Length : around 50 minutes
The following statement leads to the construction of a dairy farm model. Write the model, that you can call elevageLait_1.gms, check that it is accurate, and answer the questions below.
« M. Fourhooves has a dairy farm in the west of France. It is a 50 ha farm, 30 ha of which is arable land and 20 ha is permanent grassland. The farmer can grow wheat and/or maize (for selling purposes) on the arable land. Maize is exclusively used as silage to feed the dairy cows. The permanent grassland cannot be tilled (CAP greening regulations). Every grassland hectare is grazed then mowed to produce hay and wrapped bales.
The cows graze for 6 months (beginning of April – end of September). During the winter period, they are fed maize silage (produced on the farm) and a supplement of soya-bean cakes (bought). The heifers are fed by grazing, hay and wrapped bales.
The average production per dairy cow is 7000 litres (L) of milk per year.
Plant production
| Yields (t /ha) | Operational costs (€ /ha) | ||||
| Wheat | 6 | 600 |
| Fodder | Fodder yields (t /ha) | Operational costs (€ /ha) | |||
| Maize | Silage maize | 11.0 | 620 | ||
| Grassland | Grazing | 5 | 300 |
||
| Hay | 4.5 | ||||
| Wrapped bales | 2.2 | ||||
Maize is only used for the animals, wheat is sold. Every hectare is grazed then mowed in order to produce hay and wrapped bales.
Animal activities
Certain technico-economic indicators concerning the organization of the herd are available : herd management parameters and herd food needs.
Herd management, breeding, culling
Every cow gives birth to 0,9 calves per year which are either sold (CALF), or kept for herd renewal (HEIFER). Every year, 0,3 cows are sold as cull cows and replaced by selected heifers. The culling rate is equal to the renewal rate.
Food needs
The animals are fed using both food produced on the farm and purchased food. Food needs were estimated for cows and heifers.
“Grass-based” feeding diet
| Fodder products | Cows | Heifers (what heifers need until they give birth) |
| Hay | 2 t /year | |
| Wrapped bales | 2,6 t /year | |
| Grazing | 1.5 t/year | 2.5 t/year |
| Maize | 1 t / year | |
| Supplements bought | Cows | Heifers |
| Soya | 0,45 t /year |
Technico-economic data
The price of milk is 0,35 €/L. The price of wheat is 150€ per ton (T). The livestock farmer is subject to milk quotas which limit annual milk production to 300 00 L.
The price of purchased soya is 360 €/t
Price of meat products :
Cows : 800€/cow
Calf : 170 €/ calf
Other expenses including structural costs (depreciation of buildings, …) are estimated at 1300 € per cow and per year regardless of the diet.
The objective of the farmer is to maximize his income.
If you need help and/or to make sure you use the same names for the data : 
Part A : Writing the model
Write the model below. Check whether the results seem coherent :
- Does the number of animals meet the constraints linked to the renewal and culling rates ?
- Are the food needs covered by the production of cultivated areas ?
- Does the farmer respect the dairy quota ?
Then check the solution : 
Part B : Analysing the solutions
Analyse the solutions and answer the following questions :
- What are the binding constraints ?
- How does the farmer choose between wheat production and crops destined to feed the animals ?
- Calculations reveal that the farmer produces 67 tons of hay when he actually only needs 25 tons to feed his cattle. Why does he produce too much ?
Then check the solution :
