I LHJI: IIIL I'Hlllll MHNRIIIIMIINI The Ploughman lamily owns and operates a twill-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 difcult years. Stories about earlier generations overcoming hardships due to droughts, oods, etc., are an important part of the family history. However, the Ploughmans enjoy their selfreliant lifestyle and gain considerable satisfaction from continuing the lamily tradition of successfully living off the land during an era when many lamily farms are being aban- doned 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 nances. John's father, Grandpa Plou ghman, 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 alter school. The entire famiy can produce a total of 4,l]ll0 person- hours worth of labor during the winter and spring months and 4,5llll personhours 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 $5.5ll 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 Data per acre planted Winter and spring, person-hours Summer and fall, person-hours Net value 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 (LBS 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 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 deci- sion 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 $2l},llll 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 2,0(lll hens valued at $5,lllllJ. They wish to keep all this livestock and perhaps purchase more. Each new cow would cost $1,5llll, and each new hen would cost $3. Over a year's time, the value of a herd of cows will de- crease by about lll percent and the value of a ock of hens will decrease by about 25 percent due to aging. Each cow will require 2 acres of land for grazing and ill person-hours of work per month, while producing a net annual cash income of $850 for the family. The corresponding gures for each hen are: no signicant acreage, l}.l}5 personhour 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 fol- lowing 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}. Wheat ll ll? $40 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 neighbor ing farm, mimic.- living expenses of $40,0[l0 for the year). to] Identify verbally die components of a linear programming model for this problem. (b) Formulate this model. [Hillier an algebraic or a spreadsheet for mulation is acceptable.) to} Obtain an optimal solution and generate the additional output pro- vided for performing postoptimality analysis leg, the Sensitivity Report when using Excel}. What does the model predict regard- ing the family's monetary wou1h at the end of the coming year? td) Find the allowable range to stay optimal for die net value per acre planted for each of the three crops. Scenario Drought Flood Early frost Drought and early frost Flood and early frost [c] Find an optimal solution under each scenario after making me necessary adjustments to the linear programming model for mulated in part (b). In each case, what is the prediction re garding the family's monetary worth at the end of the year'l [1'] For the optimal solution obtained under each of the six sce- narios lincludin g me good weather scenario considered in parts (a) to (cm. calculate what the family's monetary wordi would be at the end of the year if each of the odter five scenarios oc cur instead. In yourjudgment, 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. The above estimates of the net value per acre planted in each of die diree crops assumes good weather conditions. Adverse weather conditions would harm the crops and greatly reduce the resulting value. The scenarios ptuticularly feared by die family are a drought, a flood, an early frost, both a drought and an early frost, and both a ood and an early frost. The estimated net values for die year under these scenarios are shown below. Not Value per Acre Planted Soybeans Corn mat -$1l] F115 0 $15 120 $10 SSI] 140 530 415 "120 -$'|0 I10 510 1 5 be used for each crop {weighting the net values under the various scenarios by the frequencies in the above table]. [gt Modify the linear programming model formulated in part [b] to fit this new approach. (II) Repeat part {c} for this modied model. [i] Use a shadow price obtained in part {it} to analyze whether it would be worthwhile for the family to obtain a bank loan with a 1D percent interest rate to purchase more livestock now beyond what can be obtained with the $210,000 from the investment fund. [ji- Por each of the three crops. use the postoptimality analysis information obtained in part {tr} to identify how much lati- tude 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 solu- tion will not chance