Use of Linear Programming model in determining optimal production mix for increasing the farm income

(Source Betters, 1988).

Case: Because of an increase in population, a village is concerned with the availability of food, fuelwood for cooking and, to a lesser extent, supplement income. In particular malnutrition points to a nutritional need for more protein in the diet. Farming system analysis with the help of linear programming will be implemented for different farm types to study the prospective of introduction of agroforestry systems to satisfy village’s food and fuelwood needs.

For a farmer with five hectare farm land, the optimal mix of two agroforestry practice has to be studied. The two agroforestry practices are Eucalyptus beans and Eucalyptus maize, which are expected to give the following production costs and returns:

• In the above case LP helps us to answer the question, how many hectares of Eucalyptus beans and Eucalyptus maize should be grown?

• The returns of beans ($0.50/kg) and maize ($0.25/kg) are based on prices on the local market. The price of (Eucalyptus) is based on the number of hours necessary to collect an equivalent amount of fuelwood from neighbouring native forests ($30/m3).

The optimal number of hectares of both practices is calculated with the help of a linear programming model, which can be specified as follows:

Max ! NPV = 551 x1 + 501x2

Subject to150x1 + 125 x2 ≤ 900 (labour hours)280x1 + 225x2 ≤ 1200 (budget dollars)15x1 + 10x2 ≥ 60 (fuelwood m3)250x1 +200 x2 ≥ 900 (protein kg)x1+ x2 ≤ 5 (land ha) In which x1 and x2 are number of hectares with Eucalyptus beans and Eucalyptus maize respectively.

The first constraint is the constraint on labour, the second on the budget and the last one on land. The third and the fourth constraints refer to minimum subsistence requirements for the farm household: at least 60 m3 of fuelwood and 900 kg protein has to be produced. All these constraints refer to a period of three years.

The LP model produces the following results

Conclusion of farm income analysis

• The result shows that both agroforestry practices will be applied if the farmer optimizes financial results. Further, it can be concluded that only the budget is a limiting factor, if the budget could be increased with one dollar, the NPV will be higher by $3.48. More land or labour could not increase NPV, given the budget constraint. The farmer therefore consider to apply for a loan.

• Lastly, the fuelwood requirements reduces the production of protein; fuelwood is only produced to the amount required by the constraint. If the constraint with respect to fuelwood production would be related with one m3, the NPV had been $28.4 higher.