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1 Operations Research, 2014/2015, list 1 List 1 1. The Apex Television Company has to decide on the number of 27 and 20-inch sets to be produced at one of its factories. Market research indicates that at most 40 of the 27-inch sets and 10 of the 20-inch sets can be sold per month. The maximum number of work-hours available is 500 per month. A 27-inch set requires 20 work-hours and a 20-inch set requires 10 work-hours. Each 27inch set sold produces a profit of $120 and each 20-inch set produces a profit of $80. A wholesaler has agreed to purchase all the television sets produced if the numbers do not exceed the maxima indicated by the market research. Formulate a linear programming model for this problem. Use the graphical method and computer software to solve the model. 2. A company produces two products, A and B. The sales volume for A is at least 80% of the total sales of both A and B. However, the company cannot sell more than 100 units of A per day. Both products use one raw material, of which the maximum daily availability is 240. The usage rates of the raw material are 2 per unit of A and 4 per unit of B. The profit units for A and B are $20 and $50, respectively. Determine the optimal production mix for the company. Use the graphical method and computer software. 3. Farmer Tom must determine how many hectares of rye and barley to plant on his 100 hectares field. He can sells no more than 560 tons of rye at $30 a ton and no more than 480 tons of barley at $50 a ton. A hectare of rye yields 10 tons and requires 12 hours of labor, a hectare of barley yields 8 tons and requires 20 hours of labor. At most 1400 hours of labor are available and one hour of labor costs $10. Formulate and solve an linear programming model that maximize Tom’s profit. 4. Steelworks manufactures steel by combining three alloys. The steel must meet the following requirements: at most 14% of C, at most 8% of Si, at least 25% of Mn, and at least 12% of P. The cost and properties of each alloy are given in the following table: Alloys I II III C 28% 14% 10% Elements Si Mn 10% 30% 12% 20% 6% 30% Cost per ton P 10% 10% 15% $200 $150 $400 Use linear programming to determine how to minimize the cost of producing of 5000 tons of steel. 5. A manufacturer produces three models I,II, and III, of a certain product using raw materials A and B. The following table gives the data for the problem: Raw material A B Minimum demand Profit per unit($) Requirements per unit I II III 2 3 5 4 2 7 200 200 150 30 20 50 Availability 4000 6000 The labor time per unit of model I is twice that of II and three times that of III. The entire labor force of the factory can produce the equivalent of 1500 units of model I. Market requirements specify the rations 3:2:5 for the production of the three respective models. Formulate the problem as a linear programming problem, and find the optimal solution.