LINEAR PROGRAMMING FOR PROFITABLE FARMING
Cecil A. Parker
Peter E. Hildebrand
IN-SERVICE EDUCATION WORKSHOP FOR TEACHERS OF VOCATIONAL AGRICULTURE
Corpus Christi, Texas
August 11, 1960
LINEAR PROGRAMMING FOR PROFITABLE FARMING
Cecil A. Parker and Peter E. Hildebrand
In-Service Education Workshop
for Teachers of Vocational Agriculture
Corpus Christi, Texas
August 11, 1960
In this era of increasing costs and decreasing prices one of
the biggest, if not the biggest, problems facing farmers and ranchers
is resource allocation and enterprise combination. This may simply
be stated as, how can a farmer or rancher divide his dollars, acres
of land and labor between crop and livestock units to get the greatest
A farmer or rancher has many alternatives for resource allocation.
To illustrate the multitude of alternatives from which a farmer or
rancher must choose let's consider one with $5,000 of capital to
use in producing two crops. If he considers all the possibilities
of allocating whole dollars between the two crops he has 5000 differ-
ent alternatives'from :which tc bhoose. If we add 2000.hours 9f labor
which can be allocated in whole hours to the production of these
two crops, with the $5,000, he now has 10,000,000 different ways
to combine the two resources, capital and labor, for the two crops.
Of course, we recognize readily that this number of alternatives is
not feasible because we know that the production of either crop
would not be of sufficient quantity to consider if only one dollar
or a few dollars were used. As shall be seen, linear programming
Respectively, Farm Management Specialist, Texas Agricultural
Extension Service and Assistant Professor, Department of
Agricultural Economics and Sociology, Texas A & M College.
is a technique that may be used to aid farmers and ranchers in de-
termining the single most profitable combination of these two resources.
Linear programming originated during World War II. It was
used by the Navy as a method for specifying routes that would min-
imize travel distance for limited shipping facilities. It was also
used for determining the best method of allocating scarce labor,
machines, tools and plant facilities to produce war goods. Since
the war it has been adapted to electronic computers and is being
used by a large number of private firms and research organizations.
It is also being used by agricultural production economists to
determine optimum organization of resources and enterprises on farms
and to suggest desirable farm adjustments.
Now let us compare linear programming to a more familiar farm
management tool budgeting. Budgeting, one of our most used tools
in farm management, is not an old tool. It came into use in the
twenties. There were many doubts about its use at that time but
we all recognize it as a useful tool today. Budgeting is the same
general technique as linear programming, but it uses different
computational methods. Budgeting has the same limitations as linear
programming. Wherever one gives a screwy answer, so will the other.
However, there are differences in the method. Budgeting
seldom finds the one unique production program, out of many which
gives maximum profits. Ordinarily, budgeting is used to determine
which one of two or only a very few producing methods or farm
organizations are best. Budgeting could be used to determine which
one of a hundred programs would give greatest profit but it is too
time consuming. Linear programming has the advantage, it can specify
the optimum program in a fraction of the time required for budgeting.
Computations must be accurate for both methods. More important
than the arithmetic computations is the assembling of input-output
coefficients and prices. Both methods require the same technical
information. If this information is available for a budgeting prob-
lem, then linear programming can be applied to the same problem.
Mistakes in input-output coefficients and resource requirements will
give wrong answers with either method.
However, there are only certain kinds of problems on which
linear programming can be used. The problem must have three attributes.
First of all there must be some objective. In farm management
the objective is usually profit maximization. However, adaptations
of this objective may be used. For example, certain enterprises
can arbitrarily be omitted if the farm operator is certain he does
not wish to engage in them. The method can also be used to show
the loss in income resulting from choosing a course of action differ-
ent from the one which maximizes profits.
The second attribute that is necessary is that there must be
a number of crop and livestock enterprises from which to achieve
the objective. On a given farm there are usually several different
crop and livestock enterprises from which an operator may choose.
The linear programming technique selects crop and livestock enter-
prises in the right proportions to maximize profits.
The third attribute is that there must be at least one resource
that is limited. However, more than one resource may be limited.
The most common limited resources are land, labor and capital. The
optimum combination of enterprises is then selected within these
The actual farm situation which we will use to illustrate
the linear programming technique has the three attributes listed.
First the objective of this farmer is to get the greatest net profit
from the resources he controls. Second there are a number of al-
ternatives to select from to achieve this objective. The most
limiting resource of this farmer is land.
Now let's examine, on a step-by-step basis the procedure
followed on this farm to obtain a combination of enterprises to
The first step is to accumulate data. A detailed inventory
of land facilities, machinery, available capital, amount and distrib-
ution of available labor and other factors that might influence
production was taken. The crop and livestock enterprises which the
farmer wants to consider are listed. Average and expected yields
and production and harvesting practices were obtained.
The resources available on this farm are:
Cropland 240 acres
Permanent pasture 165 acres
Cotton allotment 55 acres
Hog pasture 10 acres
Adequate machinery and equipment
Necessary labor available
Operating capital not limited
The objective of this farmer was to utilize the 55 acre cotton
allotment because of a share cropper he wanted to keep to assist
with other jobs on the farm. He also wanted to market as much as
possible of the feed crops through some type of livestock.
The enterprise and crop yields wanted to consider are:
Cotton, 350 pounds lint
Corn, 45 bushels
Grain Sorghum, 3000 pounds
Hay, 2 tons
Winter and summer supplemental pasture
The second step is to develop input-output coefficient. This
consists of determining the amount of resources (hours of labor,
amount of capital, etc.) required to produce one
cotton, one sow and two litters of pigs, one cow
calf, etc.) of each enterprise.
The third step is to determine the expected
commodity produced. This step in programming is
The prices used for this farm are:
Cotton lint 260 per po
Cotton seed $38 per toi
Corn $1 per busl
Grain Sorghum $1.52 per I
Slaughter calves (June 1) 210 per po
Light weight Steers (Oct.l) 23.500 per
Slaughter hogs 16.250 per
unit (an acre of
prices for each
Step number four is to prepare detailed budgets on a unit basis
for each enterprise. (Sample budgets are attached.) A summary of
the enterprise budgets is given below.
SUMMARY OF ENTERPRISE BUDGETS
Corn Gr. Sorg.
Cow- Slaughter hogs for for
Resources Amour; Calf Gr. Sorg Corn Cotto Sale Sale
Cropland 240 1.31 3.33 3.95 1 1 1
Pasture 165 4
Cotton Allot. 55 1
Hog Pasture IC .25 .25
Net operating $58.77 $151.07 $154.16 $29.62 $29.49 $19.00
This summary shows the amount of resources one unit of each
enterprise requires. Also it shows the net operating profit from one
unit of each enterprise.
Step number five is to place the data from the detailed budgets
in a computational table and set up mathematical equations. One
equation is required for each restrictive resource. The equations
are fed into the computer which makes the computations in a matter
of a few minutes and selects the combination of enterprises that
will give maximum profits within the limits of the resources considered
The final step is to interpret the results and examine them
to see how realistic they appear to be. Although the electronic
computer does not make mathematical errors, the results can be no
better than the input-output and price data fed into the machine.
If the results appear unrealistic, the data should be carefully
re-examined. The machines accuracy depends on using a accurate data.
Good judgement on the part of people involved is extremely important
in determining the data used in linear programming.
Three solutions are listed below. One solution represents the
optimum combination of enterprises for maximum profit. However,
this solution did not utilize the full cotton allotment. The next
solution shows the combination of enterprises using the full cotton
allotment. The other solution shows the results when the number of
brood sows were limited to 20.
Sows and 2 litters of pigs (G. S.) 40
Cows-Slaughter calves 41
Grain Sorghum for feed 133.0
Corn for feed 10.3
Hay for feed 12.8
Supplemental pasture 30.9
Corn for sale 0
Net Operating $10,029
Cotton Forced in Sows Stopped
The results of the optimum solution and the solution with total
cotton allotment being utilized are very similar. There is only $35
difference in the net operating profit of these two. When the number
of brood sows were reduced to 20 the farm produced excess corn for
cash sale. In this solution, however, the net operating profit was
reduced approximately $1100.
LInEAR PROGRAMING UNDER UNCERTAIN CONDITIONS OF YIELD AND PRICE
In the discussion so far, it has been assumed that both the yields of
the products and the prices of the products have been known with
certainty. For example, we assumed a yield of 3,000 pounds of grain
sorghum and 45 bushels of corn, year after year. Also, we assumed
that we knew the price of corn to be i1.00 and the rice of grain
sorghum to be $1.52. Prices and yields of course, are never known
beforehand with this certainty. Because this is true, it is desirable
to obtain solutions for the linear program for different yields and/or
While it is easier to obtain solutions by varying price, it is
also possible to obtain solutions by varying yields. In the farm dis-
cussed in this paper, you will recall that the operator desired all
feed raised to be fed through livestock if possible. In this case,
price is less important to the operator than is the yield of grain
sorghum and the yield of corn. If, for example, he can obtain much
higher grain sorghum yields relative to corn, then he probably should
feed grain sorghum. Conversely, if he can raise more corn relative
to grain sorghum, he probably will be better off feeding corn.
Using the following table as a guide, let us see the effect on
farm organization from varying the yields of grain sorghum and corn.
Looking first at Solution 1, let us see the effect of a corn yield
of 50 bushels while grain sorghum yield remains at 3,000 pounds.
It can be seen here that the organization of the farm remains very
similar to the way it is organized with a 45 bushel corn yield.
However, because corn yield is higher it is possible to feed one
additional cow-calf unit. Also, because of the higher corn yield,
net operating profit is a little bit higher than it is under the
solution with a lower corn yield. With yields of 3,000 pounds of
grain sorghum and 50 bushels of corn, grain sorghum is still a rela-
tively more efficient user of the limiting resources than is corn
so that the slaughter hogs are fed grain sorghum.
ALTERNATE SOLUTIONS, VARYING SORGHUM AND CORN YIELDS
Solution number 1. 2. 3. 4. 5.
Grain Sorghum Yld. 3000 3000 2750 2750 2500
Corn Yield 50 55 __45 ___ 50 45
Slaughter Hog 40 0 36 0 :0
Slaughter Hog O 40 0 37 33
Cow-Calf 41 41 41 41 41
Sorghum for feed 132 0 131 0 0
Corn for feed 9.3 138.4 10.3 141.0 141.0
Sorghum hay 12.8 12.8 12.8 12.8 12.8
Oats 20.6 20.6 20.6 20.6 20.6
So ghum 10.3 10.3 10.3 10.3 10.3
Corn for sale O 2.8 0 0 0
Cotton 55 55 55 55 55
Net Operating Profit $10,081 .288 477 743 126
Notice, however, in solution 2 the effect on the slaughter hog
program if corn yields are 55 bushels per acre. With grain sorghum
at 3,000 pounds and corn at 55 bushels per acre, corn is a more
efficient user of the resources than is grain sorghum. Thus, one
would shift from feeding the slaughter hogs grain sorghum, to feeding
them corn. Since there is no other use for grain sorghum, none is
grown. Notice that because the livestock program is limited by
acres of pasture and the corn yield is high, there is a little ex-
cess corn for sale. Again notice that the net operating profit is
higher because corn yields again are higher.
In solution 3, one can see the effect of a decrease in grain
sorghum production if corn yield is at 45 bushels, the original
estimate. When grain sorghum yield drops 250 pounds per acre,
notice that there is not enough sorghum to feed 40 sows and their
pigs. Thus, hog production must be cut back to only 36 sows. Notice
also, that the slaughter hog program is reduced while the cow-calf
program is not. Thus, one may conclude that the cow-calf program
is a more efficient utilizer of the scarce resources than is the
slaughter hog program on this particular farm. In comparing this
situation with that in Solution 1, notice that because the corn
yield is lower it takes one additional acre of corn to have enough
feed for the cow-calf operation. Also notice, that the ratio of
grain sorghum yield to corn yield in the two solutions is quite
similar, so that the over-all program in the two is similar. Thus,
there is a slaughter hog program on grain sorghum, and a cow-calf
operation. Most of the acreage is in grain sorghum and only a little
in corn. However, since both yieldsare less in solution 3 over solu-
tion 1, net operating profit is reduced about $600.
In solution 4, notice that with grain sorghum at 2750 pounds, one
will shift to feeding corn to the slaughter hogs if corn yields go
from 45 to 50 bushels (recall that when grain sorghum was at 3,000
pounds, corn yields needed to be 55 bushels before the hogs were
shifted from grain sorghum to corn.)
Finally, in the last solution, notice that with low yields of
both corn and sorghum, the cow-calf operation is still operated at
capacity, but the slaughter hogs fed on corn are limited to 33 sows.
Also, net operating profit is the lowest of any of those considered.
This of course, is due to the lower yields.
It is possible to do this same type of programming by using
different price relationships. It remains however, for the farmer
himself, to decide ahead of time what the price relationships would
be and what kind of yields he can anticipate in the coming year.
With this information, it is then possible, through linear programm-
ing, for him to pick an optimum program for his farm.
Thus, one can see, that while linear programming can help
answer questions concerning the organization of a farm, it does
not make the decisions for the farmer. Programming does not
predict price, and does not predict yields. If the farmer can
decide these, and if he has accurate information concerning his own
operation, then linear programming can help him in making his
Yield: 45 bushels per acre
Price: $1 per bushel
Income: 45 bu. @ $1/bu.
Plant and fertilize
TOTAL PREHARVEST HOURS
Seed 4 lbs. @ 120/lb.
Fertilizer 150 lbs. 16-20-0 (1/2 cost)
Tractor and equipment 3.97 hrs. @ 80/hr.
TOTAL PREHARVEST COST
TOTAL HOURS HARVEST
Cost of Harvesting:
Tractor and equipment 2.2 hrs. @ 800/hr.
Interest on operating capital $6.72 @ 7% for 6 months
Total gross income
Landlord's share 1/2
Total operating costs (Landlord's share)
NET OPERATING PROFIT
Slaughter Hog Budget Grain Sorghum
Yield: 15 pigs per sow market weight 210 pounds per hog 3,150 total Ibs.
Price: $16.25 / cwt.
Income: 3,150 Ibs. @ $16.25/cwt. P511.87
Feed for Sow:
Supplement 115 lbs. @ $5.50/cwt. 6.33
Grain 700 lbs. @ $1.16/cwt. 8.12
Supplement 225 Ibs. @ $5.50/cwt. 12.38
Grain 900 lbs. @ $1.16/cwt. 10.44
Winter 1/4 ac. @ $13.66/ac. 3.42
Summer 1/4 ac. @ $12.15/ac. 3.04
Feed for Pigs:
Pig starter 500 lbs. @ $6/cwt. 30.00
Supplement 1,836 lbs. @ $ 5.50/cwt. 100.98
Grain 8,364 lbs. @ l.16/cwt. 97.02
Veterinary vaccines and medicines 15.00
Boar service 3.00
Labor 45 hrs. @ 750/hrs. 33.75
TOTAL PRODUCTION COSTS $323.48
Hauling $ 7.50
TOTAL MARKETING COSTS $ 22.50
Interest on operating capital $161.74 @ 7% for 12 months $ 11.32
Interest on investment capital $50 @ 7% for 12 months 3.50
TOTAL OTHER COSTS $ 14.82
Total income 511.87
Total operating expense 360.80
NET OPERATING PROFIT $151.07
Cow Calf Budget
Yield: 95 percent calf crop 60 percent of calves will be sold as slaughter
calves weighing 550 pounds and 40 percent of calves will be fed for 105
days and sold weighing 660 pounds.
Price: 550 pound calves per cwt. $21.00
660 pound steers per cwt. $23.25
Income: 314 pounds @ $21/cwt. $ 65.94
251 pounds @ $23.25/cwt. 58.35
TOTAL INCOME $124.29
Production Requirements and Cost:
Protein supplement 100 Ibs. @ 3.25 $ 3.25
Hay 1,000 lbs. @8 10.48/T 5.24
Summer 1/4 ac/cow @ $12.15/ac. 3.04
Winter 1/2 ac./cow @ $13.66/ac. 6.83
Salt and bonemeal .70
Veterinary vaccines and medicines 1.00
Bull services 4.00
Labor 12 hrs. @ 75/hr. 9.00
Creep feed 450 Ibs. @ 1l.21/cwt. 5.45
Feed required for 105 day feeding period:
Protein 210 lbs. @ `3.25/cwt. x 40% 2.73
Grain (corn) 1,050 lbs. @ $.68/cwt. x 40% 2.86
Hay 630 lbs. @ $10.48/T x 40% 1.32
Grinding and mixing $3.40 x 40% 1.36
TOTAL PRODUCTION COSTS $46.78
Hauling $ 1.00
TOTAL MARKETING COSTS $ 4.50
Interest on operating capital $38.51 @ 7% for 8 mos. $ 1.80
$ 8.27 @ 7% for 4 mos. .19
Interest on investment capital $175 @ 7% for 12 mos. 12.25
TOTAL OTHER COSTS $14.24
Total income $124.29
Total operating expenses 65.52
NET OPERATING PROFIT