Exercise one (The diet problem) A diet requires four nutrients: calories, fat, protein, and carbohydrates. These nutrients can be supplied by five foods: poke bowl, salad, avocado, goji berries, and sushi. We have the following data: Nutrient Pok (x1) Sal (2) Avo (X3) Goj (4) Sus (5) Nutrient requirement Calories 35 50 40 60 45 2500 calories Fat 4 10 1 6 1200 calories Protein 6 3 800 calories Carbohydrates 5 JO 500 calories Unit cost 3 2 Find a minimum cost diet that meets our nutritional requirements. Exercise two Investment) Suppose there are four assets: bond market ETF, SP500 ETF, growth ETF, and money market. We have the following information: Bond (x1) SP (x2) Growth (x3) Money (x4) Target Forecast yield 0.06 0.08 0.15 0.04 0.10 Risk 0.15 0.25 ).50 0.05 Find a minimum risk portfolio that meets our yield target. Exercise three (Knapsack problem) We want to fill a knapsack (which has a weight constraint) with a combination of goods that has the highest possible value. We have the following data: Item 1 (x1) Item 2 (x2) Item 3 (x3) Item 4 (x4) Item 5 (x5) Budget Value 20 30 70 50 Weight 5 15 60 30 40 90 Find the combination of goods with the highest value. Exercise four Production planning) We consider a production planning problem over four years. In each year, we decide how much to produce (say 21, X2, 3, 24). We have the following data: Year Year 1 Year 2 Year 3 Year 4 Unit production cost 6 5 12 15 Unit holding cost 5 4 5 6 Forecast demand 1 = 200 d2 = 250 d3 = 300 d4 = 350 Also supp start with an stock 100. What is the optimal production plan