A farm occupies 200 acres of land and has £ 10000 available for investment. The family who run the farm can work for a total of 5000 work-hours during the winter and 7000 work-hours for the rest of the year. If any of these work-hours are not needed then the family will use them to help at the local village store, earning a profit of £ 9 per hour (regardless of the time of year). Past experience shows the family will not have more than 50 spare work-hours to work in the village store during the winter and not more than 80 over the summer.
The farm brings in cash for the family from three crops (lettuces, carrots, and swedes) and two types of livestock (dairy cows and hens). The family would like to maximise their cash income. The crops need no investment funds but each cow requires an initial investment of £ 1000 and each hen £ 10. Each cow needs 1.5 acres of land and the size of the barn limits the number of cows in the herd to 20. Each cow also require 100 work hours during the winter and 60 hours during the rest of the year. The hens do not need any specific land, but the farm can accommodate a maximum of 100 hens. To look after them well, each hen needs 1.1 work hours during the winter and 0.8 work hours during the rest of the year. A cow brings in a net annual income of £ 400 while a hen generates £ 3.
Crops also generate income and require work hours to tend to them. The table below sum- marises such information:
crop Lettuce Carrots Swede
winter work hours 202510
summer work hours 104010
income (per acre) £ 100 £ 130 £ 110
Formulate this scenario as a linear programming problem, making sure you clearly and fully define your decision variables, constraints, and objective function.
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