A B C D F F G H K L M N Planting Neighboring Farm Payment hours Profits Soybeans Corn Wheat total Rate/per hour winter-spring 5 Winter/Spirng 1 0.9 0.6 Rate/per hour Summer-Fall 5.5 Summer/Fall 1.4 1.2 0.6 Net Income 70 60 40 Acre planted 0 30 100 optimal number of Acres Hours-avaliable Hours required total number of person hours for Winter-Spring 4000 2400 total number of person hours for Summer-Fall 4500 2400 Live stock Cow Her Total 30 2000 Person-hour-work(winter-spring) 60 0.3 Hours Person-hour-work(summer-Fall) 60 0.3 Grainland require 2 0 acre 8 Capacity 42 5000 investmend fun 1500 3 OS 0 Value of livestocks 35000 5000 35250 $ Net annual income 850 4.25 S percentage-depreciation 0.1 0.25 Number to purchaseThe Ploughman family owns and operates a 640-acre farm the cows and the wheat also is used for chicken feed.) The hat 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- iving 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 earlier 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 loned 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- inances. 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 Wheat Winter and spring, person-hours 1.0 0.9 0.6 Summer and fall, person-hours 1.4 1.2 0.7 Net value $70 $60 $40 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 I 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 of 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){a} Identify verbally the components of a linear programming model for this problem. {It} Formulate this model. {Eidter an algebraic or a spreadsheet for mulation is acceptable.) to) Obtain an optimal solution and generate the additional output pro- vided for perforating postoptimality analysLs leg. the Sensitivity Report when using Excel}. What does the model predict regard ing the family's trainetary worth at one end of die coming year? it!) 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 to} Find an optimal solution under each scenario after making the necessary adjustments to the linear programming model for mulated in part {in}. In each case. what is the prediction re garding the family's monetaryI worth at the end of the year'.' {I} For the optimal solution obtained under each of the sis sce- narios [including the good weather scenario considered in parts [a] to [all]. calculate what the family's monetary worm would be at the end of the year if each of the other ve scenarios oc- cur instead. In yourjudgment. which solution provides the best balance between yielding a large truinetary worm 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 Regularity Good weather 40% Drought 20% Flood tort. Early frost 15% Drought and early frost 10% Flood and early frost 5% Widt these data. the family has decided to use the fol- lowing approach to making its planting and livestock deci- sions. Rather than the optimistic approach of assuming that good weather conditions will prevail [as done in parts {a} to Ml], the average net value under all weather conditions will 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 flood1 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 below. Hat: Ill'alue per Acre Planted Soybeans torn Wheat -51 U 51 5 I: 515 52D 51!: SSE! 54D 55!: -51 5 52U 51 I: 51 D 51f! ll 5 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 mode] formulated in pan {b} to t this new approach. {h} Repeat part {c} for this modified model. {I} Use a shadow price obtained in part {it} to analyze whether il would be worthwhile for the family to obtain a bank loan with a ll] percent interest rate to purchase more livestock now beyond what can be obtained with the $2Dl] from the investment fund. {J} For each of the three crops. use the postoptimality analysis information obtained in pan [h] to identify how much Eati tude for error is available in estimating die net value per acre planted for that crop without changing the optimal solution. 1Which two net values need to be estimated most carefulty'.' ll'bodt estimates are incorrect simultaneously. how close do the estimates need to be to guarantee that the optimal solu tion will not change? This problem illustrates a kind of situation that is fre- quently faced by various kinds of organizations. To describe the situation in general terms, an organization faces an uncertain future where any one ofa number of scenarios may unfold. 1|tlr'hich one will occur depends on conditions that are outside the control ofthe organization. The organization needs to choose the levels of various activities. but the unit contri- bution of each activity to the overall measure of performance is greme affected by which scenario unfolds. Under these circumstances. what is the best mix of activities? {k} Think about specic situations outside of farm management that fit dtis description. Describe one. A. Identify the components of a linear programming model for this problem verbally. The X1 - No of cows purchased per year objective is to how many acres should be planted and how many cows and chickens they X2 - No of Hens purchased per year will have in the incoming years. X3 - soybean planted in No of acre The entire Family can X4 - Corn planted in No of acre produce X5 - Wheat Planted in no of acres Winter and spring 4000 person hours $5/hour C1. X1+30 = (30+x1) C5. X5 >= 0.05(x2+2000) Cows C6. 60(30+X1) + 0.3(2000+X2) + X3 + 0.9X4 + 0.6X5) =0 x2>=0, x3>=0,x4>=0, x5>=0 x1,x2 have to be integers Over year decrease 10% Objective function Grazing land require 2 Acre 850(30+x1)+4.25(2000+x2))+(70x3+60x4+40x5)+(20000-(1500x1+3x2)))+ Person-hours 10 Per month per cow (35000 *90%)+(5000*75%))+5(4000-(60(30+X1) + 0.3(2000+X2) + X3 + 0.9X4 + 0.6X5) + work/month 5.50(4500-(60(30+X1) + 0.3(2000+X2 )+ 1.4X3 + 1.2X4 + 0.7X5))-40000 Net annual income $850 Per annum per cow Maximum cows 42 Cows Hens 2000 hens worth $5000 New hens cost $3 each Over year decrease 5% Person-hours 0.0 Per month per hens work/month Net annual income $4.25 Per annum per cow Maximum hens 5000 Investment fund $20000 The constraints are: The family needs to plant enough acres of corn to provide feed for the cows: Acres of corn > (1 acre/cow) - The family needs to plant enough acres of wheat to provide chicken feed: Acres of wheat > 0.05 acres/chicken) The total spent on new livestock must be no greater than the amount in the investment fund ($20,000) - total number of cows