Section 2.2 Applications of Linear Programming Problems
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1 Section 2.2 Applications of Linear Programming Problems Example 1: A company produces two models of clock radios. Each model A requires 15 min of work on assembly line I and 10 min of work on assembly line II. Each model B requires 10 min of work on assembly line I and 12 min of work on assembly line II. At most, 23 hr of assembly time on line I and 22 hr of assembly time on line II are available per work day. It is anticipated that the company will realize a profit of $12 on each model A and $10 on each model B. How many clock radios of each model should be produced per day in order to maximize the company s profit? MAX A B HR ASMBLY LINE I ASMBLY LINE II PROFIT c. State the Linear Programming Problem. Do not solve. Section 2.2 Applications of Linear Programming Problems 1
2 Example 2: A manufacturer makes camping tents, a standard model and a deluxe model. Each standard tent requires 1 labor-hour from the cutting department and 3 labor-hours from the assembly department. Each deluxe tent requires 2 labor-hours from the cutting department and 4 labor-hours from the assembly department. The maximum labor-hours available per week in the cutting department and the assembly department are 32 and 84, respectively. In addition, the distributor, because of demand, will not take more than 12 deluxe tents per week. If the company makes a profit of $50 on each standard tent and $80 on each deluxe tent, how many tents of each type should be manufactured each week to maximize the total weekly profit? Max Standard Deluxe hr Cutting Dept. Assembly Dept. Profit c. State the Linear Programming Problem. Do not solve. Section 2.2 Applications of Linear Programming Problems 2
3 Example 3: You re a dietician in a hospital and must arrange a special diet composed of two foods, Balanced Diet and Nutritional Goods. Each ounce of Balanced Diet contains 30 units of calcium, 10 units of iron, 10 units of vitamin A, and 8 units of cholesterol. Each ounce of Nutritional Goods contains 10 units of calcium, 10 units of iron, 30 units of vitamin A, and 4 units of cholesterol. If the minimum daily requirements are 360 units of calcium, 160 units of iron, and 240 units of vitamin A, how many ounces of each food should be used to meet the minimum requirements and at the same time minimize the cholesterol intake? What is the minimum cholesterol intake? c. State the Linear Programming Problem. Do not solve. Section 2.2 Applications of Linear Programming Problems 3
4 Example 4: The officers of a high school senior class are planning to rent buses and vans for a class trip. Each bus can transport 40 students, requires 3 chaperones, and costs $1,200 to rent. Each van can transport 8 students, requires 1 chaperone, and cost $100 to rent. The officers must plan to accommodate at least 400 students. Since only 36 parents have volunteered to serve as chaperones, the officers must plan to use at most 36 chaperones. How many vehicles of each type should the officers rent in order to minimize the transportation costs? What are the minimal transportation costs? b. Construct and fill-in a table. c. State the Linear Programming Problem. Section 2.2 Applications of Linear Programming Problems 4
5 d. Solve the problem. Step 1: Graph the feasible set. (1) (2) Step 2: Find the corner points of the feasible set. Step 3: Find where the optimal solution occurs and the optimal value. Interpret your results. Corner Points Min Section 2.2 Applications of Linear Programming Problems 5
6 Example 5: A patient in a hospital is required to have at least 84 units of drug D 1 and at least 120 units of drug D 2 each day (assume that an overdosage of either drug is harmless). Two substances, M and N, contain each of these drugs; however, in addition, both contain an undesirable drug D 3. Each gram of substance M contains 10 units of drug D 1, 8 units of drug D 2 and 3 units of drug D 3. Each gram of substance N contains 2 units of drug D 1, 4 units of drug D 2 and 1 unit of drug D 3. How many grams of substances M and N should be mixed to meet the minimum daily requirements and at the same time minimize the intake of drug D 3? c. State the Linear Programming Problem. Section 2.2 Applications of Linear Programming Problems 6
7 d. Solve the problem. Step 1: Graph the feasible set. (1) (2) Step 2: Find the corner points of the feasible set. Step 3: Find where the optimal solution occurs and the optimal value. Interpret your results. Corner Points Min Section 2.2 Applications of Linear Programming Problems 7
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