MATH308-2023-09-08-assignment1

1. Identify the half-space determined by each of the following inequalities. (a) 3 x 1 − 4 x 2 ≤ 12 (b) 2 x 1 − 5 x 2 ≥ 0 2. Consider the following linear program. Maximize 4 x 1 − x 2 Subject to: − x 1 + x 2 ≥ 0 , − 6 x 1 + x 2 ≤ 0 x 1 − 4 x 2 ≥ − 23 , 3 x 1 − 2 x 2 ≤ 8 2 x 1 + x 2 ≤ 17 , x 1 ≥ 0 , x 2 ≥ 0 (a) Sketch the feasible region of the linear program. (b) Are there any redundant constraints? Why or why not? (c) Solve the linear program using the graphical method. (d) Identify an optimal solution and the optimal objective function value. (e) Determine if alternative optimal solutions exist. If yes, identify three of them. If no, justify your answer. 3. Show that the linear program below is unbounded. Maximize 2 x 1 + x 2 Subject to: 3 x 1 + x 2 ≥ 4 x 1 + 3 x 2 ≥ 4 x 1 − x 2 ≤ 4 x 1 ≥ 0 , x 2 ≥ 0 Then, identify a direction d = ( d 1 , d 2 ) T and a feasible solution ( x 0 1 , x 0 2 ) T so that the ray ( x 0 1 , x 0 2 ) T + λ ( d 1 , d 2 ) T is a feasible solution to the LP for all positive values of λ in such a way that the objective function value improves along the direction d . Also, compute a feasible solution with objective function value (i) 50, (ii) 100, (iii) 1000. 4. A 40g serving of General Mills Total Cereal contains 0.8g of fat, 3.2g of protein, 32.8g of carbohydrate, and 4g of dietary fiber where as a 42g serving of Post Honey Bunches of Oats With Almonds contain 3g of fat, 3g of protein, 34g of carbohydrate, and 2g of dietary fiber. Annie is a 40 year old woman and her dietitian recommended that the daily diet should include at least 41 grams of protein, 248 grams of carbohydrates, 42 grams of fat, and 25 grams of dietary fiber. Annie aims to get at least 25% of these requirements through breakfast, mixing General Mills Total Cereal and Post Honey Bunches of Oats With Almonds. A 400g package of General Mills Total Cereal cost 3 while Post Honey Bunches of Oats With Almonds cost 4 per 400g package. How much General Mills 2