Linear programming for profitable farming

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Linear programming for profitable farming
Parker, Cecil A.
Texas Agricultural Experiment Station],
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Corn ( jstor )
Linear programming ( jstor )
Hogs ( jstor )

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Cecil A. Parker


Peter E. Hildebrand


Corpus Christi, Texas

August 11, 1960


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

net profit?

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

resource restrictions.

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

maximize profit.

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

Cow-slaughter calf

Sows-slaughter hogs

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

and slaughter

prices for each

extremely important.







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.


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.



Enterprises Units

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

Cotton 52.8

Net Operating $10,029

Cotton Forced in Sows Stopped
at 20
Units Units

40 20

40 41

133.0 66.6

9.9 10.3

12.3 12.8

29.7 30.9

0 64.4

55.0 55.0

$9994 $8959

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.


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

different prices.

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.


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
Grain Sorghum
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


Corn Budget

Yield: 45 bushels per acre

Price: $1 per bushel

Income: 45 bu. @ $1/bu.

Production Requirements:

Flow middles
Plant and fertilize

Times Over

Total, Hurs
Man Tractor
.80 .80
.80 .80
.50 .50
.37 .37

3.97 3.97

Cost Preharvest:
Seed 4 lbs. @ 120/lb.
Fertilizer 150 lbs. 16-20-0 (1/2 cost)
Tractor and equipment 3.97 hrs. @ 80/hr.

Corn picking
Cut stalks



Cost of Harvesting:
Tractor and equipment 2.2 hrs. @ 800/hr.

Other Costs:
Interest on operating capital $6.72 @ 7% for 6 months

Total gross income


Landlord's share 1/2
Total operating costs (Landlord's share)



$ .48
$ 6.72

$ 1.76

$ .24



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

Production Requirements:
Feed for Sow:
Gestation period:
Supplement 115 lbs. @ $5.50/cwt. 6.33
Grain 700 lbs. @ $1.16/cwt. 8.12
Lactation Period:
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

Marketing Costs:
Hauling $ 7.50
Marketing 15.00

Other expenses:
Interest on operating capital $161.74 @ 7% for 12 months $ 11.32
Interest on investment capital $50 @ 7% for 12 months 3.50

Total income 511.87
Total operating expense 360.80

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

Production Requirements and Cost:
Protein supplement 100 Ibs. @ 3.25 $ 3.25
Hay 1,000 lbs. @8 10.48/T 5.24
Supplemental Pasture
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

Marketing Costs:
Hauling $ 1.00
Marketing 3.50

Other Costs:
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 income $124.29

Total operating expenses 65.52


$ 58.77