Palouse Grains & Legumes, Inc. (PGL) is trying to determine its crop rotation schedule for the next four years. Despite the corporate-sounding name, PGL is a 160-acre family farm. The crops available to plant are chickpeas, lentils, spring and winter wheat (combined for convenience), and alfalfa. The following table provides average yields and prices:

Under the most simplistic interpretation, PGL would just choose the most profitable crop and plant it every year. However, agricultural decisions are far more complicated than that. In order to ensure long-term soil quality, etc., farms must allow each acre to go fallow (unplanted) every few years (in PGL’s case, every fourth year for each acre). In other words, both crop types and fallow years must be included in the crop rotation schedule. PGL must rotate crops so that the same crop is never planted in two consecutive years on a given acre. Additionally, to reduce risk and smooth consumption, PGL would like to earn a similar profit each year (so they can’t let all acres go fallow in a single year, even if that would be the most profitable long-run option). There is some uncertainty associated with each crop’s production, but we will ignore that for now. Using the information provided, answer the following questions:

a) What are the decision variables in this problem? Explain them carefully. [Hint: There can be more than 20 depending on how you set up the problem, but you may be able to explain them in groups.]

b) Write out the objective function.

c) Explain the constraints you would need (in words, at least).

d) Can we solve this problem using the linear programming approach? Explain your answer
 

Crop Chickpeas Lentils Wheat, combined Alfalfa Average yield (units/acre) 1,750 lbs. 1,500 lbs. 129 bushels 170 bushels Average price (per unit) $0.38/lb. $0.39/lb. $4.38/bushel $3.90/bushel Crop Chickpeas Lentils Wheat, combined Alfalfa Average yield (units/acre) 1,750 lbs. 1,500 lbs. 129 bushels 170 bushels Average price (per unit) $0.38/lb. $0.39/lb. $4.38/bushel $3.90/bushel Crop Chickpeas Lentils Wheat, combined Alfalfa Average yield (units/acre) 1,750 lbs. 1,500 lbs. 129 bushels 170 bushels Average price (per unit) $0.38/lb. $0.39/lb. $4.38/bushel $3.90/bushel Crop Chickpeas Lentils Wheat, combined Alfalfa Average yield (units/acre) 1,750 lbs. 1,500 lbs. 129 bushels 170 bushels Average price (per unit) $0.38/lb. $0.39/lb. $4.38/bushel $3.90/bushel Crop Chickpeas Lentils Wheat, combined Alfalfa Average yield (units/acre) 1,750 lbs. 1,500 lbs. 129 bushels 170 bushels Average price (per unit) $0.38/lb. $0.39/lb. $4.38/bushel $3.90/bushel Crop Chickpeas Lentils Wheat, combined Alfalfa Average yield (units/acre) 1,750 lbs. 1,500 lbs. 129 bushels 170 bushels Average price (per unit) $0.38/lb. $0.39/lb. $4.38/bushel $3.90/bushel Crop Chickpeas Lentils Wheat, combined Alfalfa Average yield (units/acre) 1,750 lbs. 1,500 lbs. 129 bushels 170 bushels Average price (per unit) $0.38/lb. $0.39/lb. $4.38/bushel $3.90/bushel