Linear Programming [Word Problems]

Linear Programming Practice Questions

  1. A manufacturer produces two types of steel trunks. He has two machines A and B. The first type of trunk requires 3 hours on machine A and 3 hours on machine B. The second type requires 3 hours on machine A and 2 hours on machine B. Machine A and B work at most for 18 hours per day respectively. He earns a profit of Rs.30 and Rs.25 per trunk of the first type and second type must he make each day to make the maximum profit?
  2. A company manufactures two types of toys A and B. Type A requires 5 minutes each for cutting and 10 minutes each for assembling. Type B requires 8 minutes each for cutting and 8 minutes for assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. The profit is Rs.50 on type A and Rs.60 on type B. How many toys of each type should the company manufactures in a day to maximise profit.
  3. A diet for a side person must contain at least 4000 vitamins, 50 units of minerals and 1400 calories. Two foods A and B are available at a cost of Rs.4 and Rs.3 per unit respectively. If one unit of A contains 200 units of vitamins, 1 unit of minerals and 40 units of calories, one unit of B contains 100 units of vitamins, 2 units of minerals and 40 calories. Find what combination of foods should be used to have the least cost.
  4. Custom cars does van conversions. The custom conversion requires 15 hours of shop time, 8 hours of painting time, and 1 hour of inspection time. The deluxe conversion requires 12 hrs of shop time, 12 hrs of painting time and 1.5 hrs if inspection time. There are 90 hours of shop time, 72 hrs of painting and 10 hrs of inspection time available during the coming two weeks. How many conversions of each type should custom Cars perform assuming that each custom conversions results in Rs.175 profit and each deluxe conversion results in Rs.225 profit. Find maximum profit.
  5. A manufacturer produces two types of soap bars using two machines A and B. A is operated for 2 minutes and B for 3 minutes to manufacture the first type, while it takes 3 minutes on machine A and 5 minutes on machine to manufacture the second type. Each machine can be operated at the most for 8 hours per day. The two types of soap bars are sold at a profit of Rs.0.25 and Rs.0.50 each. Assuming that the manufacturer can sell all bars of soap of each type should be manufactured per day so as to maximise his profit.
  6. A resourceful home decorator manufactures two types of lamps say A and B. Both lamps go through two technicians, first a cutter, second a finisher. Lamp A requires 2 hours of cutters and 1 hour finishers time. Lamp B requires 1 hour of cutter’s and 2 hours of finisher’s time. The cutter has 104 hours and finisher has 76 hours of time available each month. Profit on one lamp A is Rs.6 and on one lamp B Rs.11. Assuming that he can sell all that he can sell all that he produces, how many of each type of lamps should he manufacture to obtain the best return.
  7. A factory  produces two products A and B. Each of the product A requires 2 hours of mounding, 3 hours of grinding and 4 hours of polishing and each of product B requires 4 hours of moulding, 2 hours of grinding and 2 hours of polishing. The factory has moulding machine available for 20 hours, grinding machine for 24 hours and polishing machine available for 13 hours. The profit is Rs.5 per unit of And Rs.3 per unit of B and the factory can sell all that it produces. Formulate the problem as a linear programming problem to maximise the profit. 
  8. A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs 250 per bag contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs 200 per bag contains 1.5 units of nutritional elements A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?
  9. An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit?
  10. Horsing Around makes horse saddles in both a child size and an adult size. Each child saddle requires 20 hours of assembly and 15 hours of finishing. Each adult saddle requires 30 hours of assembly and 10 hours of finishing. The company has a maximum of 300 hours available per week in the assembly department and a maximum of 150 hours per week in the finishing department. The profit for each child saddle is 255 and the profit for each adult saddle is 375. If the company wants to maximize their profit, how many child and adult horse saddles should be produced per week? What is the maximum profit?
  11. A chef owns a vegetable field. He has determined that each year, he needs to fertilize his field with at least 1200 pounds of calcium and at least 600 pounds of magnesium to yield the best harvest. Each bag of Brand I fertilizer contains 6 pounds of calcium, 2 pounds of magnesium and 1 pound of sulfur. Each bag of Brand II fertilizer contains 3 pounds of calcium, 3 pounds of magnesium and 1 pound of sulfur. How many bags of Brand I and Brand II should be used to meet the minimum fertilizer requirements and also minimize the amount of sulfur added to the vegetable field?
  12. A body builder wants to build lean muscle using two types of supplement packets. Each packet of Brand I contains 10 grams of fat, 3 grams of carbohydrates and 15 grams of protein. Each packet of Brand II contains 5 grams of fat, 6 grams of carbohydrates and 30 grams of protein. He has decided that from these two types of packets, he would like to take in a maximum of 40 grams of fat per day and a minimum of 150 grams of protein per day. How many packets of each brand should he take per day to minimize the intake of carbohydrates? What is the minimum intake of carbohydrates?
  13. A small business enterprise makes dresses and trousers. To make a dress requires 1/2 hour of cutting and 20 minutes of stitching. To make a trousers requires 15 minutes of cutting and 1/2 hour of stitching. The profit on a dress is 40 and on a pair of trousers 50. The business operates for a maximum of 8 hours per day. Determine how many dresses and trousers should be made to maximize profit and what the maximum profit is.
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