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CASE 7.2 FARM MANAGEMENT The Ploughman family owns and operates a 640-acre farm the cows and the wheat also is used for chicken feed.) The that has been in the family for several generations. The crops are harvested during the late summer and fall. During Ploughmans always have had to work hard to make a decent the winter months, John, Eunice, and Grandpa make a deci- living from the farm and have had to endure some occasional sion about the mix of livestock and crops for the coming year. difficult years. Stories about carlier generations overcoming Currently, the family has just completed a particularly hardships due to droughts, floods, etc., are an important part successful harvest which has provided an investment fund of of the family history. However, the Ploughmans enjoy their $20,000 that can be used to purchase more livestock. (Other self-reliant lifestyle and gain considerable satisfaction from money is available for ongoing expenses, including the next continuing the family tradition of successfully living off the planting of crops.) The family currently has 30 cows valued land during an era when many family farms are being aban- at $35,000 and 2,000 hens valued at $5,000. They wish to doned or taken over by large agricultural corporations. keep all this livestock and perhaps purchase more. Each new John Ploughman is the current manager of the farm cow would cost $1,500, and each new hen would cost $3. while his wife Eunice runs the house and manages the farm's Over a year's time, the value of a herd of cows will de- finances. John's father, Grandpa Ploughman, lives with them crease by about 10 percent and the value of a flock of hens and still puts in many hours working on the farm. John and will decrease by about 25 percent due to aging. Eunice's older children, Frank, Phyllis, and Carl, also are Each cow will require 2 acres of land for grazing and 10 given heavy chores before and after school. person-hours of work per month, while producing a net annual The entire famiy can produce a total of 4,000 person- cash income of $850 for the family. The corresponding figures hours worth of labor during the winter and spring months for each hen are: no significant acreage, 0.05 person-hour per and 4,500 person-hours during the summer and fall. If any month, and an annual net cash income of $4.25. The chicken of these person-hours are not needed, Frank, Phyllis, and house can accommodate a maximum of 5,000 hens, and the Carl will use them to work on a neighboring farm for $5 per size of the barn limits the herd to a maximum of 42 cows. hour during the winter and spring months and $5.50 per hour For each acre planted in each of the three crops, the fol- during the summer and fall. lowing table gives the number of person-hours of work that The farm supports two types of livestock: dairy cows and will be required during the first and second halves of the year, laying hens, as well as three crops: soybeans, corn, and wheat as well as a rough estimate of the crop's net value (in either (All three are cash crops, but the corn also is a feed crop for income or savings in purchasing feed for the livestock). Data per acre planted Soybeans Corn Winter and spring, person-hours Summer and fall, person-hours Net value $60 $40 Wheat 1.0 1.4 0.9 1.2 0.6 0.7 $70 To provide much of the feed for the livestock, John objective is to maximize the family's monetary worth at the wants to plant at least 1 acre of corn for each cow in the end of the coming year (the sum of the net income from the coming year's herd and at least 0.05 acre of wheat for each livestock for the coming year plus the net value the crops hen in the coming year's flock. for the coming year plus what remains from the investment John, Eunice, and Grandpa now are discussing how fund plus the value of the livestock at the end of the much acreage should be planted in each of the crops and coming year plus any income from working on a neighbor- how many cows and hens to have for the coming year. Their ing farm, minus living expenses of $40,000 for the year). 10 (a) Identify verbally the components of a linear programming The above estimates of the net value per acre planted model for this problem. in each of the three crops assumes good weather conditions. (b) Formulate this model. (Either an algebraic or a spreadsheet for Adverse weather conditions would harm the crops and mulation is acceptable.) greatly reduce the resulting value. The scenarios particularly (e) Obtain an optimal solution and generate the additional output pro feared by the family are a drought, a flood, an carly frost, vided for performing postoptimality analysis (eg, the Sensitivity both a drought and an early frost, and both a flood and an ing the family's monetary worth at the end of the coming year? carly frost. The estimated net values for the year under these (a) Find the allowable range to stay optimal for the net value per scenarios are shown below. acre planted for each of the three crops. Soybeans Wheat Scenario Drought Flood Early frost Drought and early frost Flood and early frost $10 $15 $50 -$15 $10 Net Value per Acre Planted Corn -515 $20 $40 -520 $10 0 $10 $30 - $10 $5 (e) Find an optimal solution under each scenario after making the be used for each crop (weighting the net values under the necessary adjustments to the linear programming model for various scenarios by the frequencies in the above table). mulated in part (b). In each case, what is the prediction re- garding the family's monetary worth at the end of the year? (8) Modify the linear programming model formulated in part (b) (f) For the optimal solution obtained under each of the six sce- to fit this new approach. narios (including the good weather scenario considered in parts (h) Repeat part (c) for this modified model. (a) to (d), calculate what the family's monetary worth would (1) Use a shadow price obtained in part (h) to analyze whether it be at the end of the year if each of the other five scenarios oc- would be worthwhile for the family to obtain a bank loan with cur instead. In your judgment, which solution provides the best a 10 percent interest rate to purchase more livestock now beyond balance between yielding a large monetary worth under good what can be obtained with the $20,000 from the investment fund. weather conditions and avoiding an overly small monetary G) For each of the three crops, use the postoptimality analysis worth under adverse weather conditions, information obtained in part (h) to identify how much lati- tude for error is available in estimating the net value per acre Grandpa has researched what the weather conditions planted for that crop without changing the optimal solution. were in past years as far back as weather records have been Which two net values need to be estimated most carefully? kept, and obtained the following data. If both estimates are incorrect simultaneously, how close do the estimates need to be to guarantee that the optimal solu- tion will not change? Scenario Frequency This problem illustrates a kind of situation that is fre- Good weather 40% quently faced by various kinds of organizations. To describe Drought Flood 10% the situation in general terms, an organization faces an Early frost 15% uncertain future where any one of a number of scenarios may Drought and early frost 10% unfold. Which one will occur depends on conditions that are Flood and early frost 5% outside the control of the organization. The organization needs to choose the levels of various activities, but the unit contri- With these data, the family has decided to use the fol-bution of cach activity to the overall measure of performance lowing approach to making its planting and livestock deci- is greatly affected by which scenario unfolds. Under these sions. Rather than the optimistic approach of assuming that circumstances, what is the best mix of activities? good weather conditions will prevail (as done in parts (a) to (k) Think about specific situations outside of farm management (d)], the average net value under all weather conditions will that fit this description. Describe one. 20% CASE 7.2 FARM MANAGEMENT The Ploughman family owns and operates a 640-acre farm the cows and the wheat also is used for chicken feed.) The that has been in the family for several generations. The crops are harvested during the late summer and fall. During Ploughmans always have had to work hard to make a decent the winter months, John, Eunice, and Grandpa make a deci- living from the farm and have had to endure some occasional sion about the mix of livestock and crops for the coming year. difficult years. Stories about carlier generations overcoming Currently, the family has just completed a particularly hardships due to droughts, floods, etc., are an important part successful harvest which has provided an investment fund of of the family history. However, the Ploughmans enjoy their $20,000 that can be used to purchase more livestock. (Other self-reliant lifestyle and gain considerable satisfaction from money is available for ongoing expenses, including the next continuing the family tradition of successfully living off the planting of crops.) The family currently has 30 cows valued land during an era when many family farms are being aban- at $35,000 and 2,000 hens valued at $5,000. They wish to doned or taken over by large agricultural corporations. keep all this livestock and perhaps purchase more. Each new John Ploughman is the current manager of the farm cow would cost $1,500, and each new hen would cost $3. while his wife Eunice runs the house and manages the farm's Over a year's time, the value of a herd of cows will de- finances. John's father, Grandpa Ploughman, lives with them crease by about 10 percent and the value of a flock of hens and still puts in many hours working on the farm. John and will decrease by about 25 percent due to aging. Eunice's older children, Frank, Phyllis, and Carl, also are Each cow will require 2 acres of land for grazing and 10 given heavy chores before and after school. person-hours of work per month, while producing a net annual The entire famiy can produce a total of 4,000 person- cash income of $850 for the family. The corresponding figures hours worth of labor during the winter and spring months for each hen are: no significant acreage, 0.05 person-hour per and 4,500 person-hours during the summer and fall. If any month, and an annual net cash income of $4.25. The chicken of these person-hours are not needed, Frank, Phyllis, and house can accommodate a maximum of 5,000 hens, and the Carl will use them to work on a neighboring farm for $5 per size of the barn limits the herd to a maximum of 42 cows. hour during the winter and spring months and $5.50 per hour For each acre planted in each of the three crops, the fol- during the summer and fall. lowing table gives the number of person-hours of work that The farm supports two types of livestock: dairy cows and will be required during the first and second halves of the year, laying hens, as well as three crops: soybeans, corn, and wheat as well as a rough estimate of the crop's net value (in either (All three are cash crops, but the corn also is a feed crop for income or savings in purchasing feed for the livestock). Data per acre planted Soybeans Corn Winter and spring, person-hours Summer and fall, person-hours Net value $60 $40 Wheat 1.0 1.4 0.9 1.2 0.6 0.7 $70 To provide much of the feed for the livestock, John objective is to maximize the family's monetary worth at the wants to plant at least 1 acre of corn for each cow in the end of the coming year (the sum of the net income from the coming year's herd and at least 0.05 acre of wheat for each livestock for the coming year plus the net value the crops hen in the coming year's flock. for the coming year plus what remains from the investment John, Eunice, and Grandpa now are discussing how fund plus the value of the livestock at the end of the much acreage should be planted in each of the crops and coming year plus any income from working on a neighbor- how many cows and hens to have for the coming year. Their ing farm, minus living expenses of $40,000 for the year). 10 (a) Identify verbally the components of a linear programming The above estimates of the net value per acre planted model for this problem. in each of the three crops assumes good weather conditions. (b) Formulate this model. (Either an algebraic or a spreadsheet for Adverse weather conditions would harm the crops and mulation is acceptable.) greatly reduce the resulting value. The scenarios particularly (e) Obtain an optimal solution and generate the additional output pro feared by the family are a drought, a flood, an carly frost, vided for performing postoptimality analysis (eg, the Sensitivity both a drought and an early frost, and both a flood and an ing the family's monetary worth at the end of the coming year? carly frost. The estimated net values for the year under these (a) Find the allowable range to stay optimal for the net value per scenarios are shown below. acre planted for each of the three crops. Soybeans Wheat Scenario Drought Flood Early frost Drought and early frost Flood and early frost $10 $15 $50 -$15 $10 Net Value per Acre Planted Corn -515 $20 $40 -520 $10 0 $10 $30 - $10 $5 (e) Find an optimal solution under each scenario after making the be used for each crop (weighting the net values under the necessary adjustments to the linear programming model for various scenarios by the frequencies in the above table). mulated in part (b). In each case, what is the prediction re- garding the family's monetary worth at the end of the year? (8) Modify the linear programming model formulated in part (b) (f) For the optimal solution obtained under each of the six sce- to fit this new approach. narios (including the good weather scenario considered in parts (h) Repeat part (c) for this modified model. (a) to (d), calculate what the family's monetary worth would (1) Use a shadow price obtained in part (h) to analyze whether it be at the end of the year if each of the other five scenarios oc- would be worthwhile for the family to obtain a bank loan with cur instead. In your judgment, which solution provides the best a 10 percent interest rate to purchase more livestock now beyond balance between yielding a large monetary worth under good what can be obtained with the $20,000 from the investment fund. weather conditions and avoiding an overly small monetary G) For each of the three crops, use the postoptimality analysis worth under adverse weather conditions, information obtained in part (h) to identify how much lati- tude for error is available in estimating the net value per acre Grandpa has researched what the weather conditions planted for that crop without changing the optimal solution. were in past years as far back as weather records have been Which two net values need to be estimated most carefully? kept, and obtained the following data. If both estimates are incorrect simultaneously, how close do the estimates need to be to guarantee that the optimal solu- tion will not change? Scenario Frequency This problem illustrates a kind of situation that is fre- Good weather 40% quently faced by various kinds of organizations. To describe Drought Flood 10% the situation in general terms, an organization faces an Early frost 15% uncertain future where any one of a number of scenarios may Drought and early frost 10% unfold. Which one will occur depends on conditions that are Flood and early frost 5% outside the control of the organization. The organization needs to choose the levels of various activities, but the unit contri- With these data, the family has decided to use the fol-bution of cach activity to the overall measure of performance lowing approach to making its planting and livestock deci- is greatly affected by which scenario unfolds. Under these sions. Rather than the optimistic approach of assuming that circumstances, what is the best mix of activities? good weather conditions will prevail (as done in parts (a) to (k) Think about specific situations outside of farm management (d)], the average net value under all weather conditions will that fit this description. Describe one. 20%
