrel. Suppose a new product, light beer. is
proposed. Making a barrel of light beer
requires two pounds of corn, ﬁve ounces
of hops and 24 pounds of malt. How
much proﬁt per barrel must be obtained
from light beer to justify diverting re
sources from the production of beer and
ale? The imputed prices of the resources
answer the question. The total imputed
price of the ingredients in light beer is
(2 X $1) + (5 X $2) + (24 X $0), which
is equal to $12. This measures the prof
it that wo uld be lost by diverting resour
ces from the brewing of ale and beer to
the production of one barrel of light
beer. Thus for the brewing of light beer
to be worthwhile it must yield a proﬁt
of at least $12 a barrel. The role of marginal prices in judging {0, 32.0?) the wisdom of introducing a new prod
uct can help to explain how the simplex
method determines which edges of a
polytope lead from a given vertex to a
vertex with an improved value of the (0,32)
objective function. Suppose the vertex
currently under examination is the one
i at (0,32) in the brewery problem. First
one can determine the marginal price of
each resource in this production pro
gram. Malt and hops are in excess sup
ply at this point and so their marginal
" values are zero. Corn. however, is in
short supply and so its marginal value is
positive. Since only beer is being made
at (0,32), the marginal value of the in
gredients in a barrel of beer should be
equal to the proﬁt associated with beer:
$23. Since Only corn has nonzero mar
ginal value and since 15 pounds of corn
are needed for each barrel of beer, the COFIN (11.9. 23.1) [26. 14) (34. D) marginal value of corn for this pro— ductiort program is $23 divided by 15 A—> pounds. or about $1.53 per pound. w. a) What is the marginal value of the in gredients in ale measured at the (0.32) vertex? If this value is less than the proﬁt that can be realized from selling a barrel of ale, the diversion of some resources (12.38. 2133} from beer production to ale would in / crease the total proﬁt of the brewer. Hops Making a barrel of ale requires ﬁve (25.57.1453) pounds of corn. Therefore the marginal / value of the ingredients in a barrel of ale
c is (523/15) X 5. or about $7.67. As in the example of light beer. the marginal value represents the loss in proﬁt that would result from diverting resources from beermaking to the brewing of ale.
‘ Because this value is less than $13 (the proﬁt expected from selling each barrel of ale) it would be proﬁtable to brew more ale than the production program at the point (0,32) specifies. Indeed, by keeping the total consumption of corn constant and diverting corn from beer to l ale, the brewer can increase his profit by $13  $7.67. or $5.33, for each barrel of ale. Since the amount of corn is held constant, producing more ale corre— sponds to moving along the edge of the feasible region that represents the con MARGINAL INCREASES in the availabili
ty of scarce resources alter the potential prof
its of a hypothetical brewer in a predictable
Way. If one additional pound of corn were
available, the maximum proﬁt would increase
by $1, a change in the objective function that
0 is reﬂected in the movement of the maxi
mum vertex in the feasible region from the
colored dot to the colored circle {upper graph).
If there were an extra ounce of hops, the max
imnm proﬁt would increase by $2 (middle
.a graph). A change in the available malt would
not alter the attainable proﬁt: mall is already
a surplus resource (lower graph). The changes
in proﬁt associated with changes in the avail
ability of each resource are known as shadow
prices or imputed pric. They are used to dl f [26.01 13.93] / (34.03, 0} rect the algorithm from vertex to vertex. The
marginal changes are exaggerated for clarity. straint on the supply of com. This is pre
cisely how the simplex method recog
nizes that the objective function can be
improved along this edge. In this case
the edge connects the vertex (0,32) with
the vertex (12,28) of the polygon. ...
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This note was uploaded on 03/30/2008 for the course ORIE 320 taught by Professor Bland during the Fall '07 term at Cornell University (Engineering School).
 Fall '07
 BLAND

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