A PROBLEM SET 1 Feed Problem A laboratory needs to supply its research dogs a mixture of commercially available dog foods that meet the National Research Council (NRC) nutrient requirements (given in Table 1). The nutritional composition of each of the eight available foods is given in Table 2. Note that these data are given in terms of percentages. Food Must Have At Least (%) Food Must Have At Most (%) Protein 20 5 Fiber Moisture Fat Linoleic Acid Calcium Phosphorous Potassium Salt Magnesium NFE Table 1: National Research Council Nutrient Requirement (a) What would be appropriate variables for modeling this problem? (b) Set up a constraint for each of the food constituents listed in Table 1 based on the NRC requirements. (c) An additional constraint must be provided, because the requirements are given in terms of percentages. It must state that the sum of the amounts used equals 1. That is, each variable Wayne Purina Purina Purina TW Meal Chow HP Gaines Burgerbits Agway 2000 Wayne 27.0 10.5 23.8 9.4 1.6 26.0 10. 0 8 21.0 .0 0.9 23.0 7.0 1.5 1.6 1.6 1.75 1.0 Protein Fat Linoleic Acid Calcium Phosphorous Potassium Salt Magnesium 25.0 8.0 2.1 2.15 1.43 0.73 1.15 0.17 3.5 45.77 10.0 0.17 24.0 9.0 1.6 1.2 1.0 0.98 1.15 0.22 4.7 46.15 10.0 0.17 2.5 1.4 0.8 0.78 0.29 4. 3 41.83 9.0 0.17 1.03 0.71 0.64 0.27 3.7 48.10 9.0 0.16 1.6 1.2 0.9 1.1 0.15 4.0 41.45 12.0 0.21 0.8 0.5 1.0 0.036 5.0 51.76 10.0 0.20 0.8 0.5 1.5 0.05 5.0 47.15 12. 0 0.17 25.5 10.5 1.5 1.5 1.7 0.69 1.0 0.23 2.9 15.28 9 .2 0.16 Fiber NFE Moisture Cost ($/kg) Table 2: Dog Food Constituents (Percentage By Weight) represents a fraction of the total amount to be blended. Write out this constraint. (d) Set up an objective function using the cost data given in Table 2. (e) Consider an arbitrary constraint, ali + 0222 + ...08.18 > b. Show that the constraint in (1) is automatically satisfied if, for all i, ai > b. Similarly, show that (1) is impossible to satisfy if, for all i, ai b. Show that the constraint in (1) is automatically satisfied if, for all i, ai > b. Similarly, show that (1) is impossible to satisfy if, for all i, ai