Grant Miller currently manages his 640-acre family farm. Together, with siblings, children, and cousins, the entire family can produce a total of 4,000 person-hours of labor during the winter and spring months, and a total of 4,500 person-hours during the summer and fall. If any of these person-hours are not needed, Grant and his family will work on a neighboring farm for $8.50/hour during the winter and spring months and $10.00/hour in the summer and fall months. The farm supports two types of livestock, dairy cows and laying hens, as well as three crops: soybeans, corn, and wheat. All three are cash crops, but the corn is also feed crop for the cows and the wheat is also used for chicken feed. The crops are harvested during the late summer and fall. During the winter months, Grant decides on the mix of livestock and crops for the coming year. Currently, the family has just completed a particularly successful harvest that has provided an investment fund of $20,000 that can be used to purchase more livestock. Other money is available for ongoing expenses, including the next planting of crops. The family currently has 30 cows valued at $35,000 and 2,000 hens valued at $5,000. They wish to keep all this livestock and perhaps purchase more. Each new cow would cost $1,500, and each new hen would cost $7.75. Over a year’s time, the value of a herd of cows will decrease by about 10 percent and the value of a flock of hens will decrease by about 25 percent due to aging. Each cow will require two acres of land for grazing and 10 person-hours of work per month for each month of the year, while producing a net annual cash income of $850 for the family. The corresponding figures for each hen are no significant acreage, 0.05 person-hours per month, and an annual net cash income of $5.00. The chicken house can accommodate a maximum of 5,000 hens, and the size of the barn limits the herd to a maximum of 42 cows. For each acre planted in each of the three crops, Table 1 gives the number of person-hours of work that will be required during the first and second halves of the year, as well as a rough estimate of the crop’s net value (in either income or savings in purchasing feed for the livestock). To provide much of the feed for the livestock, Grant wants to plant at least one acre of corn for each cow in the coming year’s herd and at least 0.05 acre of wheat for each hen in the coming year’s flock. The Millers now are discussing how much acreage should be planted in each of the crops and how many cows and hens to have for the coming year. Their objective is to maximize the family’s monetary worth at the end of the coming year (the sum of the net income from the livestock for the coming year plus the net value of the crops for the coming year plus what remains from the investment fund plus the value of the livestock at the end of the coming year plus income from working on a neighboring farm minus living expenses of $40,000 for the year.
Winter and spring (person-hours) 1.0 (Soybeans) 0.9 (corn) 0.6 (wheat)
Summer and fall (person-hours) 1.4 (Soybeans) 1.2 (corn) 0.7 (wheat)
Net value $70 (soybeans) $60 (corn) $40 (wheat)
1. Write a problem statement to guide Grant’s management of his family farm for the next year. This should be a concise paragraph (no more than 5 sentences) that clearly states the problem(s) the Miller family must solve.
2. What are the decision variables for the Miller’s farm management problem? Be sure to clearly define each one (i.e., x1 = … )
3. Create the algebraic model for the farming problem. Be sure to use the variables you presented in Question 2 and clearly label all constraints.
4. Build the Excel model that corresponds to the algebraic model from Question 3, and use the Excel model to help Grant manage the family farm (i.e., run the model to obtain an optimal solution). What management decision(s) should the family make?
5. Generate a Sensitivity Report and comment on what it shows for at least 2 structural constraints and 2 decision variables. 6. Complete a Solver-table type analysis for one variable to which your group believes the Millers’ decision is highly sensitive. Explain why you choose that variable. How do the results influence the decision(s) the Millers should make?