Linear programming offers promise as an analytical tool for studying the economic aspects of farm management. In its present stage of development, the contribution of the technique of linear programming in agriculture is primarily that of furnishing a conceptual framework within which problems of resource allocation on farms may be more systematically studied. There have been extremely few applications of linear programming to specific farms for the purpose of obtaining an optimum production scheme for these particular farms. Data describing the technical relationships on a given farm are usually difficult to obtain. Also, the linear programming analyst will frequently wish to consider types of production which a given farmer has never employed. The size of the business unit in agriculture in relation to the computational burden of planning operations by the linear programming technique is also a factor in the direct application to specific farms.

Linear programming might be viewed simply as a technique foi
finding the logical implication of a set of assumptions. In the analysis
which follows these assumptions deal with resources available for allocation,technical

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1) This article was written while the author was a Fulbright Research Scholar in th< Department of Agricultural Economics at the Royal Veterinary and Agricultura College. Since a linear programming analysis depends to a large extent upon thi judgment exercised in the construction of a model, the author is greatly indebtec to the members of a seminar at the Royal Veterinary and Agricultural Gollegi whose intimate knowledge of Danish agriculture made possible a more realistii model than would otherwise have been possible. Members of the seminar were Chr. Jørgensen, Erik Kristensen, Carl C. Thomsen, and Emil Vestergaard-Jensen

2) Professor, University of Illinois

cation,technicalrelationships (crop yields, feed requirements for livestock,etc.) and prices and costs. Once the assumptions have been explicitlystated, the problem of obtaining a solution is purely mechanical. Consequently the results can only be interpreted in close connection with the initial set of assumptions.

The purpose of this article is to illustrate the use of the technique in an analysis of the effect of milk and swine price relationships on the optimum production plan for a farm in Zealand. Since the technical data do not come from a specific farm, but are the average experience of many farmers, the results are, of course, not intended to be a prescription or recipe for operation of any given farm. Rather, our focus is on the general relationships derived from varying certain of the initial assumptions, in this case, the prices of two commodities.

I. The Model.

Building the linear programming model may be considered in the
following steps:

1. Specification of the resources which are assumed to be fixed

2. Choosing the products to be considered as alternatives in production.

3. Estimating the quantity of each of the fixed resources required to
produce a unit of each product.

4. Developing the relationships which define the limits of production
of the production possibilities.

5. Choosing the product prices and estimating the costs of resources
not considered fixed (see 1 above), and constructing the income equation
to be maximized subject to the limits of available resources.

We shall now consider in a rather brief manner the important elements
in each of the steps outlined above.

1. An attempt was made to select the kinds of recources and to fix
them at levels which would represent conditions on an average farm in
Zealand.

Fixing the levels of specific resources is equivalent to choosing the length of the planning period. In this case the period is one in which it is contemplated that farm size, the family labor force, and livestock building space will not be altered.

We assume that 15 ha. of land are available to the farmer. However,
of this area, 1.5 ha. are assumed to be in permanent pasture, leaving
13.5 ha. available for rotation crops. A family labor supply of 500

hours per month is assumed to be available. This might consist of. for example, the farmer, his wife, and a son. Day labor map be hired during the months of June and October to help with the beet production which has a high labor requirement during these months.

Buildings are assumed to be available to house dairy animals, swine, and two horses. Space is available for a maximum of 15 mature dairy animal uits³) and for 70 units of swine production⁴). However, some of the dairy building space may be used for swine production making possible a maximum increase of swine production by 50 units (a total possible maximum of 120 units insofar as building space is concerned). Such an increase in swine production would be accompanied by a decrease in building space available for dairy production of 10 mature dairy animal units. Thus it is assumed that space for 5 mature dairy animal units is specialized for dairy production and cannot be transferred to swine production. Two horses are assumed to occupy the space available for them.

2. The following products usually produced on farms in Zealand
were selected to be included as possibilities in the model:

3. The crop yields per hectare (from which the land requirements for
each unit of crop production may be calculated) are presented in Table 1.
These are average yields for the period 1950-54.

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3) One mature dairy animal unit is equal to one cow or 2.25 young animals

4) A swine unit is 90 kg. of production.

Since it has been observed that farmers who have a high percent of root crops in their crop rotation scheme characteristically have higher grain yields per hectare, the average grain yields have been adjusted to indicate this beneficial effect of root crops. This assumes that the planning period is over a period of years and hence sufficiently long for this effect to occur.

The labor requirements for crop production are presented in Table 2. In this model only one technique or method of production for each crop is assumed (see footnotes to Table 2). For each crop, an attempt was made to select the most common technique followed in the production of that crop. A more complete model might include alternatives of both more and less mechanization than considered in this model.

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5) Landbrugsstatistik 1957, Gartneri, Skovbrug m. v. København 1958. Tables 15 and 17. Wheat and barley yields have been adjusted to reflect the percentage of root crops in the rotation according' to the relationship reported in Undersøgelser over landbrugets driftsforhold. XXXXI, 2. del, København 1958, page 12.

6) Lomme-Håndbog for Konsulenter og Landbrugslærere. 111. Driftsøkonomi m. v. 17.

7) Seed-bed preparation and harvesting with tractor, transport with horses, and winter theshing.

8) One-half fodder beets and one-half kohlrabi. For the fodder beets, seed-bed preparation with tractor and all other labor with horses. For kohlrabi, seed-bed preparation and root removal with tractor, and transport with horses.

9) One-third harvested as hay. Seed-bed preparation with tractor and all other pre paration with horses.

10) Seed-bed preparation with tractor and all other operations with horses

11) Non-critical operations which may be performed at nearly any time duing the year.

The feed requirements for livestock are presented in Table 3 and the labor requirements are in Table 4. Note that the per unit labor requirements decrease with size of herd. There is no completely satisfactory method for handling this type of relationship within a linear programming model¹²). In the calculation of the solutions, the medium-sized herd requirements were used.

4. We now turn to the more formal phase of describing the productior relationships in algebraic form. The central notion of linear program ming is that the production complex may be broken down into individua processes¹⁶). A process may be viewed simply as a list of the resoura requirements per unit of product.

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12) Svanson, Earl R. Programming Optimal Farm Plans. I Farm Size and, Outpu Research. Southern Cooperative Series Bulletin No. 56. Oklahoma Agr. Exj Sta. June 1958 pp. 47-58. The case of non-linear increasing coefficient can b handled by considering several levels of production as different „products".

13) Undersøgelser over landbrugets driftsforhold. XXXXI, 2. del. København 195J

14) Straw excluded. One crop unit (see Table 1) is equal to 100 feed units.

15) Lomme-Håndbog for Konsulenter og Landbrugslærere. TIL Driftsøkonomi m. v. 1'

16) See Danø, Sven. Linear programming i produktionsteorien I, II og 111. Natic naløkonomisk Tidskrift 93: 94-117, 205-233. 1955, 94: 47-61. 1956.

The resource requirements for each unit of the product generated

by

each process must be independent of the level of production of that
process as well as independent of the level of production of the other
processes in the model.

One may frequently encounter situations in agriculture in which this requirement is not satisfied. In the model considered here, the yields of grain are affected by the percent of root crops in the rotation. Even if this relationship were non-linear it could be approximated by defining the processes as crop combinations.

In this model the variables reprecenting the crop production processes
cansist of combinations of crops rather than single crops:

The combination of crops into a process rather than consideration of, for example, hectares of wheat as xi, hectares of barley as x<>, etc. permits the model to take into account the effect of the percentage of root crops on grain yield. Note that total grain production is permitted to range from 30 % to 60 % of the area with complete substitution of wheat and barley at the lower limit, but with limited substitutability at the upper limit. Selection of a single rotation, that is, xi, X2, etc. is not necessarily required by the model: linear combinations of the crop variables are permitted.

The livestock variables are as follows:

x9x9 = number of dairy cattle (mature animal units)

Xio = number of swine (in units of 90 kg. production)

The dairy cattle enterprise produces two products for the markat - nilk and meat. In practice, there exists a possibility to shift emphasis :_rom one product to another to respond to changing market conditions, [deally the dairy enterprise variable X9, should be separated into perhaps

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17) Wh= Wheat Ba = Barley Pa = Rotation pasture Fb — One-half fodder beets and one-half kohlrabi Sb = Sugar beets

two or more processes which would permit choices among the alternativeswithin the dairy enterprise. Data are not readily available for the consideration of such alternatives. Consequently xg represents an average type of production.

Similarly, swine production might have been more realistically represented by several alternative methods of production (e. g. feeding purchased pigs and feeding pigs produced on the farm). As represented in this model, swine production is a mixture of feeding some purchased pigs and some produced on the farm.

Barley is a common livestock feed which is rather easily bought and
sold. In order that these alternatives may be included, the following
variables are introduced:

xii = hkg. of barley sold

Xl2 = hkg. of barley purchased.

Hiring labor during June and October requires the following variables:

Xi 3 = hours of labor hired during June

Xi4 = hours of labor hired during October

Finally, a variable is needed to make space available for swine produc
tion from any decrease occurring in the number of dairy cattle:

Xl5 = dairy cattle building space transferred to swine produc
tion (in units of space required for one mature dair
animal unit).

We now turn to the statement of the relationships which define th
production possibilities for this farm situation. Land available for crop
ping is the first relationship:

Thus, the total land area occupied by rotation crops cannot excee
13.5 ha.

The family labor supply of 500 hours in each month also needs to b
considered. For example, the limit on production imposed by the Apr
labor supply is as follows:

The coefficients for each variable are the hours of labor required b one unit of production of that particular variable. For exampL one hectare of xi (40% wheat, 20% barley, 30% rotation pasture, an 10% fodder beets) requires 4.81 hours during April; one dairy un requires 10.9 hours during April, etc. The restrictions of the labor supp]

in the other months may also be calculated from the data in Tables
2 and 4.

Since the hiring of labor to help during the peak labor requirement periods of June and October is a possibility in this model, the labor restrictions have an additional variable in these months. For example, the June restriction appears as follows:

Provision must be made for the flow of feed produced by the crops to the livestock. Two relationships are needed, one for roughage and one for barley. For roughage, we assume that it may be produced, and completely fed to livestock, or that part of it may be fed and part of it simply be produced in surplus.

In terms of feed units the roughage relationship is:

The constant, 4076 feed units, represents the roughage production
From the 1.5 ha. of permanent pasture minus the feed requirement for
:he horses.

Because of its bulky nature, no provision has been made for the
purchase and sale of roughage.

In contrast, the barley relation (in units of hkg.) is an accounting
dentity permitting sale or purchase of barley:

Again, the constant 13.38 hkg. represents a fixed requirement for
mrse feed.

The building restricitions are as follows

5. In the preceding section the relationships defining the limits on roduction have been outlined. It remains to indicate the relative alues or weights which must be assigned to each variable in order that e may develop a criterion for choosing an optimum solution from the iany crop and livestock combinations that are possible with the

available resources. The criterion for choosing the optimum solution in this problem is maximization of return to the fixed resources. Thus each variable will have a coefficient indicating the „net" return, thai is, the gross revenues per unit of production minus the unit costs ol resources not considered as fixed..

The prices for crops and livestock (see Table 5) used are three-year averages (1954-55 through 1956-57). However, since interest lies in the effect of variation of milk and swine prices, the results which follow also include optimum systems of farming for milk and swine prices which deviate from the averages shown in Table 5. Livestock income per unit is presented in Table 6. These data include credit for manure production.

Production costs for crops are presented in Table 7 and those foi livestock are in Table 8. These production costs do not include charge: for land, laber and buildings, since these resources are considered a; being in fixed supply. Further, there has been no charge made foi management. This means that this resource is present in adequat< supply for the size af operation permitted by the land, labor, anc buildings. The relation to be maximized thus represents the return (ir

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18) Undersøgelser over landbrugets driftsforhold XXXX and XXXXI 1. del Table 5.

19) Undersøgelser over landbrugets driftsforhold XXX 2. del, Tables 10 and 14 XXXXI 2. del. Tables 11 and 15.

crowns) to land and buildings and the labor and management of the
farmer and his family:

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20) Undersøgelser over landbrugets driftsforhold. XXXXI, 2. del. København 1958. Tables 19, 21, 24 and 25.

21) Undersøgelser over landbrugets driftsforhold XXXXI, 2. del. København 1958. Tables 11 and 15.

The net return coefficient for each of the crop processes, xi, through XB, includes only the income from cash crops, wheat and sugar beets, while the costs represent the weighted production costs of all the crops produced in that particuler process. The income from roughage produced enters the relation to be maximized through the livestock processes, xg, and xio- Barley sold on the market appears as xn, while that purchased is xl2.x₁₂. Labor may be hired at four crowns per hour (xis and X1.4). The changes necessary to make dairy cattle space available for swine are assumed to involve no cost within the limits outlined above. The constant, 1138 crowns, represents annual fixed horse costs other than feed.

The prohlem now is to solve for values for the variables which will
maximize the net return relation presented immedeately above and also
satisfy the restrictions on production outlined in section 4.

II. Results.

1. Solutions were obtained by use of the simplex method²²). Using the three year average prices in Table 5, the highest return to fixed resources is achieved by Plan A in Table 9. This plan requires the cropping system X4, which emphasizes barley and sugar beet production. The dairy enterprise occupies only the specialized space available (5 mature units), while the swine enterprise is at its maximum permitted by the building space. The expected annual return to the fixed resources wher the three year average prices are used is 24, 110 crowns.

The effective restrictions on production are land and buildings. The family labor supply is adequate in all of the months except June anc October, when hired labor is employed. The total labor use, including hired labor, amounts to 4075 hours. Utilization of only approximate^ two-thirds of the available family labor supply seems peculiar in : country in which farmers apparently work long hours. A more complet« utilization of family labor might have occurred if processes having higl labor requirements during the slack seasons had been included. Also i is possible that some of the overhead labor is not reported and heno could not be allocated to the specific enterprises (Tables 2 and 4).

The solution to a linear programming problem also gives estimates o the increases in income which would accompany an increase in one uni of the resources which are effective in limiting production. In this in stance, an additional hectare of crop land is estimated to increase annua income by approximately 1000 crowns. Adding space for one unit o

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22) Danø, Sven. Numerisk Løsning av Lineære Programmeringsproblemer ved Simplex Metoden. Universitetets økonomiske Laboratorium. København 1957.

swine production would add approximately 52 crowns to annual income; and space for a dairy unit would add about 260 crowns to annual income. These returns might be compared with new building costs to aid in determiningthe desirability of investment.

2. In this section the influence of changes in milk and swine prices on the optium organization of production are analyzed. In Figure 1, the horizontal axis shows a range of swine prices going from approximately 20% above the threeyear average (Table 5) to 20% below this average. Similarly, on the vertical axis, milk prices vary through about the same range.

For the milk-swine price combinations lying in, for example, area A,

the farm plan A (Table 9) is the organization yielding the highest return to the fixed resources. The boundary line between, for example, areas A and B, represents milk-swine price combinations under which plans A and B would yield the same return to the fixed resources.

The very favorable position of swine production is the most out standing feature of this analysis. In terms of price relationships tha have been experienced, extremely low swine and high milk prices ar needed to have an optimum plan which does not emphasize swine.

While Plan A has the maximum possible amount of swine productioi (120 units). Plan E has the maximum number of dairy units (15.0). Plan G, D, and E require large amounts of roughage production for the dair cattle. This requires a shift in acreage from grain to rotation pasture anc in Plan E, a small amount of fodder beets. Because of their importanc in usual Danish cropping systems, the failure of fodder beets to compet successfully with the other crops requires explanation. Although th results indicate that rotation pasture is a more economical source c roughage, the advantage of a cultivated crop, such as fodder beets, i terms of weed control in succeeding years may be an important omitte element in the model. If farmers take this into account in their decision the linear programmer might also obtain an estimate of this relationshi

and the model could be modified in the fashion indicated above for the
grain yield-root crop ares relationship considered in this model (Table 1).

One might inquire concerning the reluctance of the Danish farmer to increase swine production in the direction indicated by these results. It has been suggested that his awareness of the ultimate consequences of farmers as a group also expanding production acts as a deterrent. If this is true, our model needs to be modified to embrace the total market demand schedules for the products, rather than the horizontal demand schedules assumed in this model. In such a situation, the role of the price mechanism as a reflection of consumers wants in guiding the composition of agriculturel production is somewhat different than that which usually characterizes agriculture.

Further, the competitive position of swine in this model may appear stronger than if the relative uncertainty surrounding prospective swine and milk prices were considered. (Swine prices are believed to fluctuate more than milk prices). However, it appears from this analysis that the subjective uncertainty discount would need to be substantial if this were the sole reason for refraining from an expansion of pig production. The seasonal distribution of income may also be more desirable from dairy cows than from swine. If certain minimum monthly cash income levels can be specified, this advantage of milk production can easily be incorporated into the model.

It is hoped that the rather simple analysis in this article will be suggestive
of further elaborations of the model which will yield better
insights into problems of adjustment of production in Danish agriculture.