(a) Identify verbally th 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 (c) Obtain an optimal solution and generate the additional output pro- feared by the family are a drought , a flood , an early frost , vided for performing postoptimality analysis (e.g., the Sensitivity Report when using Excel). What does the model predict regard- 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? early frost . The estimated net values for the year under these (d) Find the allowable range to stay optimal for the net value per scenarios are shown below. acre planted for each of the three crops . Net Value per Acre Planted Scenario Soybeans Corn Wheat Drought $10 $15 0 Flood $15 $ 20 $10 Early frost $ 50 $ 40 $ 30 Drought and early frost $15 $ 20 $10 Flood and early frost $10 $10 (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? (g) 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 (i) 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 (j) 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 20 % lood 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 each 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