1. Brothers Biscuit and Flour Company makes a cereal from several ingredients. Two of the ingredients, oats and rice, provide vitamins A and B. The company wants to know how many ounces of oats and rice it should include in each box of cereal to meet the minimum requirements of 48 milligrams of vitamin A and 12 milligrams of vitamin B while minimizing cost. An ounce of oats contributes 8 milligrams of vitamin A and 1 milligram of vitamin B, whereas an ounce of rice contributes 6 milligrams of A and 2 milligrams of B. An ounce of oats costs $0.05, and an ounce of rice costs $0.03. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis. 2. Addis Fertilizer Company makes a fertilizer using two chemicals that provide nitrogen, phosphate, and potassium. A pound of ingredient 1 contributes 10 ounces of nitrogen and 6 ounces of phosphate, while a pound of ingredient 2 contributes 2 ounces of nitrogen, 6 ounces of phosphate, and 1 ounce of potassium. Ingredient 1 costs $3 per pound, and ingredient 2 costs $5 per pound. The company wants to know how many pounds of each chemical ingredient to put into a bag of fertilizer to meet the minimum requirements of 20 ounces of nitrogen, 36 ounces of phosphate, and 2 ounces of potassium while minimizing cost. a. Formulate a linear programming model for this problem. 1 b. Solve this model by using Simplex analysis. 3. Shewa Bakery makes coffee cakes and Danish pastries in large pans. The main ingredients are flour and sugar. There are 25 pounds of flour and 16 pounds of sugar available, and the demand for coffee cakes is 5. Five pounds of flour and 2 pounds of sugar are required to make a pan of coffee cakes, and 5 pounds of flour and 4 pounds of sugar are required to make a pan of Danish. A pan of coffee cakes has a profit of $1, and a pan of Danish has a profit of $5. Determine the number of pans of cakes and Danish to produce each day so that profit will be maximized. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis. 4. Solve the following linear programming model by using simplex method: 5. Maximize Z: 3x1 + 6x2 Subject to: 3x1 + 2x2 18 X1 + x2 5 X1 4 X1, X2 0 Solve the following linear programming model graphically: maximize Z=5x+8x2 subject to 3x+5x250 2x + 4x2 40 x 8 10 XXNO 6. Dalol Restaurant has an ice-cream counter where it sells two main products, ice cream and frozen yogurt, each in a variety of flavors. The restaurant makes one order for ice cream and yogurt each week, and the store has enough freezer space for 115 gallons total of both products. A gallon of frozen yogurt costs $0.75 and a gallon of ice cream costs $0.93, and the restaurant budgets $90 each week for these products. The manager estimates that each week the restaurant sells at least twice as much ice cream as frozen yogurt. Profit per gallon of ice cream is $4.15, and profit per gallon of yogurt is $3.60. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis. 7. A canning company produces two sizes of cans regular and large. The cans are produced in 10,000-can lots. The cans are processed through a stamping operation and a coating operation. The company has 30 days available for both stamping and coating. A lot of regular-size cans requires 2 days to stamp and 4 days to coat, whereas a lot of large cans requires 4 days to stamp and 2 days to coat. A lot of regular-size cans earns $800 profit, and a lot of large-size cans earns $900 profit. In order to fulfill its obligations under a shipping contract, the company must produce at least nine lots. The company wants to determine the number of lots to produce of each size can (x1 and x2) in order to maximize profit. a. Formulate a linear programming model for this problem. b. Solve this model by using simplex analysis. 8. Adama textile factory makes T-shirts with logos and sells them in its chain of retail stores. It contracts with two different plants one in Oromiya and one in The Lion. The shirts from the plant in Oromiya cost $0.46 a piece, and 9% of them are defective and can't be sold. The shirts from The Lion cost only $0.35 each, but they have an 18% defective rate. Shirtstop needs 3,500 shirts. To retain its relationship with the two plants, it wants to order at least 1,000 shirts from each. It would also like at least 88% of the shirts it receives to be salable. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis. 9. Hana is establishing an investment portfolio that will include stock and bond funds. She has $720,000 to invest, and she does not want the portfolio to include more than 65% stocks. The average annual return for the stock fund she plans to invest in is 18%, whereas the average annual return for the bond fund is 6%. She further estimates that the most she could lose in the next year in the stock fund is 22%, whereas the most she could lose in the bond fund is 5%. To reduce her risk, she wants to limit her potential maximum losses to $100,000. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis. 10. Solve the following linear programming model by using simplex method and explain the solution result: maximize Z=60x + 90x2 subject to 60x + 30x2 1,500 100x100x26,000 x230 11. 3F Furniture Company produces chairs and tables from two resources labor and wood. The company has 80 hours of labor and 36 pounds of wood available each day. Demand for chairs is limited to 6 per day. Each chair requires 8 hours of labor and 2 pounds of wood, whereas a table requires 10 hours of labor and 6 pounds of wood. The profit derived from each chair is $400 and from each table, $100. The company wants to determine the number of chairs and tables to produce each day in order to maximize profit. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis. 12. Adigrat Drug Company produces a drug from two ingredients. Each ingredient contains the same three antibiotics, in different proportions. One gram of ingredient 1 contributes 3 units, and 1 gram of ingredient 2 contributes 1 unit of antibiotic 1; the drug requires 6 units. At least 4 units of antibiotic 2 are required, and the ingredients each contribute 1 unit per gram. At least 12 units of antibiotic 3 are required; a gram of ingredient 1 contributes 2 units, and a gram of ingredient 2 contributes 6 units. The cost for a gram of ingredient 1 is $80, and the cost for a gram of ingredient 2 is $50. The company wants to formulate a linear programming model to determine the number of grams of each ingredient that must go into the drug in order to meet the antibiotic requirements at the minimum cost. a. Formulate a linear programming model for this problem. b. Solve this model by using Simplex analysis. 134A35 company produces 100 tons of red ore and 80 tons of black ore each week.7% These can be treated in different ways to produce three different alloys, Soft, Hard or Strong. To produce 1 ton of Soft alloy requires 5 tons of red ore and 3 tons of black. For the Hard alloy the requirements are 3 tons of red and 5 tons of black, whilst for the Strong alloy they are 5 tons of red and 5 tons of black. The profit per ton from selling the alloys (after allowing for production but not mining costs, which are regarded as fixed) are 250, 300 and 400 for Soft, Hard and Strong respectively. 5 a. Formulate the problem? b. Decide how much of each alloy to make each week as a L.P. problem by using the Simplex method? 14. A coffee packer blends Brazilian coffee and Colombian coffee to prepare two products, super and deluxe brands. Each kilogram of super coffee contains 0.5 kg of Brazilian coffee and 0.5 kg of Colombian coffee, whereas each kilogram of deluxe coffee contains 0.25 kg of Brazilian coffee and 0.75 kg of Colombian coffee. The packer has 120kg of Brazilian coffee and 160kg of Colombian coffee on hand. If the profit one each kilogram of super coffee is 22 cents and the profit on each kilogram of Deluxe coffee is 30 cents, how many kilograms of each type of coffee should be blended to maximize profits? a. Decide how much of each alloy to make each week as a L.P. problem by using the Simplex method? 15. Solve the following LP Model by using a simplex Method and check your answer by using the graphic Method? maximise z = 3x1 + 4x2 subject to: x1 + 2x2 10 1+8 3x1 +5x2 26 1,20.