The Ploughman Family Owns and Operates a 640

1. Mathematics
2. Statistics
3. The Ploughman family owns and operates a 640-acre farm that

The Ploughman family owns and operates a 640-acre farm that

The Ploughman family owns and operates a 640-acre farm that has been in the family for several generations. The Ploughmans always accept had to piece of 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, and so forth, are an important part of the family history. Nonetheless, the Ploughmans enjoy their self-reliant lifestyle and gain considerable satisfaction from standing the family tradition of successfully living off the land during an era when many family farms are being abased or taken over by big agronomical corporations.

John Ploughman is the current director of the farm, while his married woman Eunice runs the house and manages the farm's finances.

John's father, Grandpa Ploughman, lives with them and withal puts in many hours working on the subcontract. John and Eunice's older children, Frank, Phyllis, and Carl, besides are given heavy chores before and afterward schoolhouse.

The entire family can produce a total of iv,000 person-hours' worth of labor during the winter and spring months and iv,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 $v/hour during the wintertime and leap months and $5.50/60 minutes during the summer and fall.

The subcontract supports two types of livestock, dairy cows and laying hens, besides as three crops: soybeans, corn, and wheat.

(All three are greenbacks crops, but the corn also is a feed ingather for the cows and the wheat also is used for chicken feed.) The crops are harvested during the late summertime and fall. During the wintertime months, John, Eunice, and Grandpa make a decision about the mix of livestock and crops for the coming twelvemonth.

Currently, the family has simply 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 two,000 hens valued at $5,000. They wish to keep all this livestock and peradventure purchase more. Each new moo-cow would price $1,500, and each new hen would cost $3.

Over a year's time, the value of a herd of cows will decrease by about x per centum and the value of a flock of hens will decrease past nearly 25 pct due to crumbling.

Each moo-cow will require two acres of land for grazing and 10 person-hours of work per month, while producing a net almanac cash income of $850 for the family. The respective figures for each hen are no meaning acreage, 0.05 person-hours per month, and an annual net greenbacks income of $four.25. The craven 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 adjacent table gives the number of person-hours of work that will be required during the first and second halves of the yr, as well as a rough estimate of the crop'southward net value (in either income or savings in purchasing feed for the livestock).

To provide much of the feed for the livestock, John wants to plant at least ane acre of corn for each moo-cow in the coming year'south herd and at least 0.05 acre of wheat for each hen in the coming year'south 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 twelvemonth. Their objective is to maximize the family unit's budgetary worth at the terminate of the coming yr (the sum of the internet income from the livestock for the coming year plus the net value of the crops for the coming yr plus what remains from the investment fund plus the value of the livestock at the end of the coming twelvemonth plus income from working on a neighboring subcontract minus living expenses of $40,000 for the year).

a. Identify verbally the components of a linear programming model for this trouble.
b. Display the model on a spreadsheet.
c. Obtain an optimal solution and generate the sensitivity study. What does the model predict regarding the family'southward budgetary worth at the end of the coming year?
d. Find the allowable range for the net value per acre planted for each of the 3 crops.
The above estimates of the net value per acre planted in each of the 3 crops assumes good weather conditions. Agin atmospheric condition conditions would damage the crops and profoundly reduce the resulting value. The scenarios specially feared past the family unit are a drought, a flood, an early on frost, both a drought and an early frost, and both a flood and an early frost. The estimated net values for the twelvemonth under these scenarios are shown next.

e. Find an optimal solution under each scenario afterwards making the necessary adjustments to the linear programming model formulated in part b. In each example, what is the prediction regarding the family's budgetary worth at the end of the yr?
f. For the optimal solution obtained under each of the six scenarios (including the skillful weather condition scenario considered in parts a-d ), summate what the family's monetary worth would exist at the end of the twelvemonth if each of the other five scenarios occurs instead. In your judgment, which solution provides the best rest between yielding a large monetary worth under good weather conditions and avoiding an overly small monetary worth under agin weather conditions?
Grandpa has researched what the weather conditions were in past years as far back as conditions records take been kept and obtained the information shown on the next page. With these data, the family has decided to utilise the post-obit approach to making its planting and livestock decisions. Rather than the optimistic approach of assuming that good weather conditions volition prevail (as washed in parts a-d), the average net value under all conditions conditions will be used for each crop (weighting the net values nether the various scenarios past the frequencies in the higher up table).
Scenario __________________ Frequency
Good weather ......................... xl%
Drought .................................. 20
Inundation ..................................... x
Early frost ............................... 15
Drought and early frost ............... 10
Flood and early frost .................... 5
g. Modify the linear programming model formulated in function b to fit this new arroyo.
h. Repeat function 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 involvement rate to purchase more livestock at present beyond what tin be obtained with the $20,000 from the investment fund.
j. For each of the three crops, utilize the sensitivity study obtained in part h to identify how much breadth for error is available in estimating the net value per acre planted for that crop without irresolute the optimal solution. Which 2 net values demand to be estimated most advisedly? If both estimates are wrong simultaneously, how close practise the estimates demand to be to guarantee that the optimal solution volition not modify? Use a two way parameter analysis report to systematically generate the optimal monetary worth as these two cyberspace values are varied simultaneously over ranges that get up to twice as far from the estimates as needed to guarantee that the optimal solution volition non modify.
This problem illustrates a kind of situation that is frequently faced by various kinds of organizations. To describe the situation in full general terms, an organization faces an uncertain future where whatsoever 1 of a number of scenarios may unfold. Which one volition occur depends on conditions that are outside the control of the organization. The organization needs to cull the levels of various activities, but the unit contribution of each action to the overall measure of functioning is greatly affected past which scenario unfolds. Under these circumstances, what is the all-time mix of activities?

1. Mathematics
2. Statistics

The Ploughman family owns and operates a 640-acre subcontract that

The Ploughman family owns and operates a 640-acre farm that has been in the family unit for several generations. The Ploughmans e'er have had to work difficult to make a decent living from the farm and have had to endure some occasional hard years. Stories about earlier generations overcoming hardships due to droughts, floods, then forth, are an of import office of the family unit history. Withal, the Ploughmans bask their self-reliant lifestyle and gain considerable satisfaction from continuing the family tradition of successfully living off the country during an era when many family unit farms are being abandoned or taken over by big agricultural corporations.

John Ploughman is the electric current manager of the farm, while his wife Eunice runs the house and manages the subcontract's finances.

John's father, Grandpa Ploughman, lives with them and however 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 later on school.

The entire family tin 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 summertime and fall. If any of these person hours are not needed, Frank, Phyllis, and Carl will employ them to work on a neighboring farm for $5/hour during the winter and jump months and $5.fifty/hr during the summer and fall.

The subcontract supports two types of livestock, dairy cows and laying hens, as well every bit three crops: soybeans, corn, and wheat.

(All 3 are cash crops, simply 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 autumn. During the wintertime months, John, Eunice, and Grandad make a conclusion about the mix of livestock and crops for the coming yr.

Currently, the family has just completed a particularly successful harvest that has provided an investment fund of $20,000 that tin can exist used to buy more livestock. (Other money is available for ongoing expenses, including the side by side planting of crops.) The family currently has thirty cows valued at $35,000 and ii,000 hens valued at $5,000. They wish to keep all this livestock and perhaps buy more than. Each new moo-cow would cost $1,500, and each new hen would toll $three.

Over a year's time, the value of a herd of cows will decrease by nearly ten percent and the value of a flock of hens volition decrease past nigh 25 percentage due to aging.

Each cow will require two acres of land for grazing and x person-hours of work per month, while producing a net almanac cash income of $850 for the family. The respective figures for each hen are no significant acreage, 0.05 person-hours per month, and an annual net cash income of $4.25. The chicken house tin can accommodate a maximum of v,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 3 crops, the next table gives the number of person-hours of work that volition be required during the commencement and 2nd halves of the twelvemonth, too as a rough gauge 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, John wants to constitute at least ane acre of corn for each cow in the coming year'due south herd and at to the lowest degree 0.05 acre of wheat for each hen in the coming yr'due south 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 accept for the coming year. Their objective is to maximize the family'southward monetary worth at the finish of the coming yr (the sum of the net income from the livestock for the coming year plus the net value of the crops for the coming yr plus what remains from the investment fund plus the value of the livestock at the end of the coming yr plus income from working on a neighboring subcontract minus living expenses of $twoscore,000 for the year).

a. Identify verbally the components of a linear programming model for this problem.
b. Brandish the model on a spreadsheet.
c. Obtain an optimal solution and generate the sensitivity report. What does the model predict regarding the family unit's monetary worth at the end of the coming year?
d. Find the allowable range 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 condition conditions. Adverse weather conditions would harm the crops and greatly reduce the resulting value. The scenarios particularly feared past the family are a drought, a flood, an early frost, both a drought and an early frost, and both a alluvion and an early frost. The estimated internet values for the year nether these scenarios are shown next.

due east. Find an optimal solution under each scenario later on making the necessary adjustments to the linear programming model formulated in part b. In each instance, what is the prediction regarding the family'south monetary worth at the end of the yr?
f. For the optimal solution obtained nether each of the six scenarios (including the good weather scenario considered in parts a-d ), calculate what the family'due south monetary worth would be at the terminate of the twelvemonth if each of the other five scenarios occurs instead. In your judgment, which solution provides the all-time balance between yielding a large monetary worth under good weather weather and avoiding an overly pocket-size budgetary worth under adverse weather conditions?
Granddad has researched what the weather conditions were in by years every bit far back as weather records have been kept and obtained the data shown on the next folio. With these data, the family has decided to utilise the following arroyo to making its planting and livestock decisions. Rather than the optimistic approach of assuming that skilful weather conditions will prevail (as done in parts a-d), the average internet value nether all weather conditions will be used for each ingather (weighting the net values under the diverse scenarios by the frequencies in the above table).
Scenario __________________ Frequency
Expert weather ......................... 40%
Drought .................................. 20
Overflowing ..................................... 10
Early on frost ............................... 15
Drought and early frost ............... x
Flood and early frost .................... 5
k. Modify the linear programming model formulated in role b to fit this new approach.
h. Repeat function c for this modified model.
i. Use a shadow price obtained in part h to clarify whether information technology would be worthwhile for the family to obtain a bank loan with a 10 percent interest rate to buy more than livestock now across what can exist obtained with the $20,000 from the investment fund.
j. For each of the three crops, use the sensitivity report obtained in part h to identify how much breadth for mistake is bachelor in estimating the net value per acre planted for that crop without changing the optimal solution. Which two cyberspace values need to be estimated most carefully? If both estimates are incorrect simultaneously, how close practise the estimates need to be to guarantee that the optimal solution will not change? Use a ii way parameter analysis study to systematically generate the optimal monetary worth every bit these ii cyberspace values are varied simultaneously over ranges that get up to twice as far from the estimates as needed to guarantee that the optimal solution volition not change.
This problem illustrates a kind of situation that is often faced by diverse kinds of organizations. To describe the situation in general terms, an organisation faces an uncertain future where whatsoever one of a number of scenarios may unfold. Which i will occur depends on conditions that are outside the control of the organization. The arrangement needs to choose the levels of various activities, simply the unit of measurement contribution of each activity to the overall measure of performance is greatly afflicted by which scenario unfolds. Nether these circumstances, what is the best mix of activities?

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