MP in Action‎ > ‎Other Applications‎ > ‎

Farm Management by Mathematical Programming

This article is based on a talk given by Eric Audsley of Silsoe Research Institute to the Mathematical Programming Study Group.

Farming as a Business

Farming is becoming an increasingly complex business in which decision support systems have a valuable role to play. The Decision Systems Group at Silsoe develops mathematical models of crops, animals and farm systems to aid decision making.

Much of this is concerned with describing how crops and animals grow and their dependence on various different factors such as date of sowing, quantity of chemical fertiliser applied, rainfall and sunshine.

In addition, some parts of these models are incorporated within more strategic models based on Mathematical Programming. These aim to answer questions of what farmers should do to maximize their profit, typically when faced with new crops or machinery, or changes to policy, legislation or price support. Eric Audsley described three such models.

Arable Farm Model

Arable farming consists of a cycle of activities: planting, growing and reaping. There is great variability in the amount of effort required at different times in this cycle, as shown in the chart below.

Figure 1: Hours of Work Required vs Hours Available by Soil Type

There is a very large peak around harvest time and a smaller one at planting. Other charts could be drawn up showing the requirements for the various resources on the farm: labour; tractor; drill; sprayer; harvester. Different crops have different requirements and there are further possibilities of shifting the dates of planting and harvesting, although these may affect yield. The number of hours a day that the land can be worked also varies through the year, being at a peak in the summer.

The problem which the farmer faces can thus be posed as:


  • the available area of land and resources on the farm;
  • the crops currently being grown;


  • the area of land allocated to each crop at each possible planting date;
  • the quantity of additional contract labour to employ;
  • whether to buy extra or replacement machinery

So as to

  • maximize the expected profit

The model runs on a PC and accesses an extensive database of crops and yields and how they vary with soil type, fertiliser, etc. These enable the model to be run on different farms nationwide.

Typical results from the model show the interconnectedness of the various crops and the opportunity costs of making changes, which can be far from obvious.

For instance, Winter barley is less profitable than wheat but introducing Spring rather than Winter oilseed rape causes a switch of some land from wheat to Winter barley. This is because Spring rape is harvested in August, which is when wheat is also harvested, and so the constraint on the harvester in August is hit. This then delays ploughing and drilling, which makes it more profitable to switch some land from wheat to an apparently less profitable alternative.

Brussels Sprouts Model

Brussels sprouts are an unusual crop because the market is segmented by the size of the buttons but the plants continue to grow throughout the season with the buttons becoming increasingly large until the plants shoot to flower and become waste. The yield as a function of button size and period of growth for one variety is shown on the graph. There is a wide choice of varieties which mature at different times but the shape of the graphs are similar.

Figure 2: Yield of Brussels Sprouts by Button Size and Date

There are several different types of buyer for Brussels sprouts: freezer companies, who want small buttons (15 - 25mm) between September and December; supermarkets, who want continuity of supply but with a peak in demand in the week before Christmas; and the general market trade, who will take what is available, but generally at lower prices than supermarkets.

These factors make the farmers' decisions unusually complex about which varieties to grow and how. They need to decide:

  • which varieties to grow;
  • the area of each;
  • the date of planting;
  • the method of planting (direct drill or seed-bed then transplant);
  • the spacing (closer together reduces the size of the buttons but also increases costs);
  • timing of harvesting.

The Institute has built a simulation of the growth of sprouts as a function of all these. This is then used within an optimizing model whose aim is to maximize the profit while meeting contracted requirements for sprouts of specified sizes at specified times and disposing of surpluses in the market. The result is a planting plan and a harvesting plan.

Dairy Farm Model

In running a dairy farm the key decisions are:

  • the number of cows to keep;
  • the use of the available grass: how much it is grazed or used for making silage, and if so, the pattern of cuttings;
  • the feeding level of the the cows: how much grass, silage and concentrates to feed them;
  • the calving pattern: the number of cows calving each month.

As those with lawns will know, grass does not grow uniformly through the growing season. It grows most vigorously at the beginning of the season, becoming progressively less vigorous as summer wears on and then picks up again towards autumn.

Another complication is that young grass has a higher energy content than old grass, which has more stem and is less palatable to cows, both when grazed and when consumed in silage. When grass is cut for silage it takes time to recover: a reasonable estimate is that the next cut is put back one week.

The milk yield of a cow rises to a peak soon after calving and then falls gradually over the remainder of the lactation. To a first approximation, one can work back from the milk yield to calculate the metabolisable energy required and derive a constraint on the energy content of the feed to be provided (whether as grass, silage or concentrates). If insufficient energy is provided, the cow will lose body weight beyond the standard weight changes during the lactation. This impairs the future milk yield permanently. Conversely, extra energy leads to a weight gain and an increase in future milk yield.

As well as being concerned with metabolisable energy, one needs to consider a cow's protein requirements and the quantity of dry matter which she will eat. Protein is similar to energy in that a minimum must be supplied for the cow to retain her weight. Dry matter is different: a cow will only eat a certain quantity of dry matter (grass, silage or concentrates) and this is a function of her milk yield. If the dry matter has low energy content, she may lose weight unless extra concentrates (which are expensive) are provided.

The model can be run to determine the optimum stocking level for a farm given a particular milk quota. This shows that, for instance, more cows should be kept on a given farm if the milk quota is increased, but that if the farmer is maximizing his profit, there is an upper limit to the number of cows to be kept which is irrespective of quota. The model provides confirmation of the standard practice of having the main calving in March with a secondary one in August: this really follows from the periods of peak growth of grass. Although many of the results confirm common practice, the model has particular value in showing the rational response to future changes in policy or economics.

Related articles include Wishing you a Merry Christmas (with MP). To find other articles, refer to the MP in Action page.