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THE LINEAR PROGRAMMING APPROACH IN FARM MANAGEMENT ANALYSIS*
Peter E. Hildebrand
What is Linear Programming?
Systematic budgeting.
The most widely used and familiar method of farm management analysis
is the system of budgeting. Budgeting is the process of bringing together
price and production data into an estimate of the outcome of a farming
operations Typically, in the planning of a farm organization, several
alternative plans are tested by the budget method. The alternatives
tested are those which seem most likely to maximize net income within
a framework restricted by the physical limitations of the farm, its
manager and workers; the likes and dislikes of the manager; and the
knowledge of feasible alternatives possessed by those concerned. The
number of alternatives tested is limited, by time and money, to some
relatively small, finite number. Because of this limitation, there is
no guarantee that the high profit organization chosen in this manner is
the optimum organization given the restrictions of available resources.
Linear programming, or activity analysis as it is sometimes called,
is a refinement of the budgeting process. Basically, it is a method of
systematic consideration of all the farm resources and a specified number
of enterprise alternatives, under certain restrictions determined by prior
consideration. If data are available for a complete budgeting process then
linear programming can be applied* This information consists of the res
trictions imposed by the farm and farmer; vizo, size, capital, labor and
management limitations; cropping, livestock and institutional restrictions;
A term paper submitted for Ag. Econ, 541, June 1958.
inputoutput relationships; price and market data; etc. Inputs are combined
on a per unit output basis and then added in a linear relationship. The
process will be described more fully in a later section,
Being a systematic budgeting approach, linear programming can make
use of a much larger number of resource categories than can budgeting,
For this reason, programming is particularly suited to large scale
problems where greater accuracy is desired. Further, a programming
solution is the one optimum solution of all possible solutions within
the restriction framework. That is, linear programming selects an
optimum high profit solution from among all the alternatives examined.
The Theory and Logic of Linear Programming.
Graphic presentation of the logic of linear programming can be
handled in two ways. The first method, limits the problem to two
constraints (or input categories), but any number of choice variables
(or products) can be considered. The problem is constructed with
capital as one constraint and labor as the other. For the purpose
of illustration, it is assumed that all other factors are nonlimiting.
The choice variables are corn, wheat and oatso
Relative to the other products, corn is a large user of capital
and light user of labor Oats is a relatively large user of labor and
uses less capital. Tiheat is intermediate In Figure I, OC, the corn
process ray shows the combination of capital and labor used in corn
production The wheat process ray, OW, and the oats process ray, 00,
show the combination of capital and labor used in producing these two
commodities, The point c on OC represents a given net value of corn.
3
That is, price per bushel minus cost per bushel times the number of bushels
produced by that amount of labor and capital. The points w and o' represent
net values of wheat and oats equal to c on OCe That is, in net revenue,
Oct = Ow = Oo By connecting points c w o a partial isovalue
curve or isorevenue curve is constructed Isorevenue curves of
different values can be constructed in the same way and will be parallel
between process rays. Also, the curves can be made smoother by including
a greater number of processes in the problem. These isorevenue curves,
when smoothed out would be analogous to the continuous production factor
factor model curves of curvilinear production functions.
xI
capital
/C
S00
w AisorevenuTe cur
0
labor X2
Figure I
Figure II is constructed the same way as Figure I except the
limitations on capital and labor are shown. Net revenue at o equals
4
that at it and, further, any point on the segment of the isorevenue curve
w' of has the same net revenue value as either of or n'. Since capital
and labor are both limiting, the feasible area of production is within the
rectangle OI:TPN Point P on the isorevenue segment wt of represents the
greatest net revenue obtainable within the restrictions of capital and labor.
Line PP' is drawn parallel to O17 and cuts 00 at ?'. Segment OR is cut equal
in length to ? I so that triangles ORR' and P'??" are congruent. Therefore
OS = TN and OS + OT ON, the upper limit on labor. In the production of
wheat valued at OR, OS amount of labor is used and in oat production valued
at OPt, OT labor is used. This combination of wheat and oat production,
therefore, utilizes the total labor supply.
It should be noted that OP" of oat production alone, without any wheat
production, would utilize the full amount of labor, but, in this case,
there would be a surplus of capital amounting to P"P and net revenue would
be decreased. That capital is exhausted by producing the combination of
wheat and oats represented by point P can be shown in a manner similar to
that shown for labor.
It now remains to show that the OR wheat plus OPt oats does, in fact,
produce maximum net revenue. This can be accomplished by proving triangles
ORR" and P'PO' congruent. By construction (wt'o and RR" are parallel as
are ORW and PIP) angle ROR" equals angle PP'o' and anles R"RO and o'PP
are equal. Then, since R = PIP by construction, the triangles are con
gruent (two angles and the included side are equal). Therefore, since
P is on the isorevenue curve w'o', and P'o' = OR", the amount of net
revenue given up by reducing oat production by P'o' is exactly made up by
wheat production of value PIP OR.
It should be noted that had P fallen on 00, only oats would have been
produced. Likewise, if P had fallen on OW only wheat would have been
produced. Further, it should be noted that a combination of no more than
two choice variables are needed for a solution when there exist only two
C
ww
capital
4P'
R"
S T N X
labor
Figure II
constraints. Hence, it has been illustrated that for two constraints and
two or more choice variables, a combination of two (or one) choice var
iables (products) can be found such that the constraints are entirely
exhausted and maximum net revenue is obtained.
Graphic demonstration can also be used to solve a problem containing
two processes and several input categories (the reverse of the first
problem). Again, wheat and oats can be the processes or enterprises in
volved, but, in addition to capital and labor, land shall be included
as a possibly limiting resource. (Iore constraints could be added.)
In Figure III let OA be the amount of wheat which can be produced on the
given amount of land and let OB be the amount of oats. Likewise the
6
given supply of capital permits OE units of wheat and OF units of oats.
The available labor is sufficient to produce OC units of wheat and OD
units of oats. Any point on the line AB represents combinations of oats
and wheat which can be produced with the given supply of land. Points
on the other lines represent similar combinations for the other factors.
Combinations represented by points on or within the area OE1NB are all
possible production levels. Production outside this area is limited by
at least one resource. If the amount of labor were increased so that
OCwheat and OD'oats could be produced, labor would no longer be a
limiting resource, since only points on or within the area bounded by
OEPB would be feasible.
A
Process I land
(Wheat)
C ,
> labor
E
P c capital
0 B D D F
Process 2 (Oats)
Figure III
For simplification, Figure IV has been derived from Figure III.
The points E, M, N, and B are the same as in Figure III. In Figure IV,
line CD represents a net isorevenue curve. For the relationship
between prices of wheat and oats represented by CD, net revenue would
be maximized by producing OF wheat and OH oats. If the price relation
ship were such that the isorevenue line were parallel to ?N, the optimum
7
combination would be indeterminant between M and N. If the isorevenue
line were nearly horizontal, only wheat would be produced.l/
When algebra is used in place of graphics, a large number of
alternatives can be compared relative to the resource limitations of
the farm. Linear programming, using the principles just presented,
utilizes algebra to accomplish this procedure. In the following section,
the computational procedure will be outlined.
C
Process 1
(Wheat) \
E 
I \
0 \
Process 2 H B D
(Oats)
Figure IV
1/ The two constraints multiprocess problem was based on arguments
presented in: (a) Plaunt, Darrel H., Optimum Combinations of Enterprises
on Farms in the Pennyroyal rea of Kentucky, Unpublished i.S. Thesis,
University of Kentucky, 1957. (b) Dorfnan, Robert; Samuelson, Paul A;
and Solow, R. r., Linear Programrming and Economic Analysis, McGrawHill
Book Co., New York, 1958.
The two process, multiconstraint problem was based on: McKee, Dean
E.; Heady, Earl O.; and Scholl, J. 5., Optimum Allocation of Resources
Between Pasture Improvement and Other Opportunities on Southern Iowa
Farms, Iowa Agricultural experimentt Station Research Bulletin l35, Ames,
Iowa, January, 1956.
8
Computing the Optimum
Iatrix Notation
Obtaining the optinum allocation of resources in a farm organization
by linear programming involves solving a system of linearly independent
equations which are subject to certain restrictions. Of course, linear
programming is not restricted solely to the study of economics. A
linear programming problem is any problem which can be stated in the
following form:
(1) I~aximize f(x) = clx1 + c2x2 + .... + Cnx
(2) Subject to
(a) P11x + Pl2x2 + + Plnn b1
P21X+ 22X2 + .... + P2nn= b2
PmlXl+ P2X2+ mnXn = b
(b) xj 0 for all j = 1, 2, ..., n
This problem can be restated in matrix notation as follows:
(3) Maximize f(x) = CX
(4) Subject to (a) ?X = B
(b) X Z 0 >>>>
where C is a row vector of n constants
X is a column vector of n variables
P is a matrix of constants of order mxn
TYe have here a system of m linear equations in n unknowns with n
being greater than m. The solution to be obtained must be such that no
value of x shall be less than zero, that the left hand member of the m
equations shall be exactly equal to the constants b and that the value
of f(x) shall be a maximun given the foregoing restrictions.
9
It is knoTm from one of the theorems of linear programming, which
is asserted here without proof, that a solution will be obtained which
will contain nonzero values for exactly m of the n unknowns which will
satisfy all of the above conditions. The remaining nm unknowns shall
be equal to zero; therefore, the P matrix and X vector may be partitioned
as follows:
(5) partition P into P1* and P*, selecting P such that it is
nonsingular and of order m x m.
(6) partition X accordingly into X* of m elements and X*; of nm
elements.
The original statement of the linear programming problem can then be
restated as follows:
(x* )
(7) (P*,*) (X;t) = B
(8) P X* P**X, = B
Solving equation (8) for X, we obtain,
(9) X* = P*I3 PIP,'1
Setting the elements of X'*, equal to zero, then
(10) X* P ;1l
Equation (10) shall be referred to as the original solution or basis.
This solution will satisfy the side restrictions stated in (2a) and (2b)
above, but may not necessarily result in the maximum value of f(x). The
original solution is then systematically revised by interchanging elements
of X3f* with elements of X until a combination of m unknowns having values
greater than zero is obtained which will satisfy all conditions of the
problem, the remaining nm unknowns being equal to zero. In making the
TABLE 2*
SOLUTION FOR THE EXAMPLE PROBLEM BY THE SIMPLEX METHOD
Prices, Resources, and Processes
Ci
Kinds of April June
Res. & Prod, Quantities Land Labor Labor Capital Mheat Corn .Row Sum R
Pj PO P3 Ph P P6 P P2 Check
0 P3 Land 140 1 0 0 0 .04 .02 4106 3,500
H 0 P4 April labor 105 0 1 0 0 .00 .03 106,03 Unlimited
g 0 P5 June labor 120 0 0 1 0 .04 .00 121o04 3,000
0 P6 Capital 2,250 0 0 0 1 .50 .50 2.252,00 l4,500
    
Szj (C0) 0 0 0o 0 0 .00
Z. Ci 0 0 0 0 0 1.50 *83 2.33
0 P Land 20 1 0 1 0 0 .02 20.02 1,000
C 0 P4 April labor 105 0 1 0 0 0 o03 106.03 3,500
g 1,50 Pi Wheat 3,000 0 0 25. 0 1 o00 3,026.00 Unlimited
0 P6 Capital 750 0 0 12.5 1 0 050 739.00 1,500
 
SZj (OC) 4,500 0 0 37.5 0 1.50 0 4,539.00
Zj Cj (MR) 4,500 0 0 37.5 0 0 .83 4,536.67
.83 P2 Corn 1,000 50 0 50, 0 0 1 1,001.00 
S 0 P April labor 75 1.5 1 1.5 0 0 ro 76o00 50.00
S150 P Wheat 3,000 0 0 25. 0 1 0 3,026,00 120,00
S 0 Capital 250 25 0 12,5 1 0 0 238.50 20,00

Zj (OC) 5,330 U1.o50 0 4 0 1.50 .83 5,369,83
Zi C1 5,330 41.50 0 4 0 0 5367o50
S.83 P2 Corn 2,000 50. 0 0 0 1 1,955o00
E 0 P4 April labor 45 1.5 1 0 .12 0 0 47.38
. 5 1.50 PI Wheat 2,500 50. 0 0 2.00 1 0 2,549.00
0 P0 June labor 20 2 0 1 08 0 0 19.08
.&   ,L~
SZ (OC) 5,410 33.50 0 0 .32 1.50 .83 5,146. 15
Z C (MR) 5, 10. 33,.0 0 0 32 0 0 5l443.82
STaken from Plaunt, opo cit,, p. 67. One copy can be extracted to use for following the argument.
10
necessary revisions in the original solution an interactive procedure
is followed which has become known as the "simplex" method.2/
It is necessary to introduce additional notation at this point.
The elements of X* shall be denoted by xi and the corresponding C values
by ci, the elements of X;: as xj and the corresponding C values by cj,
P*1B which is a column vector of m elements shall be denoted by S and
its elements by si, P*lP*whichis a matrix of order m x (nm) shall be
denoted by R and the coefficients of this matrix by rij. The original
solution will be denoted by f(x)* and the next revised solution or
iteration by f(x).
Equation (9) can then be rerritten as,
(11)  = S RXH;
The effect on the value of f(x)* of revising the solution by substituting
an element of X;* (viz., cjA) for an element of X* can be shown as
follows:
(12) (a) f(x) = C:: = xi = 2 cisi
(b) 7i = si rijxj
(c) setting xj = A
(d) f(x)* = Zci(si rijA ) + cj A
(e) f(x)** = Zcis Zcirij A+ cj A
(f) f(x):* = Zcisi + A(cj Zcirij)
Since f(x)* = Z cisi, the change in the value from
the initial to the alternative solution is f(x)** 
f(x) = A(cj Zcirij). Therefore,
(g) if (cj Zcjrij)> o, f(x)* will be increased by
replacing xi in the solution by xj.
2/ George D. Danzig, Maximization of a Linear Function of Variables
Subject to Linear Inequalities in ctivity Analysis of Production and
Allocation, Tjalling, C. Koopmans, editor, John 'iley and Sons, Inc.,
New York, 1951 pp. 339347.
11
If for any xj the condition stated in equation 12(g) holds true it
is possible to increase the value of f(x)* by revising the solution.
The iterative procedure is continued until the condition stated in 12(g)
no longer holds true for any xj. At that point the optimum solution
has been obtained and further iterations are no longer necessary.
The "simplex" method.
The actual computational technique used in many linear programming
problems is arranged so that the work can be carried out by a clerk or
electronic computer. This technique is knovm as the "sinplex" method
of computation. Once the problem is set up and the original figures
entered in a table, the actual solution is purely mechanical. This is
an advantage of programming over budgeting where each new step must be
carefully thought out and weighed by the researcher himself. However,
since the computations are strictly mechanical, it becomes more important
that the problem be set up correctly.
The algebraic notation of the proceeding section demonstrated that
f(x) or net revenue over variable cost is maximized by a series of
substitutions into and elii:inations from the original solution. The
simplex method is used to systematically substitute the next higher
yielding alternative until the maximum is reached. Table 2 is an example
of a simplified problem solved by this method. The mechanical computations
actually involved will not be covered completely in this paper since this
is available elsewhere._/ The logic of the computations are important,
3/ The proceeding discussion drew heavily from: icKee, Dean E., The
Use o6 IB for Linear Programming, Agricultural Economics Yimeo 652,
iichigan State University, June, 1956.
h/ See for example: Heady, Earl 0., "Si.mplified Presentation and
Logical Aspects of Linear Programing Technique", JF.S., Dec., 1954.
TABLE 2*
SOLUTION FOR THE EXAMPLE PROBLEM BY THE SIMPLEX METHOD
Prices, Resources, and Processea
Coj /. so .9 _
Kinds of April June
Res. & Prod, Quantities Land Labor Labor Capital Wheat Corn Row Sum R
Pj P0 P Ph P Ph P P P1 P2 Check
0 P3 Land 140 1 0 0 0 .o0 .02 11.06 3,500
H 0 P4 April labor 105 0 1 0 0 .00 .03 106,03 Unlimited
go P5 June labor 120 0 0 1 0 .o0 .00 121,04 3,000
0 O P6 Capital 2.250 0 0 0 1 .50 .50 2.252,00 4,500
oZj (OC) 0 0 0 0 0 0 0 .00
Zi C, 0 0 0 0 0 1.50 .83 2.33
0 P3 Land 20 1 0 1 0 0 .02 20.02 1,000
S 0 P, April labor 105 0 1 0 0 0 .03 106.03 3,500
g 1,50 P Wheat 3,000 0 0 25. 0 1 .00 3,026.00 Unlimited
S 0 P6 Capital 750 0 0 12.5 0 50 739.00 1,500
0   
Zj (oc) 4,500 0 o 37.5 c 150 0o ,539o00
Zj Cj (MR) 4,500 0 0 37.5 0 0 .83 4,536.67
.83 Po Corn 1,000 50 0 50. 0 0 1 1,001.00 
" 0 Ph April labor 75 1.5 1 1.5 0 0 ro 76.00 50.00
g 1.50 P 4Wheat 3,000 0 0 25. 0 1 0 3,026.00 120.00
S0 P6 Capital 250 25 0 12.5 1 0 0 238,50 20,00
I zj (CC) 5,33000 l.o o 4 C 1.50 .83 5,369,83
Zj Cj 5,330 L1,50 0 4 0 0 0 5,367.50
S.83 P2 Corn 2,000 50. 0 0 h 0 1 1,955.00
0 P4 April labor hS 1.5 1 0 .11. 0 0 47.38
1.50 PI Wheat 2,500 50. 0 0 2,00 1 0 2,549.00
O 0 P June labor 20 2 0 1 .08 0 0 19,08
Sz (oc) 5,41o 33.50 o o0 3 1.50 .83 5,h46.15
Z C (MR) 5, 10 33.5o0 0 0 .32o o 5.h43.82
STaken from Plaunt, op. cit,s p. 67. One copy can be extracted to use for following the argument.
however, in understanding the mechanics of the solution and will be
discussed.
The table is composed of four sections, the first of which contains
the original solution to the problem. Each subsequent section is an in
teraction of the proceeding solution. Section 1 of the table is com
posed of a list of the limiting resources on the farm, together with
the coefficients of production of each resource and a statement of
the first solution to the problem. Ordinarily, it would be more
efficient to use the graphic method for the solution to a two process
problem, but for purposes of illustration, it will be solved here by the
simplex method.
The farmer is faced with a decision concerning the production of
wheat and corn. By following a procedure similar to that of Figure III,
it has been determined that the limiting resources are land, April labor,
June labor, and capital. The farm has available 140 acres of land, 105
hours of April labor, 120 hours of June labor and $2,250 of capital.
The kinds of resources and the quantities of each are listed in the
second and third columns of the table. In the eighth column, under
wheat P1, the coefficients of the resources in wheat production are
listed. The first number, .Oh, indicates that 4% of 1 acre is required
to produce one bushel of wheat. In other words, land is figured to
produce 25 bushels of wheat per acre. April labor does not limit
wheat production as indicated by .00 in the column. Four hundretha (.0O)
hours of June labor is needed to produce 1 bushel of wheat, Finally,
the coefficient of capital is $.50 per bushel of wheat. The coefficients
for corn are listed in the ninth column under corn P2,
13
It is necessary to introduce a disposal or nonuse process so that
all need not be exhausted for a solution. The complete disposal process
will represent the first solution to the problem. When this solution
is in effect it means that all resources remain idle, i.e., nothing is
produced. The coefficients for the land disposal process are one for
land and zero for the other resources since it takes one acre of land
to "not use" one acre and none of the other resources are used. The
coefficients for the land disposal process are listed in the fourth
column of Table 2 under land P3 Similarly, the coefficients of the
disposal processes for the other resources are listed in the next
three columns, P P5, P6.
The next step is to find the process with highest net returns per
unit, which will become the first process examined. In the case of the
first solution, this is simply the highest priced product wheat, in
our example. (The case for subsequent solutions will be discussed later.)
After this choice has been made, the last column of the first section
can then be completed. This column, labeled R, indicates the total
production of wheat possible when considering each resource independently.
For example, by dividing 140 (acres) by the coefficient of land in
wheat production, .04, it is seen that 3500 bushels of wheat can be
produced. Alternately, 25 bushels per acre on 140 acres equals 3500
bushels. Since April labor is not limited in wheat production, an
unlimited amount can be produced when considering this resource alone.
The available June labor is capable of producing 3000 bushels and the
capital limits wheat production to 4500 bushels. It can be seen from
this column that June labor is the most limited resource in wheat
production, and that by using all of this resource, 3000 bushels can be
produced. Thus, the first iteration will be the substitution of 3000
14
bushels of wheat for 120 hours of June labor. The effect of this
substitution is shown in section 2 of Table 2.
The production of 3000 bushels of wheat utilizes all but 20 acres
of land and 7750 of the capital, Since no April labor is used i wheat
production, the total amount, 105 hours, still remains unused. These
quantities then become the amounts in column Po section 2. Row P1
(wheat) is substituted for P5 (June labor) in section 2. The elements
of row P are computed by dividing each element of row P in section 1
by the June labor coefficient for wheat, row PS column P1 section 1,
or .04. In column P5 row P1 section 2,25 = 1/.04. The number 25
represents the number of bushels of wheat which would have to be given
up in order to withdraw 1 hour of June labor from wheat production and
substitute it in the production of another crop. (If any June labor
were used in the production of corn, the element, row P column P 
section 2 would have a value. If this value were, for example, 0.5,
it would mean that 0.5 bushels of wheat would be forgone if enough
June labor were transferred from wheat to produce 1 bushel of corn. If
the value were 0.5 then the per bushel requirement for corn of June
labor would have to be .02, That is, since .04 hours are required to
produce one bushel of wheat, only half the labor would have to be with
drawn that it takes to produce a whole bushel of wheat. Therefore,
only one half bushel of wheat is forgone to produce one bushel of
corn with the June labor.)
The elements of rows P3, P4 and P6 of section 2 are formed by the
equation:
a'ij = aij (arj/ark) aik (i k)
where k indicates the crop enterprise coming into the production plan
(wheat); r is the activity being removed (June labor); j stands for any
one of the column headings and i stands for the row headings. The "prime"
indicates that the number to which it is attached belongs to the new
section being formed./
In order to demonstrate the meaning of the computation, a portion
of Table 2 is extracted and presented as Table 3 below, with the letters
of the equation and the subscripts added in parentheses. The first
number on the right hand side of the equation (aij) is the capital
13.
Table 3
An extract from Table 2
June
Labor
P5 (j)
P Land 0
P4 April labor 0
P5 June labor (r) 1 (a j)
P Capital (i) 0 (a j)
Z (oc) 0
Zj cj 0
P3 Land 1
P4 April labor 0
P1 Wheat 25
P6 Capital (i) 12,5 (a'ij)
z (oc) 37.5
Z c 37.5
Wheat
P1 (k)
.0o
100
.04 (ark)
.50 (aik
0
1,50
0
0
1
0
1.50
0
SBowlen, Bernard and Heady, Earl O., Optimum Combinations of Com
etitive Crops at Particular Locations, Iowa'Agricultural Experiment
station Research Bul~ 426, Ames, Iowa, April, 1955.
*
15
expense, P6, per hour of June labor in the disposal process, P In
the example, this value is zero. The quantity of the equation in
brackets (arj/ark) is the marginal rate of substitution of June labor
(column P5) for wheat (column P1) as specified by the relative require
ment of each for June labor (row P ). The last quantity (aik) is the
capital expense required per bushel of wheat. The product (arj/ark) aik
is the amount of capital which is released from wheat production as each
hour of June labor is reduced from (since column PS is a disposal process)
the production plan. The rate of substitution is specified by the relative
requirement of wheat (col. P1) and June labor (col. P5) for June labor
(row P ), i.e., aj /a bushels of wheat being given up for each hour
5 rj rk
of June labor subtracted from the enterprise. The quantity (arj/ark) a
rj rk aik
is subtracted from a. the capital expense per hour of June labor in
the disposal process, to give the "net" expenditure out of available
capital ($750) per hour of June labor going into a new enterprise. Thus,
the equation is:
12.5 = 0 ( ) .50 0 (25) .50 = 12,5
.o0
where the capital expense per hour of June labor in the disposal process
is 0; the marginal rate of substitution of June labor for wheat is 25
bu,; the capital expense required per bushel of luheat is $.50; and the
"net" expenditure out of available capital per hour of June labor
going out of wheat production is $12.50, i.e., an addition to
capital. The remainder of the top portion of section 2 excluding
column R is filled out in this manner with appropriate meanings being
given to each element.
17
The next step in the simplex solution is to determine the process with
the next highest net returns per unit. The rows Zj and Zj C_ contain
the answer Since C. is given (i.e., the prices of the products), Zj must
be determined. The element appearing in column P2 row P1 Section 2
is the amount of wheat (P1) which must be given up if one bushel of corn
(P2) is added to the production plan. Since, in this example, corn and
wheat are not competitive for resources, the figure is .00, That is, no
wheat needs to be forgone to increase corn production one bushel, For
corn then, Z. C .C = $.830 If the element referred to above, is
multiplied by the price of wheat, the product is the value of wheat given
up to produce one more bushel of corn. The value Zj Cj is called the
"net marginal revenue" (considering the opportunity cost) of one unit of
the product in the column it appears. In the proceeding example, since
Zj = opportunity cost is zero, the "net marginal revenue" of 1 bushel of
corn is minus its price.
As was the case in section 1, the process with the largest negative
value of Z Cj is chosen for the next process to be substituted into the
production plan. In section 2, corn is chosen. The R column is then com
puted exactly as it was in section 1. Land is found to be the most limit
ing resource, so in section 3, corn replaces land, the amount of corn
being such that land is completely exhausted. It is found in section 3
that it would be profitable to substitute the $250 of unused capital for
20 hours of June labor. This action will, of course, decrease the amount
of wheat produced and increase the amount of corn. In the Zj C row of
section 4, no negative values appear. This indicates that the optimum or
ganization has been reached. For maximum profit, the corn enterprise
should be operated at the 2,000 bushel level and wheat at 2,500 bushels.
18
There will be 45 hours of unused April labor and 20 hours of unused June
labor. Land and capital are exhausted. These figures are found in the
Po column of section 4.
One more relationship in Table 2 should be mentioned, viz. the Z
figure in the Po column. It is computed by multiplying the quantity of
product produced (in the particular section) by its price and summing
over each product. For example, in section 3, 5330 = 1000 x $.83 4
3000 x $1.50, Since the price of each product is a net price per unit,
the Zj figure is a "profit" figure. It should be noted that profit" in
creases for each new production plan*
Linear Programming in Farm Management
Assumptions
The name, linear programming, comes from the assumption of linear
ity. For a given process, resources combine in a constant proportion and
the ratio of input to output is constant and independent of the level at
which the process is used. The concept of diminishing returns can be
included by calling each different point along the function a different
process with a diminishing ratio of fixed to variable resources. The
curve can be made as smooth as desired by increasing the number of pro
cesses considered,
It is further assumed that resources and products are divisible
into small fragments or into unit levels. This same assumption is made
in production function analysis. The assumption of additivity states
that the combined output of two or more products will be the sum of the
production from each process. Likewise, the combined use of resources
will be the sum of the consumption of the individual processes, and will
not exceed the total available supply. Complementarity of products can
19
be handled by including each different combination as a different process.
Different rations, for example, can be classed as different processes,
so that the additivity assumption is still valid.
Only a finite number of available alternatives can be considered in
order that the problem can be handled in a reasonable length of time and
with a reasonable expenditure. For this reason, linear programming solu
tions result in optimum programs from among only those alternatives exam
ined rather than from all possible alternatives. Curvilinear functions
are capable of examining more minute changes in inputs than is linear pro
gramming, but, in these, too, there is a definite limitation on the num
ber of alternatives which can be included in any one problem.
Inputoutput coefficients and prices of both are regarded as
single valued. Perfect knowledge in this respect is also assumed by
other farm management techniques. In production function analysis,
however, inputoutput coefficients are not assumed, but are an end
product of the analysis
Setting up the problem
"In developing a farm planning model for linear programming, we
start with a specification of the relevant restricting resources.
Thus, we specify the quantities (and seasonal distribution) of
the labor supply, land, capital and other resources that may
limit production. Next we select the alternative enterprises
(activities) to be considered and we assemble data on the
resource requirements for each enterprise. Then we establish
other activities to provide for the flow of certain inter
mediate inputs from a primary to a secondary enterprise, e.g.,
hay fed to livestock. In addition, we add buying and selling
activities to permit the direct sale of primary products
(grain) as well as their purchases in the event the supply
produced proves inadequate for the requirements of the
secondary enterprise (livestock). Finally we choose prices
and construct the profit equation, which is maximized subject
to the specified restraints, some of which may be inequalities
(e.g., resources in fixed supply need not be fully utilized)."/
Swanson, Earl R,, "Application of Linear Programming in Agricultural
Production Economics Research", paper presented to Cowles Commission Seminar,
University of Chicago, February 10, 1995.
20
Since the computations in linear programming are strictly mechanical,
it is extremely important that the problem be set up properly and accur
ately, The amounts of resources on any particular farm determine the level
of production, but it .is their relative proportions which determine the
kinds of products appearing in the solution. The resources can be rough
ly grouped into three categories. The first group contains those which
are available in any quantity. Included are those items which can be
readily purchased such as, gas, oil, fertilizer and seed. Actually, the
amount of these items available is limited by the capital restriction,
but, they themselves are seldom considered as limiting resources. In
the second group of resources are those which arc limited in quantity
in the time span under consideration. Changing the time span affects
the number of items in this category. Land, labor, capital and build
ing space are commonly in this group, Often irrigation water, govern
ment allotments and personal preference restrictions appear in this cate
gory, Minimum restrictions, too, can be included. The third category
includes those intermediate products which are used as inputs in a ver
tically integrated enterprise. These items often appear in buying and
selling activities, also.
An important consideration when grouping resources into single
input categories is to separate those which are complements, since one
or the other may be limiting. If, however, they are nearly perfect
complements for all activities they may enter and both are limited, they
could be grouped together if given the lower limit.
Setting up the alternative processes merits further consideration.
It should be noted that each point on a smooth curvilinear production
function represents a different process. If more than one cultural or
21
feeding practice is to be considered, each should be included as a
separate activity. High roughage rations represent a different point
on a production function than high concentrate rations. That is, the
roughage concentrate ratio changes along the function. Similarly,
different crop rotations represent different activities. For any
single item, a selling activity should be priced below a buying
activity, i.e. price is less when selling than when buying an inter
mediate product such as grain, in order to prevent the purchasing
and selling of it without utilizing it in another process. In ad
dition, for each limiting resource there should be a disposal activity
so that it is not necessary to fully exhaust each of them. It would be
possible to obtain a solution by utilizing all resources, but this would
not necessarily be the optimum program.
There is no clearcut limit to the number of activities and the num
ber of restrictions which can be included in a linear programming prob
lem. Where an electronic computer is available, the limit of the ma
chine will usually be the limit imposed However, if time and money
are available, there is virtually no limit to the size of a problem
which can be calculated by clerks A system containing a large number
of variables can be reduced to manageable proportions by computing pre
liminary problems based on more general information and then eliminating
those groups not involved in the optimum solution. For example, high,
medium and low roughage rations for a feeding enterprise could be com
Z/Peterson, G. E., op. cit., po 5h8.
!/Heady, Earl O., op. cit., p. 100O.
22
puted in a preliminary problem. Then more detailed rations from within
the group involved in the preliminary optimum could be examined further,
The Solution
The divisibility assumption and the arithmetic involved in the com
putations lead to fractional activities in the solution. The true opti
mum solution would, in fact, be composed of fractional activities. Di
rect application of the program usually requires an adjustment of the
optimum solution. Table 4 is an example of a nonadjusted, optimum solu
tion to a programming problem.
Table 4
Optimum Combinations of Enterprises on a Mechanized
Farm with 50 Acres of Cropland and With Family
Labor Only, Southern Piedmont Area,
North Carolina/
Items Units Net Revenue
Crops Acres Dollars
Cotton lol 196
Corn 23 9 2,006
Oats & lespedeza 158 748
Alfalfa 9.2 458
Livestock Numb er
Hens 2050 4,629
2 Sutherland, J. Gwyn and Bishop, C. E., Possibilities for Increas
ing Production and Incomes on Small Commercial Farms, Southern Piedmont
Area, North Crolina. North Carolina Agricultural SEperiment Station
Tech Bulo No. 117, December, 1955, p. 27.
23
Adjusting an optimum solution, again, requires the personal
attention of the researcher r manager. Acreages may need to be ad
justed to field size, In some instances, only a very small amount
of some activity may be contained in the solution. A decision will
have to be made whether to drop the activity entirely or increase it
to applicable proportions. Since the preferences of the farmer are,
or can be, included in the programming solution, adjustments will be
based on feasibility and profitableness, Of course, any adjustment
away from the optimum will decrease net revenue to some extent. Prob
lems of this type are usually resolved by partial budgeting. By this
method, the most profitable of several alternative adjustments can be
determined. It should not be assumed that linear programming yields a
final answer that would be put into practice without adjustment,
Adaptations and Limitations
In the field of farm management, linear programming is best adapted
to solving farm organizational problems. Aggregative analysis of a group
of farms raises the same problem as other methods of analysis. Special
ized restrictions such as personal preferences cannot be handled in the
aggregate. Analysis of individual farm organizations, however, present
problems well suited to programming.
Linear programming is also adaptable to enterprise analysis and
several studies have been conducted on this basis l/ Detailed resource
allocation can be obtained when using programming for this purpose,
/ For example see: G ervais, Michel A., Optimal Utilization of
Limited Floor Space in a Laying Enterprise. Unpublished M, S. thesis,
Michigan State University, East Lansing, Michigan, 1957.
24
However, the accuracy of the information obtained is highly dependent upon
the accuracy of the inputoutput data which goes into the problem. One
of the more efficient methods of obtaining this inputoutput data is by
fitting functions of the CobbDouglas type to production data. Functional
analysis, however, is also efficient in analyzing whole enterprises. In
many situations, it will be a superior method to linear programming for
enterprise analysis. Input category limitations are more restrictive in
the functional analysis. This leads to more easily applied answers from
programming, since the input categories are more meaningful. The continu
ousness of the production function, though, produces a more exact optimum
solution if this is desired. The data available and the purpose for which
the analysis is to be used will determine the best method.
Functional analysis is quite well adapted to analyzing area prob
lems. At times, this method precedes a more detailed programming analy
sis of particular farms or typesoffarms in the area. In this combined
use the complementarity of the two methods is well exhibited. The comple
mentarity is also expressed in problems where data for programming is
obtained by functional analysis, as described above,
Programming cannot be used for studies such as the analysis of fer
tilizer responses of crops or feeding rates on livestock. It is this
type of information which is assumed in linear programming. Because of
the linear nature of programming, the marginal and average physical prod
ucts derived by functional analysis are assumed equal. These productivity
coefficients, which are assumed in programming, are productivities for a
particular process. The marginal value products or Zj Cj values obtained
in a programming solution are the MVP's of the resources to the farm as a
whole unit. Also characteristic of programming is the fact that no MVP's
are assigned to nonlimiting resources, since nothing is gained nor lost
if one unit is added or subtracted.
"Perhaps one of the important values of the linear programming
technique is the rigor which it enforces in thinking about the
farm business. In a sense, the technique transfers judgement
and intuition from the actual process of comparing alternative
organizations to the establishment of a set of realistic con
ditions and relations... .Another important consequence of
farm planning by linear programming is the dramatic way in
which we discover deficiencies in our inventory of technical
coefficients, In so far as the programming model parallels
reality, these deficiencies should prove to be valuable guides
to technical workers for needed research."l_/
/Swanson, Earl R., op. cit., p. 4o
