Two crops sugarcane and wheat

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Reference no: EM133853652

Question

A Queensland farmer is deciding how to allocate their limited land between two crops: sugarcane and wheat. The farmer wants to maximize total profit, where each plot of sugarcane generates $20,000 in profit and each plot of wheat generates $10,000 in profit. Growing on the arid land in Queensland can be tough, so both crops will need both fertiliser and irrigated water, but they have different requirements. Sugarcane is nutrient-hungry, so it needs 3 units of fertiliser per plot, whereas wheat is more hardy so it only needs 1 unit of fertiliser per plot. On the other hand, sugarcane can grow in drier environments, so it only requires 2 units of water per plot, whereas wheat needs a lot of water: 7 units per plot. The farmer cares about two things: the environment, and making a livelihood by selling crops. Fertilizer runs off into the local waterways, harming coral reefs and fish. So, the farmer wants to limit their total use of fertiliser to 3 units. Queensland has a long history of droughts, so the farmer is very water-wise, wanting to only use 14 units of water in total. The birds in the farmer's region prefer to be in natural forests, but if they have to live in cropland then they prefer wheat. Knowing this, the farmer wants to make sure that the number of plots planted with sugarcane does not exceed the number of plots planted with wheat by more than | plot. The farmer wants to maximise the total profit from selling both crops while staying within the envi- ronmental limits. How many plots of sugarcane and wheat should the farmer plant?

(a) Write out the linear programming problem described above, including the objective and all constraints.

(b) Solve using the graphical method and make a recommendation to the farmer.

(c) Identify all the basic solutions (list coordinates). Please describe whether each solution is feasible or infeasible.

(d) Use graphical sensitivity analysis to determine how much the profitability of either crop can change before the optimal solution changes.

Reference no: EM133853652

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