CASE 6.2 FARM MANAGEMENT

The Ploughman family owns and operates a 640-acre farm that has been in the family for several generations. The Ploughmans always have had to work hard to make a de- cent living from the farm and have had to endure some occasional difficult years. Sto- ries about earlier generations overcoming hardships due to droughts, floods, etc., are an important part of the family history. However, the Ploughmans enjoy their self- reliant lifestyle and gain considerable satisfaction from continuing the family tradition of successfully living off the land during an era when many family farms are being abandoned or taken over by large agricultural corporations.

John Ploughman is the current manager of the farm while his wife Eunice runs the house and manages the farm's finances. John's father, Grandpa Ploughman, lives with them and still puts in many hours working on the farm. John and Eunice's older children, Frank, Phyllis, and Carl, also are given heavy chores before and after school.

The entire famiy can produce a total of 4,000 person-hours worth of labor during the winter and spring months and 4,500 person-hours during the summer and fall. If any of these person-hours are not needed, Frank, Phyllis, and Carl will use them to work on a neighboring farm for $5 per hour during the winter and spring months and $5.50 per hour during the summer and fall.

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 also is a feed crop for the cows and the wheat also is used for chicken feed.) The crops are harvested during the late summer and fall. During the winter months, John, Eunice, and Grandpa make a decision about the mix of livestock and crops for the coming year.

Currently, the family has just completed a particularly successful harvest which has provided an investment fund of $20,000 that can be used to purchase more live- stock. (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 $3.

CASE 6.2 FARM MANAGEMENT 305

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 2 acres of land for grazing and 10 person-hours of work per month, while producing a net annual cash income of $850 for the family. The corre- sponding figures for each hen are: no significant acreage, 0.05 person-hour per month, and an annual net cash income of $4.25. The chicken house can accommodate a max- imum 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, the following table gives the num- ber 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 sav- ings in purchasing feed for the livestock).

Data per acre planted

Wheat

Winter and spring, person-hours 0.6 Summer and fall, person-hours 0.7 Net value $40

To provide much of the feed for the livestock, John wants to plant at least 1 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.

John, Eunice, and Grandpa 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 any income from working on a neighboring farm, minus living expenses of $40,000 for the year).

(a) Identify verbally the components of a linear programming model for this problem.

(b) Formulate this model. (Either an algebraic or a spreadsheet formulation is acceptable.)

(c) Obtainanoptimalsolutionandgeneratetheadditionaloutputprovidedforperformingpostop-

timality analysis (e.g., the Sensitivity Report when using Excel). What does the model pre-

dict regarding the family's monetary worth at the end of the coming year?

(d) Find the allowable range to stay optimal for the net value per acre planted for each of the

three crops.

The above estimates of the net value per acre planted in each of the three crops assumes good weather conditions. Adverse weather conditions would harm the crops and greatly reduce the resulting value. The scenarios particularly feared by the family are a drought, a flood, an early frost, both a drought and an early frost, and both a flood and an early frost. The estimated net values for the year under these scenarios are shown on the next page.

Soybeans

Corn

1.0 1.4 $70

0.9 1.2 $60

306 6 DUALITY THEORY AND SENSITIVITY ANALYSIS

Net Value per Acre Planted Scenario Soybeans Corn

Drought $10 Flood $15

Early frost

Drought and early frost Flood and early frost

(e) Find an optimal solution under each scenario after making the necessary adjustments to the linear programming model formulated in part (b). In each case, what is the prediction re- garding the family's monetary worth at the end of the year?

(f) For the optimal solution obtained under each of the six scenarios [including the good weather scenario considered in parts (a) to (d)], calculate what the family's monetary worth would be at the end of the year if each of the other five scenarios occur instead. In your judgment, which solution provides the best balance between yielding a large monetary worth under good weather conditions and avoiding an overly small monetary worth under adverse weather conditions.

Grandpa has researched what the weather conditions were in past years as far back as weather records have been kept, and obtained the following data.

Wheat

$15 0 $20 $10 $50 $40 $30 $15 $20 $10 $10 $10 $5

Scenario

Frequency

Good weather 40% Drought 20% Flood 10% Early frost 15% Drought and early frost 10% Flood and early frost 5%

With these data, the family has decided to use the following approach to making its planting and livestock decisions. Rather than the optimistic approach of assuming that good weather conditions will prevail [as done in parts (a) to (d)], the average net value under all weather conditions will be used for each crop (weighting the net val- ues under the various scenarios by the frequencies in the above table).

(g) Modify the linear programming model formulated in part (b) to fit this new approach. (h) Repeat part (c) for this modified model.

1. (i)Use a shadow price obtained in part (h) to analyze whether it would be worthwhile for the
2. family to obtain a bank loan with a 10 percent interest rate to purchase more livestock now
3. beyond what can be obtained with the $20,000 from the investment fund.
4. (j)For each of the three crops, use the postoptimality analysis information obtained in part (h) to identify how much latitude for error is available in estimating the net value per acre planted for that crop without changing the optimal solution. Which two net values need to be esti- mated most carefully? If both estimates are incorrect simultaneously, how close do the esti-

mates need to be to guarantee that the optimal solution will not change?

CASE 6.3 ASSIGNING STUDENTS TO SCHOOLS (REVISITED) 307

This problem illustrates a kind of situation that is frequently faced by various kinds of organizations. To describe the situation in general terms, an organization faces an uncertain future where any one of a number of scenarios may unfold. Which one will occur depends on conditions that are outside the control of the organization. The or- ganization needs to choose the levels of various activities, but the unit contribution of each activity to the overall measure of performance is greatly affected by which sce- nario unfolds. Under these circumstances, what is the best mix of activities?

(k) Think about specific situations outside of farm management that fit this description. De- scribe one.

304 6 DUALITY THEORY AND SENSITIVITY ANALYSIS it for a simultaneous and equal change for both sulfur oxides and hydrocarbons in the opposite direction from particulates. (g) Letting \u000e denote the percentage increase in all the policy standards given in Table 3.12, formulate the problem of analyzing the effect of simultaneous proportional increases in these standards as a parametric linear programming problem. Then use your results from part (e) to determine the rate at which the total cost of an optimal solution would increase with a small increase in \u000e from zero. (h) Use the simplex method to find an optimal solution for the parametric linear programming problem formulated in part (g) for each \u000e \u0002 10, 20, 30, 40, 50. Considering the tax incentive offered by the city, use these results to determine which value of \u000e (including the option of \u000e \u0002 0) should be chosen to minimize the company's total cost of both pollution abatement and taxes. (i) For the value of \u000e chosen in part (h), repeat parts (e) and ( f ) so that the decision makers can make a final decision on the relative values of the policy standards for the three pollutants. CASE 6.2 FARM MANAGEMENT The Ploughman family owns and operates a 640-acre farm that has been in the family for several generations. The Ploughmans always have had to work hard to make a decent living from the farm and have had to endure some occasional difficult years. Stories about earlier generations overcoming hardships due to droughts, floods, etc., are an important part of the family history. However, the Ploughmans enjoy their selfreliant lifestyle and gain considerable satisfaction from continuing the family tradition of successfully living off the land during an era when many family farms are being abandoned or taken over by large agricultural corporations. John Ploughman is the current manager of the farm while his wife Eunice runs the house and manages the farm's finances. John's father, Grandpa Ploughman, lives with them and still puts in many hours working on the farm. John and Eunice's older children, Frank, Phyllis, and Carl, also are given heavy chores before and after school. The entire famiy can produce a total of 4,000 person-hours worth of labor during the winter and spring months and 4,500 person-hours during the summer and fall. If any of these person-hours are not needed, Frank, Phyllis, and Carl will use them to work on a neighboring farm for $5 per hour during the winter and spring months and $5.50 per hour during the summer and fall. 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 also is a feed crop for the cows and the wheat also is used for chicken feed.) The crops are harvested during the late summer and fall. During the winter months, John, Eunice, and Grandpa make a decision about the mix of livestock and crops for the coming year. Currently, the family has just completed a particularly successful harvest which 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 $3. CASE 6.2 FARM MANAGEMENT 305 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 2 acres of land for grazing and 10 person-hours of work per month, 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-hour per month, and an annual net cash income of $4.25. 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, the following table 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). Data per acre planted Winter and spring, person-hours Summer and fall, person-hours Net value Soybeans Corn Wheat 1.0 1.4 $70 0.9 1.2 $60 0.6 0.7 $40 To provide much of the feed for the livestock, John wants to plant at least 1 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. John, Eunice, and Grandpa 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 any income from working on a neighboring farm, minus living expenses of $40,000 for the year). (a) Identify verbally the components of a linear programming model for this problem. (b) Formulate this model. (Either an algebraic or a spreadsheet formulation is acceptable.) (c) Obtain an optimal solution and generate the additional output provided for performing postoptimality analysis (e.g., the Sensitivity Report when using Excel). What does the model predict regarding the family's monetary worth at the end of the coming year? (d) Find the allowable range to stay optimal for the net value per acre planted for each of the three crops. The above estimates of the net value per acre planted in each of the three crops assumes good weather conditions. Adverse weather conditions would harm the crops and greatly reduce the resulting value. The scenarios particularly feared by the family are a drought, a flood, an early frost, both a drought and an early frost, and both a flood and an early frost. The estimated net values for the year under these scenarios are shown on the next page. 306 6 DUALITY THEORY AND SENSITIVITY ANALYSIS Net Value per Acre Planted Scenario Drought Flood Early frost Drought and early frost Flood and early frost Soybeans Corn Wheat \u0006$10 $15 $50 \u0006$15 $10 \u0006$15 $20 $40 \u0006$20 $10 0 $10 $30 \u0006$10 $ 5 (e) Find an optimal solution under each scenario after making the necessary adjustments to the linear programming model formulated in part (b). In each case, what is the prediction regarding the family's monetary worth at the end of the year? (f) For the optimal solution obtained under each of the six scenarios [including the good weather scenario considered in parts (a) to (d )], calculate what the family's monetary worth would be at the end of the year if each of the other five scenarios occur instead. In your judgment, which solution provides the best balance between yielding a large monetary worth under good weather conditions and avoiding an overly small monetary worth under adverse weather conditions. Grandpa has researched what the weather conditions were in past years as far back as weather records have been kept, and obtained the following data. Scenario Good weather Drought Flood Early frost Drought and early frost Flood and early frost Frequency 40% 20% 10% 15% 10% 5% With these data, the family has decided to use the following approach to making its planting and livestock decisions. Rather than the optimistic approach of assuming that good weather conditions will prevail [as done in parts (a) to (d)], the average net value under all weather conditions will be used for each crop (weighting the net values under the various scenarios by the frequencies in the above table). (g) Modify the linear programming model formulated in part (b) to fit this new approach. (h) Repeat part (c) for this modified model. (i) Use a shadow price obtained in part (h) to analyze whether it would be worthwhile for the family to obtain a bank loan with a 10 percent interest rate to purchase more livestock now beyond what can be obtained with the $20,000 from the investment fund. (j) For each of the three crops, use the postoptimality analysis information obtained in part (h) to identify how much latitude for error is available in estimating the net value per acre planted for that crop without changing the optimal solution. Which two net values need to be estimated most carefully? If both estimates are incorrect simultaneously, how close do the estimates need to be to guarantee that the optimal solution will not change? CASE 6.3 ASSIGNING STUDENTS TO SCHOOLS (REVISITED) 307 This problem illustrates a kind of situation that is frequently faced by various kinds of organizations. To describe the situation in general terms, an organization faces an uncertain future where any one of a number of scenarios may unfold. Which one will occur depends on conditions that are outside the control of the organization. The organization needs to choose the levels of various activities, but the unit contribution of each activity to the overall measure of performance is greatly affected by which scenario unfolds. Under these circumstances, what is the best mix of activities? (k) Think about specific situations outside of farm management that fit this description. Describe one. CASE 6.3 ASSIGNING STUDENTS TO SCHOOLS (REVISITED) Reconsider Case 4.3. The Springfield School Board still has the policy of providing bussing for all middle school students who must travel more than approximately 1 mile. Another current policy is to allow splitting residential areas among multiple schools if this will reduce the total bussing cost. (This latter policy will be reversed in Case 12.4.) However, before adopting a bussing plan based on parts (a) and (b) of Case 4.3, the school board now wants to conduct some postoptimality analysis. (a) If you have not already done so for parts (a) and (b) of Case 4.3, formulate and solve a linear programming model for this problem. (Either an algebraic or a spreadsheet formulation is acceptable.) (b) Generate a sensitivity analysis report with the same software package as used in part (a). One concern of the school board is the ongoing road construction in area 6. These construction projects have been delaying traffic considerably and are likely to affect the cost of bussing students from area 6, perhaps increasing them as much as 10 percent. (c) Use the report from part (b) to check how much the bussing cost from area 6 to school 1 can increase (assuming no change in the costs for the other schools) before the current optimal solution would no longer be optimal. If the allowable increase is less than 10 percent, re-solve to find the new optimal solution with a 10 percent increase. (d) Repeat part (c) for school 2 (assuming no change in the costs for the other schools). (e) Now assume that the bussing cost from area 6 would increase by the same percentage for all the schools. Use the report from part (b) to determine how large this percentage can be before the current optimal solution might no longer be optimal. If the allowable increase is less than 10 percent, re-solve to find the new optimal solution with a 10 percent increase. The school board has the option of adding portable classrooms to increase the capacity of one or more of the middle schools for a few years. However, this is a costly move that the board would consider only if it would significantly decrease bussing costs. Each portable classroom holds 20 students and has a leasing cost of $2,500 per year. To analyze this option, the school board decides to assume that the road construction in area 6 will wind down without significantly increasing the bussing costs from that area