Precision Farming

Farmer Ben from DS City needs help deciding what crops to plant and what treatments to use. He is thinking about next year as well, when he will need money to buy new seeds and also make some repairs to his machines. Given the uncertainty in yields and selling prices, what would be your recommendation for him?

Ben owns 100 acres of land and has experience in growing corn and soybeans. He can spend $12,000 for seeds and treatments. Seeding an acre costs $100 for corn and $120 for soy. For corn, the expected yield is 25 tons per acre with a standard deviation of 1.5 tons (we assume normal distributions everywhere in this puzzle). He expects to sell a ton of corn for $225 with a standard deviation of $22.50. For soy, the expected yield is 10 tons per acre with a standard deviation of 2 tons. The expected sale price for soybeans is $500 per ton with a standard deviation of $10.

There are two different kinds of treatments: fertilizers, which increase the expected yield, and pesticides, which lower the variance in yield. Fertilizing an acre of corn costs $10 and increases the expected yield by 2 tons. Fertilizing an acre of soy costs $30 and increases the expected yield by 0.2 tons. The pesticides lower the standard deviation in yield by 0.2 tons at a cost of $60 for corn and by 1.5 tons at a cost of $10 for soy. When Ben applies both fertilizer and pesticide for corn, the cost for both treatments is $50 per acre, a $20 discount over what both treatments would cost individually. For soy, the discount is $10, bringing the total cost to $30 for applying both treatments.

Ben's fixed costs are $445,000 for the year, which include labor, insurance, and gas. Ben is rather risk-averse and wants to make sure that his revenue will not only cover his fixed costs but also potential repairs of $7,000 for one of his tractors plus the money needed to start next year's seed, which he estimates at 10% above the costs for seeds and treatments spent in this year. He therefore requests that the net profit for this year's plan exhibit a conditional value at risk of 5% of at least 1.1 times this year's investment in seeds and treatments plus $7,000 for future repairs.

What should Ben plant, and what treatments should he buy?

If you have found a plan that meets Ben's requirements, what would you recommend regarding Ben's aversion to risk? Is there a better plan for him?

In this puzzle, we need to find a linear combination of different products of normal distributions with maximum expected value while exhibiting a CVaR-5% that exceeds the desired threshold. In total, we have 8 normal distributions for the yield, four for each choice of crop, which we can opt to fertilize and/or protect from insects, times the respective price distribution.

Using this analysis, we would recommend that Ben plant corn on 85 acres using only fertilizer but no pesticides and soybeans on the remaining 15 acres using both treatments. This incurs upfront costs of $11,600, within the budget of $12,000 that Ben has set aside. After subtracting all fixed costs and upfront investments, we expect Ben to make a net profit of $136,225, with a CVaR-5% of $19,763, which slightly exceeds the 1.1*$11,600 + 7,000 that he set as a threshold.

Ben should be advised to use even more land to plant the more profitable corn, though. If he uses 92.3 acres for that and only 7.7 acres for soybeans, his expected profit rises to $143,500, while his CVaR-5% only reduces slightly to $18,000. That is, by increasing his risk tolerance to accept only $1,800 less in profit (as a one-in-20-years average), he can expect to increase his profits by over $7,000.

Operations Uncer Uncertainty

Many companies must make decisions like Ben, with only incomplete knowledge of the future. They require a tool that can help them make these decisions on an operational basis. One such tool is InsideOpt Seeker. The complete model for today's puzzle is given below. As you can see, we ask Seeker to work with 1,000 scenarios in this case. Moreover, we exploit Seeker's unique multi-objective optimization capabilities to find a favorable tradeoff between the CVaR-5% and the expected profits.