Note:- Please do not send the solution as provided in HP WILLIAM'S BOOK (MODEL BUILDING IN MATHEMATICAL PROGRAMMING)

12.8 Farm planning

A farmer wishes to plan production on his 200 acre farm over the next five years. At present, he has a herd of 120 cows. This is made up of 20 heifers and 100 milk-producing cows. Each heifer needs 2/3 acre to support it and each dairy cow 1 acre. A dairy cow produces an average of 1.1 calves per year. Half of these calves will be bullocks, which are sold almost immediately for an average of 30 each. The remaining heifers can be either sold almost immediately for 40 or reared to become milk-producing cows at two years old. It is intended that all dairy cows be sold at 12 years old for an average of 120 each, although there will probably be an annual loss of 5% per year among heifers and 2% among dairy cows. At present, there are 10 cows each aged from newborn to 11 years old. The decision of how many heifers to sell in the current year has already been taken and implemented. The milk from a cow yields an annual revenue of 370. A maximum of 130 cows can be housed at the present time. To provide accommodation for each cow beyond this number will entail a capital outlay of 200 per cow. Each milkproducing cow requires 0.6 tons of grain and 0.7 tons of sugar beet per year. Grain and sugar beet can both be grown on the farm. Each acre yields 1.5 tons of sugar beet. Only 80 acres are suitable for growing grain. They can be divided into four groups whose yields are as follows:

Group 1	20 acres	1.1 tons per acre
Group 2	30 acres	0.9 tons per acre
Group 3	20 acres	0.8 tons per acre
Group 4	10 acres	0.65 tons per acre

Grain can be bought for 90 per ton and sold for 75 per ton. Sugar beet can be bought for 70 per ton and sold for 58 per ton. The labour requirements are as follows:

Each heifer	10 h per year
Each milk-producing cow	42 h per year
Each acre put to grain	4 h per year
Each acre put to sugar beet	14 h per year

Other costs are as follows:

Each heifer	50 per year
Each milk-producing cow	100 per year
Each acre put to grain	15 per year
Each acre put to sugar beet	10 per year

Labour costs for the farm are at present 4000 per year and provide 5500 h labour. Any labour needed above this will cost 1.20 per hour. How should the farmer operate over the next five years to maximise profit? Any capital expenditure would be financed by a 10-year loan at 15% annual interest. The interest and capital repayment would be paid in 10 equally sized yearly instalments. In no year can the cash flow be negative. Finally, the farmer would neither wish to reduce the total number of dairy cows at the end of the five-year period by more than 50% nor wish to increase the number by more than 75%.

Please provide the LP Solver file for the above model or provide all the algebraic equations which can be inserted in LP solver for all the constraints in a simplified manner.

Please do not send the solution as provided in HP WILLIAM'S BOOK (MODEL BUILDING IN MATHEMATICAL PROGRAMMING)

For Reference providing you the optimal solution value

The optimal plan results in a total profit of 121719 over the five years